Universität Ulm
Fakultät für Mathematik und
Wirtschaftswissenschaften

The Impact of Stochastic Volatility on
Pricing, Hedging and Hedge Efficiency of
Guaranteed Minimum Withdrawal Benefits
for Life Contracts
Diplomarbeit
in Wirtschaftsmathematik

vorgelegt vonFrederik Ruez
September 2009

Gutachter
Prof. Dr. Hans-Joachim Zwiesler
Prof. Dr. Rüdiger Kiesel

Contents

Contentsi

List of Figures

iv

List of Tablesv

1 Introduction1

1.1

Motivation of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 Liability Framework6

2.1

High-level description of the considered insurance contracts . . . . . .

6

2.2

Model of the liabilities . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1

Transition at a policy calculation date . . . . . . . . . . . . .

10

2.2.2

Computation of the guaranteed withdrawal amount W guar . .

10Contract valuation framework . . . . . . . . . . . . . . . . . . . . . .

11

2.3

3 Financial Market and Numerical Analysis Framework
3.1

3.2

16

Models of the financial market . . . . . . . . . . . . . . . . . . . . . .

16

3.1.1

Black-Scholes-Merton model . . . . . . . . . . . . . . . . . . .

17

3.1.2

Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Change of measure: the equivalent martingale measure . . . . . . . .

18

3.2.1

18

Black-Scholes-Merton model . . . . . . . . . . . . . . . . . . .
i

CONTENTS

3.2.2
3.3

3.4

ii

Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Numerical analysis framework . . . . . . . . . . . . . . . . . . . . . .

22

3.3.1

Monte-Carlo simulations . . . . . . . . . . . . . . . . . . . . .

22

3.3.2

Computation of sensitivities (Greeks) . . . . . . . . . . . . . .

26

3.3.3

Valuation of European standard options via Fourier inversion .

26

Black-Scholes-Merton Implied Volatility surface . . . . . . . . . . . .

29

4 Hedging34

4.1

Hedge portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.2

Dynamic hedging strategies . . . . . . . . . . . . . . . . . . . . . . .

36

4.2.1

No active hedging . . . . . . . . . . . . . . . . . . . . . . . . .

36

4.2.2

Delta hedging . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

4.2.3

Delta and Vega hedging . . . . . . . . . . . . . . . . . . . . .

38

4.2.4

Additional semi-static hedge . . . . . . . . . . . . . . . . . . .

41

5 Software Solution

42

5.1

Functional specification . . . . . . . . . . . . . . . . . . . . . . . . . .

42

5.2

Architecture and implementation . . . . . . . . . . . . . . . . . . . .

44

5.3

User interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

5.4

Used libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

5.5

Usage example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

6 Contract Analysis
6.1

54

Determination of the fair guaranteed withdrawal rate . . . . . . . . .

54

6.1.1

Results for the Black-Scholes-Merton model . . . . . . . . . .

55

6.1.2

Results for the Heston model . . . . . . . . . . . . . . . . . .

57

6.2

Distribution of withdrawals . . . . . . . . . . . . . . . . . . . . . . .

59

6.3

Distribution of the guarantee’s trigger time . . . . . . . . . . . . . . .

61

6.4

The Delta of the GLWB rider option . . . . . . . . . . . . . . . . . .

64

CONTENTS

iii

7 Analysis of Hedge Efficiency

66

7.1

Objective and risk measures . . . . . . . . . . . . . . . . . . . . . . .

66

7.2

Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

7.2.1

Comparison of hedge results . . . . . . . . . . . . . . . . . . .

68

7.2.2

Distribution of hedge portfolio’s value and insurer’s balance .

71

7.2.3

Unmodified Vega hedge . . . . . . . . . . . . . . . . . . . . . .

75

8 Conclusion

77

8.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

8.2

Outlook and future research . . . . . . . . . . . . . . . . . . . . . . .

80

List of Figures
3.1

Implied volatility surface generated by the Heston model. . . . . . . .

32

3.2

Implied volatility surfaces generated by the Heston model with different start values for the instantaneous variance process. . . . . . . . .

33

Screenshot showing the Microsoft Office Excel Function Wizard withan exemplary C++ function. . . . . . . . . . . . . . . . . . . . . . . .

48

Development over time of the empirical percentiles of the guaranteed
withdrawal amount. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Distribution of the guarantee’s trigger time for each of the productdesigns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Development over time of the empirical percentiles of the delta of apool of GLWB rider options. . . . . . . . . . . . . . . . . . . . . . . .

64

Development over time of the empirical percentiles of the insurer’s
hedge portfolio value. . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Development over time of the empirical percentiles of the insurer’sbalance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.1

6.1

6.2

6.3

7.1

7.2

iv

List of Tables
3.1

Overview of Euler schemes for simulation of the local volatility process
under the Heston model. . . . . . . . . . . . . . . . . . . . . . . . . .

24

Hedge ratios for the different strategies if the Black-Scholes-Merton
model is used as hedging model. . . . . . . . . . . . . . . . . . . . . .

40

Hedge ratios for the different strategies if the Heston model is used as
hedging model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

6.1

Assumed fee structure for all considered contracts. . . . . . . . . . . .

55

6.2

Assumed deterministic surrender rates. . . . . . . . . . . . . . . . . .

