# Le management du risque pour les compagnies d'assurance : une approche marchés financiers

Saad Mouti

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Saad Mouti. Le management du risque pour les compagnies d'assurance: une approche marchés financiers. Mathématiques [math]. Sorbonne Université, 2017. Français. NNT: 2017PA066744. tel-04021249

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École doctorale n° 386 : Sciences Mathématiques de Paris Centre

## THÈSE

pour obtenir le grade de docteur délivré par l'

## Université Pierre et Marie Curie

Spécialité : Mathématiques Appliquées

présentée et soutenue publiquement par

Saad MOUTI

le 6 Décembre 2017

# Le management du risque pour les compagnies d'assurance : une approche marchés financiers

## Composition du jury :

**Directeurs :**
Nicole EL KAROUI Université Pierre et Marie Curie
Mathieu ROSENBAUM Ecole Polytechnique

**Encadrant entreprise :** Aymeric KALIFE AXA

**Rapporteurs :**
Giorgia CALLEGARO Université de Padoue
Jim GATHERAL Baruch College
Caroline HILLAIRET ENSAE

**Examinateurs :**
René AID Université Paris-Dauphine
Romuald ELIE Université Paris-Est Marne-la-Vallée
Gilles PAGES Université Pierre et Marie Curie
Christian ROBERT ISFA
## Remerciements

Je tiens en premier lieu à exprimer ma gratitude envers mes directeurs de thèse, Nicole El Karoui et Mathieu Rosenbaum, pour leur disponibilité et les précieux conseils qu'ils m'ont donnés tout au long de cette thèse. Ça a été un honneur pour moi que Nicole accepte de codiriger cette thèse. Je la remercie pour son temps, le partage de sa connaissance et de son expérience mais aussi pour sa bienveillance. Je ne remercierai jamais assez Mathieu sans qui ce projet de recherche n'aurait jamais abouti. Il m'a énormément apporté sur le plan professionnel et humain, et a su me rassurer pendant les moments de doute. Sa pédagogie et son coté humain dépassent mes attentes.

L'aventure a commencé chez AXA et je dois beaucoup à mon encadrant en entreprise Aymeric Kalife qui m'a accueilli au GRM et grâce à qui j'ai eu le financement pour la thèse. Je le remercie de m'avoir accordé sa confiance, pour les sujets intéressants qu'il m'a proposés et pour ses conseils, échanges et collaborations enrichissantes.

Je remercie Giorgia Callegaro, Caroline Hillairet, ainsi que Jim Gatheral d'avoir accepté de rapporter ma thèse. Je suis très honoré par leur lecture attentive de ce manuscrit et leur intérêt pour mon travail. Je remercie également René Aïd, Romuald Elie, Gilles Pagès et Christian Robert d'examiner ma thèse et de participer à ma soutenance.

Je suis très reconnaissant envers mes professeurs du Master à Dauphine et plus particulièrement Idriss Kharroubi, Christian Robert et Nicola Choppin qui ont contribué à la poursuite de cette thèse.

Les travaux qui forment cette thèse ont été en grande partie en collaboration. Je remercie donc Ludovic Goudenège, Giulia Livieri, Gabriela López Ruiz, Andrea Pallavicini, Xiaolu Tan et Lihang Wang pour ces collaborations instructives.

Je remercie tous mes collègues du GRM qui ont rendu mon quotidien plus agréable et avec qui j'ai partagé de très bons moments. J'ai une pensée particulière pour les anciens thésards du GRM, Fatima, Medeleine-Sophie et Nabil, mais aussi tous les collègues et personnel d'AXA avec qui j'ai pu intéragir pendant cette thèse. Je cite Claire, Grace, Jamila, Marion, Mildraid, Mouna, Rachida, Rizlaine, Ahmed, Bernard, Berthold, Cyril, Eric, Johny, Jérémie, Mohamed, Pierre, Philipe et Tom, la liste étant non-exhaustive.

Je remercie également mes frères d'armes du LPMA de la même subdivision Pamela, Omar, Thibault, Jiatu et Weibing, mes collègues de bureau Alice, Candia, Nina, Eric et Yi pour ces années enrichissantes.

Je tiens à remercier tout le personnel du secrétariat, Florence, Serena et Josette pour leur servabilité et leur agréable compagnie dans la tour 16/26, et le personnel de l'école doctorale pour leur aide administrative.

Pour finir, je remercie mes proches, amis et familles de leur patience. Un grand merci à mes frères et mes sœurs ainsi qu'à mon épouse Fatima Ezzahra de m'avoir rassuré et soutenu pendant cette thèse. Je resterai éternellement reconnaissant envers mes parents pour leur soutien inconditionnel pendant cette thèse et bien avant.
Résumé

Cette thèse traite plusieurs aspects des risques financiers liés aux contrats d'assurance vie. Elle étudie trois sujets distincts et est composée de six chapitres qui peuvent être lus indépendamment.

Le comportement de l'assuré est un risque majeur pour les assureurs dans le cadre de produits d'assurance vie comme les annuités variables. Ainsi, nous nous penchons dans les premiers chapitres sur le comportement optimal pour deux classes de produits commercialisés. Nous traitons le cas du rachat total pour les "guaranteed minimum account benefits" (GMAB), et le retrait optimal dans le cadre des "guaranteed minimum income benefit" (GMIB). Le troisième chapitre est dédié au management et à la couverture d'une classe de produits à unité de compte également commercialisés par les assurances.

Le quatrième chapitre traite l'exécution optimal d'un large portfeuille d'options. En effet, les produits d'assurance vie sont partiellement couverts statiquement par la détention d'options vanilles. Nous considérons le cas où la taille des trades affecte le prix des options et cherchons à définir la stratégie optimale permettant de minimiser le coût de l'acquisition de ce portefeuille de couverture en prenant en compte l'impact de marché.

Enfin, le dernier thème de la thèse étudie le processus de volatilité. A cet effet, nous utilisons deux types d'estimateurs. En l'absence de données haute fréquence, les estimateurs dit de "range" permettent de revérifier que la volatilité est rugueuse. Ensuite, en utilisant les prix d'options, l'estimateur volatilité implicite court-terme et une version raffinée de cette dernière permettent encore une fois d'aboutir à la même conclusion.

Abstract

This thesis tackles several aspects of financial risks encountered in the life insurance industry and particularly in a class of products insurers offer; namely variable annuities and unit-linked products. It consists of three distinct topics and is split into six chapters that can be read independently.

In variable annuities, policyholders' behavior is a major risk for the insurer that affects the life insurance industry in almost every aspect. The first two chapters deal with policyholders' optimal behavior for two classes of these products. We address the rational lapse behavior in the guaranteed minimum account benefit, and optimal withdrawals in the guaranteed minimum income benefit. The third chapter is dedicated to a class of unit-linked products from a managing and hedging point of view.

The third chapter addresses the optimal execution of a large book of options. Typically, life insurance products are partially hedged using vanilla options. We consider the case where trades are affected by the traded quantity, and seek to find an optimal strategy based on two criteria; expected cost and mean-variance.

Finally, in the last topic we study the volatility process using two different proxies. First, range-based estimators that rely on the asset price range data allow us to double-check that volatility is a rough process in the sense that it has a scaling parameter H less than 1/2. Then, using short time-to-maturity implied volatilities as a proxy for the spot volatility, and a refined version of it, we are able to confirm that the rough aspect of volatility is universal based on different proxies.
**Keywords**: GMAB, Variable Annuity, rational lapse strategy, stochastic interest, PDE, ADI, high-dimensional regression, GMIB, Variable Annuities, rational behavior, optimal withdrawals, PDE, dynamic programming, CPPI, dynamic multiplier, jump processes and gap risk, vanilla and gap options, Market impact, option pricing, optimal execution, mean-variance, stochastic control, HJB equation, range-based volatility, Garman-Klass estimator, Parkinson estimator, volatility scaling, rough volatility, fractional Brownian motion, fractional Ornstein-Uhlenbeck, RFSV, volatility forecasting, implied volatility, Medvedev-Scaillet approximation.
List of papers being part of this thesis

* A. Kalife, S. Mouti and L. Wang, *Financial risk management and the rational lapse strategy in life insurance policies*, Insurance Markets and Companies, 4(2), 2013.

* A. Kalife, L. Goudenege and S. Mouti (2014), *Managing gap risks in iCPPI for life insurance companies : a risk return cost analysis*, Insurance Markets and Companies, 5(2), 2014.

* A. Kalife and S. Mouti, *On Optimal Options Book Execution Strategies with Market Impact*, Market Microstructure and Liquidity, 2(3), 2016.

* S. Mouti *Range-based proxies and rough volatility*, pre-print, 2017.

* G. Livieri, S. Mouti, A. Pallavicini and M. Rosenbaum, *Rough volatility : evidence from option prices*, submitted, 2017.
# Table des matières

## Table des matières ix

## Introduction 1

1. Chapter 1 - Financial risk management and the rational lapse strategy in life insurance policies 6

2. Chapter 2 - Optimal behavior strategy in Guaranteed Minimum Income Benefit 8

3. Chapter 3 - Managing gap risks in iCPPI for life insurance companies : a risk return cost analysis 10

4. Chapter 4 - Optimal execution of options book under market impact 15

5. Chapter 5 - Range-based proxies and rough volatility 19

6. Chapter 6 - Volatility is rough : evidence from option price data 21

## I Life insurance products 23

### 1 Financial risk management and the rational lapse strategy in life Insurance policies 25

1.1 Introduction 25

1.2 Description of the contract 27

1.3 Valuation of a GMAB with zero lapse 28

1.4 Valuation a GMAB with rational lapse assumption 29

1.5 Life insurance policy pool 31

1.6 Numerical tests 32

1.7 Conclusion 35

### 2 Optimal behavior strategy in the GMIB product 43

2.1 Product description 43

2.2 A brief review of the literature 50

2.3 Formulation and basic notations 51

2.4 Contract valuation 56

2.5 Results 64

2.6 Conclusion 73

### 3 iCPPI and Gap Risk 75

3.1 Introduction 75

3.2 Methodolody 78

3.3 Mitigating downside risk : Preventing from breaching the floor 83

3.4 Conclusion 87
II Market impact and volatility 95

4 On Optimal Options Book Execution Strategies with Market Impact 97

4.1 Introduction 97

4.2 Market impact model : A transaction costs approach 100

4.3 The optimal execution problem 107

4.4 Adding the agent risk aversion : A mean-variance framework 110

4.5 Numerical solution and results 116

4.6 Extension to a local volatility model : A numerical method for the general case 123

5 Range-based proxies and rough volatility 129

5.1 Introduction 129

5.2 Overview on range-based volatility estimation 131

5.3 Range-based volatility as spot volatility proxy : empirical results 134

5.4 RFSV model validation using range-based proxies 139

5.5 Forecasting range-based volatility using the RFSV model 147

5.6 Conclusion 152

6 Rough volatility : evidence from option prices 153

6.1 Introduction 153

6.2 At-the-money implied volatility with short maturity as spot volatility proxy 155

6.3 A refined implied volatility based proxy for the spot volatility 158

6.4 On the upward bias when estimating the Hurst parameter 162

6.5 Conclusion 167
# Introduction

## Preliminary: General review of the life insurance industry

The world's older population is growing rapidly. According to data published in 2015 by the United Nations, there was a substantial increase of 48% (from 607 to 901 million) of people aged 60 or over between 2000 and 2015. And by 2050, the population aged 60 and over might reach nearly 2.1 billion. Moreover, the "oldest-old" (aged 80 or over) population accounted for 14% of old population (aged 60 or over) in 2015, and is expected to triple 2015's value by 2050, see [133].

As a result of these demographic shifts, longer life expectancy, increasing lifestyle and health-care costs, the idea that individuals and households need to plan for their own retirement is gaining a lot of attention. On the other hand, low interest rates are putting pressure on the insurance sector, pushing providers and consumers alike to look for ways to make the most of their assets. This situation has led to a significant reduction in the demand for traditional life insurance products and an increase in the demand for annuities and other financial planning products, see [66].

Insurance companies offer a range of savings products indexed on financial assets (stocks, funds, government bonds...). These so-called unit linked products constitute a major segment of insurance companies' earnings. One reason is that they allow the transfer of the financial risk to the insured, thus reducing the required capital for the insurance company's solvency. However, in order to make these products more attractive, insurers started to offer complex guarantees that got closer and closer to banks' financial options.

Life insurance guarantees introduced numerous risks for the insurer which need to be hedged using different hedging mechanisms (static or dynamic hedging, reinsurance,...). The relatively long maturity, i.e. at least 10 years, of these products adds an additional complexity compared to classical financial products, and necessitates robust methods to estimate and hedge the resulting risks. Unsuitable risk management combined with important products sales could lead to disasters, as was seen for Equitable Life<fnref n="1" /> in 2000.

In this thesis, we analyze some of the financial risks related to the life insurance business, and present some of the potential solutions. It is split into two major parts. The first part focuses on some of the commercialized products and the behavioral risk linked to them, as well as an alternative to products with option guarantees. The second part covers financial risk in a general perspective. In the following introduction, we present a general review of the products we are concerned about, followed by an overview of the content of each chapter.

<footnote n="1">1. Equitable Life is a life insurance company in the United Kingdom found in 1762 and closed to new business in December 2000 due to large unhedged liabilities and high guaranteed fixed returns to investors without provision for adverse market changes.</footnote>
# Life insurance products: Unit linked products and variable annuities

## Unit linked products

Unit linked policies generally consist in insurance products which accumulate capital. They are usually taken with the purpose of accumulating a financial benefit to the policyholder at a future point in time. The policyholder can typically choose from investment funds or individual stocks. The account value is given by the number of units acquired, multiplied by the price of one unit. Unit linked products can be extended by complementary types of investment guarantees, and can also be based on asset management strategies.

For unit linked products, the amount available for annuitization after an accumulation phase depends on the development of a mutual fund. In the 1995's, equity-index annuities were introduced in the U.S. as another form of unit linked products. Their returns rate is determined according to a formula that takes into account changes in an equity index, e.g. the S&P500 or a basket of equities or mutual funds. Furthermore, these products exhibit a minimum guarantee on the premium, e.g. 90% of the premium paid and additionally 3% annual interest rate. Other variations of unit linked products include, among others, flexible premium payments during the contract term, partial withdrawals during the deferral phase, or the possibility of shortening or extending the accumulation phase.

## Annuities products & variable annuities

In the last decade, equity-linked policies have become more and more popular, exposing policyholders to financial markets and providing them with different ways to consolidate investment performance over time as well as protection against mortality-related risks. The best examples of such contracts are variable annuities.

The origin of annuities dates back to the Roman empire. The term annuities comes from the Latin word "annua" which means annual income. *Annua* were ancient Roman contracts which provided an individual with a stream of payments for a specified period of time, or for life, in exchange for an upfront payment. Gnaeus Domitrius Annius Ulpianis, a roman speculator and jurist credited with creating the very first actuarial life table, is cited as one of the earliest dealers of these annuities. Annuities were also used to compensate roman soldiers for military service. Nowadays, the most famous annuities products are "variable annuities".

Variable annuities are unit linked or managed fund vehicles, which offer optional guarantee benefits for the customer. They can be purchased either by a single payment, or a series of payments. This amount constitutes the principal of the contract and, apart from some upfront costs, is entirely invested into a reference portfolio. The investment options (subaccounts) offered by the insurance companies are typically mutual funds of stocks, bonds, money market instruments, or some combination of the three.

Depending on the terms and conditions specified by the contract, the insurer promises to make periodic payments to the policyholder on predefined future dates. These payments are usually determined as percentage of the invested premium and deducted from the contract value. Policyholders can choose the date when the payout phase begins. The retirement date is often recommended to initiate annuitization. Although clients can annuitize later than this, insurers usually specify a maximum annuitization date. For example, it could be the later of the policyholder's 95th birthday or the 15th anniversary of the contract.
Upon annuitization, there are a number of possibilities on the duration of the payments :

— The lifetime of the policyholder.

— The lifetime of the named beneficiary.

— A specified period such as 20 years.

— The longer of the policyholder's lifetime and a certain period.

Another feature that contributed to make these products attractive is the presence of tax incentives introduced by governments, particularly in the U.S., to support the development of individual pension solutions and contain public expenditure. But the main feature of variable annuities remains the possibility to benefit from various guarantees against investment and mortality/longevity risks. These guarantees are usually referred to as GMxB, where x stands for the class of benefits involved. The most famous GMxB riders on the market are :

— **Guaranteed Minimum Accumulation Benefits (GMAB)**: Also known as maturity guarantees, see [39], GMABs are one of the earliest products in the family of GMxB riders on variable annuities. The original form of this put-like rider is purchased for a fixed term (e.g. 10 years). The GMAB contract gives policyholders the ability to protect their retirement investments against downside market risk. They allow the policyholder to receive the greater of the account value and the benefit base at maturity, and may include some specific features.

— **Guaranteed Minimum Death Benefits (GMDB)**: Introduced in the market in the 1990's, GMDBs are guarantees in case of policyholder's death. Upon his death during the term of the contract, a specific monetary amount is passed on to a person of the policyholder's choosing, i.e. usually spouse or children. There are several variations for the death benefit. It may simply be the original premium when combined with other riders for example, or may accumulate deposits at a fixed rate.

— **Guaranteed Minimum Income Benefits (GMIB)**: GMIB riders, also launched in the 1990's, provide policyholders the right to convert the benefit base at the end of deferral period into annuities for life with a constant rate fixed at inception. Generally speaking, the value of the benefit base is not less than the initial account value paid by policyholders. Due to enduring competitions, most insurers currently add some "features" for these guarantees. For example, the benefit base can be reset to the high-water mark of the account value on anniversary dates (step-up or ratchet) when the market has performed well, or can roll up with a fixed percentage (known as roll-up rate, e.g. 2%), regardless of market conditions.

— **Guaranteed Minimum Withdrawal Benefits (GMWB) and Guaranteed Lifelong Withdrawal Benefits (GLWB)**: GMWB are a relatively recent innovation in the life insurance market. As their predecessors GMIBs, GMWB riders also promise a minimal annuity level from an initial investment capital, regardless of the performance of the underlying account value. The most important difference between GMIB and GMWB is the policyholder's surrender right after the end of the deferred period. While the policyholder has to give up all the asset invested once the contract enters into payment period for GMIB contract, in case of GMWB product, the insurer deducts the annuity and some charge fees. GMWB contract sets a limited payment period at inception. Thus, the insurer will stop paying annuities and return the remaining account value (if not exhausted) to the policyholder. GLWB contracts, also known as GMWB for life (GMWB-L), on the other hand, pay annuities until the policyholder's death.

— **Combo variable annuities**: In order to attract more investors, insurance companies propose combined variable annuities that offer more than one guarantee in a single contract. A typical example is the GMIB-DB, which combines an income benefit if the policyholder is alive at the contract maturity and a death benefit if he dies before.
Compared to fixed annuities, variable annuities are designed to protect against the effect of inflation on fixed income over the long run. Based on numbers from the Life Insurance and Market Research Association (LIMRA)<fnref n="2" />, Figure 1 illustrates individual sales in variable annuities from the last decade in the U.S. In particular, sales were particularly high in 2006 and 2007 due to attractive guaranteed lifetime payments like GLWB offerings. However, the financial crisis brought about a strategic reset for many insurance carriers as interest rates were lowered and equity volatility rose. We also see from Fi-

<figure><img src="image_2.png" /><figcaption>FIGURE 1 – Annuity Sales Estimates in the U.S. in 2006-2015. Source : LIMRA Secure Retirement Institute U.S. individual annuities sales survey</figcaption></figure>

gure 1 a noticeable superiority of variable annuities in sales with respect to fixed annuities for almost a decade. Nevertheless, there is also an evident decline on variable annuities sales in recent years which can be explained by market volatility, growing popularity of indexed annuities, and a looming labor department, see [103]. Table 2 shows the top 20 sales leaders for total, variable and fixed annuities, as well as the top manufacturers of indexed annuities in 2015 published by LIMRA Secure Retirement on March 2016. AXA is ranked 8th in the total annuities sales and 5th in the variable annuities sales.

Variable annuities business seems to struggle to restart after 2008 crisis. In fact, years 2009 and beyond have been marked by the use of more dynamic asset allocation. Dynamic asset allocation comprises the use of "portfolio insurance" strategies and risk-control strategies. They allow insurers to tailor products that necessitate less capital requirement, and attract new costumers. In the last section, we will see one of these strategies and the product based upon it : the individualized constant proportion portfolio insurance product.

## Individualized Constant Proportion Portfolio Insurance products

Constant proportion portfolio insurance is a dynamic asset allocation strategy between two pools of assets : a risky basket that is intended to provide the returns, and a safe basket that provides some level of a predefined capital protection. The percentage allocated to each depends on the "cushion" value, defined as (current portfolio value – floor value), and a multiplier coefficient, where a higher number denotes a more aggressive strategy.

<footnote n="2">2. LIMRA is an organization that conducts research on distribution systems for the life and health insurance products on behalf of its member companies.</footnote>
<table><caption>FIGURE 2 – 2015 year-end U.S. Individual Annuity Sales given in thousands of dollars. Source: LIMRA Secure Retirement Institute U.S. Individual Annuities Sales Survey</caption><thead><tr><th>Rank</th><th>Company name</th><th>Total</th><th>Company name</th><th>Variable</th><th>Company name</th><th>Fixed</th></tr></thead><tbody><tr><td>1</td><td>Jackson National Life</td><td>24,491,828</td><td>Jackson National Life</td><td>23,109,447</td><td>Allianz Life of North America</td><td>8,773,123</td></tr><tr><td>2</td><td>AIG Companies</td><td>19,999,606</td><td>TIAA</td><td>12,752,518</td><td>New York Life</td><td>8,644,037</td></tr><tr><td>3</td><td>Lincoln Financial Group</td><td>14,638,405</td><td>Lincoln Financial Group</td><td>11,507,596</td><td>AIG Companies</td><td>8,508,110</td></tr><tr><td>4</td><td>TIAA</td><td>12,752,518</td><td>AIG Companies</td><td>11,491,496</td><td>American Equity Investment Life</td><td>7,083,967</td></tr><tr><td>5</td><td>New York Life</td><td>12,015,254</td><td>AXA US</td><td>9,848,026</td><td>Forethought Annuity</td><td>5,258,481</td></tr><tr><td>6</td><td>Allianz Life of North America</td><td>10,783,660</td><td>Prudential Annuities</td><td>8,722,770</td><td>Symetra Financial</td><td>4,085,009</td></tr><tr><td>7</td><td>MetLife</td><td>10,149,277</td><td>Transamerica</td><td>7,786,784</td><td>Great American</td><td>4,061,020</td></tr><tr><td>8</td><td>AXA US</td><td>9,875,961</td><td>MetLife</td><td>7,046,118</td><td>Nationwide</td><td>3,281,000</td></tr><tr><td>9</td><td>Prudential Annuities</td><td>9,539,028</td><td>Nationwide</td><td>5,374,000</td><td>Lincoln Financial Group</td><td>3,130,809</td></tr><tr><td>10</td><td>Nationwide</td><td>8,655,000</td><td>RiverSource Life Insurance</td><td></td><td>MetLife</td><td>3,103,160</td></tr><tr><td>11</td><td>Transamerica</td><td>7,882,188</td><td>Pacific Life</td><td>3,655,793</td><td>Midland National</td><td>3,056,870</td></tr><tr><td>12</td><td>American Equity Investment Life</td><td>7,083,967</td><td>New York Life</td><td>3,371,217</td><td>Pacific Life</td><td>2,931,193</td></tr><tr><td>13</td><td>Pacific Life</td><td>6,586,985</td><td>Thrivent Financial for Lutherans</td><td>3,312,568</td><td>Athene Annuity &amp; Life</td><td>2,477,932</td></tr><tr><td>14</td><td>Forethought Annuity</td><td>6,452,977</td><td>Allianz Life of North America</td><td>2,010,537</td><td>Principal Financial Group</td><td>2,272,371</td></tr><tr><td>15</td><td>RiverSource Life Insurance</td><td>5,522,156</td><td>Fidelity Investments Life</td><td>2,007,503</td><td>Fidelity &amp; Guaranty Life</td><td>2,059,554</td></tr><tr><td>16</td><td>Symetra Financial</td><td>4,113,791</td><td>Ohio National Life</td><td>1,952,352</td><td>Security Benefit Life</td><td>1,963,292</td></tr><tr><td>17</td><td>Great American</td><td>4,094,123</td><td>Northwestern Mutual Life</td><td>1,475,214</td><td>North American Company for Life and Health</td><td>1,951,044</td></tr><tr><td>18</td><td>Thrivent Financial for Lutherans</td><td>3,862,528</td><td>Forethought Annuity</td><td>1,194,495</td><td>EquiTrust Life</td><td>1,945,253</td></tr><tr><td>19</td><td>Midland National</td><td>3,436,721</td><td>Protective Life</td><td>1,155,228</td><td>Voya Financial</td><td>1,802,508</td></tr><tr><td>20</td><td>Principal Financial Group</td><td>3,232,160</td><td>Massachusetts Mutual Life</td><td>1,064,262</td><td>Western Southern Group</td><td>1,662,688</td></tr><tr><td></td><td>Top 20</td><td>$185,168,133</td><td></td><td>$124,066,116</td><td></td><td>$78,051,421</td></tr><tr><td></td><td>Total industry</td><td>$236,677,000</td><td></td><td>$133,000,000</td><td></td><td>$103,677,000</td></tr><tr><td></td><td>Top 20 share</td><td>78%</td><td></td><td>93%</td><td></td><td>75%</td></tr></tbody></table>

In rising markets, the asset allocation mechanism (which is typically algorithmic and fully deterministic) is designed to either i) maintain maximum permitted exposure to the risky basket or ii) increase its exposure whenever the maximum permitted allocation has not been reached. In falling markets, the asset allocation mechanism will allocate more to the safe basket and in extreme markets will "monetize", i.e. allocate 100% of the portfolio value to the safe basket to ensure the guarantee at maturity.