56

6.3

Fair guaranteed withdrawal rates xW L under the Black-Scholes-Merton
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

6.4

Benchmark parameters for the Heston model. . . . . . . . . . . . . .

57

6.5

Q -parameters for different choices of the market price of volatility
risk factor λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

6.6

Fair guaranteed withdrawal rates xW L under the Heston model. . . .

59

7.1

Hedging results for different hedging strategies and different datagenerating models. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Hedging results for the unmodified Vega hedge. . . . . . . . . . . . .

76

4.1

4.2

7.2

v

Chapter 1
Introduction
1.1

Motivation of the thesis

Variable Annuities are fund-linked annuities where typically the policyholder pays asingle premium that is then – after deduction of acquisition fees – invested in one orseveral mutual funds. Usually, the policyholder may choose from a variety of different mutual funds, which in most cases consist of positions in bonds and equities.
Such products were introduced in the 1970s in the United States. Two decades later,in the 1990s, insurers started to offer certain guarantee riders on top of the basic
structure of the Variable Annuity policies, including so-called guaranteed minimum
death benefits (GMDB), as well as guaranteed minimum living benefits, which can beclassified into three main subcategories: guaranteed minimum accumulation benefits(GMAB), guaranteed minimum income benefits (GMIB) and guaranteed minimum
withdrawal benefits (GMWB). The GMAB type of guarantee provides the policyholder with some guaranteed value at the maturity of the contract, while the GMIB
type provides a guaranteed annuity benefit, starting after a certain deferment period.
However, the currently most popular type of guaranteed minimum living benefits isthe third one, the GMWB rider.
Under certain conditions, the policyholder may withdraw money from their account,even if the value of the account has dropped to zero since policy inception. Suchwithdrawals are guaranteed as long as both, the amount that is withdrawn withineach policy year and the total amount that has been withdrawn over the term of thepolicy stay within certain limits.
Recently, insurers started to include additional features in GMWB type of products.
The most prominent is called “GMWB for Life” (also known as guaranteed living
1

CHAPTER 1. INTRODUCTION

2

withdrawal benefits, GLWB): guaranteed lifelong withdrawals. Within this guarantee type, the total amount of withdrawals is unlimited. However, the annual amountthat may be withdrawn by the policyholder while the insured is still alive must notexceed some maximum value, or otherwise the guarantee will be affected.
The withdrawals made by the policyholder are deducted from their account value –as long as it has not been depleted. Afterwards, the insurer has to compensate forthe guaranteed withdrawals for the rest of the insured’s life.
In contrast to a conventional annuity, in which the single premium paid by thepolicyholder is annuitized, the fund assets of the contract remain accessible to thepolicyholder within a “GMWB-for-Life” or “GLWB” type of policy. The policyholder may access the remaining fund assets at any time by (partially) surrendering
the contract.
In case of death of the insured, any remaining fund value is paid out to the insured’s
beneficiaries.
In return for this guarantee, the insurer receives guarantee fees that are usuallydeducted as a fixed annual percentage from the policyholder’s fund assets (but ofcourse only as long as there are any assets left).
Therefore, from an insurer’s point of view, these products contain a combination ofseveral risks from policyholder behavior, financial markets, and longevity that makesthese kind of guarantees difficult to hedge.
Owing to the significant financial risk that is inherent within the insurance contracts
sold, in general risk management strategies such as dynamic hedging are applied.
However, during the recent financial crisis, insurers have suffered from inefficienthedging strategies within their books1 .
Among other effects, the financial crises led to a significant increase in actual andimplied equity volatility, and thus to a tremendous increase in the value of moststandard and non-standard equity-linked options, including the value of the soldand offered Variable Annuity guarantee riders. In particular for insurers with no or
no sufficient hedging concept against the risk of changing volatilities (in particularincreasing volatility levels), the hedge portfolio did not compensate for the increase in
the option’s value, leading to a loss for existing business and less attractive conditions
for new contracts, i.e. the same guarantees come at higher guarantee fees.
1

Cf. e.g. different articles and papers in “Life and Pensions”: “A challenging environment”(June 2008), “Variable Annuities - Flawed product design costs Old Mutual 150m” (September2008), “Variable annuities - Milliman denies culpability for clients’ hedging losses” (October 2008),“Variable Annuities - Axa injects $3bn into US arm” (January 2009).

CHAPTER 1. INTRODUCTION

3

There already exists some literature on the pricing of different guaranteed minimum
benefits and in particular on the pricing of GMWB rider options: Valuation methods
have been proposed by e.g., Milevsky and Posner (2001, [26]) for the GMDB option,
Milevsky and Salisbury (2006, [27]) for the GMWB option, Bacinello et al. (2009,[3]) for life insurance contracts with surrender guarantees, and Holz et al. (2007,[17]) for GMWB for Life riders.
Bauer et al. (2008, [2]) introduced a general model framework that allows for thesimultaneous and consistent pricing and analysis of various Variable Annuity guarantees. We also refer to their paper for a comprehensive analysis of non-pricing related
literature on Variable Annuities.
To our knowledge, however, there is only little literature on the efficiency of different
strategies for hedging against the market risk inherent in Variable Annuity guarantees. Coleman et al. (2005, [7], and 2007, [8]) provide such analyses for death
benefit guarantees under different hedging and data-generating models. Again to
our knowledge, the performance of different hedging strategies for GLWB contracts
with varied product designs has not been analyzed under stochastic equity volatility
yet, nor have the accompanying inherent model risks been evaluated.
The aim of this thesis is to fill this gap.

1.2

Outline of the thesis

The remainder of the thesis is organized as follows.
First, we give a high-level description of the Guaranteed Living Withdrawal Benefits(GLWB) rider options and explain their general functionality in chapter 2, where wealso present the product designs of the GLWB rider options that will be analyzed
in the numerical section of the thesis. The considered product designs differ in theratchet (or step-up) and bonus feature that is implemented in the respective design.
We also describe the model framework for insurance liabilities that is used for ouranalyses and which is akin to the one presented by Bauer et al. (2008, [2]).
In the last part of chapter 2, we describe the general pricing framework that is used
for the evaluation of the GLWB rider option, which is necessary in order to find the“fair” guaranteed withdrawal rate. We also present the assumptions that we usedregarding the policyholder behavior and the mortality of the insured.
In chapter 3, we present the models of the financial market that we will use for different purposes within our analyses. We describe in detail both equity models that weCHAPTER 1. INTRODUCTION4

use for comparison, the well-known Black-Scholes-Merton model (with deterministic
equity volatility) as reference model and the Heston model as a model that allows forstochastic equity volatility. We show how for both models an equivalent martingale
measure can be derived, which is necessary for the evaluation of the GLWB options
within the general pricing framework of chapter 2.
In the second part of chapter 3 we present and explain the numerical methods that
we use later for the analyses within the numerical part of this thesis, including thenumerical valuation of European standard options via Fourier inversion techniques
under the Heston model.
Further, we introduce the notion of Black-Scholes-Merton Implied Volatility andanalyze the implied volatility surfaces generated by the Heston model for different
parameter sets.
The hedging strategies whose hedge efficiency we will analyze later in the numericalpart of the thesis are presented in chapter 4, where we introduce different types ofdynamic hedging strategies that may be applied by the insurance company in orderto reduce the risk that originates from selling GLWB rider options. The strategies
differ mainly in the model used for hedging and in the hedge instruments used within
the strategies. Finally, for each hedging strategy, we give a clear summary of theused hedge ratios in table form.
Chapter 5 deals with the design, architecture and implementation of the software
solution that we used to conduct the analyses of the numerical part of this thesis. We
also give details on how we use Microsoft Excel as an user interface to our software
solution and we list and discuss the open-source libraries that we made use of. We
conclude the chapter with a detailed usage example of the final software solution.
In the first chapter of the numerical part of the thesis, chapter 6, we present the first
set of analyses regarding the pricing of the GLWB rider options, i.e. the determination of the “fair” guaranteed withdrawal rate under different product designs andunder different assumptions regarding the financial market and policyholder behavior.
We also analyze the characteristics of the considered product designs that were introduced in chapter 2. To this end, we analyze the risk-return profiles of the considered
contracts from the viewpoint of the policyholder, and analyze the respective distribution of the GLWB guarantee’s “trigger time”, i.e. the specific point in time, when,
for the first time, guarantee payments from the insurer to the policyholder are due.
Furthermore, we present our results regarding the development of the sensitivity of apool of identical policies with respect to changes in the underlying fund’s price (theCHAPTER 1. INTRODUCTION5so-called “Delta”) over time and for each product design.
In the second chapter of the numerical part of this thesis, chapter 7, we analyze
the performance and behavior of the different hedging strategies that were presented
in chapter 4 under different stochastic scenarios. The results include analyses ofthe distribution of the hedge portfolio’s value, the insurer’s cumulative profit/loss
and certain risk measures hereof. Additionally, we quantify the model risk that isinherent in the use of the considered hedging strategies. To this end, we use different
models of the financial market for calculation of the hedge positions and for data
generation within the simulation. A particular focus lies on examining the effects if
the model used for hedging differs from the data-generating model.
We conclude with a summary of the presented work in the final chapter 8, where wealso give an outlook of possible future research work.

Chapter 2
Liability Framework
In Bauer et al. (2008), a general framework for modeling and a valuation of variableannuity contracts was introduced. Within this framework, any contract with one orseveral living benefit guarantees and/or a guaranteed minimum death benefit can be
represented. In their numerical analysis however, only contracts with a rather shortfinite time horizon were considered. Within the same model framework, Holz et al.
(2008) describe how GMWB for Life products can be included in this model. In
what follows, we introduce this model framework focusing on the peculiarities of thecontracts considered within our numerical analyses. We refer to Bauer et al. (2008)
as well as Holz et al. (2008) for the explanation of other living benefit guarantees
and more details on the model.