In managing the portfolio assets in this manner, the CPPI asset allocation mechanism aims (but does not guarantee) to provide returns via the risky assets subject to meeting the predefined capital protection constant. Such mechanism, however, is also subject to major risks, such as high transaction costs, market liquidity risk, discontinuous price process, or unexpected changes in the volatility of the underlying stocks, which might imply a failure of the strategy, see [148]. This also includes the risk of monetization, which implies that a reallocation into the risky asset and a further participation in market upturns are no longer possible.

Individualized CPPI (iCPPI) employs the exact same mechanism as CPPI to the individual policyholder level rather than a pooled set of policies invested in the same CPPI fund, thereby allowing a life insurance company to provide protected / guaranteed solutions fully tailored to the individual policyholder. Furthermore, the market risk associated with providing such guarantees can be precisely and fully hedged out to a third party (typically an investment bank with iCPPI capabilities), leaving the life company with the residual actuarial risk that it is best positioned to manage.
# Part I : Policyholder behavior in variable annuities and iCPPI gap risk

In variable annuities, policyholders' behavior is a major risk for the insurer, and a complex issue that affects life insurance industry in almost every aspect; product design, pricing, marketing and distribution, financial reporting and risk management. Insurers' concern about policyholders behaviors risk is not new, and is still in the early stages of understanding and modeling. The first two chapters of this first part deal with this risk. We address the rational lapse behavior in GMABs, and optimal withdrawals in GMIBs. The third chapter is dedicated to the iCPPI product from a managing and hedging point of view.

## 1 Chapter 1 - Financial risk management and the rational lapse strategy in life insurance policies

In Chapter 1, we address the problem of pricing a GMAB contract under rational lapse assumption. A rational lapse assumes the policyholder behaves rationally through maximizing a certain criterion. In our case, we take the expected value.

We consider a policyholder possessing a GMAB contract defined in the preliminary section. The payoff of the guarantee is expressed as follows :

$$H(T, A(T), G) = \max(A(T), G) = A(T) + (G - A(T))^{+},$$

where $A(T)$ is the account value at maturity, $T$ denotes the maturity of the contract, and $G$ the guaranteed minimum return, also known as the benefit base.

We assume that the account value $A$ is invested in a single underlying asset, denoted by $S = (S(t))_{t \ge 0}$, following the Black-Scholes framework. The short term interest rate $r = r(t)_{t \ge 0}$ is driven by the one factor Hull and White model :

$$\begin{cases} dS(t) = r(t)S(t)dt + \sigma S(t)dW(t) \\ dr(t) = a(\theta(t) - r(t))dt + \sigma_r dZ(t), \quad Z := (1 - \rho^2)^{\frac{1}{2}}W^{\perp} + \rho W, \end{cases}$$

where $a$ and $\sigma_r$ are positive constants, $\theta$ is a deterministic Lebesgue-integrable function, $\sigma$ is the instantaneous volatility of the asset return and $Z$ and $W$ are standard Brownian motions with correlation $\rho$.

We assume that annual fees $\alpha$ are deducted continuously from the policyholder's account value $A$. It is given by $A(t) = e^{-\alpha t}S(t)$ and can be expressed through the following stochastic differential equation :

$$dA(t) = (r(t) - \alpha)A(t)dt + \sigma A(t)dW(t),$$

where $\alpha$ is a constant corresponding to the total fees deducted from the account value.

If we assume that the policyholder does not lapse the contract prior to its maturity $T$, the guarantee becomes a European-style option. In this case, we can use the formula given in [102] to express the forward liability $\tilde{v}^E$ of GMAB contract as :

$$\tilde{v}^E(t, F^T(t)) = e^{-\alpha(T-t)}F^T(t) + GN(-d_2) - e^{-\alpha(T-t)}F^T(t)N(-d_1), \quad d_{1,2} = \frac{\log(F^T(t)/G) - \alpha(T-t)}{\Gamma} \pm \frac{\Gamma}{2}.$$
where $F(t)^T$ is the forward value of $A(t)$ at time $T$ observed at time $t$, $\Gamma = \sqrt{\int_0^{T-t} \omega_s^2 ds}$, $\omega_s^2 = \sigma^2 + v^2 B_r^2(s) + 2\rho\sigma v B_r(s)$, $B_r(s) = \frac{1-e^{-as}}{a}$ and $N(.)$ is the cumulative distribution function of the standard normal density.

Unlike European-style options, GMAB contracts allow the policyholder to get his account value before the maturity. As a consequence, a zero lapse assumption is not consistent with market practice. We observe that the lapse rate changes significantly in different market conditions (equity and interest rate level). Such variability has a notable impact on the liability value and the insurer's hedging strategy. This is known as the rational lapse strategy, and is similar to the optimal early-exercise strategy of classical Bermudian options, see [36].

**Question.** Assuming the policyholder can lapse the contract at the end of each policy year, how can we define a rational strategy?

Let $0 = t_0 < \dots < t_n < \dots < t_N = T$, $n = 1, 2, \dots$, $N$ be the policy years. Lapses can take place at each anniversary date but not between two successive ones. We denote by $t_n^-$ the time right before an anniversary date $t_n$, i.e. before the policyholder's decision to stay in the contract or lapse.

Due to the similarity to the optimal early-exercise strategy of classical Bermudian options, the value of the Bermudian-style liability $\tilde{v}^B$ under the forward measure $\mathbb{Q}^T$ satisfies the following equation :

$$ \tilde{v}^B(t_n^-, F^T(t_n^-)) := \max(F^T(t_n), \mathbb{E}^{\mathbb{Q}^T}[\tilde{v}^B(t_{n+1}^-, F^T(t_{n+1}^-)) | \mathcal{F}_{t_n}]), $$

where the terminal condition is

$$ \tilde{v}^B(T, F^T(T)) = F^T(T) + (G - F^T(T))^+. $$

**Result 1.** Assuming the benefit base $G$ is fixed at inception, $\tilde{v}^B(t, f)$ is convex and nondecreasing w.r.t the forward price $f$, thus, for each $t_n$ $n = 1, \dots, N$ there exists a real number $f^*(t_n^−)$ such that

$$ \begin{aligned} 0 \le f < f^*(t_n^-) &\Rightarrow \tilde{v}^B(t_n^-, f) > f \quad (\text{No Lapse}), \\ f \ge f^*(t_n^-) &\Rightarrow \tilde{v}^B(t_n^-, f) = f \quad (\text{Lapse}). \end{aligned} $$

$f^*(t_n^-)$ is referred to as the "critical forward account value" since the policy should be lapsed as soon as the forward account value increases to this level at time $t_n$. The existence of such critical value results in an increase of the value of the liability compared to the European-style one. As a consequence, the rational lapse case is the worst case scenario the insurer can face, therefore, it is crucial to take it into account.

To evaluate the liability $\tilde{v}$ and the critical forward value $f^*$ of a single GMAB rider, we use two numerical methods : PDE schemes and Monte Carlo simulations. To our knowledge, this study is the first to apply these methods for a GMAB contract. Numerical tests, detailed in the chapter, show not only the consistency between the PDE and Monte Carlo methods, but also the precision of both methods. We also find that PDE method is faster and more precise than Monte Carlo. Moreover, it can calculate the price and other important Greeks for different $f$ and $t$ at the same time. On the other hand, Monte Carlo method is more flexible, easier to implement and can be extended to other high-dimensional problems.

Insurers are aware that they can neither rely on zero lapse strategy, nor consider a full rational behavior. In fact, they deal with an inhomogeneous pool of policies that can be terminated for various reasons which are not necessarily rational, and others that remain until their expiration. In this case, the lapse strategy can be represented by the frequency of the policies that are early terminated at a given anniversary date $t_n$. The goal is to find a way to estimate a "reasonable" lapse rate.
**Question.** How can we use the rational strategy results to estimate a "reasonable" lapse frequency?

Assume we want to predict policyholders lapses at a future anniversary date $t_n$. We can either use past observations and a regression model to predict future lapses, or use the results based on rational lapse strategy given present market observations. The first approach can underestimate lapse risk, while the second one focuses on market information and might be too extreme.

We denote by $p(t_n)$ the proportion of lapses at $t_n$. The rational lapse strategy allows us to express the proportion $p(t_n)$ as a deterministic function $h$ of the forward account value $F^T(t_n)$, that is $h(F^T(t_n)) = \mathbb{1}_{\{F^T(t_n) \ge f^*(t_n^-)\}}$. This implies that all policyholders lapse the contract once $F^T(t_n)$ touches the critical boundary $f^*(t_n^-)$, and hold the policy otherwise.

**Result 2.** Inspired by the mortgage prepayment models and evaluation approaches of surrender options for other life insurance products, we assume the lapse function is nonincreasing piece-wise linear of the value $F^T(t_i)$. When $F^T(t_i) < f^*(t_i^-)$, the lapse rate is not zero due the policyholder circumstances, and when $F^T(t_i) \ge f^*(t_i^-)$, some rational lapses never occur. Therefore we present a reasonable lapse frequency given by Figure 3.

<figure><img src="image_3.png" /><figcaption>FIGURE 3 – Comparing the rational lapse function with the reasonable lapse function.</figcaption></figure>

*The four parameters F₁, F₂, Pmin and Pmax are chosen by the insurer to match some empirical tests.*

By adopting this framework, insurers are able to incorporate policyholders rationality into their lapse assumptions, instead of limiting their modeling to empirical estimations, which happen to be quite reckless.

**2 Chapter 2 - Optimal behavior strategy in Guaranteed Minimum Income Benefit**

Rational behavior for GMAB products is limited to lapsing the contract. Other products are subject to optimal behaviors that include withdrawing money without necessarily terminating the contract, i.e. partial withdrawals. These so-called optimal withdrawals are for example experienced in GMWBs and GMIBs, i.e. see the preliminary section for details on these products. Many authors address this problem in the case of GMWBs, see for example [61, 110, 131, 143, 149]. In this chapter, we are interested in GMIBs which has got less attention because they were thought to be "safer". However, due to increasing guarantees and features, GMIBs became riskier and more exposed to optimal behaviors.

Consider an *x*-year old policyholder possessing a GMIB rider with an income benefit in case the insured is alive and the possibility of a death benefit in case he dies. Again, the account value A of the GMIB is invested in a single underlying asset, denoted by $S_t$ following the Black-Scholes framework. Due to the complexity of the contract, we consider the simple case of constant interest rate.
Let $0 = t_0 < \dots < t_n < \dots < t_N = T$, $n=1,2,\dots,N$ be the policy years, where $t_0$ is the contract inception and $t_N = T$ its maturity. All events related to the contract are considered to take place at these dates and not between two consecutive ones, i.e. evolution of the benefit base, withdrawals, lapse, payments, etc...

The value of the GMIB contract at time $t$ is determined by three main state variables :

- The account value $A_t$ which represents the wealth of the policyholder and evolves according to $S_t$ performance.

- The benefit base $G$ which is updated at anniversary dates by following a roll-up, i.e. $G_{t_n} = (1 + \eta)G_{t_{n-1}}$, a ratchet, i.e. $G_{t_n} = \max(G_{t_{n-1}}, A_{t_n})$, or by combining the two, i.e. $G_{t_n} = \max((1+\eta)G_{t_{n-1}}, A_{t_n})$. The roll-up rate $\eta$ is a constant fixed at inception by the insurer.

- A discrete state variable $I_n = \{0, 1\}$ which informs if the policyholder died between $(t_{n-1}, t_n]$ or is still alive at time $t_n$. Therefore, we denote the death probability during $(t_{n-1}, t_n]$ as $q_n = P(I_n = 0 | I_{n-1} = 1)$, and the probability of being alive at time $t_n$ as $p_n$. Note that $q_n$ and $p_n$ depend on the age of the contract holder at time $t_n$ and thus on the age $x$ at $t_0 = 0$.

At time 0, $A_0$ and $G_0$ are set equal to the upfront premium. We denote by $\gamma_n$ withdrawals at time $t_{n=1,\dots,N-1}$, and the values of the benefit base (resp. account value) just before and after events take place at $t_n$ by $G_{t_n^-}$ and $G_{t_n^+}$ (resp. $A_{t_n^-}$ and $A_{t_n^+}$).

Let $t_n$ be an arbitrary policy year. We consider the following modeling notations and assumptions :

- Fees deducted from both the account value and benefit base between $t$ and $t+dt$ for $t \in (t_{n-1}, t_n]$ are expressed as :

$$ \alpha_{tot}(A_t, G_t)dt = (\alpha^A A_t + \alpha^G G_{t_{n-1}^+})dt, $$

where $\alpha^A$ and $\alpha^G$ are constants defined by the insurer at inception.

- At an arbitrary anniversary date $t_n$, the contract guaranteed withdrawal is a proportion of the benefit base at time $t_n^-$, i.e. $\gamma_n^{gua} = \eta G_{t_n^-}$, where $\eta$ is the guaranteed rate (which is the same as the roll-up rate).

- Upon withdrawing an amount $\gamma_n$, the account value and benefit base are reduced. Their evolution depends on both their values before the withdrawal and the withdrawn amount. We express those jump conditions as :

$$ A_{t_n^+} := h^A(A_{t_n^-}, G_{t_n^-}, \gamma_n), $$

$$ G_{t_n^+} := h^G(A_{t_n^-}, G_{t_n^-}, \gamma_n), $$

where the withdrawal amount $\gamma_n$ belongs to an admissible space $\mathcal{A}_n(A_{t_n^-}, G_{t_n^-})$, and $h^A(\cdot)$ (resp. $h^G(\cdot)$) is some function that determines the change in the account value (resp. benefit base) subject to withdrawing $\gamma_n$. The change in $A$ and $G$ upon withdrawing $\gamma_n$ may include penalties for excess withdrawals, i.e. withdrawals that exceed the guaranteed amount.

- The income benefit can be activated at any time between a starting date fixed by the insurer $t_1$, and the maturity $T$, i.e. typically the policyholder is given the choice to start his income period between the 10th anniversary of the contract and the 85th or 95th policyholder's birthday.

- If the policyholder dies during $(t_{n-1}, t_n]$, the contract allows for a payout of the death benefit $D(t_n, A_{t_n^-}, G_{t_n^-})$ to the beneficiary at $t_n$.

- The cash flow received by the policyholder at $t_n$ is a function of the current account value, benefit base, and withdrawn amount $f_n(A_{t_n^-}, G_{t_n^-}, \gamma_n)$. It is usually equal to the withdrawn amount.
The specifications vary across different proposed products and different insurers and are quite painful to extract from the long product specifications document. Moreover, the academic literature usually presents different specifications for GMIBs than those commercialized.

We are interested in the worst case scenario from an insurer point of view in terms of policyholders behavior. This corresponds to the strategy that maximizes the expected value of their discounted future cash flows. Therefore, we will consider this pricing strategy to define a fair price and extract the optimal withdrawals at each anniversary date for a given state, i.e. level of the account value and benefit base. The following assumptions are considered :

- Financial risk can be eliminated by continuous hedging.

- Mortality risk is fully diversified via selling the contract to many people of the same age.

- Financial risk and mortality risk are independent.

By exploiting the Markovian property of the state variables, and taking the expectation w.r.t mortality, we can calculate the price under the optimal strategy. This is justified by the fact that the financial risk and mortality are independent<fnref n="3" />, and that the policyholder's decision does not affect mortality.

**Result 3.** The value function $\Phi$ which solves the optimal withdrawals problem is given by the following explicit recursion

$$ \Phi(t_n^+, A, G) = \mathbb{E}_{t_n^+} [B_{n,n+1} \Phi(t_{n+1}^-, A, t_{n+1}^-, G, t_{n+1}^-) | A, G], \\ \Phi(t_n^-, A, G) = \max_{\gamma_n \in \mathcal{A}_n} \left( \Phi(t_n^+, h^A(A, G, \gamma_n), h^G(A, G, \gamma_n)) + p_n \gamma_n + p_{n-1} q_n D(t_n^+, A, G) \right), $$

where $B_{n,n+1}$ is the actualization factor between $t_n$ and $t_{n+1}$ and $\mathcal{A}_n$ is the set of admissible strategies.

**Remark 1.** The first equation translates the transition between $t_{n+1}^-$ and $t_n^+$ backwards, while the second one provides the jump condition upon choosing between starting the income benefit, or staying in the contract. In the second case, the withdrawal is weighted by the probability that the insurer is alive at time $t_n$, i.e. $p_n$, and the death payout by the probability that he has lived up to $t_{n-1}$ and dies in the interval $(t_{n-1}, t_n]$, i.e. $p_{n-1}q_n$. A linear search of the optimum is performed over the set of withdrawal strategies, which allows to find the optimal strategy $\gamma_n^*$ at each time $t_n$ for each account value A and benefit base G.

The solution of the optimal withdrawals problem allows the insurer to mitigate policyholders' behavior risk related to GMIB contracts. Our analysis points out the following observations :

- The optimal withdrawal strategy of the policyholder is limited to 4 possibilities; zero withdrawal, guaranteed withdrawal, income benefit election and lapse. Intermediate withdrawals do not seem to be among those decisions.

- Given the fees levels, the contract seems to be underpriced under the optimal withdrawal strategy for most cases. Either increasing the fees or adjusting the roll-up rate can be a solution to overcome this issue.

# 3 Chapter 3 - Managing gap risks in iCPPI for life insurance companies : a risk return cost analysis

In order to diversify their products, improve their attractiveness, and reduce their capital requirement, insurance companies offer a range of savings other than variable annuities. The individualized

<footnote n="3">3. It is a common hypothesis in life insurance industry to consider that financial and demographic risks are independent to price variable annuities.</footnote>
constant proportion portfolio insurance (iCPPI) is one of them.

As mentioned in the preliminary section, an iCPPI product is based on the CPPI mechanism. It consists in decomposing the portfolio value $V_t$, at each time $t$, into a sum of a risky and non-risky exposure. Given a multiplier $m$, the amount invested in the risky asset is leveraged, such that, at each time $t$, the risky exposure $e_t$ is equal to $m(V_t - F_t)$, where the "floor" $F_t$ is the actualized guarantee G at time $t$, i.e. $F_t = Ge^{-r(T-t)}$. When the portfolio value "breaches" the floor, i.e. $V_t \le F_t$, all its assets are switched to the non-risky part. Ideally, by doing so, the insurer is guaranteed to recover G at maturity.

The portfolio rebalancing occurs at discrete times, i.e. on a daily or weekly basis, and the value of the fund can be defined using a recursive formula :

$$V_{t_{k+1}} - F_{t_{k+1}} = \begin{cases} (V_{t_k} - F_{t_k}) \left( \frac{S_{t_{k+1}}}{S_{t_k}} - (m-1)e^{r \frac{T}{N}} \right) & \text{if } V_{t_k} - F_{t_k} > 0 \\ (V_{t_k} - F_{t_k})e^{r \frac{T}{N}} & \text{if } V_{t_k} - F_{t_k} \le 0. \end{cases}$$

We define the probability of breaching the floor as the probability that the portfolio final value falls below the guarantee, i.e. $P^{BF} := \mathbb{P}(V_T \le G)$. This probability is null for continuous rebalancing but is strictly positive in practice. Moreover, it depends on the multiplier $m$, the change in the asset price between two rebalancing dates, the rebalancing frequency, and the interest rate.

If the underlying asset has jumps, we can no longer control the probability of breaching the floor, and the insurer can be subject to "gap risk", which occurs when the underlying asset has a negative jump. We study two solutions :

- Adjusting the multiplier $m$ to reduce the probability of breaching the floor.

- Hedging strategies in case the underlying asset has jumps.

**Question.** How can the insurer reduce the probability $P^{BF}$ of breaching the floor in case of volatile market?

This issue suggests choosing a time-varying multiplier that adapts to market conditions, namely the volatility, drift and interest rate level. A few studies were held in this sense to define a dynamic multiplier. Among them, the long-term risk sensitive portfolio optimization, see [93], gives a multiplier under which the strategy is optimal. This multiplier is equal to the excess return of the strategy divided by the variance. Following Merton's optimum consumption and portfolio rules, see [125], one can define a multiplier based on the optimal certainty equivalent returns (CERs) approach using HARA utilities. These multipliers are, however, very small compared to the ones used in practice, reducing the upside potential of the risky asset. Our approach suggests choosing a value-at-risk based multiplier in which the investor defines his risk tolerance.

**Result 4.** Based on the "value-at-risk based portfolio insurance" (VBPI) introduced in [106], our dynamic multiplier is expressed by

$$m_t = \frac{1}{1 - \exp\left(\left(\mu - r - \frac{1}{2}\sigma^2\right)(T-t) - z_p\sigma\sqrt{T-t}\right)},$$

where $z_p = \phi^{-1}(p)$ is the quantile function of the standard normal distribution.

Such multiplier allows the manager to adjust the drift and volatility in this expression to the market risk and return. As shown through backtestings, dynamic-CPPI performs better than a classical CPPI with fixed multiplier.
However, the dynamic multiplier does not totally eliminate the risk of sudden negative jumps observed in the market. To incorporate jumps in the asset price dynamic, we use Lévy processes. For this analysis we consider the Kou model introduced in [112].

The dynamic of the asset price $S_t$ is given by :

$$ \frac{dS_t}{S_t} = \mu dt + \sigma dW_t + d\left(\sum_{i=1}^{N_t} (e^{Y_i} - 1)\right), $$

where $\mu$ and $\sigma$ are constants, $W_t$ is a standard Brownian motion, $N_t$ is a Poisson process with rate $\lambda$, and $Y_i$ a sequence of independent identically distributed (i.d.d.) non-negative random variables with a double exponential distribution with density :

$$ f_Y(y) = q\eta^+ e^{-\eta^+ y} \mathbb{1}_{y \ge 0} + p\eta^- e^{-\eta^- y} \mathbb{1}_{y < 0}, $$

where $\eta^+ > 1$, $\eta^- > 0$ and $p, q, p+q=1$, represent the probabilities of upward and downward jumps.

**Question.** How can the manager deal with rapid downside movements?

**Result 5.** Rapid downside market movements, known as "gaps", can be hedged using :

- Semi-static hedging with vanilla put options.

- Static hedging with gap options.

These solutions were analyzed in [57]. The first one consists in buying vanilla puts at each rebalancing date $t_k$, maturing the following one $t_{k+1}$ with strike $(1-1/m)e^{rT/NS_{t_k}}$. The insurer can deduct the hedging costs from the portfolio final value and, in this case, they represent the sum of all the put options required for the whole period :

$$ \bar{c} = \sum_{k=0}^{n-1} m \frac{C_{t_k}}{S_{t_k}} \mathbb{E}^{\mathbb{Q}} \left[ \left( (1-1/m)e^{rT/n} S_{t_k} - S_{t_{k+1}} \right) \right], $$

where the notation $C_{t_k}$ is used for the cushion $V_{t_k} - F_{t_k}$ at time $t_k$.