2.1

High-level description of the considered insurance contracts

Variable Annuities are fund-linked products. At inception of the contract, the policyholder pays a single premium P , which is then – after deduction of acquisition
fees – invested in one or several mutual funds. We call the value of the insured’s
individual portfolio the account value and denote it by AVt . During the runtimeof the contract, all running fees are taken from the account value by cancellationof fund units. Furthermore, the policyholder has the possibility to surrender thecontract, which is the same as withdrawing the whole account value, or, of course,
to withdraw just a portion of the account value. Products with a GMWB (”Guaranteed Minimum Withdrawal Benefits”) option give the policyholder the possibility of6

CHAPTER 2. LIABILITY FRAMEWORK

7

guaranteed withdrawals during the lifespan of the contract, i.e. the policyholder may
withdraw a guaranteed amount at prespecified points in time, even if the value of theportfolio has dropped to zero in the meantime. The initially guaranteed withdrawal
amount is usually (or may be expressed as) a certain percentage xW L of the single
premium P .
For our analyses, we focus on the case where such withdrawals are guaranteed lifelong(”GMWB for life” or ”Guaranteed Lifetime Withdrawal Benefits”, GLWB), whichmeans that, if the account value of the policy drops to zero while the insured is still
alive, the insured can still continue to withdraw the guaranteed amount until death.
The insurer charges a fee for this guarantee which is usually a prespecified annual
percentage of the account value. In case the insured dies before the account value
was depleted and/or the contract was surrendered, the remaining account value ispaid to the beneficiary as death benefit.
Depending on the type of the GLWB option, the amount guaranteed for withdrawal
may increase during the policy lifespan if the fund’s assets perform well, allowing thepolicyholder to withdraw a higher amount than the initially guaranteed amount. This
increase may either be permanent (withdrawal “step-up” or “ratchet”) or be effective
just for the single withdrawal (“surplus distribution” or “performance bonus”). In
our numerical analyses in sections 6 and 7, we have a closer look on four different
product designs that can be observed in the market:• No Ratchet: The first and simplest alternative is one where no ratchets or
surplus exists at all. In this case, the guaranteed withdrawal amount is constantand does not depend on market movements.
• Lookback Ratchet: The second alternative is a ratchet mechanism where awithdrawal benefit base at outset is given by the single premium paid. During the contract term, on each policy calculation date the withdrawal benefit
base is increased to the account value if the account value exceeds the previous
withdrawal benefit base. The guaranteed withdrawal is increased accordingly
to xW L multiplied by the new withdrawal benefit base. This effectively meansthat the fund performance needs to compensate for policy charges and annual
withdrawals in order to increase the guaranteed withdrawals. With this product design, increases in the guaranteed withdrawal amount are permanent,
i.e. over time, the guaranteed withdrawal amount may only increase, neverdecrease.

CHAPTER 2. LIABILITY FRAMEWORK

8• Remaining WBB Ratchet: With the third ratchet mechanism, the withdrawal benefit base at outset is also given by the single premium paid. The
withdrawal benefit base is however reduced by every guaranteed withdrawal.
On each policy calculation date where the current account value exceeds thecurrent withdrawal benefit base, the withdrawal benefit base is increased tothe account value. The guaranteed annual withdrawal is increased by xW L
multiplied by the difference between the account value and the previous withdrawal benefit base. This effectively means that the fund performance needs tocompensate for policy charges only but not for withdrawals in order to increase
guaranteed withdrawals. This ratchet mechanism is therefore c.p. somewhat“richer” than the Lookback Ratchet. As with the Lookback Ratchet design,
increases in the guaranteed amount are permanent.
• Performance Bonus: For this alternative the withdrawal benefit base isdefined similarly as in the Remaining WBB ratchet, but with the difference,that in this design the withdrawal benefit base is never increased. Insteadof permanently increasing the guaranteed withdrawal amount, on each policy
calculation date where the current account value is greater than the current
withdrawal benefit base, 50% of the difference is paid out immediately as aso-called “Performance Bonus”, additionally to the base guaranteed amount of
xW L · P . In contrast to the previous two designs, the minimum amount of theforthcoming guaranteed withdrawals remains unchanged in the Performance
Bonus design. For the calculation of the withdrawal benefit base, only thebase guaranteed amount xW L · P is subtracted from the benefit base and not
the performance bonus payments.

2.2

Model of the liabilities

For the sake of simplicity, we assume that all payments made to and by the policyholder, including fee payments, withdrawals, surrender benefits and death benefits,
may only occur at prespecified points in time t̃i i≥0 , expressed in terms of years
from contract inception. We will refer to these dates as contract (or policy) calculation dates. Let the filtration (Ft )t∈{t̃i }represent the corresponding informationi≥0

structure, where Ft is the information available at time t .
As a further simplification, we restrict the options that the policyholders have totwo choices: At a contract calculation date t̃k , they may either withdraw the thenCHAPTER 2. LIABILITY FRAMEWORK

9

guaranteed withdrawal amount Wt̃guar , or they may fully surrender the contract andkwithdraw all of the remaining fund assets.
At outset, the percentage ϕacq is deducted from the single premium P as acquisition
fee and the remaining amount is invested in the selected funds, hence it holds for the
initial account value AV0 = P · (1 − ϕacq ) .
Let AVt− denote the account value at time t after deduction of fees, but before
withdrawals. Accordingly, AVt+ denotes the value of the account at time t after
both, deduction of fees and withdrawals. We assume that at each end of a policy
period administration and guarantee fees, in terms of percentages ϕadm and ϕguar ,are deducted from the policyholder’s account value. With St denoting the spot
price of the portfolio’s underlying fund at time t , the transition of the account value
between two calculation dates t̃k and t̃k+1 is given byAVt̃− = AVt̃+ ·k+1

kSt̃k+1 −ϕadm −ϕguar
·e
St̃k

(2.1)

While being still alive, the policyholder withdraws an amount Wt from the accountat each of the contract calculation dates following the outset of the contract. The
random variable Wt is Ft -measurable. Note that, although (depending on theguarantee of the contract) withdrawals may exceed the current account value, theaccount value itself may not drop below zero and is therefore given by
AVt+ = max AVt− − Wt , 0 .

(2.2)

At each contract calculation date t , the maximum amount allowed to be withdrawn
by the policyholder is the maximum out of the account value AVt− and the guaranteed withdrawal amount Wtguar , which are both Ft -measurable random variables.
Therefore, a (full) surrender of the policyholder at a given contract calculation date t
may be expressed as Wt > Wtguar ∧ Wt = AVt− , with all of the following withdrawals
being set to zero. The guarantee payments Gt due to be made by the insurer at time
t are then given by the shortage resulting from the difference between guaranteed
withdrawal amount and the remaining account value:
Gt := Wtguar − AVt−+
= max Wtguar − AVt− , 0 .

(2.3)

CHAPTER 2. LIABILITY FRAMEWORK

2.2.1

10

Transition at a policy calculation date

At a policy calculation date t̃k , we have to distinguish the following cases:

The insured has died within the previous periodt̃k−1 , t̃k: If the insured

has died within the previous policy period, the account value is paid out to thebeneficiary as death benefit. With the payment of the death benefit, the insurance
contract matures. Thus, the account value and the guaranteed withdrawal amountare set to zero.

The insured has survived the previous policy period and withdraws anamount within the limits of the withdrawal guarantee at time t̃k : If theinsured has survived the previous period, no death benefit is paid. For the withdrawal
amount Wt̃k , it holds 0 ≤ Wt̃k ≤ Wt̃guar . The amount withdrawn is deducted fromkthe account value as far as the funds suffice. After the account value has dropped
to zero, the insurer bears the exceeding part of the guaranteed withdrawal amount.
Thus, AVt̃+ = max(AVt̃− − Wt̃k , 0) .
k

k

The insured has survived the previous policy year and at the policy anniversary withdraws an amount exceeding the limits of the withdrawal
guarantee : In this case again, no death benefits are paid. For the sake of brevity,
we only give the formulae for the case of full surrender, i.e. the case in whichWt̃k = AVt̃− > Wt̃guar , since partial surrender is not analyzed in what follows. Inkkcase of full surrender, the complete account value is withdrawn, therefore we set
AVt̃+ = 0 and the contract terminates.
k

2.2.2

Computation of the guaranteed withdrawal amountW guarFor all three product designs that make use of the withdrawal benefit base as an
auxiliary variable for computing the guaranteed withdrawal amount, we have twovariables at time t : W BBt− , the withdrawal benefit base before the withdrawalof Wt takes place, and the variable W BBt− , which denotes the withdrawal benefit
base right after the deduction of Wt . At outset, the withdrawal benefit base is set
to the single premium P paid by the policyholder, i.e. W BB0+ = P , and the initial

CHAPTER 2. LIABILITY FRAMEWORK

11

guaranteed withdrawal amount is set to W0guar = xW L · P .
At a contract calculation date t̃k > 0 , the computation of the guaranteed withdrawal
amount Wt̃guar works as described below for each ratchet mechanism:k• No Ratchet:Wt̃guar = xW L · P

(2.4)

k

• Lookback Ratchet:
W BBt̃− = max(W BBt̃+ , AVt̃− )(2.5)

Wt̃guar =xW L · W BBt̃−(2.6)

W BBt̃+ = max(W BBt̃− − Wt̃k , 0)

(2.7)

k

k−1

k

k

k

k

k

• Remaining WBB Ratchet:
W BBt̃− = max(W BBt̃+ , AVt̃− )
k

k−1

(2.8)

k

Wt̃guar =Wt̃guar + xW L · max(AVt̃− − W BBt̃+ , 0)

(2.9)

W BBt̃+ = max(W BBt̃− − Wt̃k , 0)(2.10)

k

k−1

k

k

k−1

k

• Performance Bonus:
W BBt̃− =W BBt̃+k(2.11)k−1

1max(AVt̃− − W BBt̃+ , 0)
k
k−1
k
2
+
−
W BBt̃ = max(W BBt̃ − xW L · P, 0)
Wt̃guar =xW L · P +
k

2.3

k

(2.12)(2.13)

Contract valuation framework

The valuation framework in this section follows in some parts the one used in
Bacinello, Biffis and Millossovich (2009, [3]) and in others Bauer, et. al. (2008,[2]). We take as given a filtered probability space (Ω, F, F, P) , in which P is thereal-world (or physical measure) and F = (Ft )t≥0 is a filtration with F0 = {Ø, Ω}and Ft ⊂ F ∀ t ≥ 0 . We assume that trading takes place continuously over time andwithout any transaction costs or spreads. Furthermore we assume that the price processes of the traded assets in the market are adapted and of bounded variation. With