In practice, the portfolio is self-financed and the price of the put used for the hedge is deducted directly from the portfolio's value. This is expressed through an adjusted cushion $\tilde{C}_t$ given by the following recursive formula :

$$ \tilde{C}_{t_{k+1}} = e^{-rT/N} \tilde{C}_{t_k} \frac{\left( m \frac{S_{t_{k+1}}}{S_{t_k}} + (1-m)e^{rT/N} \right)^+}{\mathbb{E}^{\mathbb{Q}} \left[ \left( m \frac{S_{t_{k+1}}}{S_{t_k}} + (1-m)e^{rT/N} \right)^+ \mid \mathcal{F}_t \right]} $$

In practice, using options can be hardly applicable for the following reasons :

- The maturity is of 1 to 5 days and options reaching maturity are very volatile or illiquid.

- The insurer usually uses "exotic" underlying assets like funds or funds of funds, or detains a mixed portfolio of different funds, indexes, bonds etc...

- The cost of the strategy can be very high due to the number of options needed for the whole period.
Gap options are derivatives which allow for a protection against sudden significant downside moves of an underlying asset. If a gap event occurs between two consecutive dates, i.e. the asset performance is below a certain level called trigger, the option is exercised and the buyer receives the difference between the performance of the underlying asset, and a fixed threshold. These options are particularly useful in the case of iCPPI products. Indeed, through dynamic allocation strategy, the insurer's protection only fails if $m \frac{S_{t_{k+1}}}{S_{t_k}} + (1-m)e^{rT/N} < 0$. By choosing a put gap option, i.e. with payoff $f(x) = (K-x)^+$, and a strike $K = (1-1/m)e^{rT/N}$, the insurer is protected against gap events and ensured to recover the guarantee at maturity.

Our numerical analysis is based on the Kou model under some mild assumptions. The gap option price has the following reduced closed form :

$$G_h \approx \frac{\lambda p \eta^{-}}{1 + \eta^{-}} K^{1+1/\eta^{-}} \frac{1 - e^{-T(r+\lambda p e^{\beta/\eta^{-}})}}{r + \lambda p e^{\beta/\eta^{-}}}$$

where $p$ is the probability that a jump is negative, $\eta^{-}$ its intensity, $\lambda$ the Poisson process rate, $h = T/N$ is the time step, and $\beta$ the log return level which triggers the gap option.

The comparison between hedging using vanilla and gap options reveals that, even though both are equivalent, the second choice is less costly. Indeed, gap options are priced over-the-counter based on very rare events. Furthermore, while put options need to be bought at each rebalancing date with the following one as their maturity, and use a floating strike, gap options need to be acquired at time 0 for the whole period with a fixed strike.

## Part II: Focus on market impact and volatility

Insurance products like variable annuities are characterized by three main features : long-term duration, large volumes, and significant market risk exposure. Large volumes means a greater risk of impacting the market when executing a large order, for hedging purposes for example. Market risk exposure and long-term duration induce an important volatility risk, which, in light of all the recent developments in volatility modeling, suggests a need to revisit some of the fundamentals of volatility and its properties.

In this second part of the thesis, we address these two problems; market impact and volatility. Before exposing the analyses and results for each of these subjects, we give a brief review and motivate the choice for these problematics. In Part II-A, which addresses the optimal execution of a large book of options, we give a review of the current practices in hedging variable annuities and long-term equity-linked products.

## Part II-A: Hedging life insurance products and the market impact dilemma

Guarantees in variable annuities are similar to options. The underlying asset price is the fund value, and the insurer plays the role of the option writer. However, unlike options where the premium is paid upfront, the costs of variable annuities are paid periodically as a percentage of the account value throughout the life of the contract. The fees collected should then be partially used to hedge the provided guarantees. Hedging variable annuities can be done via dynamic hedging, static hedging or semi-static hedging.
- **Dynamic hedging**: It is the most common approach to hedge financial risks in variable annuities embedded guarantees, see [109] and [110]. The principal of dynamically hedging a guarantee that depends on a tradable asset $S_t$, is to hold, at any time $t$, $\Delta_t$ shares of the underlying asset. In a complete market with continuous rebalancing and no transaction costs, the guarantee can be perfectly hedged in a self-financing manner.

- **Static hedging**: Investigated by [96] and [123], it suggests replicating the embedded guarantees with a static position in put options. The agent takes position, at inception, in a portfolio of financial instruments available in the market, so that the variable annuities cash flows can be replicated and match the cash flows of the hedge portfolio. Once the position set, static hedging assumes no intermediary costs between the contract's inception and maturity. Consequently, static hedging is considered to be highly robust, model independent, and does not involve any rebalancing throughout the life of the product. Unfortunately, this approach is far from being perfect. In fact, long-term options with maturities of at least 10 years are not available, illiquid, or subject to counterparty risk. Moreover, most guarantees are path-dependent and vanilla options are not suitable to hedge them. Last but not least, variable annuities fees are collected periodically, which makes it hard to match future fees and the amount borrowed at the inception to purchase the hedge.

- **Semi-static hedging**: Similar to static hedging, semi-static hedging exploits available financial instruments to hedge the guarantee. Instead of fixing the hedging portfolio from the contract inception until its maturity, here the insurer constructs a hedging portfolio at each rebalancing date by following an optimal hedging strategy for some optimality criterion. Several authors have studied this approach, see [25, 53, 54, 111]. In particular, static-hedging is investigated for a GMDB contract with ratchet features in [53, 54]. Under certain assumptions, the GMDB rider with a ratchet feature can be seen as a lookback option. They show that using vanilla options to hedge the guarantee can be significantly more effective than delta-hedging. [111] propose to hedge a path-dependent option by taking a position in an optimally chosen European option. They present results of a simulation study of a version of the GMWB product and use local risk minimization as the optimality criterion for each hedging date.

## A review of derivatives holdings in the U.S. insurance industry

In its review of derivatives holdings and exposure trends in the U.S. insurance industry, the NAIC's Capital Markets Bureau<fnref n="4" /> reports a $2 trillion total notional value of derivatives over the year-end 2014. An overwhelming 94% of the total notional value is used for hedging purposes. Out of the 94%, 49% is related to interest rate hedges, while 25% is aimed at hedging equity risk. Swaps accounted for the largest share (50%) of total notional value, followed by options (44%), futures (3%) and forwards (3%). After several consecutive years of increase, U.S. insurer's derivatives leveled off in 2015 in terms of notional value.

As given in Figure 4, derivatives exposure in BACV as of Dec. 31, 2015, totaled $55 billion, accounting 1% of total cash and invested assets, and representing a decrease of 4% from year-end 2014. In the life industry segment, only 18% of insurance companies had derivatives exposure. However, those involved with derivatives are larger, accounting for $3.28 trillion, or 87% of the segment total. Derivatives positions can be quite large; the average position size was $26.4 million. The largest single position

<footnote n="4">4. The NAIC's Capital Markets Bureau monitors developments in the capital markets globally and analyzes their potential impact on the investment portfolios of U.S. insurance companies. They published several reports concerning derivatives. These reports provide insight into exposure trends, credit default swaps, hedging, reporting requirements, and market developments. In particular, they review U.S. insurer's derivatives holdings and exposure trends. A list of archived Capital Markets Bureau Special Reports is available via: http://www.naic.org/capital_markets_archive_index.htm</footnote>
open at Dec. 31, 2015, in terms of notional value, was $10 billion corridor option<fnref n="5" /> that expires in 2021, which was purchased as an interest rate hedge for a company's fixed income portfolio.

<figure><img src="image_4.png" /><figcaption>FIGURE 4 - U.S. insurance industry derivatives exposure as of Dec. 31, 2015. Source: NAIC</figcaption></figure>

# 4 Chapter 4 - Optimal execution of options book under market impact

Market impact is defined as the effect that a market participant has when he buys or sells a given volume of an asset. In that respect, it is important that insurance companies, who trade large quantities of options for hedging purposes, take into account this effect when trading options by splitting trades intelligently to minimize its cost.

Inspired by the large literature on equities, and particularly the work of Almgren and Chriss [7], we tackle in this chapter the optimal execution of a book of options under market impact. To do so, we consider a universe with two agents: the option market maker who sells the option, and the end-user who buys the option to hedge an external risk.

Following the steps of usual optimal execution problems with market impact, our approach is three steps; first we need to define the option price under market impact constraints. Second, we choose the risk criterion and derive the optimal execution problem. Finally, depending on the complexity of the formulation, we use an appropriate method to solve the optimization problem.

**Question.** How can we define an option price that incorporates the execution quantity?

Our approach is inspired by the feedback effect and option replication with transaction costs see [78] and [115]. The market maker who dynamically hedges the option to the first order, i.e. delta-hedging, moves the underlying asset price accordingly. Our assumption is based on a linear impact function, i.e. buying x shares drives up the asset price by a cost proportional to x, and selling x shares drives it down by the same proportion cost.

Following Leland's transaction costs framework, see [115], we can derive a price for the option under market impact constraints. For simplicity, we assume that interest rates are null. Furthermore, we assume that the asset price is modeled by a geometric Brownian motion in the absence of market impact.

<footnote n="5">5. A corridor option is a derivative whose payoff at maturity depends on the amount of time a specified spot rate remains within a specified range during the option's life.</footnote>
**Result 6.** The option execution price under market impact constraints can be expressed through a Black-Scholes like PDE with an enlarged volatility $\tilde{\sigma}$ such that:

$$ \tilde{\sigma}^2 = \sigma^2 + f(t, \dot{x}_t, x_t, \sigma), $$

where $f$ is the market impact function which depends on time $t$, volatility $\sigma$, inventory $x_t$ and trading speed $\dot{x}_t$.

The previous formulation can be generalized to local volatility by following Lepinette's arguments, see [117]. We can also use a simple Taylor approximation and Black-Scholes closed formula to rewrite the option execution price as the sum of the option price without market impact and a linear impact term:

$$ \tilde{P}(t, S_t, \dot{x}_t, x_t) = P(t, S_t) + \frac{1}{2} \{\tilde{\eta}\dot{x}_t + \tilde{\gamma}(x_t - x_0)\} \sigma S_t^2 (\hat{T} - t)^{3/2} \Gamma(t, S_t), $$

where $\tilde{\eta}$ and $\tilde{\gamma}$ are constant parameters associated with the market impact temporary and permanent components$^6$, $\hat{T}$ is the option maturity and $\Gamma$ its second derivative w.r.t asset price.

Note that the option gamma is known to explode close to maturity. However, our framework assumes the trades are performed far from the option expiry, i.e. $T \ll \hat{T}$, which removes any undesirable irregularities in the option price or the impact function.

Let us consider a trade execution strategy in which an initial long position of X options, with fixed strike K and maturity $\hat{T}$, is liquidated by a fixed time horizon $[0,T]$, where $T \ll \hat{T}$ is the end time. The asset position $x_t$ is nonincreasing with $x_0 = X < 0$ and $x_{T^+} = 0$ for a pure buy strategy. The optimal strategy is based on the mean-variance criterion $E[\mathcal{C}(x)] + \lambda \forall ar[\mathcal{C}(x)]$. It is reduced to the simple expected cost for $\lambda = 0$, which has an explicit solution under our hypotheses.

**Result 7.** Assuming only temporary impact, i.e. $\tilde{\gamma} = 0$, and under the Black-Scholes framework, the optimal strategy $x^*$ resulting in minimizing the expected cost is given by:

$$ \dot{x}^*(t) = \frac{K_1}{(\hat{T} - t)^{3/2}} \\ x^*(t) = \frac{K_1}{(\hat{T} - t)^{1/2}} + K_2 $$

$$ \text{where } K_1 = \frac{X}{2(\hat{T}^{-1/2} - (\hat{T}-T)^{-1/2})} \quad \text{and} \quad K_2 = -2K_1(\hat{T} - T)^{-1/2}. $$

We recall that the expected cost optimal strategy for the equity case is characterized by having a constant trading rate $x_t^* = -\frac{X}{\hat{T}}$, as shown in [27] in a discrete setting. In our case, the trading speed is an increasing convex function of time.

For $\lambda > 0$, the mean-variance minimization problem necessitates to establish a proper stochastic dynamic programming framework. We parameterize the trading strategies $x$ by their speed of trading $\alpha_t = -\dot{x}_t$ and introduce $\mathcal{A}(T,X)$ the set of admissible strategies such that the parametrized strategy $x^\alpha$ satisfies necessary conditions.

We restrict our framework to Markovian controls and thus, solving the optimal stochastic control problem at time 0 is brought to a more general case where the agent starts buying at any arbitrary time

<footnote n="6">6. The temporary impact component reflects the instantaneous effect of the trade, while permanent component transmits the information of buy/sell impact to the market on the long run.</footnote>
$t \in [0, T]$ with an initial quantity $x$ without losing the optimality. We then define the value function $U(t, S, x)$ for the mean-variance framework as :

$$U(t, S, x) = \inf_{\alpha \in \mathcal{A}(T,X)} \mathbb{E}_t \left[ \int_t^T \left\{ \alpha_u^2 S_u^2 (\hat{T} - u)^{3/2} \Gamma(u, S_u) + \lambda \sigma^2 (x_u^\alpha)^2 S_u^2 \Delta^2(u, S_u) \right\} du \right].$$

Note that the value function $U(t, S, x)$ satisfies the so-called finite-fuel condition. A state with a non-zero option position with no time left for its liquidation, means that the liquidation task has not been performed, and thus should receive an infinite penalty. We replace this constraint by a finite terminal condition with large penalty and denote by $U_\epsilon$ the value function in this case. By applying the classical framework to derive the Hamilton-Jacobi-Bellman equation, we find :

**Result 8.** Let $U_\epsilon^*$ be a regular function which solves the PDE :

$$\left\{ \begin{array}{l} \partial_t U_\epsilon^* + \frac{1}{2} \sigma^2 S^2 \partial_{SS} U_\epsilon^* + \lambda x^2 \sigma^2 S^2 \Delta^2(t, S) - \frac{(\partial_x U_\epsilon^*)^2}{4(\hat{T}-t)^{3/2} \Gamma(t, S)} = 0 \\ U_\epsilon^*(T, S_T, x_T) = \frac{1}{\epsilon} \psi(x_T^\alpha). \end{array} \right.$$

Then $U_\epsilon^*$ is the unique solution to the optimal execution problem. Moreover, the optimal execution rate $\alpha_t^* = -\dot{x}_t^*$ is given by :

$$\alpha_t^* = \frac{\partial_x U_\epsilon^*(t, S_t, x_t^*)}{4(\hat{T} - t)^{3/2} S_t^2 \Gamma(t, S_t)}.$$

The PDE satisfied by $U_\epsilon^*$ is quadratic in its first derivative w.r.t $x$, and without knowing the dependence of $U_\epsilon^*$ to $x$ it is difficult to solve it accurately. Fortunately, we are able to reduce the problem's dimension by either reparameterizing the state variable, or using an ansatz to separate the spacial variables. We write $U_\epsilon(t, s, x) := x^2 u_\epsilon(t, s)$ where $u_\epsilon$ is the reduced problem, and use the rate of trading $\kappa_t = -x_t \frac{dx_t}{dt}$ as a control variable. It follows the result :

**Result 9.** Let $u_\epsilon^*$ be a regular function verifying the following PDE

$$\left\{ \begin{array}{l} \partial_t u_\epsilon^* + \frac{1}{2} \sigma^2 S^2 \partial_{SS} u_\epsilon^* + \lambda \sigma^2 S^2 \Delta^2(t, S) - \frac{1}{(\hat{T}-t)^{3/2} S^2 \Gamma(t, S)} u_\epsilon^2 = 0 \\ u_\epsilon^*(T, s) = \frac{1}{\epsilon}. \end{array} \right. \qquad (1)$$

Then $u_\epsilon^*$ is the unique solution to the reduced optimization problem (4.29). The optimal trading rate $\kappa_t^*$ is defined by :

$$\kappa^*(t, S) = \frac{u_\epsilon(t, S)}{(\hat{T} - t)^{3/2} S^2 \Gamma(t, S)}.$$

In this case $u_\epsilon$ solves a nonlinear PDE with quadratic term of order 0. This is much simpler than the previous PDE. Adding appropriate boundary conditions, the problem can be solved numerically by using finite differences schemes. In Figure 5, we present the trading rate $\kappa_t = -\frac{\dot{x}_t}{x_t}$ surface as a function of time and asset price for $\lambda = 100$. We can see that it depends on the asset level and that it increases as function of time.
<figure><img src="image_5.png" /><figcaption>FIGURE 5 – The rate of trading $κ$ as a function of the underlying price $S$ and time $t$ for $λ = 100$. The strike of the option $K = S_0$ is fixed at time 0.</figcaption></figure>

## Part II-B : Volatility models

In the insurance industry, variable annuities and other savings and investment products are subject to market risk like any other financial product. Therefore, it is important to be able to make quick adjustments and build hedging strategies in response to market changes.

Volatility is one of the main drivers of market prices and a key parameter to assess risk, and a large literature tackles its modeling. Since the seminal work of Black and Scholes [30], the most classical way to model the behavior of an asset price $S_t$ is to use continuous semi-martingale dynamic of the form

$$d\log S_t = \mu_t dt + \sigma_t dW_t,$$

where $\mu_t$ a drift process and $W_t$ a Brownian motion. The coefficient $\sigma_t$ is the so-called volatility process.

Following the pioneering approach of [30], practitioners have first considered the case where the process $\sigma_t$ is constant or deterministic, that is the Black and Scholes model. However, in the late eighties, it became clear that such specification for the volatility is inadequate. In particular, the Black and Scholes model is inconsistent with observed prices for liquid European options. Indeed the implied volatility, that is the volatility parameter that should be plugged into the Black-Scholes formula to retrieve a market option price, depends in practice on the strike and maturity of the considered option, whereas it is constant in the Black-Scholes framework.

Hence more sophisticated models have been introduced. A first possible extension, proposed by Dupire [69] and Derman and Kani [63], is to take $\sigma_t$ as a deterministic function of time and asset price. Such models, called local volatility models, enable us to perfectly reproduce a given implied volatility surface. However, its dynamic is usually quite unrealistic under local volatility. Another approach is to consider the volatility $\sigma_t$ itself as an Ito process driven by an additional Brownian motion, typically correlated to $W$. Doing so one obtains less accurate static fits for the implied volatility surface but more suitable dynamics. Among the most famous of these stochastic volatility models are the Hull and White model [101], the Heston model [97] and the SABR model [95]. More recent market practice is to use the so-called local-stochastic volatility models which both fit the market exactly and generate
reasonable dynamics.

In all the Brownian volatility models mentioned above, the smoothness of the sample path of the volatility is the same as that of a Brownian motion, namely $1/2 - \epsilon$ Hölder continuous, for any $\epsilon > 0$. However, it is shown in [88] that in practice, spot volatility is much rougher than this. This result in [88] is based on a statistical analysis of historical data using sophisticated high frequency estimation methods. More precisely, it is established in [88] that the dynamic of the log-volatility process is very close to that of a fractional Brownian motion with Hurst parameter smaller than $1/2$. Recall that a fractional Brownian motion $W^H$ with Hurst parameter $H \in (0, 1)$ is a centered Gaussian process with stationary increments such that

$$ \mathrm{Cov}[W_t^H, W_s^H] = \frac{1}{2} (|t|^{2H} + |s|^{2H} - |t-s|^{2H}). $$

The Hölder regularity of $W^H$ is $H - \epsilon$ for any $\epsilon > 0$ and for $H = 1/2$ we retrieve the classical Brownian motion. Therefore, models where the volatility is driven by a fractional Brownian motion with $H < 1/2$ are called rough volatility models. Beyond fitting almost perfectly historical volatility time series, rough volatility models enable us to reproduce important stylized facts of liquid option prices that local/stochastic volatility models typically fail to generate. In particular, the exploding term structure when maturity goes to zero of the at-the-money skew (the derivative of the implied volatility with respect to strike) is readily obtained, see [20, 79]. Other developments about rough volatility models can be found in [22, 23, 70, 71, 74, 80, 94, 105, 134].

In this part of the thesis we revisit the finding in [88] using range-based and option-based data. Indeed in [88], the authors work with realized volatility based on high frequency data to estimate spot volatility. Access to high frequency data is sometimes costly and/or unavailable for certain assets. Therefore, other proxies are used to estimate daily volatility. Here we use two spot volatility proxies.

In Chapter 5, we use range-based estimators are based on open, high, low and close daily prices. We particularly focus on Garman-Klass and Parkinson estimators, see [86, 139]. In Chapter 6 we use a spot volatility proxy which is not based on historical price data, but on implied volatility. More precisely, we approximate the spot volatility by the implied volatility of an at-the-money liquid option with short maturity (or a refined version of it). This idea can be justified by the fact that in most models, the at-the-money implied volatility tends to the spot volatility as maturity goes to zero, see for example [132].

Our main results are a confirmation of that in [88] : when using alternate spot volatility measurement methods, we can still conclude that volatility is rough.

# 5 Chapter 5 - Range-based proxies and rough volatility

**Question.** What is the regularity of the volatility based on range-based data?

We look at the measure $m(q, \Delta)$ defined by :

$$ m(q, \Delta) = \mathbb{E}[|\log(\sigma_{t+\Delta}) - \log(\sigma_t)|^q] $$

By plotting $log(m(q, \Delta))$ against $log(\Delta)$ for different values of $q$, from 1 day to a few months, we try to find a form for the scaling function $\zeta$ such that :

$$ m(q, \Delta) = K_q \Delta^{\zeta_q}. $$
**Result 10.** The increments of the log-volatility behave as that of a fractional Brownian motion with small Hurst exponent H for a very large range of time scales :

$$\zeta_q \approx Hq,$$

where H < 0.1 for all assets<fnref n="7" /> we performed the study on.

We also find that log-volatility increments over a lag Δ have a similar distribution than a centered normal distribution with variance Δ$^H$. As a result, log-volatility increments can be modeled by the increments of a fBm :

$$\sigma_t = \sigma_0 e^{\nu W_t^H}.$$

This model is unfortunately not stationary. Stationarity being a desirable property in time series, volatility can be modeled through a fractional Ornstein-Uhlenbeck volatility model with Hurst exponent H < 1/2, and a mean-reverting parameter α such that 1/α (the mean-reversion time scale) is very large compared to the time scales of interest. This so-called fractional rough volatility model (RFSV), see [88], is fundamentally different than Compte and Renault fractional volatility model (FSV), see [56]. Indeed, FSV considers H > 1/2 and α ≫ 1.

More precisely, we have :

$$d \log(\sigma_t) = -\alpha(\log(\sigma_t) - m) + \nu dW_t^H,$$

with H < 1/2 and α ≪ 1.

**Question.** Can we validate the RFSV model? Doesn't the FSV model exhibit the same behavior?

**Result 11.** Empirical results and simulations allow to exclude the FSV model as a data consistent model. The RFSV model, on the other hand did not fail the tests we performed.

To perform our tests, we simulate the FSV and FRSV models with known parameters. We then estimate realized and range-based volatility proxies. By looking at the behavior of log(m(q, Δ)) versus log Δ, we find that FSV model can be misleading. In fact, as illustrated by the sketch in Figure 6, log(m(q, Δ)) exhibits three types of behaviors :

- For small Δ, the estimated smoothness parameter H is affected by noise and is thus close to 0.

- For intermediate Δ, one can approach the true value of H but it can be relatively difficult to find it with good accuracy.

- For large Δ, the stationarity (through the mean-reversion) governs the value of the estimated H, which is again very small compared to the real one.

For RFSV model, the estimated smoothness for the RFSV is consistent whatever the lag Δ. Hence, the FSV model does not seem to be consistent with observed data, meanwhile RFSV simulated volatility is much closer to observations. We conclude the analysis by studying the prediction power of RFSV. Using result by Nuzman and Poor, see [135], we can use the conditional expectation of the fractional Brownian. As a result, the conditional expectation of log-variance and variance w.r.t past information $\mathcal{F}_t$ satisfies

$$\mathbb{E}[\log(\sigma_{t+\Delta}^2) | \mathcal{F}_t] = \frac{\cos(H\pi)}{\pi} \Delta^{H+1/2} \int_{-\infty}^{t} \frac{\log(\sigma_s^2)}{(t-s+\Delta)(t-s)^{H+1/2}},$$

<footnote n="7">7. We estimated the scaling of the volatility for liquid and less liquid assets based on the Garman-Klass and Parkinson proxies</footnote>
<figure><img src="image_6.png" /><figcaption>FIGURE 6 – Observed behavior for $log(m(q, \Delta))$ as a function of $log(\Delta)$ using the realized-volatility proxy</figcaption></figure>

and

$$ \mathbb{E}[\sigma_{t+\Delta}^2 | \mathcal{F}_t] = \exp \left( \mathbb{E}[\log(\sigma_{t+\Delta}^2) | \mathcal{F}_t] + 2cv^2\Delta^{2H} \right), $$

$$ \text{where } c = \frac{\text{Gamma}(3/2-H)}{I^{(H+1/2)} I^{(2-2H)}}. $$

**Result 12.** RFSV outperforms other times series models such as AR, HAR and GARCH in predicting future log-volatility and variance.

# 6 Chapter 6 - Volatility is rough: evidence from option price data

Instead of focusing on volatility proxies recovered from the underlying historical price, we wish to use option price based data. Our goal is again to study the instantaneous volatility smoothness. To do so, we exploit asymptotic properties of the implied volatility, and particularly, the fact that at-the-money implied volatility converges to spot volatility as time-to-maturity goes to zero.

**Result 13.** Using the same procedure as in Garman-Klass and realized volatility, we find that the scaling of at-the-money short-term implied volatility is fractional, i.e. :

$$ \zeta_q = qH, $$

with $H \simeq 0.32$, meaning volatility is rough.

**Question.** Why is the value large compared to realized and Garman-Klass volatilities? Can we improve the result?

One reason for this relatively high value, is that our options have a significant remaining time to maturity of one month. This induces a smoothing phenomenon in the estimation of the Hurst parameter. This effect is of the same nature as that described and explained in [88], caused by the discrepancy between spot and integrated volatility over a short time interval.

To improve the result, one solution would be to use asymptotic methods to estimate implied volatility as time-to-maturity tends to zero, i.e. $\tau \to 0$, and raw option data for the estimation. We choose the methodology by Medvedev and Scaillet who give an asymptotic formula for implied volatility, see Chapter 6 and [124].

**Result 14.** The Medvedev-Scaillet estimator of spot volatility is rough, with $H \simeq 0.3$.
The value of the smoothness parameter is still high compared to H found for realized-volatility and Garman-Klass, even though we used an asymptotic approach to approximate close-to-maturity implied volatility. We believe this is still due to the remaining time-to-maturity as the option data we used for the Medvedev-Scaillet proxy is still of at least 15 days time-to-maturity.

To understand this phenomenon, we perform a simulation study where spot volatility is rough (H = 0.04). Under mild assumptions, we find that 1 day ATM-IV has H = 0.06, while 20 days ATM-IV has H around 0.27.

**Result 15.** Implied volatility regularity increases as time-to-maturity increases as given by Figure 7.

<figure><img src="image_7.png" /><figcaption>FIGURE 7 – Estimated values of the Hurst parameter using implied volatilities as a function of time to maturity.</figcaption></figure>

To provide a more quantitative understanding of such upward bias, we consider the crude approximation of the Black-Scholes implied volatility :

$$ \hat{v}^{\tau}(t) := (\sigma_{imp}^{\tau}(t))^2 = \frac{1}{\tau} \int_{t}^{t+\tau} \mathbb{E}[\sigma_{t}^{2} | \mathcal{F}_{t}] du, $$

using the simplified rough volatility model :

$$ \sigma_u^2 = \sigma_0^2 + \nu W_u^H. $$

**Result 16.** The scaling based on the moment of order 2 is given by equation :

$$ \mathbb{E}[(\hat{v}^{\tau}(\Delta) - \hat{v}^{\tau}(0))^2] \propto f(\tau/\Delta)\Delta^{2H}, $$

where $f(\tau/\Delta) \xrightarrow{\tau/\Delta \to 0} 1$.

This means that the same scaling relationship, as that associated to the spot volatility, is approximately satisfied when considering implied volatilities with small enough times to maturity. Otherwise, one should add a multiplicative factor.
Première partie

Life insurance products
<figure><img src="image_8.png" /></figure>
# Chapitre 1

## Financial risk management and the
rational lapse strategy in life Insurance
policies

***Abstract***— Over the past decade, Variable Annuities have experienced tremendous growth accounting for half of the life insurance industry, as unit-linked products offering both participation in equity market and guarantees at key life moments (retirement, death).

The recent Quantitative Impact Study (QIS 5) of the Solvency II framework showed that lapse risk is the most important risk among life underwriting risks for Variable Annuities, as illustrated by solvency issues experienced by the policyholder run in the late 1980’s. Thus research on lapse rates is crucial to a proper calibration of regulatory standard models and internal risk models.

Usually the lapse behavior has been modeled by historical or backward looking statistical regressions which have empirically underestimated the risk due to the scarcity of extreme scenario samples and the inability to dynamically extrapolate the observed behavior to various market conditions. In contrast, a "rational" lapse strategy valuation is a prudent forward looking approach where policyholders lapse in a way that maximizes the net present value of the future cash-flows, depending on key drivers. Empirically consistent with herd behavior as experienced in the last financial crisis, this approach is illustrated on a GMAB VA product using two alternatives numerical schemes (PDE and Monte Carlo).

However, as policyholders cannot be expected to lapse all at the same time, this rational lapse framework is slightly amended by introducing a proportion of lapses among the contract still active, which translates into the notion of "reasonable" lapse more consistent with empirics.

**Keywords** : GMAB; Variable Annuity; rational lapse strategy; stochastic interest; PDE; ADI; high-dimensional regression.

### 1.1 Introduction

The VA product, a popular retirement savings vehicle in the US, is starting to emerge as a viable option in other markets, including Europe and Asia. The GMAB riders written on VAs (also known as Maturity Guarantees, see [40]) provide policyholders a guaranteed amount at a fixed expiration date, so this kind of products have some similar properties as long-term vanilla puts. One important attractiveness of GMAB products is that this guarantee gives policyholders the ability to protect their
retirement investments against downside market risk by allowing the policyholder to receive the greater of the account value and the benefit base at the maturity. The benefit base can either step up to the high-water mark of the account value at the end of each policy year (annual ratchet), or can roll up with a fixed percentage (the roll-up rate, e.g. 2%), regardless of the market conditions. Thanks to these new product characteristics, the guarantee not only protects policyholders against investment losses, but also allows customers taking advantage of upside gain from the market. In exchange for this benefit, the policyholder pays a charge fee each year.

The recent Quantitative Impact Study (QIS 5) of the Solvency II framework showed that lapse risk is the most important risk among life underwriting risks for Variable Annuities, as illustrated by solvency issues experienced by the policyholder run in the late 1980's. Thus research on lapse rates is crucial to a proper calibration of regulatory standard models and internal risk models.

The dynamic behavior is essentially a selection process of the policyholders against the VA writer, where an increase leaves fewer insured to ultimately make a claim on the guarantees but reduces the fees the insurer can collect. The general pattern is that more policies will lapse when the capital market is up, and fewer policies will lapse when the capital market is down.

- In an up market, the value of the minimum guarantee diminishes as the account value is likely to exceed the minimum guarantee values. As such, surrendering the policy does not create much loss to the policyholder.

- On the other hand, a down market can result in the surrender value being less than the guarantee value causing the policy to be in-the-money. If the policyholder surrounds at this time then he or she can only get the reduced surrender value, forfeiting the added value from the guarantee rider. The result is that there is strong incentive for the policyholder to keep the in-the-money VA contract in force.

As the lapse assumption may impact significantly the profitability of GMAB riders, a rigorous modeling framework of the lapse rate is necessary for both pricing and hedging purpose. During the last decade, the literature on pricing and risk management of these guarantees has been evolving.

- Tradionnally the lapse behavior has been modeled by historical or backward looking statistical regressions which have empirically underestimated the risk due to the scarcity of extreme scenario samples for these new products and the inability to dynamically extrapolate the observed behavior to various market conditions.

- In contrast, a "rational" lapse strategy valuation is a prudent forward looking approach where policyholders lapse in a way that maximizes the net present value of the future cash-flows, depending on key drivers. This reflects a potential extreme policyholder behavior, as experienced in the last market crash, with an initial immediate and sustained fall in lapses right after the crash, before an abrupt recovery consistent with the interest rates. In contrast, dynamic lapses modeling are usually unable to provide such empirical dynamics.

This approach is illustrated on a GMAB VA product using two alternatives numerical schemes (PDE and Monte Carlo), as the valuation of a Bermudan-style contingent claim for the insurer, where the contingency is closely related to equity market conditions and the interest rate level, see [35].

The price evaluated by this approach can be interpreted as the fair value of the policy if all policyholders use the same rational lapse strategy, which is similar with the optimal early-exercise strategy of Bermudan options. However, as policyholders cannot be expected to lapse all at the same time,
this rational lapse framework is slightly amended by introducing a proportion of lapses among the
contract still active, which translates into the notion of "reasonable" lapse more consistent with em-
pirics. Note that this is only an interpretation, and that the critical aim is to make sure the lapse risk
can be hedged no matter which strategy the holders use.

The remainder of this paper is organized as follows. Firstly, in Section 2 the GMAB policy is explai-
ned in full details. Section 3 introduces the modeling framework to evaluate the liability of GMAB
policies in no-lapse assumption. The rational lapse strategy and critical lapse boundaries are studied
in Section 4. Section 5 addresses the pooling of lapse risks. In Section 6 we implement two numerical
methods, the PDE approach and the high-dimensional regression method (Monte Carlo, see [37]) to
calculate the no-arbitrage price of GMABs in the Hull-White interest rate model. Numerical results of
these two methods are shown in Section 6 and conclusions in Section 7.

**1.2 Description of the contract**

In practice, most GMAB policies are purchased in a lump sum. We assume that a single premium is
paid at inception of the contract and denoted by A(0) = 100$ the initial account value at time zero
after the upfront fees have been paid. The account value is invested in a portfolio consisting mainly
of equities and bonds. At the end of each policy year tᵢ, the insurer deduct a charge fee ᾱA(tᵢ) on the
account value, where ᾱ = 2% is the annual charge rate. The life time of the policy is T = 10 years if there
is neither early termination nor rollover.

For a contract that is held until the maturity, there is a guaranteed minimum return paid to the policy-
holder. We represent this guarantee to the policyholder as G, which is called the benefit base for insu-
rers. In other words, at the maturity, the policyholder has the right to receive a cash payment equal to
either A(T) or to the benefit base G. Consequently, at maturity, the value of the policy is max(A(T), G).
This payoff can be decomposed to the sum of the account value A(T) and a vanilla put (G − A(T))⁺ (the
strike level is G). The benefit base G is fixed at inception, which is equal to A(0)(1 + r̄)ᵀ, where r̄ is the
roll-up rate. In most cases, r̄ is approximately equal to the yield of zero-coupon bonds maturing at
T. We assume that one GMAB policy is purchased in 2000 and hold until 2010, and the account value

<figure><img src="image_9.png" /><figcaption>FIGURE 1.1 – The illustration of the account value over time compared with the US 10-year note.</figcaption></figure>

is invested in S&P 500 at inception. The roll-up rate is set at 6.67%, which is equal to the yield of US
10-year notes in January 2000. All other parameters are the same as those mentioned above. Figure 1.1
plots over time the account value A(t). By comparing the net return of 10-Year bonds with that of the
roll-up GMAB rider, we can see that the roll-up benefit base can not only protect policyholders from
catastrophes in stock market, but also from risks of the persistent decrease of the interest rate <fnref n="1" />.

<footnote n="1">1. In fact, the US 10-year bond yield was 6.67% in January 2000, while it was 3.61% ten years later.</footnote>
1.3 Valuation of a GMAB with zero lapse

Firstly we establish the general modeling framework to evaluate the liability of GMAB with zero lapse (European GMAB). From now on, we let $(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t \le T}, \mathcal{Q})$ denote a complete filtered probability space supporting two independent standard one dimensional Brownian motions $W$ and $W^\perp$. Here $T > 0$ is a fixed time horizon. We assume that the filtration $\mathbb{F}$ is the completion of the rough filtration generated by $(W, W^\perp)$, so that any martingale $(\mathcal{Q}, \mathbb{F})$-martingale can be represented as a stochastic integral with respect to $(W, W^\perp)$.

During the last decade the literature on pricing variable annuities has evolved, but many evaluation approaches proposed (e.g. [18, 129]) are still based on the assumption of deterministic interest rates. Such an assumption is harmless in most situations since the interest-rates variability is usually negligible when compared to the variability observed in equity markets. While pricing a long-maturity securities such as VA guarantees, however, the volatile feature of interest rates can have stronger impacts on the liability of GMAB. In such case it is therefore advisable to use stochastic interest rate models.

In this paper, we assume that the short term interest rate $r = (r(t))_{t \ge 0}$ is driven by the one factor Hull and White model, and the underlying asset $S = (S(t))_{t \ge 0}$ in which the account value is invested follows a Black and Scholes type dynamics, namely :

$$
\begin{equation}
\begin{cases}
dS(t) = r(t)S(t)dt + \sigma S(t)dW(t) \\
dr(t) = a(\theta(t) - r(t))dt + \sigma_r dZ(t), \quad Z := (1 - \rho^2)^{\frac{1}{2}} W^{\perp} + \rho W
\end{cases}
\tag{1.1}
\end{equation}
$$

Here, $a$ and $\sigma_r$ are positive constants, $\theta$ is a deterministic Lebesgue-integrable function, $\sigma$ is the instananeous volatility of the asset return, and $\rho$ is the correlation$^2$ between the account value and the interest rate. Note that the above financial market is complete whenever $S$ and a zero-coupon bond with maturity $T$ can be freely traded, and that $\mathcal{Q}$ is the only martingale (risk neutral) measure.

For the account value, a charge fee is deducted at a rate $\alpha$ continuously, where $\alpha = -\log(1 - \bar{\alpha})$. This
means that $A(t)$ evolves according to

$$
dA(t) = (r(t) - \alpha)A(t)dt + \sigma A(t)dW(t) \quad (1.2)
$$

Since $(r, A)$ is a Markov process, the European-style liability $V^E$ of a single GMAB rider can be identified
to a deterministic liability function $v^E$ by :

$$
V^E(t) := v^E(t, r(t), A(t)) = \mathbf{E}^Q [D_t^T \max(A(T), G) | \mathcal{F}_t] = \mathbf{E}^Q [D_t^T (A(T) + (G-A(T))^+) | \mathcal{F}_t] \quad (1.3)
$$

where $D_{t_1}^{t_2}$ represents the stochastic discount factor between $t_1$ and $t_2$

$$
D_{t_1}^{t_2} := \exp\left(-\int_{t_1}^{t_2} r(s) ds\right).
$$

Equation (1.3) shows that the European-style liability of GMAB riders can be considered as the sum of a forward contract of the account value ending at T and a vanilla put with the maturity T and the strike level G. In the Hull-White interest rate model, this liability value can be easily calculated analytically.

However, it does not always exist some closed formula of the liability value, especially when the early-
lapse premiums are taken into account. Thus in practice, we need to use some numerical methods,

<footnote n="2">2. For VAs, the correlation is often negative, as most portfolio contains fixed income assets, such as bonds.</footnote>
such as PDE or Monte-Carlo based algorithms to evaluate the policies. For GMAB riders, it is sometimes more convenient to price the liability under the so-called forward measure rather than the risk-neutral measure $\mathbb{Q}$. Because in $\mathbb{Q}^T$, we can reduce the number of dimensions of the liability evaluation problem (1.3) from three to two and the pricing process can be significantly accelerated.

To facilitate the following study, we evaluate the GMAB riders in the forward measure $\mathbb{Q}^T$. Firstly, we introduce the forward value of $A(t)$ at $T$ observed at date $t$, denote by $F^T(t) = A(t)/Z_t^T$, where $Z_t^T$ is the price of a zero-coup bond maturing at $T$. Applying Ito's lemma to $F^T(t)$, we get the dynamics of the forward account value:

$$ \frac{dF^T(t)}{F^T(t)} = (\nu^2 B_r^2(u) + \rho \nu \sigma B_r(u) - \alpha) dt + \sigma dW(t) + \nu B_r(u) dZ(t), $$

where $u = T - t$ is the time to maturity and the function $B_r(u) = (1 - e^{-au})/a$. By doing the following transformations of the Brownian motions from $\mathbb{Q}$ to $\mathbb{Q}^T$:

$$ \begin{aligned} dZ(t) &\rightarrow dZ^T(t) - \nu B_r(u) dt \\ dW(t) &\rightarrow dW^T(t) - \rho \nu B_r(u) dt \end{aligned} \quad (1.4) $$

we have that, under $\mathbb{Q}^T$, the dynamics of $F^T(t)$ can be written as:

$$ \begin{aligned} dF^T(t) &= -\alpha F^T(t) dt + \sigma F^T(t) dW^T(t) + \nu B_r(u) F^T(t) dZ^T(t) \\ &= -\alpha F^T(t) dt + \omega_u F^T(t) d\tilde{W}^T(t) \end{aligned} \quad (1.5) $$

where $\omega_u^2 = \sigma^2 + \nu^2 Br^2(u) + 2\rho\sigma\nu Br(u)$ and $\tilde{W}^T$ is a Brownian motion in $\mathbb{Q}^T$. The results above allow us to simplify the pricing problem of GMAB riders. Instead of computing the liability under risk-neutral measure $\mathbb{Q}$ (as in (1.3)), we evaluate the forward liability $\tilde{v}^E(t, f)$ in $\mathbb{Q}^T$:

$$ \tilde{v}^E(t, F^T(t)) := \frac{\nu^E(t, r(t), A(t))}{Z_t^T} = E^{\mathbb{Q}^T} [F^T(T) + (G - F^T(T))^+ | \mathcal{F}_t] \quad (1.6) $$

Equation (1.5) and (1.6) show that the European-style forward liability $\tilde{v}^E$ can be evaluated by the following analytical formula:

$$ \tilde{v}^E(t, F^T(t)) = e^{-\alpha u} F^T(t) + GN(-d_2) - e^{-\alpha u} F^T(t) N(-d_1), \quad d_{1,2} = \frac{\log(F^T(t)/G) - \alpha u}{\Gamma} \pm \frac{\Gamma}{2} \quad (1.7) $$

where $\Gamma = \sqrt{\int_0^{T-t} \omega_s^2 ds}$ and $N(\cdot)$ is the cumulative distribution function of the standard normal distribution.

## 1.4 Valuation a GMAB with rational lapse assumption

In the previous section, we have formulated the pricing issue of GMAB riders under the no-lapse assumption. If policyholders are not allowed to lapse contracts before maturity, the liability of GMAB riders can be calculated analytically by (1.7). However, in practice we can not assume the lapse rate to be zero or some other constant, as we observe that the lapse rate does change significantly in different market conditions (equity market and interest rate level) and this fluctuation of lapse rate has notable impacts on the liability value and insurer's hedging strategy.

As explained in the introduction, we consider the pricing problem of liabilities with lapse options as the valuation of a Bermudan-style contingent claim, see [42]. Because the rational lapse strategy
discussed here are similar with the optimal early-exercise strategy of classic Bermudan options. We
assume that the policyholder can lapse the contract at the end of each policy year, noted as $t_i$, $i =$
1, 2, ..., N, where $t_N$ = T. According to the definition, we have

$$
\begin{align}
v^B(T, r(T), A(T)) &= \max(A(T), G) \nonumber \\
v^B(t_i-, r(t_i-), A(t_i-)) &= \sup_{\tau_i \in \mathcal{T}_i} \mathbf{E}^Q [D_i^{\tau_i} A(\tau_i) + \mathbf{1}_{\{\tau_i=T\}} D_i^T \max(A(T), G) | \mathcal{F}_{t_i}] \tag{1.8}
\end{align}
$$

where $\mathcal{T}_i$ is the set of all stopping times taking values in $\{t_i, t_{i+1}, \dots, T\}$ and $\tau_i$ denotes the stopping time of the rational lapse strategy since time $t_i$. According to the assumption, the policyholder is not authorized to lapse the contract between two purchase anniversaries $t_i$ and $t_{i+1}$, so the process $V^B$ of the liability should evolve in the same way as $V^E$ for $t_i \le t < t_{i+1}$. Applying the fact that $r(t)$ and $A(t)$ are all Markov processes, we have,

$$
\forall i < N, \quad v^B(t_i, r(t_i), A(t_i)) = \mathbf{E}^Q [D_i^{i+1} v^B(t_{i+1}-, r(t_{i+1}-), A(t_{i+1}-)) | \mathcal{F}_{t_i}] \quad (1.9)
$$

Similarly with Bermudan-style options, at discrete time points $t_i$, the policyholder is supposed to compare the account value with the value of the liability to decide whether lapse or not. If the account value is bigger than the liability, policyholders surrender the contract and get back $A(t_i)$. Otherwise, they continue to hold the policy. That is to say, at time $t_i$, the Bermudan-style liability should evolve as following

$$
v^B(t_i-, r(t_i-), A(t_i-)) = \max(A(t_i), v^B(t_i, r(t_i), A(t_i))) \quad (1.10)
$$

Equation (1.10) reflects the fact that the liability before the annuity payment is equal to the greater of
the current account value $A(t_i^-)$ and the value of continuation. To simplify the pricing process of the
Bermudan-style liability, we can calculate the expectations under the forward measure $\mathbb{Q}^T$ instead of
the risk neutral measure $\mathbb{Q}$. That is to say, we write (1.10) as

$$
\begin{align}
\tilde{v}^B(t_i-, F^T(t_i-)) &:= \frac{v^B(t_i-, r(t_i-), A(t_i-))}{Z_i^T} = \max(F^T(t_i), \tilde{v}^B(t_i, F^T(t_i))) \nonumber \\
&= \max(F^T(t_i), \mathbf{E}^{\mathbb{Q}^T}[\tilde{v}^B(t_{i+1}-, F^T(t_{i+1}-)) | \mathcal{F}_{t_i}]) \tag{1.11}
\end{align}
$$

where the boundary condition at maturity is

$$
\tilde{v}^B(T, F^T(T)) = F^T(T) + (G - F^T(T))^+ \quad (1.12)
$$

Another important issue related with the evaluation problem of the Bermudan-style liability is the determination of the rational lapse strategy to be followed. As the benefit base G is fixed at inception, according to (1.11) and (1.12), we have that $\tilde{v}^B(t, f)$ is a convex, nondecreasing function of $f$. In addition, it is also a positive function on $(t, f) \in [0, T) \times [0, \infty)$, for $\tilde{v}^B(t, f) > \tilde{v}^E(t, f) > 0$. Finally, for the charge fee $\alpha > 0$, (1.11) and (1.12) also imply that $\lim_{f \to \infty} (f - \tilde{v}^B(t_i, f)) > 0$. It follows from the previous arguments that, for each $t \in \{t_0, \dots, t_i, \dots, t_{n+1}\}$, there exists a real number $f^*(t_i-)$,

$$
\begin{align}
0 \le f < f^*(t_i-) &\Rightarrow \tilde{v}^B(t_i-, f) > f \quad \text{(Not Lapse)} \tag{1.13} \\
f \ge f^*(t_i-) &\Rightarrow \tilde{v}^B(t_i-, f) = f \quad \text{(Lapse)} \nonumber
\end{align}
$$

In this paper, $f^*(t_i-)$ is referred to as the "critical forward account value" since the policy should be
lapsed as soon as the forward account value increases to this level at time $t_i$. As it is shown by (1.13),
thanks to the change of measure, the critical boundary here depends only on the forward account va-
lue, rather than on both the interest rate and the account value level. However, it is not always possible
to do this kind of simplifications when we evaluate Bermudan-style options, because sometimes the
intrinsic payoff (such as $f$ for $\tilde{v}^B(t, f)$) is not a linear function of the underlying (e.g. American vanilla
options).
The objective now is to evaluate the liability $\tilde{v}^B$ and the critical boundary $f^*$ of a single GMAB rider.
Although many analytical approximations exist in academy literatures, see [129], most of them are not
sufficiently precise due to the long maturity property. In the present paper, we propose two numeri-
cal methods : PDE and Monte Carlo schemes (see Appendix C for the description of the numerical
schemes and numerical tests for the results), to calculate both the Bermudan-style liability $v^B$ and
the critical lapse surface. As we have mentioned at the beginning, the PDE method is precise for low-
dimensional problems ($< 3$), while the Monte Carlo is more efficient when there are more than three
dimensions in the pricing problem (e.g. multi-asset account value or stochastic volatility models).

**1.5 Life insurance policy pool**

The analysis above is focused on the rational lapse strategy and the no-arbitrage value of a single GMAB policy. The liability $v^B$, which takes into account the lapse risks, allows the insurer to hedge the uncertain customer behavior no matter what the lapse strategy of the policyholder is. But in practice, the insurers often need to estimate the lapse risks of a pool of life insurance policies, and in this case, the lapse strategy can be represented by the frequency $p(t_i)$ of the policies that are early terminated at time $t_i$, see [3]. Actually, the common sense and experience tells us that not all policyholders will lapse the contract at the same time, so we need to slightly change the function $p(t_i)$ to estimate the real lapse rate of a policy pool.