CHAPTER 2. LIABILITY FRAMEWORK

12

the assumed absence of arbitrage, a probability measure Q∗ exists which is equivalent to P and under which the gain from holding a traded asset is a Q∗ -martingale
after discounting by the chosen numéraire process, the money-market account. We
call Q∗ the equivalent martingale measure (EMM). Details on the derivation andthe existence of the EMM Q∗ subject to the used financial market model are given
in chapter 3.
Let τX denote the date of the insured’s death, expressed in terms of years since
policy inception. Similarly, let iX denote the index of the contract calculation date
immediately following the death of the insured. Let H denote the filtration generated
by the process Nt = 1τX ≤t , which is zero as long as the insured is alive and whichjumps to one in the moment the insured dies. With G being the enlargement of
the filtration F in order to include H , i.e. G = F ∨ H , it follows that τX is a
G -stopping time, since{ω ∈ Ω : τX (ω) < t} ⊂ Gt

(2.14)

holds. Assuming independence between financial markets and mortality as well as
risk-neutrality of the insurer with respect to mortality risk, we are able to use theproduct measure of the risk-neutral measure of the financial market and the mortality measure. In what follows, we denote this product measure by Q and use theenlargement (Ω, G, G, Q) of the filtered probability space (Ω, F, F, Q∗ ) .
We denote by x0 the insured’s age (in full years) at the time of the start of thecontract, and with X = τX + x0 the age of the insured at the time of their death.
Further denote t px0 the probability under Q for a x0 -year old to survive the next t
years, s qx0 +t the probability under Q for a x0 +t -year old to die within s years, andlet ω be the limiting age of the mortality table, i.e. the age beyond which survival
is assumed to be impossible. The probability that an insured aged x0 at inception
passes away within the period (t, s] is thus given by the product t px0 · s qx0 +t . The
limiting age ω allows for a finite time horizon T = ω − x0 . The policy calculation
dates are therefore given by 0 = t̃0 < t̃1 < . . . < t̃M = T , with M ∈ N .
We restrict the policyholder behavior to the simple case where the policyholder (while
being still alive) has two possibilities to chose from at each policy calculation datet:
• either they withdraw exactly the guaranteed amount Wtguar• or they surrender the contract, i.e. they withdraw AVt− > Wtguar .

CHAPTER 2. LIABILITY FRAMEWORK

13This means we do not allow for partial surrender nor a withdrawal amount lower
than the guaranteed amount. We denote the policy calculation date at which thepolicyholder surrenders with τS , where τS = ∞ is to be interpreted as though thepolicyholder did not chose to surrender the contract while the contract was still inforce. Let iS denote the corresponding index of the policy calculation date τS , suchthat τS = t̃iS holds, given τS < ∞ .

The contract calculation date at which the account value of the policyholder falls
below the guaranteed withdrawal amount for the first time and thus the date at
which the insurer has to compensate for the guaranteed withdrawals for the firsttime, may then be expressed via the random variable τG , which is given by theformulano MτG := inf t ∈ t̃i i=0 t < τX , t < τS , AVt− < Wtguar .

(2.15)As usually, the infimum of the empty set is defined as +∞ , which in that case
implies that the guarantee of the contract does not trigger while the insured is still
alive or before they chose to surrender. Let iG denote the corresponding index of
the contract calculation date τG , such that τG = t̃iG holds, given τG < ∞ . In thefollowing, we will sometimes refer to τG as the “trigger time” of the guarantee. Afterthe guarantee has triggered, for the policyholder there is no reason to surrender thecontract, hence it holds τS < ∞ ⇒ τS < τG . However, for the sake of simplicity, weallow for (theoretical) surrender after the guarantee has triggered and will interpret
this as if the policyholder did not surrender. Both, τS and τG , are stopping times
with respect to the filtration G = (Gt )t∈{t̃i }, i.e. it holds ∀ i ∈ N
i≥0ω ∈ Ω : τG (ω) < t̃i ⊂ Gt̃i

(2.16)ω ∈ Ω : τS (ω) < t̃i ⊂ Gt̃i .

(2.17)

With Bt denoting the value at time t of the cash-bond (the chosen numéraire), thetime- t value GPt of the deflated future guarantee payments Gt (see formula 2.3) of

CHAPTER 2. LIABILITY FRAMEWORK

14

the insurer is given by

GPt =

=


0

, τG = ∞

Bt ·


0

PiX

i

i=iG Bt̃i· 1{t̃i >t}(2.18), τG < ∞, τG = ∞



Bt ·

Gt̃

)
(Wτguar
−AVτ−
G
G
BτG

1{τG >t} +Wt̃guari

PiX

i=iG +1Bt̃i1{t̃i >t}, τG < ∞. (2.19)

The time- t value GFt of all future guarantee fees deducted from the account value
can be computed similarly. We assume that the fraction of the total amount ofrunning fees (for administration and guarantee) that is assigned for covering the
guar

ϕ
1
costs of the guarantee is given by ϕadm
. Thus, GFt takes the form
+ϕguarϕguar·
GFt =Bt · ϕadm
+ϕguarmin(iG ,iX ,iS )

·

X
i=1



−(ϕadm +ϕguar )

1−e



St̃i
1
· AVt̃+
·
·1
i−1
Bt̃i
St̃i−1 {t̃i >t}(2.20)


ϕguar−(ϕadm +ϕguar )=Bt · ϕadm
·
1
−
e
guar
+ϕ
min(iG ,iX ,iS )

·

X
i=1

1
adm
guar
· AVt̃−· eϕ +ϕ· 1{t̃i >t}
i
Bt̃i

ϕguar
=Bt · ϕadm·
+ϕguarϕadm +ϕguare(2.21)

G ,iX ,iS ) min(iX1· AVt̃−−1 ·· 1{t̃i >t} .
i
Bt̃i
i=1

(2.22)We define the time- t value VtG of the GLWB option as the value of all (deflated)
future guarantee payments occurring after time t , less the value of all (deflated)
future guarantee fees deducted from the policyholder’s account value, again only
those occurring after time t . In formulas,VtG = EQ GPt Gt − EQ GFt Gt .