For a pool of GMAB policies, we denote by $p(t_i)$ the proportion of lapses at date $t_i$ among the contracts still active in the pool. According to the rational lapse strategy, we can express $p(t_i)$ as a deterministic function $h$ of the forward account value $F^T(t_i)$, that is $h(F^T(t_i)) = 1_{\{F^T(t_i) \ge f^*(t_i-}\}}$. This lapse function implies that once $F^T(t_i)$ touches $f^*(t_i-)$, all policyholders lapse the contract and otherwise everybody hold the policy.

Inspired by the mortgage prepayment models, see [137], and evaluation approaches of surrender options for other life insurance products, see [3], we assume that $h$ is a nondecreasing piece-wise linear function of the variable $F^T(t_i)$. When $F^T(t_i) < f^*(t_i -)$, the lapse rate is not zero due to policyholders' personal circumstances (including liquidity and death), which is independent of financial considerations. These "irrational" lapses are analogous to noneconomic prepayment on low-rate mortgages. While when $F^T(t_i) \ge f^*(t_i -)$, some rational lapses never occur, and a reasonable specification of $p(t_i)$ may be illustrated as that in Figure 1.2 :

<figure><img src="image_10.png" /><figcaption>FIGURE 1.2 – Comparing the rational lapse function with the reasonable lapse function.</figcaption></figure>

The four parameters $F_1$, $F_2$, $P_{\min}$ and $P_{\max}$ are determined by insurers according to some empirical
tests. To be consistent with the rational lapse assumption, $F_1$ should be very close to $f^*(t_i)$ and $P_{\max}$
should be set high enough (normally $\ge 50\%$). Under the reasonable lapse assumption, once the cri-
tical lapse level and the reasonable lapse function are determined, the GMAB liability can be simply
evaluated as an European-style option, see [3]. However, it is worthy to mention that, unlike the rational lapse approach, the reasonable lapse assumption makes the insurer partially exposed to the risk of lapses in the future.

## 1.6 Numerical tests

In this section, we use two numerical methods (PDE and Monte Carlo) introduced above to evaluate the Bermudan-style liability of one standard GMAB policy. Our final results show not only the consistency between these two methods, but also the efficiency and precision of both methods. In addition, the option value of GMAB (defined later), the forward delta and the "critical boundary" found by these two methods are also compared.

For the tested policy, The account value is supposed to evolve according to Hull-White model, where the principle model inputs are listed in Table 1.1. The initial equilibrium short-rate curve $\theta(t)$ is sup-

<table><caption>TABLEAU 1.1 – Hull-White Model Inputs</caption><thead><tr><th>σ</th><th>r<sub>0</sub></th><th>θ</th><th>a</th><th>σ<sub>r</sub></th><th>ρ</th></tr></thead><tbody><tr><td>0.2</td><td>0.02</td><td>0.02</td><td>0.03</td><td>0.01</td><td>0</td></tr></tbody></table>

posed to be flat ($\theta$ constant) and the short rate at inception is denoted by $r_0$. All other parameters of product properties will be clarified later.

For simplicity, we also assume that the policyholder is alive at the maturity of GMAB policies. Although we are focused on the liability of a single policy in the following numerical tests, the methodology we propose here can be easily extended to evaluate a GMAB policy pool by adding up policies of different maturities with a proper weight indicated by mortality rate assumptions.

### 1.6.1 Bermudan-style GMAB Liability

Firstly, we calculate the liability of a standard GMAB policy. The principle product parameters are listed in Table 1.2, where the charge fees rate is $\alpha = 2\%$, the maturity is 10 years and the benefit base, fixed at inception, is 100$ for one policy. For simplicity, we assume that the policyholders are allowed to lapse the contract only at one specific date of each policy year.

Figure 1.3 shows the forward Bermudan GMAB liability $\tilde{v}^B(t, f)$ computed by the PDE scheme for different forward account values through time. In addition, the intrinsic value of GMAB policies, which is equal to the instantaneous forward account value, is also recorded in Figure 1.3. It is worthy to mention that at the dates when lapses are allowed, once we have $\tilde{v}^B(t, f) = f$ for $f$ big enough, policyholders should lapse the contract immediately. This phenomena is consistent with our intuitive, as the higher the account value is, the less the GMAB guarantees worth and the more likely that policy-

<table><caption>TABLEAU 1.2 – Product Parameters of the GMAB Policy</caption><thead><tr><th>α</th><th>G</th><th>T</th><th>Lapse Date Frequency</th></tr></thead><tbody><tr><td>2%</td><td>100$</td><td>10 years</td><td>1/year</td></tr></tbody></table>

holders lapse the contract and get back the intrinsic value immediately.
<figure><img src="image_11.png" /><figcaption>FIGURE 1.3 – The forward Bermudan liability ($\tilde{v}^B$) of a standard GMAB policy for different forward account values and time points, compared with the intrinsic value at all exercisable dates.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIGURE 1.4 – The forward early lapse premium ($\tilde{v}^B - \tilde{v}^E$) of a standard GMAB policy for different forward account values and time points.</figcaption></figure>

To be further protected from potential lapse waves or other financial risks, the insurers can charge the policyholders an up-front fee, which is equal to $Z_0^T(\tilde{v}^B(0, f_0) - f_0)$ (the difference between the liability and the asset), to make sure the balance sheet is in equilibrium at inception, see [35].

Figure 1.4 shows the early lapse premium (the difference between $\tilde{v}^B$ and $\tilde{v}^E$) calculated by the PDE scheme. In this figure, we observe that the Bermudan liability is almost equal to the European-style liability when the account value falls to very low levels. Because in this case, the probability that policyholders lapse the contract before the maturity is extremely small. While when the forward account value increases to very high levels, the early lapse premium grows almost linearly with $f$. This is due to the fact that when $f \gg f^*(t)$, $\tilde{v}^B(t, f) = f$ and $\tilde{v}^E(t, f) \approx e^{-\alpha(T-t)}f$. Finally we observe that, like other Bermudan contingent claims, the early lapse premium of GMAB policies reduces gradually to 0 at the expiration date.

### 1.6.2 Option value of the GMAB policy

Firstly, we define the forward option value, denoted as $\tilde{w}^B(t, f)$, of the Bermudan-style GMAB policy,

$$ \tilde{w}^B(t, f) := \tilde{v}^B(t, f) - f \qquad (1.14) $$

The notation $\tilde{w}^B$ implies that the option value has many similar properties as a vanilla put<fnref n="3" />, see Appendix B. In fact, $\tilde{w}^B(t, f)$ can be simply interpreted as the difference between the liability $\tilde{v}^B(t, f)$ and the asset $f$ of GMAB policy issuers. That is to say, $\tilde{w}^B(t, f)$ is the option that the insurers should replicate in practice.

In Figure 1.5, we provide the numerical results obtained by the PDE scheme for different forward account values and different dates from the inception to the expiration of the policy. In this figure, we observe that the forward option value $\tilde{w}^B(t, f)$ evolves similarly as a vanilla put, see Appendix B. In addition, when the forward account value is significantly higher than the critical lapse boundary, the option value becomes negative between two discrete exercisable dates. This is due to the charge fees that policyholders are obliged to pay to insurers.

<footnote n="3">3. For European GMAB policies, we know that the option value is in fact a vanilla put (see (1.7)).</footnote>
<figure><img src="image_13.png" /><figcaption>FIGURE 1.5 – The forward option value ($\tilde{w}$) of a standard GMAB policy for different forward account values and dates from the inception to the expiration of the policy.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIGURE 1.6 – Comparing the numerical results of the option value $\tilde{w}(0, f)$ computed by two different methods : PDE and Monte Carlo.</figcaption></figure>

Figure 1.6 compares the numerical results of $\tilde{w}^B(0, f)$ computed by two methods : PDE and Monte Carlo. For the Monte Carlo method used here, we simulate 10,000 scenarios with the step length of 0.1 year. To evaluate $\tilde{w}^B(t, f)$ in this example, the average computing time of Monte Carlo method is about 1 to 2 seconds. Figure 1.6 shows that the prices calculated by Monte Carlo-S1 method is consistent with the results of PDE method.

We also compare the numerical results of the forward delta of the option value computed by PDE and Monte Carlo-S1 in Figure 1.8. We observe that the forward delta jumps up to 0 very quickly when $f$ approaches to the critical boundary. This is easy to understand, as in this case all policyholders are supposed to lapse the contract and the insurers have no more need to hedge their liabilities. Fortunately, this difficulty of hedging lapse risks near the critical boundary can be partly overcome by diversifying the portfolio of GMAB policies (e.g. different maturities and benefit base levels).

Figure 1.9 compares the critical lapse boundary for different discrete dates computed by the two methods introduced above. We observe that the numerical results of Monte Carlo method becomes more and more instable from the expiration to the inception of the policy. This is because of the accumulation of pricing errors caused by linear regressions backward through time.

In summary, we find that in our specific example here, the PDE method is faster and more precise, especially for $f$ nearing the critical boundary, than Monte-Carlo based methods. In addition, this method can calculate the price and other important Greeks for different $f$ and $t$ at the same time. However, compared with the PDE method, the Monte Carlo method is much more flexible and easier to be implemented. Moreover, unlike PDE based methods, the Monte Carlo method can be extended to other high-dimensional problems, such as path-dependent payoffs, stochastic volatility models or basket account values, see [35, 37].
<figure><img src="image_15.png" /><figcaption>FIGURE 1.7 – The forward delta of $\tilde{w}(t, f)$ of a standard GMAB policy for different forward account values and dates from the inception to the expiration of the policy.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>FIGURE 1.8 – Comparing the numerical results of the forward delta of the option value computed by two different methods : PDE and Monte Carlo-S1.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>FIGURE 1.9 – Comparing the critical lapse boundary computed by two methods : PDE and Monte Carlo.</figcaption></figure>

**1.7 Conclusion**

In this paper, we introduce a framework to evaluate the liability of GMAB polices under rational lapse assumption. We study in full details not only the financial sensitivities, but also the rational lapse strategy of GMAB products in the stochastic interest rate model. Two numerical methods, the PDE and Monte Carlo, are implemented to price the policy and also to determine the critical lapse boundary. Moreover, we find a semi-analytical formula to approximate the lapse premium of the GMAB. Inspired by the rational lapse assumption, we finally introduce the reasonable lapse assumption to help insurers to measure the lapse risks of a policy pool.

**Appendix A: American-style GMAB**

To the best of our knowledge, there is no closed formula to evaluate the Bermudan-style liability $\tilde{v}^B(t, f)$. However, we can use some semi-closed formulas to approximate the liability if we assume that the policyholders can lapse the contract at any time. In this case, we denote the GMAB liability as $\tilde{v}^A(t, f)$, which is in fact an American-style contingent claim.

In the past twenty years, many analytical approaches for evaluating American-style options in the Black-Scholes model are published, such as [15, 28, 46, 107], etc. However, most of them are not flexible for different payoff functions. In this paper, we find that the BAW and JZ approaches, see [15] and [107], can be extended to estimate $\tilde{v}^A(t, f)$ in the one factor Hull-White model. Numerical tests show that these two approximations are efficient and precise for GMAB policies with 20 years maturity.
Firstly, we show how the BAW method can be applied directly to estimate the American-style liability $\tilde{v}^A(t, f)$. It is obvious that $\tilde{v}^A(t, f)$ is the solution of (1.29),

$$ \frac{\partial \tilde{v}^A}{\partial t} - c f \frac{\partial \tilde{v}^A}{\partial f} + \frac{w_{T-t}^2 f^2}{2} \frac{\partial^2 \tilde{v}^A}{\partial f^2} = 0 \quad (1.15) $$

and is subject to the boundary conditions $\tilde{v}^A(t-, f) = \max(f, \tilde{v}^A(t, f))$ for $0 \le t \le T$. The key insight of BAW approximation, see [15], is that if both American options and European options are solutions of (1.29), then the early exercise premium $\psi(t, f)$ of GMAB policies, which is equal to $\tilde{v}^A(t, f) - \tilde{v}^E(t, f)$, is also a solution of (1.29). Defining $\tau = T - t$ and changing the variable of $\psi$ from $t$ to $\tau$, we have that $\psi(\tau, f)$ is the solution of the following equation,

$$ - \frac{\partial \psi}{\partial \tau} - c f \frac{\partial \psi}{\partial f} + \frac{1}{2} \omega_{\tau}^2 f^2 \frac{\partial^2 \psi}{\partial f^2} = 0 \quad (1.16) $$

In practice, it is very difficult to find the solution of (1.16) analytically. So the authors of [15] developed an approximation method to estimate $\psi(\tau, f)$.

According to the BAW method, the early exercise premium can be approximated by the function $\bar{\psi}(\tau, f) = h(\tau)u(h, f)$, where $h(\tau) = 1 - e^{-g\tau}$ (numerical tests show that $g = -\log(Z_t^T)/\tau$ could be a good choice) and $u(h, f)$ is a function to determine. Replacing $\psi(\tau, f)$ by $\bar{\psi}(\tau, f)$ in (1.16) and neglecting the term $\partial u/\partial h$, we have:

$$ - \frac{g e^{-g \tau}}{1 - e^{-g \tau}} u - c f \frac{\partial u}{\partial f} + \frac{1}{2} \omega_{\tau}^2 f^2 \frac{\partial^2 u}{\partial f^2} = 0 \quad (1.17) $$

The general solution $u(h, f)$ of (1.17) is:

$$ u(h, f) = A_1 f^{\lambda_1} + A_2 f^{\lambda_2}, \quad \text{where } \lambda_{1,2} = \frac{\omega_{\tau}^2 + 2c \pm \sqrt{(\omega_{\tau}^2 + 2c)^2 + 8g e^{-g\tau} \omega_{\tau}^2 / h(\tau)}}{2\omega_{\tau}^2} $$

As $\lambda_2 < 0$ while the early exercise premium is worthless when the asset price drops to zero, the coefficient $A_2$ must be zero. Thus when $f < f^*(t)$ at time $t$, the American-style liability can be approximated by:

$$ \tilde{v}^A(t, f) \approx \tilde{v}^E(t, f) + h(\tau)A_1 f^{\lambda_1} \quad (1.18) $$

It remains the coefficient $A_1$ and the critical forward account value $f^*(t)$ to find. In fact, (1.13) implies that at $f^*(t)$, $\tilde{v}^A(t, f^*)$ is equal to the forward account value, that is:

$$ f^*(t) = \tilde{v}^E(t, f^*(t)) + h(\tau)A_1 f^*(t)^{\lambda_1} \quad (1.19) $$

and the slope of the exercisable value, which is the forward account value, is set equal to the slope of $\tilde{v}^A(t, f)$ at $f^*(t)$, that is,

$$ 1 = \frac{\partial \tilde{v}^E(t, f)}{\partial f} \Big|_{f=f^*(t)} + h(\tau) \lambda_1 A_1 f^*(t)^{\lambda_1 - 1} \quad (1.20) $$

Solving (1.19) and (1.20) by the algorithm of Newton-Raphson, see [15], we can find both $f^*(t)$ and $A_1$ at time $t$.

The numerical tests show that the pricing error of BAW method is tiny if the forward account value $f$ is not too small. However, when the GMAB policies are deep in the money, the BAW approximation becomes less precise. To improve the precision in this case, we extend the method developed by Ju and Zhong (JZ method, see [107]) to our specific evaluation problem here. In fact, the authors of
[107] proposed to add a perturbation term to the function $\overline{\Psi}(\tau, f)$ to improve the precision of the early exercise premium. This corrected function, denoted by $\overline{\Psi}_j$, is defined as :

$$ \overline{\Psi}_j(\tau, f) := (1 + \epsilon(h, f)) \overline{\Psi}(\tau, f) = (1 + \epsilon(h, f)) h(\tau) u(h, f) \quad (1.21) $$

where $\epsilon(h, f)$ is a function to determine. Replacing $\psi$ by $\overline{\Psi}_j$ in (1.16) and applying (1.17), we obtain an equation for $\epsilon(h, f)$ at time 0,

$$ -\frac{\partial h}{\partial \tau} \frac{\partial u}{\partial h}(1+\epsilon) - u \frac{\partial h}{\partial \tau} \frac{\partial \epsilon}{\partial h} + (\omega_{\tau}^2 f^2 \frac{\partial u}{\partial f} - c f u) \frac{\partial \epsilon}{\partial f} + \frac{1}{2} \omega_{\tau}^2 f^2 u \frac{\partial^2 \epsilon}{\partial f^2} = 0 \quad (1.22) $$

After a series of approximations, see [107], we get the corrected approximation to the American-style liability $\tilde{v}^A(t, f)$:

$$ \tilde{v}^A(t, f) \approx \tilde{v}^E(t, f) + \frac{d}{1 - bx^2 - \alpha x} \left( \frac{f}{f^*(t)} \right)^{\lambda_1} \quad (1.23) $$

where $x = \log(f/f^*(t))$ and $a, b, c$ and $d$ are four parameters to be determined by (1.22). Figure 1.10 compares the American liability $v^A$ computed by (1.18) and (1.23) with the numerical results of PDE

<figure><img src="image_18.png" /><figcaption>FIGURE 1.10 – Comparing the liability calculated by two approximation methods with the numerical results of PDE.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIGURE 1.11 – Comparing the empirical approximation methods with the numerical results of PDE.</figcaption></figure>

scheme 4, which is considered as the benchmarks here. We observe that the approximation methods are precise for a wide range of initial account values.

In addition, we find an empirical relationship between the Bermudan GMAB liabilities and the American ones, which can be simply written as

$$ v^B(0, f) \approx v^A(0, f) - (v^A(0, f) - v^E(0, f)) \frac{\Delta t}{T} \quad (1.24) $$

where $T$ is the maturity and $\Delta t$ is the interval between two exercisable dates of Bermudan GMAB liabilities. Figure 1.11 verifies the empirical approximation (1.24) by the numerical results of PDE scheme. In fact, our numerical tests show that (1.24) is also applicable for other long-term Bermudan contingent claims (e.g. vanilla puts with maturities longer than 5 years).

## Appendix B: Option value of GMAB policies

<footnote n="4">4. In this numerical test, the discrete time step of the PDE scheme is 0.01 year.</footnote>
The forward option value $\tilde{w}^B(t, f)$ of GMAB policies, defined by (1.14), is what the insurers should replicate in practice once they write GMAB contracts. According to the definition, we can decompose $\tilde{w}^B(t, f)$ into two parts :

$$ \tilde{w}^B(t, f) = \tilde{v}^B(t, f) - f = [\tilde{v}^B(t, f) - e^{-\alpha\tau}f] - (1 - e^{-\alpha\tau})f \quad (1.25) $$

For simplicity, we define $\tilde{u}^B(t, f) = \tilde{v}^B(t, f) - e^{-\alpha\tau}f$. Applying (1.11), it is easy to verify that $\tilde{u}^B(t_i-, F^T(t_i-))$ evolves as

$$ \begin{aligned} \tilde{u}^B(t_i-, F^T(t_i-)) &= \max((1 - e^{-\alpha(T-t_i)})F^T(t_i), \tilde{u}^B(t_i, F^T(t_i))) \\ &= \max((1 - e^{-\alpha(T-t_i)})F^T(t_i), E^{Q^T}[\tilde{u}^B(t_{i+1}-, F^T(t_{i+1}-))|\mathcal{F}_{t_i}]) \end{aligned} \quad (1.26) $$

and at the maturity, we have $\tilde{u}^B(T, F^T(T)) = (G - F^T(T))^+$. Therefore, we can interpret $\tilde{u}^B(t, f)$ as a Bermudan put option with the exercisable value $(1 - e^{-\alpha\tau})f$ at dates $t_i$. Some insurers call $\tilde{u}^B(t, f)$ as the forward value of claims, and $(1 - e^{-\alpha\tau})f$ as the forward value of charges, for this term is in fact the expectation of forward charge fees insurers can receive if the policyholder holds the contract to the maturity. According to (1.25), the forward option value $\tilde{w}^B(t, f)$ is equal to the difference between the forward value of claims and the forward value of charges.

## Appendix C: Numerical schemes

It follows from the definition of the forward Bermudan-style liability $\tilde{V}^B(t)$ with an optimal stopping time $\tau \in \{t_1, t_2, \dots, T\}$, that the process of the forward liability $\tilde{V}^B(t)$ satisfies the backward programming equation, for $0 < t_i < T$

$$ \tilde{V}^B(t_i-) = \max\{F^T(t_i), E^{Q^T}[\tilde{V}^B(t_{i+1}-)|\mathcal{F}_{t_i-}]\} \quad (1.27) $$

and at the maturity, we have $\tilde{V}^B(T) = \max(G, F^T(T))$.

Thanks to the martingale property of $\tilde{V}^B$ on the interval $[t, \hat{\tau}_i)$, we have for $0 < t_i < T$,

$$ \tilde{V}^B(t_i-) = E^{Q^T}[F^T(\hat{\tau}_i) + I_{\{\hat{\tau}_i=T\}} \max(G, F^T(T))|\mathcal{F}_{t_i-}] \quad (1.28) $$

where the optimal stopping time $\hat{\tau}_i$ is defined as : $\hat{\tau}_i := \inf\{t_j \ge t_i : \tilde{V}^B(t_j-) = F^T(t_j-)\}$.

To the best of our knowledge, it is difficult to find precise analytical formulas to evaluate the Bermudan-style contingent claims in practice. In this paper, we extend the traditional semi-analytical methods, see [15, 107], to estimate the spot Bermudian-style liability $\tilde{v}^B(t, f)$ of GMAB policies, see Appendix A. However, the approximation method introduced here is not as flexible as numerical approaches, especially for high dimensional problems. Thus in most cases, we need to use numerical methods, such as PDE and Monte Carlo, to calculate the Bermudan liability $\tilde{v}^B(t, f)$.

### 1. PDE scheme

In this paper, we transform the evaluation problem (1.11) of Bermudan-style liability $\tilde{v}^B(t, f)$ into a free-boundary partial differential equation, for which $\tilde{v}^B(t, f)$ is the solution. For GMAB policies, the Bermudan-style liability $\tilde{v}^B(t, f)$ is represented as a function of two variables : the time $t$ and the forward account value $f$. Applying Itô's lemma and the martingale representation theorem together, we
know that the liability $\tilde{v}^B(t, f)$ is the solution of a one dimensional PDE. By adding the free-boundary constraint implied by equation (1.11) to this PDE, we have

$$ \frac{\partial \tilde{v}^B}{\partial t} - c f \frac{\partial \tilde{v}^B}{\partial f} + \frac{w_{T-t}^2 f^2}{2} \frac{\partial^2 \tilde{v}^B}{\partial f^2} = 0 \qquad (1.29) $$

on $\{(t, f) : t_{i-1} \le t < t_i, f > 0\}$, subject to the boundary conditions at time points $0 < t_i < t_{n+1}$

$$ \tilde{v}^B(t_i-, f) = \max(f, \tilde{v}^B(t_i, f)) \qquad (1.30) $$

and at the maturity T, we have

$$ \tilde{v}^B(T, f) = \max(f, G) \qquad (1.31) $$

On each of the intervals $[t_{i-1}, t_i)$, the PDE (1.29) can be calculated numerically by using the Crank-Nicolson method, see [59], for $f \in [0, F)$, where F is the upside boundary of the numerical solution. While at discrete time points $t_i$, the critical lapse surface $f^*(t_i)$ can be easily found by the free-boundary constraint indicated in (1.30). On the boundary, we impose the zero-convexity conditions<fnref n="5" />:

$$ \tilde{v}^B(t, f)|_{f=0} = G; \qquad \frac{\partial^2 \tilde{v}^B}{\partial f^2}|_{f=F} = 0 $$

In fact, according to [14], the precision of the final solution is not very sensible to the error on boundaries if the solution domain of parabolic equation is large enough. So in most cases, the practitioner can choose other boundary conditions instead of those we propose here<fnref n="6" />.

## 2. Monte-Carlo scheme

The liability $\tilde{v}^B(0)$ is estimated as the conditional expected value of the forward liability based on Monte-Carlo simulation.