(2.23)

At outset, we call the contract (in an actuarial sense) fair from an insurer’s point
1

This is motivated by the fact that
1 − exp(−(ϕadm + ϕguar )T ) .

RT0ϕguar · exp(−(ϕadm + ϕguar )t)dt =ϕguar
ϕadm +ϕguar·

CHAPTER 2. LIABILITY FRAMEWORK

15

of view, if the GLWB option’s value V0G is zero. That is, to make the contract fair,
the following equation must hold at inception of the contract:
 
  !
EQ GP0 = EQ GF0 .

(2.24)

Finally, we assume that the policyholder surrenders the contract at each contract
calculation date t with a certain probability pSt conditional to the insured being
alive, the contract being still in force and an account value above zero. The probability that the policyholder surrenders at the contract calculation date t̃i (again
under the assumptionof an untriggered guarantee and the insured being still alive)
Qi−1
S
S
S
is therefore given by
j=0 (1 − pt̃j ) pt̃i . We will denote this probability by p̃i .
Then the computation of the time-0 value of the GLWB option, V0G , allows for thefollowing conditionings of the expected value under Q :
V0G = EQ GP0 − GF0 = EQ EQ GP0 − GF0 τX
  
 
= EQ EQ EQ GP0 − GF0 τS τX
  
 
= EQ EQ EQ GP0 − GF0 τS iX
=

=

M
X
i=1
M
X

t̃i−1 px0 ·t̃i −t̃i−1 qx0 +t̃i−1 · EQt̃i−1 px0 ·t̃i −t̃i−1 qx0 +t̃i−1 ·(2.26)(2.27)(2.28)

EQ GP0 − GF0 τS iX = ix

" i −1x
X(2.29)

p̃Sis · EQ GP0 − GF0 iS = is , iX = ix

is =0

i=1

+



(2.25)

1−

iX
x −1
is =0

!
p̃Sis

#


· EQ GP0 − GF0 τS = ∞, iX = ix.

(2.30)

Chapter 3
Financial Market and Numerical
Analysis Framework
3.1

Models of the financial market

For our analyses we assume two primary traded assets: the fund’s underlying, whose
spot price at time t we denote by St , and the money-market account, whose value attime t is denoted by Bt . The price processes B = (Bt )0≤t≤T and S = (St )0≤t≤T are
assumed to be adapted, right-continuous with left-limits (“càdlàg”1 ) and strictly positive semi-martingales on a filtered probability space (Ω, F, F, P) with F = (F)0≤t≤T
and a fixed time horizon T ∈ (0, ∞) . Furthermore, we assume the market to be
free of arbitrage, in the sense of being equivalent (cf. Bingham and Kiesel, 2004, [5])to the existence of an equivalent martingale measure (EMM). However, we do notassume the market to be complete in all cases, i.e. there may be contingent claims
that are non-attainable by self-financing strategies.
We assume the interest rate spread to be zero and the money-market account toevolve at a continuously compounded risk-free rate r , which remains constant overtime. Thus, the dynamics of the money-market account are given by
dBt = rBt dt(3.1)

and thus
Bt = B0 exp(rt) .
1

(3.2)

French “continue à droite, limitée à gauche”.

16CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 17

For the dynamics of the process of the underlying spot price, S = (St )0≤t≤T , weuse two different models: first we assume the equity volatility to be deterministic
and constant over time, and hence use the Black-Scholes-Merton (Black and Scholes,
1973, [6], Merton, 1973) model. Second, to allow for a stochastic equity volatility
modeling, we use the Heston (1993, [16]) model, in which both, the underlying itself
and its instantaneous (or local ) variance, are modeled by stochastic processes.

3.1.1

Black-Scholes-Merton model

In the Black-Scholes-Merton framework (Black and Scholes, 1973, [6], Merton, 1973),
the underlying’s spot price S follows a geometric Brownian motion whose dynamics
under the real-world measure P are given by the following stochastic differential
equation (SDE)dSt = µSt dt + σBS St dWt ,S0 ≥ 0(3.3)

where µ is the (constant) drift rate of the underlying, σBS ≥ 0 its constant volatility
and W = (Wt )0≤t≤T denotes a P -Wiener process. By Itō’s lemma, S has thesolution
St = S0 exp

3.1.2

σ2
µ − BS2t + σBS Wt ,S0 ≥ 0.