The forward liability satisfies two conditions:

— the backward programming equation:

$$ \left\{ \begin{array}{l} \tilde{V}^B(t_i-) = \max\{F^T(t_i), \mathbf{E}^{\mathbf{Q}^T}[\tilde{V}^B(t_{i+1}-)|\mathcal{F}_{t_i-}]\} \\ \tilde{V}^B(T) = \max(G, F^T(T)) \end{array} \right. \qquad (1.32) $$

— the martingale property of $\tilde{V}^B$ on each $[t, \tilde{\tau}_i)$ translated by:

$$ \left\{ \begin{array}{l} \tilde{V}^B(t_i-) = \mathbf{E}^{\mathbf{Q}^T}[F^T(\hat{\tau}_i) + \mathbf{1}_{\{\hat{\tau}_i=T\}} \max(G, F^T(T))|\mathcal{F}_{t_i}] \\ \hat{\tau}_i := \inf\{t_j \ge t_i : \tilde{V}^B(t_j-) = F^T(t_j-)\} \end{array} \right. \qquad (1.33) $$

As pointed out in [37], these equations ((1.27) and (1.28)) lead to two algorithms, referred to as S1 and S2 hereafter.

The first algorithm S1 computes the optimal stopping time to lapse in three steps:

1. Simulate N discrete scenarios of the forward account value, denoted as $F^T(k)$ ($0 \le i \le n+1$ and $0 < k \le N$), according to (1.5).

<footnote n="5">5. This assumption is based on the fact that the gamma of the liability is small on the boundary.</footnote>

<footnote n="6">6. In the specific case here, the first or second order derivative boundary condition is preferred to the Dirichlet condition. As the latter could lead to significant errors on the boundary.</footnote>
2. Set the forward Bermudan-style liability at maturity for each scenario : $\tilde{V}_{[1]}^{B(k)}(T) = F^T(T)$.

3. Apply (1.27) from $t_n$ to $t_0$. For $i=n$ to 0 :

$$
\text{if } F^{T(k)}(t_i -) < B: \quad \tilde{V}_{[1]}^{B(k)}(t_i -) = \tilde{V}_{[1]}^{B(k)}(t_{i+1} -)
$$

$$
\text{if } F^{T(k)}(t_i -) \ge B: \quad \tilde{V}_{[1]}^{B(k)}(t_i -) = \max\{F^{T(k)}(t_i -), \tilde{\mathbf{E}}^{\mathbb{Q}^T}[\tilde{V}_{[1]}^{B}(t_{i+1} -) | F^{T(k)}(t_i -)]\}
$$

From step 3 of scheme S1, we can identify the estimated rational lapse time $\tilde{\tau}_0^{(k)}$ as the first time for the k-th scenario where the liability equals the account value. Once $\tilde{\tau}_0^{(k)}$ is recorded for each path, we can estimate the Bermudan-style liability by scheme S2 where we regress the cash flows on a set of basis functions.

This latest computes the corresponding liability following four steps :

1. Simulation : Use the same N simulated scenarios as in S1.

2. Initialization : Set the rational lapse time $\tilde{\tau}_0^{(k)} = t_{n+1}$, for $0 < k \le N$.

3. Backward induction : For $i=n$ to 0, $\tilde{\tau}_i^{(k)} = i \mathbf{1}_{\{(k) \in \mathcal{L}_i\}} + \tilde{\tau}_{i+1}^{(k)} \mathbf{1}_{\{(k) \in \mathcal{L}_i\}}$. (where $\mathcal{L}_i := \{(k) : V_{[1]}^{B(k)}(t_i -) = F^{T(k)}(t_i -)\}$ and $\mathcal{L}_i^c := \{(k) : V_{[1]}^{B(k)}(t_i -) > F^{T(k)}(t_i -)\}$ its complement)

4. Price estimator at 0: $\tilde{V}_{[2]}^{B}(0) := \frac{1}{N} \sum_{k=1}^{N} [F^{T(k)}(\tilde{\tau}_{0}^{(k)}) + \mathbf{1}_{\{\tilde{\tau}_{0}^{(k)}=T\}} \max(G, F^{T(k)}(T))]$.

In [37], the authors find the following relation with the two estimators $\tilde{V}_{[2]}^{B}$ and $\tilde{V}_{[1]}^{B}$ computed above :

$$
\mathbf{E}[\tilde{V}_{[2]}^{B}(0)] \le \tilde{V}^{B}(0) \le \mathbf{E}[\tilde{V}_{[1]}^{B}(0)] \quad (1.34)
$$

In the numerical tests in the section below, we calculate both $\tilde{V}_{[1]}^{B}(0)$ and $\tilde{V}_{[2]}^{B}(0)$ to construct confidence intervals $[\tilde{V}_{[2]}^{B}(0), \tilde{V}_{[1]}^{B}(0)]$ for the true value $\tilde{V}^{B}(0)$.

**Appendix D : Linear regression vs global polynomial regression**

We now introduce the scheme used to calculate the conditional expected value of continuation for scenarios such that $F^{T(k)}(t_i -) > B$. Here we use the local linear regression approach proposed in [37] to calculate this value, as opposed to the global polynomial regression method developed in [120]. The reason for this is that the latter can lead to some instability in the regression process for high dimensional and long maturity problems, see [37].

For our specific problem, we have only one dimension : the forward account value $F^T$. The idea is to use, at each time step $t_i$, a set of functions $\psi_d$ having local hypercube supports $D_l$, where the space is cut into I regions, $l = 1$ to I and $\{D_d\}$ is a partition of $[\min_{\{k=1,N\}} F^{T(k)}(t_i), \max_{\{k=1,N\}} F^{T(k)}(t_i)]$. The index $\cdot$)($^k$) denotes the k-th simulated scenario. On each support $D_l$, we define a linear function $\Psi_l$ with 2 degrees of freedom, which are represented by a constant and $F^T$. Our goal now is to regress the future cash flow of liability on the function $\Psi_l$ to estimate the relevant conditional expectation. The two regression basis of $\Psi_l$, noted as $(\psi_l^0, \psi_l^1)^7$.

For simplicity, we define the function $G^N(t_i, F^T(t_i))$ as the conditional expectation at time $t_i$, we have :

$$
G_i^{N(k)} : G^N(t_i - , F^{T(k)}(t_i -)) = \mathbf{E}^{\mathbb{Q}^T}[\tilde{V}_{[1]}^{B}(t_{i+1} - , F^{T(k)}(t_{i+1} -)) | F^{T(k)}(t_i -)]
$$

<footnote n="7">7. The two regression basis correspond to the constant and the forward account value.</footnote>
where $F_i^{N(k)}$ is the conditional expectation associated with the $k$-th path at time $t_i$. In the context of S1, the numerical procedure to calculate $G_i^{N(k)}$ reads as follows :

Scheme $S_c$: estimator of $G_i^{N(k)}(0 \le i \le n)$ with regression :

1. At time $t_i$, realize a quick-sort of $F_i^{T(k)}(t_i)$ for N scenarios and identify the support $D_l$ of the functions $\Psi_l$ so that each support contains approximately the same number of scenarios.

2. For each scenario $0 < k \le N$, set the three regression basis of $\Psi_l : (\psi_l^0, \psi_l^1)$, where $\psi_l^0(\cdot) = 1$, $\psi_l^1(F_i^{T(k)}(t_i -)) = F_i^{T(k)}(t_i -)$.

3. On each support $D_l$, regress $\{V_{[1]}^{B(k)}(t_{i+1}-)\}_{k \le N}$ on $\Psi_l$. In other words, for $\forall l$, we calculate the co-efficients $(\alpha_l^0, \alpha_l^1)$ that minimize $\sum_{k=1}^N |V_{[1]}^{B(k)}(t_{i+1}-) - \sum_{m=0}^1 \alpha_l^m \Psi_l^m(\cdot)^{(k)}|^2$, and set $G_i^{N(k)} = \sum_{m=0}^1 \alpha_l^m \Psi_l^m(F_i^{T(k)}(t_i^-))$.
# Chapitre 2

## Optimal behavior strategy in the GMIB product

**Abstract**—This chapter falls within the scope of quantitative studies done at AXA Group Risk Management. It focuses on a variation of the guaranteed minimum income benefit with and without a death benefit. This variable annuity rider offers the policyholder the possibility to convert the benefit base into annuities for life. The income rate is fixed by the insurer at inception, depending on the insured age at election. From the insurer's point of view, this product embeds the policyholder "optimal behavior". In this chapter we study such behavior from the policyholder's perspective who maximizes the expectation of his future cash flows. We develop this analysis in a dynamic programming framework. Using convenient scaling properties of the contract value, we reduce the dimension of the problem. We policyholder's decision as a function of the contract moneyness. Furthermore, we analyze the sensitivity of the product to different drivers like the volatility, interest rate and roll-up rate. In particular, we find that the contract is usually underpriced under optimal behavior.

**Keywords** : GMIB; Variable Annuities; rational behavior; optimal withdrawals; PDE; dynamic programming.

### 2.1 Product description

Guaranteed Minimum Income benefit (GMIB) product appeared in the market in 1996. This guarantee enables policyholders to make annual partial withdrawals (typically 4% to 7%) of their guaranteed protection amount and ensures an analogous percentage of the GMIB benefit base for their entire lifetime, no matter how the investments in the sub-accounts perform. It combines longevity protection with withdrawal flexibility, hence it is seen as a "second-generation" guarantee. The guarantee can concern one or two lives (typically spouses). Each annual withdrawal does not exceed some maximum value, but it is evident that the total amount of withdrawals is limited only to the exhaustion of the client's account value. Annual withdrawals of about 5% of the (single initial) premium are commonly guaranteed for insured aged 60+. In case of death, any remaining fund value is paid to the insured dependents. To satisfy the new needs of an ageing population, insurance companies have started offering a lifetime benefit feature with GMIB.

While the GMIB product was defined in the introduction, its form Retirement Cornerstone® commercialized by AXA will be explained and illustrated in this section.
## 2.1.1 Retirement Cornerstone

Retirement Cornerstone® is a long-term GMIB-type product designed for retirement purposes. This deferred variable annuity is commercialized by AXA Equitable life insurance company in the U.S. since 2011. It offers tax-deferred growth potential and living and death benefits as optional features.

### Benefit Base and partial withdrawals

To describe the benefit features of this GMIB clearly, some key terms must be addressed: GMIB benefit base and guaranteed withdrawal amount (GWA).

- **GMIB benefit base:** The GMIB benefit base is an amount used to determine the guaranteed annual withdrawal amount and lifetime payments. The GMIB benefit base is created and increased by allocations and transfers to the account value, as well as annual withdrawn amounts. This percentage is known as roll-up rate and deferral roll-up rate. It must be noticed that the GMIB benefit base is a "fictive" amount, i.e., it can not be considered as an account nor cash value, only as a reference in the computation of lifetime payments and guaranteed withdrawal amounts.

- **(Annual) guaranteed withdrawal amount (GWA):** The "annual guaranteed withdrawal amount" is the withdrawal amount suggested by the insurer. It is equal to the annual roll-up rate in effect on the first day of the contract year, multiplied by the current benefit base. It is also the maximum amount upon which the benefit base is reduced without penalty, in contrast to excess withdrawals.

In Table 2.1, we illustrate a concrete example of the calculations of a GMIB benefit. BB designates the benefit base and AV refers to the account value. The contract initial premium is $100,000. The policyholder does not make any withdrawal till the 6th contract year and once he/she does, all withdrawals stay within the boundaries of the GWA. Therefore, the GMIB benefit base does not diminish. The effect of excess withdrawals will be discussed later. Till the 5th contract anniversary, the defer-

<table><caption>TABLEAU 2.1 – GMIB benefit base evolution for an allocation of $100,000 with partial withdrawals that does not affect the value of the guaranteed account.</caption><thead><tr><th>Year</th><th>Deferral/ Roll-up rate</th><th>GMIB BB</th><th>WA</th><th>Percentage of GMIB BB</th><th>GMIB BB after withdrawal</th></tr></thead><tbody><tr><td>0</td><td>-</td><td>$100,000</td><td>$0</td><td>0%</td><td>$100,000</td></tr><tr><td>1</td><td>4.8%</td><td>$104,800</td><td>$0</td><td>0%</td><td>$104,800</td></tr><tr><td>2</td><td>4.3%</td><td>$109,830</td><td>$0</td><td>0%</td><td>$109,830.40</td></tr><tr><td>3</td><td>5.2%</td><td>$114,553</td><td>$0</td><td>0%</td><td>$114,553.11</td></tr><tr><td>4</td><td>5.4%</td><td>$120,510</td><td>$0</td><td>0%</td><td>$120,509.87</td></tr><tr><td>5</td><td>5.0%</td><td>$127,017</td><td>$0</td><td>0%</td><td>$127,017.40</td></tr><tr><td>6</td><td>4.7%</td><td>$133,368</td><td>$6,268.31</td><td>4.7%</td><td>$127,099.96</td></tr><tr><td>7</td><td>5.2%</td><td>$133,368</td><td>$6,935.15</td><td>5.2%</td><td>$126,433.12</td></tr><tr><td>8</td><td>5.4%</td><td>$133,368</td><td>$7,201.89</td><td>5.4%</td><td>$126,166.39</td></tr><tr><td>9</td><td>6.0%</td><td>$133,368</td><td>$5,334.73</td><td>4.0%</td><td>$128,033.54</td></tr><tr><td>10</td><td>7.3%</td><td>$136,036</td><td>$5,441.43</td><td>4.0%</td><td>$130,594.21</td></tr></tbody></table>

ral roll-up rate is used to calculate the amount that is credited to the benefit base each year since no withdrawal has been made. For example, the GMIB benefit base in the 5th year is computed by taking the value of the previous year and adding the corresponding 5%, i.e. $120,510+5.4%×$120,510 = $120,510+6,507.54=$127,017.54.
Once the client proceeded to his/her first withdrawal, the roll-up rate determines the evolution of the GMIB benefit base and the annual GWA. If the maximum guaranteed quantity is withdrawn, the benefit base remains unchanged, as shown in years 6 to 9, i.e. the same quantity withdrawn from the benefit base is added by the roll-up amount. In case of withdrawing less than the annual GWA, a greater value for the benefit base is obtained. In the 10th year, the policyholder only withdraws 4 % of the GMIB guaranteed value. Consequently, the benefit base becomes $133,368-$5,334.73+$133,368×6.0% = $128,033.54+$8002.08=$136,035.62.

The step-up option enables the policyholder to reset the guaranteed withdrawal balance to the current higher account value when investment performance is strong. By choosing to reset the benefit base, the policyholder is able to increase the total benefit amount and the annual guaranteed withdrawal amount. The option may reduce the inflation effect on incomes when the account value goes up and the step-up option is available. Accordingly, the period over which lifetime payments can begin is extended of 10 years. Nevertheless, at policyholder's 95th birthday, the lifetime payments are set to automatically begin no matter how many times the reset option has been chosen. In Figure 2.1 we illustrate the evolution of the protected account value, and the different options for the GMIB benefit base. The contract is issued at the policyholder's 50th birthday. Partial withdrawals from this annuity

<figure><img src="image_20.png" /><figcaption>FIGURE 2.1 – Behavior of the account value and the different options for the GMIB benefit base when no withdrawals have been performed.</figcaption></figure>

contract are taxable as ordinary income and, if made prior to age 59 1/2 may be subject to an additional 10% federal tax and withdrawal charges. All amounts invested in an annuity's portfolios are subject to fluctuation in value and market risk, including loss of principal. The account value may be reduced due to fees and charges such as operations and sales charges, administrative fees, and optional benefits additional charges.

## Fees and charges

The fee structure has an impact on the GMIB price. The GMIB charge is deducted from the contract value periodically. It is usually presented as a percentage of the current account value, although it can also be a percentage of the initial premium, a percentage of the remaining guaranteed benefit amount, or the greater of these two. The annual charge ranges for Retirement Cornerstone® is comprised between 20 and 75 basis points depending on the nature of the benefit.
Lifetime payments

Lifetime GMIBs provide guaranteed annual income until death. Policyholders are also able to access potentially increased account values, and control the asset allocation in ways that the traditional variable annuitization normally does not allow. Lifetime GMIBs usually have two options : single life or joint spousal life. For the single life option, the benefit payments end at the death of the person covered. For the joint spousal life option, the benefit payments end when the remaining spouse dies. The fee rate for the single life option ranges from 25 to 55 basis points, while the spousal life option tends to be 10-20 basis points higher.

For the single life option, a spouse continuation option is available upon the first death with the same charge, but the account value and the benefit amount may be adjusted. For the joint spousal life, there will be no recalculation of the benefit amount when the first death occurs. The annual benefit payment amount is a percentage of the initial guaranteed benefit amount. The older the policyholder is, the larger the value of the lifetime payments will be. For example, the guaranteed factor to compute the lifetime payment for the legacy product Accumulator 7 is 5.3% if the annuity starts at age 73 and 7.1% if the attained age is 83. The factor used to compute the lifetime payments is given by the insurer and depends on the policyholder's age at inception.

Annual lifetime payments in GMIB products begin as follows :

(i) The next contract year following the date the account value falls to zero.

(ii) The contract date anniversary following the policyholder's 95th birthday.

(iii) The policyholder's election to exercise the GMIB.

Similarly to GLWB product, GMIB is subject to a waiting period which begins on the date when the account value is first fund, and it ranges from 10-15 years depending on the policyholder's age.

If an excess withdrawal, i.e. withdrawal superior to the guaranteed withdrawal amount, reduces the account value to zero, the GMIB will be terminated. Even if an excess withdrawal does not cause the contract to terminate, it can greatly reduce the GMIB benefit base and the value of the benefit since it is done in pro rata basis.

In Table 2.2 cash flows for a maximum annual guaranteed withdrawal strategy are shown. The current and guaranteed factors<fnref n="1" /> are illustrated for the case of a 60 year old male that acquires the contract in 2016. These factors take into account the age of the policyholder when the contract is issued, his/her gender and probability of survival. In the particular case of the current annuity factor, the market interest rates play a key role. A constant net return of 3 % and fees of 4.5% has been considered to facilitate comprehension. The factors need to be recalculated when they are faced to changing market interest rates.

In the case illustrated in Table 2.2, the policyholder takes the annual guaranteed withdrawal amount till his account value turns to zero at age 76. At that moment, lifetime payments begin and the owner of the policy faces two options : to annuitize the GMIB benefit base or the account value. Annual payments will be based on the guaranteed or current factor depending on the policyholder's choice. This is possible till the client's 85th anniversary, otherwise he will lose the possibility to transform the contracts benefit base into annual income payments. In the described scenario and taking the discount factors into account, the policyholder will obtain $86,981.33, meaning he will not recover the

<footnote n="1">1. The current (resp. guaranteed) factor is the annual income rate when the income benefit is based on the account value (resp. benefit base).</footnote>
<table><caption>TABLEAU 2.2 – Protected account value and GMIB behavior given a static withdrawal strategy and lifetime payments.</caption><thead><tr><th>Contract Year</th><th>AV</th><th>Roll-up rate</th><th>WA</th><th>GMIB BB</th><th>GMIB factor</th><th>Current factor</th><th>Annuitization BB</th><th>Annuitization AV</th><th>Lifetime payments</th></tr></thead><tbody><tr><td>0</td><td>$100,000</td><td></td><td>$0</td><td>$100,000</td><td>4.2%</td><td>5.4%</td><td>$4,204.79</td><td>$5,367.16</td><td>-</td></tr><tr><td>1</td><td>$99,500</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.3%</td><td>5.5%</td><td>$4,269.15</td><td>$5,444.66</td><td>-</td></tr><tr><td>2</td><td>$93,032.50</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.3%</td><td>5.6%</td><td>$4,336.57</td><td>$5,193.53</td><td>-</td></tr><tr><td>3</td><td>$86,597.34</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.4%</td><td>5.7%</td><td>$4,407.24</td><td>$4,935.07</td><td>-</td></tr><tr><td>4</td><td>$80,194.35</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.5%</td><td>5.8%</td><td>$4,481.31</td><td>$4,668.45</td><td>-</td></tr><tr><td>5</td><td>$73,823.38</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.6%</td><td>6.0%</td><td>$4,558.97</td><td>$4,392.83</td><td>-</td></tr><tr><td>6</td><td>$67,484.26</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.6%</td><td>6.1%</td><td>$4,640.42</td><td>$4,107.29</td><td>-</td></tr><tr><td>7</td><td>$61,176.84</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.7%</td><td>6.2%</td><td>$4,725.87</td><td>$3,810.86</td><td>-</td></tr><tr><td>8</td><td>$54,900.96</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.8%</td><td>6.4%</td><td>$4,815.53</td><td>$3,502.52</td><td>-</td></tr><tr><td>9</td><td>$48,656.45</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>4.9%</td><td>6.5%</td><td>$4,909.63</td><td>$3,181.14</td><td>-</td></tr><tr><td>10</td><td>$42,443.17</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>5.0%</td><td>6.7%</td><td>$5,008.38</td><td>$2,845.48</td><td>-</td></tr><tr><td>11</td><td>$36,260.95</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>5.1%</td><td>6.9%</td><td>$5,111.95</td><td>$2,494.23</td><td>-</td></tr><tr><td>12</td><td>$30,109.65</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>5.2%</td><td>7.1%</td><td>$5,220.49</td><td>$2,126</td><td>-</td></tr><tr><td>13</td><td>$23,989.10</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>5.3%</td><td>7.3%</td><td>$5,334.10</td><td>$1,739.42</td><td>-</td></tr><tr><td>14</td><td>$17,899.16</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>5.5%</td><td>7.4%</td><td>$5,452.86</td><td>$1,333.15</td><td>-</td></tr><tr><td>15</td><td>$11,839.66</td><td>6%</td><td>$6,000</td><td>$100,000</td><td>5.6%</td><td>7.7%</td><td>$5,576.81</td><td>$905.97</td><td>-</td></tr><tr><td>16</td><td>$5,810.46</td><td>6%</td><td>$5,810.46</td><td>$100,189.54</td><td>5.7%</td><td>7.9%</td><td>$5,716.80</td><td>$456.79</td><td>-</td></tr><tr><td>17</td><td>$0</td><td>6%</td><td>$0</td><td>$0</td><td>5.8%</td><td>8.1%</td><td>$0</td><td>$0,00</td><td>$5,716.80</td></tr><tr><td>18</td><td>$0</td><td>6%</td><td>$0</td><td>$0</td><td>6.0%</td><td>8.3%</td><td>$0</td><td>$0,00</td><td>$5,716.80</td></tr><tr><td>19</td><td>$0</td><td>6%</td><td>$0</td><td>$0</td><td>6.1%</td><td>8.5%</td><td>$0</td><td>$0,00</td><td>$5,716.80</td></tr><tr><td>20</td><td>$0</td><td>6%</td><td>$0</td><td>$0</td><td>6.3%</td><td>8.7%</td><td>$0</td><td>$0</td><td>$5,716.80</td></tr></tbody></table>

initial premium invested on the contract.

## Death benefit

The Retirement Cornerstone® also offers the possibility of combining the GMIB with Guaranteed Minimum Death Benefit (GMDB) when the contract is purchased. This is not unusual since variable annuities typically provide a guarantee if the policyholder dies before receiving any income.

The death benefit often equals the greater of the account value and total premiums paid less any withdrawals. For example, a person had paid premiums totaling $100,000, and had made withdrawals equaling $15,000. The account value stands at $80,000 because of these withdrawals and investment losses. If he were to die, his beneficiary would receive the aforementioned quantity. Within Retirement Cornerstone® product this type of death benefit is known as "Return of Principal" and is considered without any extra charges.

Retirement Cornerstone® also offers optional death benefits in the form of roll-up or annual ratchet and reset with extra charges. These options are:

- **Highest anniversary value death benefit**- The "Highest anniversary value death benefit" is an optional guaranteed minimum death benefit in connection with the account value. The death benefit is calculated using the highest value of the account on the contract date anniversary.

- **Roll-up to age 85 benefit base**- The "Roll-up to age 85 benefit base" is equal to the GMIB benefit base, i.e., it is reduced dollar-for-dollar in the case of partial withdrawals being done within the limits of the guaranteed withdrawal amount and pro-rata, when an excess withdrawal has been made by the policyholder. It is favored by the roll-up amount till the policyholder's 85th birthday. This option is tied only to "Greater of" death benefit, i.e. it can not be chosen individually.

- **"Greater of" death benefit**- The "greater of" death benefit is an optional guaranteed minimum
death benefit in connection with the protected benefit account value only. The death benefit is calculated using the greater of two benefit bases- the greater of the roll-up to age 85 benefit base and the highest anniversary value benefit base. There is an additional charge for the "greater of" death benefit under the contract.

Once the lifetime payments corresponding to the GMIB start, the policyholder loses the possibility of keeping the GMDB. This right is lost at policyholder's 95th anniversary since the lifetime payments start automatically. The return of principal, highest anniversary value, and "greater of" guaranteed minimum death benefits will terminate without value if the account value falls to zero as a result of withdrawals or payment of any applicable charges. This will happen whether the policyholder elects the GMIB or receive lifetime GMIB payments or not.

Policyholders can elect the optional death benefit guarantees between age 20 and 68, implying that this product targets a "younger" sector of the population compared to the "return of principal" which can be chosen till age 80.

Some numerical examples will be presented to illustrate the evolution of the GMIB and GMDB under partial withdrawals. A premium of $100,000-dollar is considered for a policyholder aged 60, with no additional contributions, and no transfers. Throughout these examples, no charges are deducted from the account value and there is a fixed roll-up rate of 4%. The assumed returns do not follow any market trends and were chosen to serve the purposes of illustrating two types of scenarios.

We define the following notation :

GMIB BB : Guaranteed minimum income benefit base

RP BB : Return of principal benefit base

RU BB : Roll-up to age 85 benefit base

HA BB : Highest anniversary value benefit base

GO BB : "Greater of" benefit base

Guaranteed minimum death benefit

<table><thead><tr><th>Year</th><th>Net Return</th><th>AV</th><th>WA</th><th>Roll-up rate</th><th>GMIB BB</th><th>RP BB</th><th>HAV BB</th><th>RU BB</th><th>GO BB</th></tr></thead><tbody><tr><td>0</td><td></td><td>$100,000</td><td></td><td></td><td>$100,000</td><td>$100,000</td><td>$100,000</td><td>$100,000</td><td>$100,000</td></tr><tr><td>1</td><td>3%</td><td>$103,000</td><td>$0</td><td>4%</td><td>$104,000</td><td>$100,000</td><td>$103,000</td><td>$104,000</td><td>$104,000</td></tr><tr><td>2</td><td>4%</td><td>$107,120</td><td>$0</td><td>4%</td><td>$108,160</td><td>$100,000</td><td>$107,120</td><td>$108,160</td><td>$108,160</td></tr><tr><td>3</td><td>6%</td><td>$113,547.20</td><td>$0</td><td>4%</td><td>$113,547.20</td><td>$100,000</td><td>$113,547.20</td><td>$113,547.20</td><td>$113,547.20</td></tr><tr><td>4</td><td>6%</td><td>$120,360.03</td><td>$0</td><td>4%</td><td>$120,360.03</td><td>$100,000</td><td>$120,360.03</td><td>$120,360.03</td><td>$120,360.03</td></tr><tr><td>5</td><td>7%</td><td>$128,785.23</td><td>$0</td><td>4%</td><td>$128,785.23</td><td>$100,000</td><td>$128,785.23</td><td>$128,785.23</td><td>$128,785.23</td></tr></tbody></table>

Alternative 1 : annual guaranteed withdrawal amount (dollar-for-dollar)

<table><tbody><tr><td>6</td><td>-5%</td><td>$122,345.97</td><td>$5,151.41</td><td>4%</td><td>$128,785.23</td><td>$95,789.47</td><td>$123,633.82</td><td>$128,785.23</td><td>$128,785.23</td></tr><tr><td>7</td><td>1%</td><td>$118,418.02</td><td>$5,151.41</td><td>4%</td><td>$128,785.23</td><td>$91,622.45</td><td>$118,482.42</td><td>$128,785.23</td><td>$128,785.23</td></tr><tr><td>8</td><td>-2%</td><td>$116,049.66</td><td>$5,151.41</td><td>4%</td><td>$128,785.23</td><td>$87,555.36</td><td>$113,331.01</td><td>$128,785.23</td><td>$128,785.23</td></tr><tr><td>9</td><td>2%</td><td>$118,370.66</td><td>$5,151.41</td><td>4%</td><td>$128,785.23</td><td>$83,745.01</td><td>$108,179.60</td><td>$128,785.23</td><td>$128,785.23</td></tr><tr><td>10</td><td>2%</td><td>$120,738.07</td><td>$5,151.41</td><td>4%</td><td>$128,785.23</td><td>$80,171.94</td><td>$103,028.19</td><td>$128,785.23</td><td>$128,785.23</td></tr></tbody></table>

Alternative 2 : excess withdrawal (pro-rata)

<table><caption>TABLEAU 2.3 – GMIB and GMDB behavior given policyholder's withdrawals when the account value is less than the GMIB benefit base at the time of the first withdrawal.</caption><tbody><tr><td>6</td><td>-5%</td><td>$122,345.97</td><td>$7,000</td><td>4%</td><td>$126,839.35</td><td>$94,278.52</td><td>$121,765.78</td><td>$126,839.35</td><td>$126,839.35</td></tr><tr><td>7</td><td>3%</td><td>$119,016.35</td><td>$7,000</td><td>4%</td><td>$124,786.30</td><td>$88,733.49</td><td>$114,803.39</td><td>$126,839.35</td><td>$126,839.35</td></tr><tr><td>8</td><td>-2%</td><td>$109,636.02</td><td>$4,991.45</td><td>4%</td><td>$124,786.30</td><td>$83,742.03</td><td>$109,811.94</td><td>$126,839.35</td><td>$126,839.35</td></tr><tr><td>9</td><td>2%</td><td>$106,837.29</td><td>$4,991.45</td><td>4%</td><td>$124,786.30</td><td>$78,750.58</td><td>$104,820.49</td><td>$126,839.35</td><td>$126,839.35</td></tr><tr><td>10</td><td>2%</td><td>$103,982.59</td><td>$4,991.45</td><td>4%</td><td>$124,786.30</td><td>$73,759.13</td><td>$99,829.04</td><td>$126,839.35</td><td>$126,839.35</td></tr></tbody></table>
Table 2.3 shows that the account value is reduced dollar-for-dollar by the withdrawal amount before considering market behavior no matter the size of withdrawals. In alternative 1, when the owner withdraws the annual guaranteed withdrawal amount [4% (roll-up rate) × $128,785 (the roll-up benefit bases as of the 6th contract anniversary)] the GMIB and roll-up to age 85 benefit bases neither decrease nor increase. The return of principal benefit base is reduced prorata as follows: since the withdrawal amount of $5,151 equals 4.21% of the account value ($5,151 = 4.21\% \times $122,346), the return of principle (RP) benefit base is also reduced by 4.21%; while the highest anniversary value (HAV) benefit base is reduced dollar-for-dollar, i.e., $128,785 (HA BB as of the last contract date anniversary) - $5,151 = $123,634 for the 6th contract year.

In the case of an excess withdrawal, as it is the case of contract years 6 and 7 of the second scenario, the Return of principal is reduced in the same way: since the withdrawal amount of $7,000 equals 5.721% of the account value in 6th year ($7,000 divided by $122,346 = 5.721\%), the RP benefit base is reduced by 5.721%. The pro-rata reduction of the roll-up benefit bases is as follows: $7,000 (the amount of the withdrawal, including any applicable withdrawal charge) - $5,151 (GWA) = $1,849 ("excess") which represent 1.511% of the account value, there is a decrease of 1.511% in the Roll-up benefit bases. The highest anniversary value benefit base is reduced dollar-for-dollar and pro-rata, as follows: $128,785 (HAV BB as of the last contract date anniversary) - $5,151 (GWA) = $1,868 (($128,785 - $5,151) × 1.511%) = $121,766.

In Table 2.4, the account value is reduced dollar-for-dollar by the withdrawn amount no matter

<table><caption>TABLEAU 2.4 – GMIB and GMDB behavior given policyholder's withdrawals when the account value is greater than the GMIB benefit base at the time of the first withdrawal.</caption><thead><tr><th colspan="10">Guaranteed minimum death benefit</th></tr><tr><th>Year</th><th>Net return</th><th>AV</th><th>WA</th><th>Roll-up rate</th><th>GMIB BB</th><th>RP BB</th><th>HAV BB</th><th>RU BB</th><th>GO BB</th></tr></thead><tbody><tr><td>0</td><td></td><td>$100,000</td><td></td><td></td><td>$100,000</td><td>$100,000</td><td>$100,000</td><td>$100,000</td><td>$100,000</td></tr><tr><td>1</td><td>3%</td><td>$103,000</td><td>$0</td><td>4%</td><td>$104,000</td><td>$100,000</td><td>$103,000</td><td>$104,000</td><td>$104,000</td></tr><tr><td>2</td><td>4%</td><td>$107,120</td><td>$0</td><td>4%</td><td>$108,160</td><td>$100,000</td><td>$107,120</td><td>$108,160</td><td>$108,160</td></tr><tr><td>3</td><td>6%</td><td>$113,547.20</td><td>$0</td><td>4%</td><td>$113,547.20</td><td>$100,000</td><td>$113,547.20</td><td>$113,547.20</td><td>$113,547.20</td></tr><tr><td>4</td><td>6%</td><td>$120,360.03</td><td>$0</td><td>4%</td><td>$120,360.03</td><td>$100,000</td><td>$120,360.03</td><td>$120,360.03</td><td>$120,360.03</td></tr><tr><td>5</td><td>7%</td><td>$128,785.23</td><td>$0</td><td>4%</td><td>$128,785.23</td><td>$100,000</td><td>$128,785.23</td><td>$128,785.23</td><td>$128,785.23</td></tr><tr><td colspan="10">Alternative 1: annual withdrawal amount</td></tr><tr><td>6</td><td>5%</td><td>$135,224.50</td><td>$5,151.41</td><td>4,00%</td><td>$130,073.09</td><td>$96,190.48</td><td>$130,073.09</td><td>$130,073.09</td><td>$130,073.09</td></tr><tr><td>7</td><td>3%</td><td>$133,975.28</td><td>$5,202.92</td><td>4%</td><td>$130,073.09</td><td>$92,454.92</td><td>$128,772.36</td><td>$130,073.09</td><td>$130,073.09</td></tr><tr><td>8</td><td>-2%</td><td>$126,196.91</td><td>$5,202.92</td><td>4%</td><td>$130,073.09</td><td>$88,643.14</td><td>$123,569.43</td><td>$130,073.09</td><td>$130,073.09</td></tr><tr><td>9</td><td>2%</td><td>$123,413.86</td><td>$5,202.92</td><td>4%</td><td>$130,073.09</td><td>$84,906.09</td><td>$118,366.51</td><td>$130,073.09</td><td>$130,073.09</td></tr><tr><td>10</td><td>2%</td><td>$120,575.16</td><td>$5,202.92</td><td>4%</td><td>$130,073.09</td><td>$81,242.32</td><td>$115,372.24</td><td>$130,073.09</td><td>$130,073.09</td></tr><tr><td colspan="10">Alternative 2: excess withdrawal</td></tr><tr><td>6</td><td>5%</td><td>$135,224.50</td><td>$7,000</td><td>4%</td><td>$128,224.50</td><td>$94,823.42</td><td>$128,224.50</td><td>$128,224.50</td><td>$128,224.50</td></tr><tr><td>7</td><td>3%</td><td>$132,071.23</td><td>$7,000</td><td>4%</td><td>$125,071.23</td><td>$89,797.62</td><td>$125,071.23</td><td>$125,071.23</td><td>$125,071.23</td></tr><tr><td>8</td><td>-2%</td><td>$122,569.81</td><td>$5,002.85</td><td>4%</td><td>$117,566.96</td><td>$86,132.41</td><td>$120,068.38</td><td>$117,566.96</td><td>$120,068.38</td></tr><tr><td>9</td><td>2%</td><td>$119,918.30</td><td>$4,802.74</td><td>4,00%</td><td>$115,115.56</td><td>$82,682.80</td><td>$115,265.65</td><td>$115,115.56</td><td>$115,265.65</td></tr><tr><td>10</td><td>2%</td><td>$117,417.87</td><td>$4,610.63</td><td>4%</td><td>$112,807.25</td><td>$79,436.11</td><td>$112,807.25</td><td>$112,807.25</td><td>$112,807.25</td></tr></tbody></table>

the size of withdrawal: in year 7 the AV is computed as $135,224.50 [AV as of the last contract date anniversary] - $7,000 (the amount of the withdrawal, including any applicable withdrawal charge) × (1+0.03 [assumed net return for the 7th contract anniversary]) = $128,224.50 × 1.03 = $132,071.23. When the owner limits himself to making only annual guaranteed withdrawals [4% (roll-up rate) × $128,785 (the roll-up benefit bases as of the 6th contract anniversary)] the GMIB and roll-up to age 85 benefit are reduced dollar-for-dollar but since AV after withdrawal is greater than the aforementioned quantity, they are automatically set to $130,073.
As a result of the GMIB benefit base increase in contract year 6, the annual withdrawal amount in contract year 7 is $5,203 [4% (roll-up rate) x $130,073 (the roll-up benefit bases as of the sixth contract anniversary)]. The return of principal benefit base is reduced pro-rata as shown in the previous examples and the highest anniversary value benefit base is reduced dollar-for-dollar as follows: $128,785 (highest anniversary value benefit base as of the 5th contract date anniversary)-$5,151 = $123,634. The highest anniversary value benefit base is reset to the protected benefit account value after withdrawal ($130,073).

In the case of an excess withdrawal and similar to the previous example, the roll-up bases will be reduced in the same percentage as the excess. Taking the 6th contract anniversary as an example, it is reduced by 5.177% ($7,000 divided by $135,224) which gives $121,944. The roll-up to age 85 benefit base and GMIB benefit base are then set to the protected account value after withdrawal $128,224 since this value is clearly superior that of the benefit bases after the pro-rata reduction.

The RP benefit base continues to be reduced in pro-rata basis. The highest anniversary value benefit base is reduced dollar-for-dollar and pro-rata, as follows : $128,785 (highest anniversary value benefit base as of the 5th contract anniversary)-$5,151 (annual withdrawal amount) -$1,690 (($128,785 - $5,151) × 1.367%) = $121,944. Here 1.367% represents the percentage of the excess in withdrawal with respect to the GWA ($1,849 divided by $135,224 = 1.367%). The highest anniversary value benefit base is also reset to the protected account value after withdrawals ($128,224).

In the following section, we will set up the mathematical formulation of the product for the purpose of studying its valuation. For the sake of simplicity, we will limit our study to a single benefit base.

## 2.2 A brief review of the literature

There is a large literature on pricing and hedging variable annuities guarantees. Most of it addresses on individual variable annuities contracts. Milevsky and Posner, see [127], price a GMDB contract using the usual risk-neutral valuation theory. Gerber and Shiu, see [91], exploit the closed-form solution of European look-back options to price complex guarantees embedded in some equity-linked annuities. Milevsky and Salisbury, see [128], study the impact of policyholder behavior on the cost and value of the GMWB rider and argue that the current pricing is not sustainable. An analysis of the design of general equity-indexed annuities from the investor's perspective and a generalization of the conventional design are proposed in [38].

The optimal behavior approach in a GMWB valuation was formalized by Dai and co-authors in [61]. They develop a singular stochastic control problem in a continuous framework, and also construct discrete pricing formulation that models withdrawals on discrete dates. In [19], the authors develop an extensive and comprehensive framework, to price any of the common guarantees available with VAs, using Monte Carlo simulations in deterministic withdrawals scenarios. On the other hand, in [50], the authors explore the effect of various modeling assumptions on the optimal withdrawal strategy of the policyholder, and examine the impact on the guarantee value under sub-optimal withdrawal behavior. Shah and Bertsimas, see [149], analyze the GLWB option in a time continuous framework considering simplified assumptions on population mortality, and adopting different asset pricing models. In [11], a number of guarantees under a more general financial model with stochastic interest rates, volatility, and mortality are considered. A utility-based approach, see [82], is used to study the valuation of the GMDB rider.

Holz and co-authors, see [98], price GLWB contracts for different product designs and model para-
meters under the geometric Brownian Motion dynamic. They consider various policyholders behaviors assumptions including deterministic, probabilistic and stochastic models. The GMIB is studied in [62] under a local volatility framework. The authors argue that an appropriate volatility modeling is important to the long-dated guarantees. Finally, Dai and Yang, see [156], develop a tree model to price the GMWB rider embedded in deferred life annuity contracts. Other papers investigate the impact of volatility risk, or assess the mortality risk in GLWB, or analyze equity and systematic mortality risks, see for example [81, 92, 143]. Recently, the work by Shevchenko and co-authors, see [150], provides a useful general framework to price different living and death guarantees. They use a direct integration method to solve the problem and compare it to PDE-based methods. In the following we will focus on the PDE method for the GMIB product. Our goal is to be able to analyse the impact of different market drivers and product design on policyholders' behaviors and the value of the contract.

## 2.3 Formulation and basic notations

In summary, GMIB contracts promise a policyholder an income stream at maturity for the rest of his life. Before the contract maturity, the insured is allowed to withdraw a certain amount on a yearly basis, called a withdrawal. If the GMIB contract contains a death benefit (GMIB-DB), then a certain amount is paid to the beneficiaries in case the policyholder dies during the term of the contract.

To formulate our problem, we consider an x-year old policyholder possessing a GMIB contract. At inception, an initial endowment is invested in a risky asset $S_t$. The specifications of the contract include a set of dates $0 = t_0 < t_1 < ... < t_n < ... < t_N = T$, where $t_0 = 0$ is the contact inception and $t_N = T$ its maturity. These so-called contract anniversaries are the dates in which events can take place, i.e. bonuses, withdrawals, payments, etc...

### 2.3.1 The contract assumptions

#### The financial market

Variable annuities pricing is based on the common pricing literature which assumes the existence of a risk neutral measure $\mathcal{Q}$ under which future cash flows can be valued as their expected discounted values. The existence of such measure implies an arbitrage free financial market. Moreover, the derivative's payoff can be replicated by a self-financing strategy, which allows the insurer to hedge the liabilities.

We assume that the risky asset $S_t$, which serves as an underlying mutual fund for the variable annuity, follows a Geometric Brownian motion with constant coefficients under $\mathcal{Q}$:

$$dS_t = rS_t dt + \sigma S_t dW_t,$$

where $\sigma$ is the volatility or the risky asset, $r$ the risk-free rate and $W$ a standard Brownian motion under $\mathcal{Q}$.

The money market evolves with risk-free interest rate, and the numeraire process $B_t$ is given by:

$$dB_t = rB_t dt.$$

Under the risk neutral probability measure, the discounted asset process $B_t^{-1}S_t$ is a martingale.
## The mortality assumption

It is common practice among insurers to use deterministic mortality rate to evaluate and replicate their policy pool. We also use this assumption in this chapter by considering future mortality rates as a deterministic curve. Moreover, we make the common assumption that financial markets and bio-metric events are independent. Let us introduce the mortality notations as:

- $x_0$: the policyholder's age at the contract inception.

- $q_n$: the probability that the policyholder, aged $x_0$ at inception, dies between time $t_{n-1}$ and $t_n$.

- $p_n$: the probability that the policyholder, aged $x_0$ at inception, is alive at time $t_n$.

- $\omega$: the limiting age beyond which survival is impossible.

According to the definition, we have $p_n = (1 - q_n)p_{n-1}$, where $n \in \{1, 2, ..., N\}$. From the insurer's perspective, the percentage of active contracts in a large policy pool of policyholders aged $x = x_0 + t_n$ at a given time $t_n$ is thus given by $p_n$.

## The contract state variables

At a given anniversary date $t_n$, the value of a GMIB contract, purchased by an $x_0$-year old policyholder at inception, is determined by three main state variables: the account value, the benefit base, and the a two-states variable determining if he is alive or dead at time $t_n$.

- **Account value** $A_t$: the value of the investment account which is indexed on the asset value $S_t$, and reduced by withdrawals and fees.

- **Benefit base** $G_t$: also referred as the guarantee account, is an "imaginary" wealth upon which annuities, guaranteed withdrawals and benefits are calculated. However, if the insured wants to lapse the contract, he will not be able to get this wealth.

- **Death Process** $I_n$: a two states variable in $\{0, 1\}$ informing if the policyholder died during $(t_{n-1}, t_n]$, or is still alive at $t_n$. The death probability in the interval $(t_{n-1}, t_n]$ is given by $q_n = P(I_n = 0 | I_{n-1} = 1)$, which depends on the policyholder's age at inception.

More state variables need to be included if one needs to incorporate stochastic interest rate and/or volatility, take into account taxation or consider different benefit bases, i.e. evolving differently or for different riders.

We restrict our analysis to single premium contracts $A_0 = G_0$, i.e. one premium at inception with no additional contributions. The policyholder can either withdraw money or exercise the income benefit. Withdrawals include "zero" withdrawals, guaranteed ones, i.e. up to a limited amount fixed by the insurer, excess withdrawals, i.e. withdrawals that exceed the guaranteed withdrawal amount, or completely surrender the contract, i.e. lapse.

For the sake of simplicity, we assume that the policyholder can take withdrawals each policy anniversary $t_n$, and denote by $\gamma_n$ the withdrawal amount. The income benefit also starts at anniversary years and, in case of a death benefit, the latter is paid out at these dates as well. Thus, the state variables described above may have discontinuities at times $t_1, ..., t_N$. Therefore, for a state variable $Y$, we distinguish between its value $Y_{t_n^-}$ before and $Y_{t_n^+}$ after events take place at the anniversary date $t_n$.
**Development between two policy years** ($t_{n-1}$, $t_n$]

Assuming that an annual guarantee fee $\alpha$ is continuously charged by the issuer, the value of the account value $A_{t_n}$ evolves as :

$$A_{t_n} = A_{t_{n-1}} \times \frac{S_{t_n}}{S_{t_{n-1}}} \exp(-\alpha \Delta t), \quad n = 1, 2, \dots, N,$$

where $\Delta t = t_n - t_{n-1}$, and $S_t$ follows a geometric Brownian and has the closed formula :

$$S_{t_n} = S_{t_{n-1}} \exp\left(\left(r - \frac{1}{2}\sigma^2\right)\Delta t + \sigma\sqrt{\Delta t}z_n\right),$$

where $z_1, \dots, z_N$ are independent and identically distributed standard Normal random variables.

In practice, the guaranteed fee is charged discretely and proportional to the account value that can easily be incorporated into the wealth process. Denoting the discretely charged fee with the annual basis as $\tilde{\alpha}$, the wealth process becomes

$$A_{t_n} = A_{t_{n-1}} \frac{S_{t_{n-1}}}{S_{t_n}} (1 - \tilde{\alpha}) \Delta t$$

The benefit base remains constant between two policy years, i.e :

$$G_{t_n} = G_{t_{n-1}}.$$

**Remark 2.** For continuously charged fees, the evolution of the account value can be rewritten in the form of an SDE :

$$dA_t = (r - \alpha)A_t dt + \sigma A_t dW_t,$$

where $W_t$ is the risky asset Brownian motion, $\sigma$ its volatility, and $r$ is the risk free rate.

In Retirement Cornerstone ®, some of the fees are actually proportional to the benefit base. We denote by $\alpha^A$ (resp. $\alpha^G$) fees proportional to the account value (resp. benefit base). In this case, we rewrite the account dynamic between $t_{n-1}^+$ and $t_n^-$ as :

$$dA_t = (r - \alpha^A)A_t dt - \alpha^G G_{t_{n-1}^+} dt + \sigma A_t dW_t, \quad (2.1)$$

**Transition at a policy year $t_n$**

As mentioned earlier, the contract events take place at the discrete policy years. In the following, we denote by $\gamma_n^{gua}$ the guaranteed withdrawal amount, and $\tilde{f}_n$ the cash flow at time $t_n$. The guaranteed withdrawal amount at $t_n$ is typically proportional to the the benefit base at time $t_{n-1}^+$ (or $t_n^-$), by a rate $\eta$ fixed by the insurer at inception, i.e. $\gamma_n^{gua} = \eta G_{t_n^-}$. Each policy year can exhibit the following scenarios :

1. **The insured has died within the previous year ($t_{n-1}$, $t_n$] :**

If the insured has died within the previous year and no death benefit has been set in place we have $A_{t_n^+} = 0$, $G_{t_n^+} = 0$, $\gamma_n^{gua} = 0$ and $\tilde{f}_n = 0$.

2. **The insured has survived the previous policy year and does not withdraw any money from the account at time $t_n$ :**

Different ratchet and roll-up mechanisms can be applied to the benefit base at $t_n$, thus changing the value of the guaranteed withdrawal amount. The different parameters develop as follows :

$$\text{Roll-up only : } G_{t_n^+} = (1 + \eta)G_{t_n^-}$$
— Ratchet : $G_{t_n^+} = \max(G_{t_n^-}, A_{t_n^-})$

— Reset : $G_{t_n^+} = \max((1 + \eta)G_{t_n^-}, A_{t_n^-})$

Here $\eta$ represents the roll-up rate which determines the quantity credited annually to the benefit base. This quantity is also used to compute the guaranteed withdrawal amount each year by $\gamma_n^{gua} = \eta \cdot G_{t_n^-}$. If no withdrawals are made from the contract, i.e $W_{t_n} = 0$, we have $A_{t_n^+} = A_{t_n^-}$ and the cash flows $\tilde{f}_n = 0$.

### 3. The insured has survived the previous policy year and at the policy anniversary withdraws an amount within the limits of the guaranteed withdrawal amount:

Any withdrawal up to the guaranteed annual withdrawal amount reduces the account value by the withdrawn amount. Of course, we do not allow for negative policyholder account values and thus get $A_{t_n^+} = \max(0, A_{t_n^-} - \gamma_n)$ and $\tilde{f}_n = \gamma_n$. The transformations discussed in 2) occur simultaneously with the withdrawals resulting in:

— Roll-up only : $G_{t_n^+} = (1 + \eta)G_{t_n^-} - \gamma_n$.

— Ratchet : $G_{t_n^+} = \max(G_{t_n^-} - \gamma_n, A_{t_n^-})$.

— Reset : $G_{t_n^+} = \max((1 + \eta)G_{t_n^-} - \gamma_n, A_{t_n^-})$.

The guaranteed annual amount $\gamma_n^{gua}$ needs to be recalculated using the formula presented immediately above. Note that if the annuity owner withdraws the maximum quantity $\gamma_n^{gua}$, the level of the benefit base remains stable when the roll-up is taken into account:

$$G_{t_n^+} = (1 + \eta)G_{t_n^-} - \gamma_n^{gua} = (1 + \eta)G_{t_n^-} - \eta \cdot G_{t_n^-} = G_{t_n^-}.$$

### 4. The insured has survived the previous policy year and the policy anniversary, and withdraws an amount exceeding the limit of the withdrawal guarantee:

In this case the account value is again reduced by the withdrawal amount $A_{t_n^+} = \max(0, A_{t_n^-} - \gamma_n)$. The benefit base as of the last contract anniversary date is reduced pro-rata by the percentage of the excess withdrawal w.r.t the account value, i.e. $\frac{\gamma_n - \gamma_n^{gua}}{A_{t_n^-}} G_{t_n^-}$. Therefore we have:

$$G_{t_n^+} = G_{t_n^-} \left( 1 - \frac{\gamma_n - \gamma_n^{gua}}{A_{t_n^-}} \right)$$

We then apply the ratchet if there is any, i.e $G_{t_n^+} = \max(G_{t_n^+}, A_{t_n^-} - \gamma_n)$.

### 5. The insured has survived the previous policy year and decides to activate the GMIB rider:

In this case, the contract matures and lifetime payments begin the following policy anniversary date taking into account the state of the variables at time $t_n$. Details on annuitization are given in the following section.

We summarize the previous cases into the following:

— Roll-up only case:

$$ 
\begin{align} 
A_{t_n^+} &= h^A(A_{t_n^-}, G_{t_n^-}, \gamma_n) := \max(0, A_{t_n^-} - \gamma_n) \nonumber \\ 
G_{t_n^+} &= h^G(A_{t_n^-}, G_{t_n^-}, \gamma_n) := \begin{cases} \max(0, (1+\eta)G_{t_n^-} - \gamma_n) & \text{if } \gamma_n \le \gamma_n^{gua} \\ G_{t_n^-} \left(1 - \frac{\gamma_n - \gamma_n^{gua}}{A_{t_n^-}}\right) & \text{if } \gamma_n > \gamma_n^{gua} \end{cases} \tag{2.2} 
\end{align} 
$$

— Ratchet only case:

$$ 
\begin{align*} 
A_{t_n^+} &= h^A(A_{t_n^-}, G_{t_n^-}, \gamma_n) := \max(0, A_{t_n^-} - \gamma_n), \\ 
G_{t_n^+} &= h^G(A_{t_n^-}, G_{t_n^-}, \gamma_n) := \begin{cases} \max(A_{t_n^-} - \gamma_n, G_{t_n^-} - \gamma_n) & \text{if } \gamma_n \le \gamma_n^{gua} \\ \max(A_{t_n^-} - \gamma_n, G_{t_n^-} \left(1 - \frac{\gamma_n}{A_{t_n^-}}\right)) & \text{if } \gamma_n > \gamma_n^{gua} \end{cases}. 
\end{align*} 
$$
— Reset (roll-up + ratchet) case :

$$
\begin{align*}
A_{t_n^+} &= h^A(A_{t_n^-}, G_{t_n^-}, \gamma_n) := \max(0, A_{t_n^-} - \gamma_n), \\
G_{t_n^+} &= h^G(A_{t_n^-}, G_{t_n^-}, \gamma_n) := \begin{cases}
\max(A_{t_n^-} - \gamma_n, (1+\eta)G_{t_n^-} - \gamma_n) & \text{if } \gamma_n \le \gamma_n^{\text{gua}} \\
\max(A_{t_n^-} - \gamma_n, G_{t_n^-} \left(1 - \frac{\gamma_n - \gamma_n^{\text{gua}}}{A_{t_n^-}}\right)) & \text{if } \gamma_n > \gamma_n^{\text{gua}}.
\end{cases}
\end{align*}
$$

**Remark 3.** The guaranteed rate is usually set equal to the ratchet rate, i.e. $\gamma_n^{\text{gua}} = \eta G_{t_n^-}$ at time $t_n$. (ii) The ratchet case can easily be deduced from the reset case by setting the ratchet and guaranteed rate to 0.

The income and death benefit

At maturity, the holder of a GMIB contract can select to take a lump sum of the account value $A_{t_N}$, annuitize this amount at an "actual" annuitization rate or annuitize the benefit base at pre-specified guaranteed annuitization rate. Annuity factors which give the annuitization rates, denoted by $\ddot{a}_{t_N}^{act}$ for the actual, and $\ddot{a}_{t_N}^{gua}$ for the guaranteed, are defined as the price of an annuity paying one dollar each year with either a the market's rates curve, or an internal guaranteed rates defined by the insurer. The calculations of the annuity factors takes into account the probability that the insurer is alive in the future. They are given by :

$$
\ddot{a}_{t_N}^{(.)} = \sum_{t_i=t_N}^{\omega-x_0} p_i e^{-r_{t_i}^{(.)} (t_i - t_N)},
$$

where $r^{(.)}$ are risk-free interest rate in case of annuitizing the account value, and based on hypothesis
fixed by the insurer in case of annuitizing the benefit base. Therefore, annuitizing the account value is
equivalent to a lump sum, and annuitizing a benefit base G is equivalent to the amount $G \frac{\ddot{a}_{t_N}^{gua}}{\dot{a}_{t_N}^{act}}$.

For GMIB contracts analyzed in this thesis, annuitization is not restricted to the maturity $t_N$. Indeed, $t_N$ is actually the last anniversary date in which the insured is allowed to annuitize. Typically, the policyholder can exercise his income benefit starting the 10th year of the contract. An annuity factor is then defined for each date $t_n \in \{t_{10}, t_N\}$. These factors are increasing since an older insurer will likely to have less annuities than a younger one.

Thus, the cash flow of the income benefit, based on a financially rational acting customer, is given
by :

$$
P(t_n, A_{t_n^{-}}, G_{t_n^{-}}) = \max(A_{t_n^{-}}, G_{t_n^{-}} \frac{\ddot{a}_{t_n}^{\text{gua}}}{\dot{a}_{t_n}^{\text{act}}}),
$$

otherwise $P(t_n, A_{t_n}^{-}, G_{t_n}^{-}) = 0$, $P$ denotes the income benefit, $A_{t_n}^{-}$ the level of the account value, and $G_{t_n}^{-}$ the level of the benefit base.

The policyholder can subscribe to a GMDB along with the GMIB. In this case, if the policyholder's death occurs before or at the contract maturity, a death benefit is provided to the beneficiaries. Assuming the policyholder dies during ($t_{n-1}$, $t_n$], the beneficiaries will receive the amount $D(t_n, \dots)$ at $t_n$. There are several types of death benefits. The most famous one is the so-called return of premium death benefit (return of principle for Retirement Cornerstone ®AXA product). In the Retirement Cornerstone ®AXA product, the death benefit can also consist of the greater of the annual ratchet benefit base and the current account value. Insurers typically charge tenth of potential market growth for this
additional rider. In the case of the Retirement Cornerstone ®13, this additional charge is 0.35% for this particular death benefit.

At any case, we resume the death benefit cash flow $D(t_n, \dots, t_n)$ at $t_n$ by:

$$D(t_n, A_{t_n}, G_{t_n}) = \begin{cases} A_0, & \text{return of principal death benefit (type 0),} \\ A_{t_n}, & \text{account value death benefit (type 1),} \\ G_{t_n}, & \text{benefit base the death benefit (type 2),} \\ \max(A_{t_n}, G_{t_n}), & \text{greater of the two death benefit (type 3).} \end{cases}$$

Our numerical analyses will be based on type 2, where the benefit base evolves as described in previous section.

## 2.4 Contract valuation

To model mortality, the standard way is to use official life tables to estimate the death probability $q_n = P(I_n = 0 | I_{n-1} = 1)$ during $(t_{n-1}, t_n]$. They provide annual death probabilities for each age and gender in a given country. Some adjustments can be applied to these tables. In addition to life tables, other approaches can be considered such as the stochastic benchmark Lee-Carter model, see [114], which forecasts the required death probabilities accounting for systematic mortality risk.

For pricing purposes, we consider a pool of policyholders who hold identical contracts and in which each insured has the same age, gender and thus same probabilities of life and death<fnref n="2" />. We assume the number of policyholders to be large enough such that the assumption that deaths occur exactly according to probability $q_n$ is justified. Given this set of conditions, mortality risk is fully diversified.

In the following, we set up the pricing framework of the GMIB contract. In particular, we are interested in the rational policyholder behavior which maximizes the expected value of his future cash flows. We will address a stochastic control problem as formulated in [150].

### The stochastic control problem

Let $\mathbf{\gamma} = (\gamma_1, \dots, \gamma_N)$ be a withdrawal strategy, $\mathbf{G} = (G_{t_0}, \dots, G_{t_N})$ the state variable corresponding to the benefit base, $\mathbf{A} = (A_{t_0}, \dots, A_{t_N})$ the account value, and $\mathbf{I} = (I_0, \dots, I_N)$ the death state. We introduce the state vector before the withdrawal as $X_n = (A_{t_n}, G_{t_n}, I_n)$ at time $t_n$ and $\mathbf{X} = (X_1, \dots, X_N)$. The present value of the overall payoff of the GMIB contract is defined as:

$$H_0(\mathbf{X}, \mathbf{\gamma}) = B_{0,N}H_N(X_N) + \sum_{n=1}^{N-1} B_{0,n}f_n(X_n, \gamma_n), \quad (2.3)$$

where:

$$H_N(X_N) = P(t_N, A_{t_N}, G_{t_N}) \times \mathbb{1}_{I_N=1} + D(t_N, A_{t_N}, G_{t_N}) \times \mathbb{1}_{I_N=0}, \quad (2.4)$$

is the cash flow at maturity and

$$f_n(X_n, \gamma_n) = \bar{f}_n(A_{t_n}, G_{t_n}, \gamma_n) \times \mathbb{1}_{I_n=1} + D(t_n, A_{t_n}, G_{t_n}) \times \mathbb{1}_{I_n=0}, \quad (2.5)$$

<footnote n="2">2. Other criterion can be taken into account to compose an "homogeneous pool". Some Life Tables consider smoking, the policy value etc... However, these criterion are specific to other insurance products. To our best knowledge, the mortality table of variable annuities is restricted to age. The resulting death probability is a weighted average of same age male and female holding the policy.</footnote>
is the cash flow at time $t_n$. Here $1_{(.)}$ is the indicator function, and $B_{i,j}$ is the discount factor from $t_j$ to $t_i$

$$B_{i,j} = \exp(-r(t_j - t_i)), \quad t_j > t_i$$

To simplify notations, we drop the mortality state variable $I_n$ when the policyholder is alive, i.e. $I_n = 1$, in the function argument. We define $V(t_n, A, G)$ the price of the contract with a guarantee at the policy year $t_n$ when $A_{t_n} = A$, $G_{t_n} = G$. We assume that the financial risk can be eliminated via continuous hedging, i.e. complete and frictionless market, and that mortality risk is fully diversified via selling the contract to a large number of insured of the same age. Thus, the average of the contract payoffs of M policyholders $H_0(\mathbf{X}, \mathbf{\gamma})$ converges to $\mathbb{E}_{t_0}^{\mathbb{I}}[H_0(\mathbf{X}, \mathbf{\gamma})]$ as $M \to \infty$, where $\mathbb{I}$ is the real probability measure corresponding to the mortality process $I_1, I_2, \dots, I_N$. Then the price under the given withdrawal strategy $\mathbf{\gamma}$ can be calculated as:

$$V(t_0, A_{t_0}, G_{t_0}) = \mathbb{E}_{t_0}^{\mathbb{I}, \mathbb{Q}}[H_0(\mathbf{X}, \mathbf{\gamma})], \quad (2.6)$$

where $\mathbb{E}_{t_0}^{\mathbb{I}, \mathbb{Q}}[H_0(\mathbf{X}, \mathbf{\gamma})]$ denotes the expectation w.r.t the state vector $\mathbf{X}$, conditional on information available at time $t_0$, i.e w.r.t both the financial risky asset process under $\mathbb{Q}$, and the mortality process under the real probability measure $\mathbb{I}$.

**Remark 4.** The fair fee $\bar{\alpha} = \alpha^*$ is defined as the fees charged so that the value of the contract at time $t_0$ is equal to the premium, i.e. $V(0, A_0, G_0) = A_0$. It is important to note that the strategy $\gamma$ can change for different realizations of underlying wealth process and the control variable $\gamma_n$ at $t_n$ affects the transition law of the underlying wealth process from $t_n$ to $t_{n+1}$, i.e calculating the contract price in this case is reduced to solving an optimal stochastic control problem.

The withdrawal strategy $\gamma$ can depend on the information available at time $t_n$ through the state variable $\mathbf{X}$ at $t_n$ and is assumed to be given when the price of the contract is calculated in (2.6). Withdrawal strategies are classified into three categories: *static*, *optimal* and *suboptimal*.

- **Static case.** Under a static strategy $\gamma$, the policyholder's decisions are deterministic, fixed at the beginning of the contract, and independent the state variable value. Under this strategy, the price of the contract can be calculated as:

$$V(t_0, A_{t_0}, G_{t_0}) = \mathbb{E}^{\mathbb{I}, \mathbb{Q}}[H_0(\mathbf{X}, \mathbf{\gamma})].$$

- **Optimal case.** Under the optimal withdrawal strategy, the withdrawal amount $\gamma_n$ depends on the information available at time $t_n$ through the state variable $X_n$. The optimal strategy is the strategy $\gamma$ under which the contract price is maximized, i.e worst case scenario for the insurer/best case scenario for the insured:

$$\gamma^*(\mathbf{X}) = \underset{\gamma \in \mathcal{A}}{\operatorname{argsup}} \mathbb{E}^{\mathbb{I}, \mathbb{Q}}[H_0(\mathbf{X}, \mathbf{\gamma})] \quad (2.7)$$

where the supremum is taken over all admissible strategies $\gamma$ and denoted by the set $\mathcal{A}$. That is for each time $t_n$ we have $\gamma_n \in [0, A_{t_n}]$.

- **Suboptimal case.** Any other strategy $\gamma$ different from $\gamma^*$ is called suboptimal. It can also depend on the state variable.

In the following we will be interested in the optimal case. Given that the state variable $\mathbf{X} = (X_1, \dots, X_N)$ is a Markov process, and the contract payoff is represented by Formula (2.3), the calculation of the contract value under optimal strategy given by Equation (2.7), is brought to a more general problem whereby the policyholder starts at an arbitrary time $t_n$. This falls within the framework of standard
optimal stochastic control problems for a controlled Markov process. Note that the control variable $\gamma_n$ depends on the account value A and benefit base G.

Finding the contract value $V(t_n, x)$ at time $t_n$ when $X_n = x$ for $n = N-1, ..., 0$ is done via a backward Bellman equation. Since the account value A evolves between two anniversary dates, and the benefit base is a constant piecewise function (i.e. changes at anniversary dates only), the required backward recursion is written between $t_{n+1}^-$ and $t_n^+$ as

$$ 
\begin{aligned}
V(t_n^+, A, G) &= \mathbb{E}^I [B_{n,n+1} (1_{I_{n+1}=1} \times \mathbb{E}_{t_n^+}^Q [V(t_{n+1}^-, A_{t_{n+1}^-}, G_{t_{n+1}^-}) | A, G] \\
&\quad + 1_{I_{n+1}=0} \times \mathbb{E}_{t_n^-}^Q [D(t_{n+1}^-, A_{t_{n+1}^-}, G_{t_{n+1}^-}) | A, G])] \\
&= (1 - q_{n+1}) \mathbb{E}_{t_n^+}^Q [V(t_{n+1}^-, A_{t_{n+1}^-}, G_{t_{n+1}^-}) | A, G] + q_{n+1} \mathbb{E}_{t_n^-}^Q [D(t_{n+1}^-, A_{t_{n+1}^-}, G_{t_{n+1}^-}) | A, G],
\end{aligned}
$$

with jump condition

$$ V(t_n^-, A, G) = \max_{\gamma_n \in \mathcal{A}_n} (\tilde{f}_n(A, G, \gamma_n) + V(t_n^+, h^A(A, G, \gamma_n), h^G(A, G, \gamma_n))) $$

The recursion starts from the maturity condition $V(t_N^-, A, G) = P(t_N^-, A, G)$ goes backwards for $n = N-1, N-2, ..., 0$.

**Remark 5.** Given that the mortality and financial asset processes are assumed independent, and the withdrawal decision does not affect the mortality process, we have:

$$ \sup_{\mathbf{Y}} \mathbb{E}_{t_0}^{\mathbb{Q}, I} [H_0(\mathbf{X}, \mathbf{Y})] = \sup_{\mathbf{Y}} \mathbb{E}_{t_0}^{\mathbb{Q}} [\mathbb{E}_{t_0}^{I} [H_0(\mathbf{X}, \mathbf{Y})]]. $$

One can calculate the expected value of the payoff (2.3) w.r.t the mortality process:

$$ \tilde{H}_0(\mathbf{A}, \mathbf{G}) = \mathbb{E}_{t_0}^{I} [H_0(\mathbf{X}, \mathbf{Y})], $$

and then calculate the price under the given strategy $\mathbb{E}_{t_0}^{\mathbb{Q}}[\tilde{H}_0(\mathbf{A}, \mathbf{G})]$, or under the optimal strategy $\sup_{\mathbf{Y}} \mathbb{E}_{t_0}^{\mathbb{Q}}[\tilde{H}_0(\mathbf{A}, \mathbf{G})]$.

Therefore we have:

$$ 
\begin{aligned}
\tilde{H}_0(\mathbf{A}, \mathbf{G}) = B_{0,N} & \left( P(t_N, A_{t_N^-}, G t_N^-) \times \mathbb{E}_{t_0}^I [1_{I_N=1}] + D(t_N, A_{t_N^-}, G t_N^-) \times \mathbb{E}_{t_0}^I [1_{I_N=0}] \right) \\
& + \sum_{n=1}^{N-1} B_{0,n} \left( \tilde{f}_n(A_{t_n^-}, G_{t_n^-}, \gamma_n) \times \mathbb{E}_{t_0}^I [1_{I_n=1}] + D(t_n, A_{t_n^-}, G_{t_n^-}) \right)
\end{aligned}
$$

Moreover, since $\mathbb{E}_{t_0}^I [1_{I_n=1}] = \mathbb{P}(\tau > t_n | \tau > t_0) = p_n$, and $\mathbb{E}_{t_0}^I [1_{I_n=0}] = \mathbb{P}(t_{n-1} < \tau < t_n | \tau > t_0) = p_{n-1}q_n$ for random death time $\tau$, i.e. $p_n = p_{n-1}(1-q_n)$, we can rewrite $\tilde{H}_0(\mathbf{A}, \mathbf{G})$ as the following:

$$ 
\begin{aligned}
\tilde{H}_0(\mathbf{A}, \mathbf{G}) = B_{0,N} & \left( p_N \mathbb{P}(t_N, A_{t_N^-}, G_{t_N^-}) + q_N p_{N-1} \mathbb{D}(t_N, A_{t_N^-}, G_{t_N^-}) \right) \\
& + \sum_{n=1}^{N-1} \left( p_n \tilde{f}_n(A_{t_n^-}, G_{t_n^-}, \gamma_n) + p_{n-1} q_n \mathbb{D}(t_n, A_{t_n^-}, G_{t_n^-}) \right).
\end{aligned}
\quad (2.8) $$

Note that, previously we defined $q_n = \mathbb{P}(t_{n-1} < \tau \le t_n | \tau > t_{n-1})$.

The payoff (2.3) has the same general form as the payoff (2.8). Thus, the optimal stochastic control
problem $\Phi(t_0, A_0, G_0) = \sup_{\gamma} \mathbb{E}[\tilde{H}_0(\mathbf{A}, \mathbf{G})]$ can be solved using Bellman equation. We describe the optimization problem at each policy anniversary date recursively by the two following equations

$$ \Phi(t_n^+, A, G) = \mathbb{E}_{t_n^+}^{\mathcal{Q}} [B_{n,n+1} \Phi(t_{n+1}^-, A, G, G_{t_{n+1}}) | A, G], \quad (2.9) $$

and

$$ \Phi(t_n^-, A, G) = \max_{\gamma_n \in \mathcal{A}_n} \left( p_n \tilde{f}_n(A, G, \gamma_n) + p_{n-1} q_n D(t_n, A, G) \right. \\ \left. \qquad + \Phi(t_n^+, h^A(A, G, \gamma_n), h^G(A, G, \gamma_n)) \right), \quad (2.10) $$

for $n = N-1, N-2, ..., 0$, starting from the final condition:

$$ \Phi(t_N^-, A, G) = p_N P(t_N^-, A, G) + p_{N-1} q_N D(t_N^-, A, G). \quad (2.11) $$

As a consequence, the recursion leads to the same solution $\Phi(t_0, A, G) = V(t_0, A, G)$, and the same optimal strategy $\gamma$. Moreover, for each $t_n$ we have $\Phi(t_n, A, G) = p_n V(t_n, A, G) + p_{n-1} q_n D(t_n, A, G)$.

### 2.4.1 Numerical scheme for the discrete withdrawal model

Realistic VA riders with discrete events such as ratchets, bonuses as set-up options and optimal withdrawals have no closed form solutions. Their fair price needs to be calculated numerically, even for a standard Brownian motion with constant interest rates and volatility.

The numerical solution of the backward recursion (2.9)-(2.10) is accomplished using PDEs, direct integration or regression-type Monte Carlo methods. Under the static strategy, one can always use standard Monte-Carlo to simulate state variables forward in time till the contract maturity or the policyholder death and average the payoff cash flows over a large number of independent realizations.

In the case of discrete withdrawal, following the procedure of deriving the Hamilton-Jacobi-Bellman (HJB) equations in stochastic control problems, the value of the annuity under optimal withdrawal is found to be governed by a one-dimensional PDE, similar to the Black-Scholes equation, with jump conditions at each withdrawing date to link the prices at the adjacent periods.

In the following, we provide detailed description of the algorithm used to compute the fair value of the VA riders and the optimal strategy.

#### General algorithm

The algorithm starts from a final condition for the contract value at $t_N^-$. Subsequently, solving the PDE gives solution for the contract value at $t_{N-1}^+$. The PDE used to compute the expected value (2.9) under the assumed risk-neutral process for the risky asset $S_t$ is easily derived using Feynman-Kac theorem. When the risky asset follows a geometric Brownian motion process, the governing PDE right after a withdrawal decision $t_n^+$ to right before the following one $t_{n+1}^-$ for $n = N-1, N-2, ..., 0$ is expressed as the following

$$ \partial_t \Phi + \frac{1}{2} \sigma^2 A^2 \partial_{AA} \Phi + (r - \alpha^A) A_t \partial_A \Phi - \alpha^G G_{t_n^+} \partial_A \Phi - r \Phi = 0, \quad (2.12) $$

to which we add boundary conditions given in the next section.

Note that the benefit base changes only at the anniversary dates and is a constant parameter between