(3.4)

Heston model

There are various extensions to the Black-Scholes-Merton model that aim at a more
realistic modeling of the underlying’s volatility. We use the Heston (1993, [16])model in our analyses where the instantaneous (or local) variance of the asset ismodeled stochastic. Under the Heston model, the market is assumed to be driven
by two stochastic processes: the underlying’s price S = (St )0≤t≤T , and its instantaneous (local) variance V = (Vt )0≤t≤T , which is assumed to follow a one-factor
mean-reverting square-root process identical to the one used in the Cox-IngersollRoss (1985, [10]) interest rate model. The dynamics of the two processes under the
real-world measure P are given by the following system of stochastic differential
equations,pp
Vt St ρdWt1 + 1 − ρ2 dWt2 ,pdVt = κ (θ − Vt ) dt + σV Vt dWt1 ,dSt = µSt dt +S0 ≥ 0(3.5)

V0 ≥ 0,(3.6)

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 18

where µ again is the drift of the underlying, Vt is the instantaneous (or local )variance of the underlying’s spot price process at time t , κ is the speed of mean
reversion, θ is the long-term average variance, σV is the so-called vol of vol, or(more precisely) the volatility of the variance, ρ denotes the correlation
between
1/2

the processes of the underlying and the variance, and W 1/2 = Wt
independent P -Wiener processes.

are two0≤t≤T

The condition 2κθ ≥ σV2 ensures that the variance process will remain strictly
positive almost surely (cf. Cox, Ingersoll, Ross, 1985 [10]).
To our knowledge, there is no analytical solution for S available, thus numericalmethods are used in the simulation, which are presented in section 3.3.1.

3.2

Change of measure: the equivalent martingale
measureIn order to determine the values (i.e. the risk-neutral expectations) of the assets
in our model, we need to transform the real-world measure P into its risk-neutral
counterpart Q∗ , i.e. into an equivalent measure (that is, both measures share thesame null sets, i.e. same things possible and same things impossible) under which
the process of the deflated (by the natural numeréraire B ) underlying’s spot priceis a (local) martingale. While the transformation to such a measure is unique underthe Black-Scholes-Merton model, it is not under the Heston model.

3.2.1

Black-Scholes-Merton model

If no dividends are paid on the underlying,by Itō’s formula, the dynamics of thedeflated underlying’s price S̃ = S̃t

0≤t≤T, S̃t := St /Bt , under the Black-Scholes-

Merton model is given by the following SDE (cf. Bingham and Kiesel, 2004, [5]):dS̃t = (µ − r)S̃t dt + σBS S̃t dWt , S0 ≥ 0(3.7)

where r denotes the risk-free rate of return, µ the return of the underlying and
W = (Wt )0≤t≤T is a Wiener process under the real-world measure P .
Let γ = (γ(t))0≤t≤T be a measurable, adapted process withRT
0

γ(t)2 dt < ∞ almost

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 19

surely. We then define the process L = (L(t))0≤t≤T via

 Z t
Z
1 t
2
0
γ(s) ds .
γ(s) dWs −L(t) = exp −
2 0
0

(3.8)

Now assume γ fulfills Novikov’s condition (again cf. Bingham and Kiesel, 2004, [5]),
i.e.
 Z t

1
2
E exp
γ(s) ds
<∞
2 0


(3.9)

holds, which implies that L is a (continuous) martingale. Girsanov’s theorem thenRtshows that the process W̃t := Wt + 0 γ(s)ds, 0 ≤ t ≤ T , is a Wiener process underthe equivalent probability measure P̃ (defined on (Ω, FT )) with the Radon-NikodýmderivativeP̃= L(T ).
P

(3.10)

If F = (Ft )0≤t≤T is the Brownian filtration (the filtration generated by the Wiener
process), any pair of equivalent probability measures Q∗ ∼ P on FT is a Girsanvov
pair, i.e.
Q∗
= L(t).
P Ft(3.11)

With that, the dynamics for S̃ under the equivalent measure Q∗ areS̃t = (µ − r − σBS γ(t)) S̃t dt + σBS S̃t dW̃t , S0 > 0.

(3.12)

Since S̃ has to be a (local) martingale under Q∗ , the drift part of the SDE must be
zero, i.e.
!

(µ − r − σBS γ(t)) = 0,

0 ≤ t ≤ T,

(3.13)

which implies that γ(t) is a constant and given by the following formula (also known
as the “market price of risk”
γ(t) ≡µ−r.
σBS(3.14)We then arrive at the following Q∗ -dynamics for the undiscounted price of theCHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 20

underlying, S :St = rSt dt + σBS St dW̃t , S0 > 0,

(3.15)

with W̃ being a Q∗ -Wiener process.

3.2.2

Heston model

Under the Heston model, as there are two sources of risk, W 1 and W 2 , there arealso two “market price of risk” processes, denoted by γ1 and γ2 (corresponding toW 1 and W 2 ).
It is known (for all of the following we refer to Wong and Heyde, 2006, [31]) thatthe (class of) equivalent martingale measures Q∗ , if existent, can be considered interms of the Radon-Nikodým derivativeQ∗= L(T )
P FT
(
= exp

(3.16)
Z T
−
−

1
2

γ1 (u)dWu1 +

Z T

0

0

Z T

Z T

γ12 (u)du +0γ2 (u)dWu2).
γ22 (u)du(3.17)(3.18)

0

A necessary condition to ensure that the discounted underlying price is a local martingale and therefore a necessary condition for an equivalent local martingale measure
to exist, is given by the equation

p 
p
2
µ − r = Vt ργ1 (t) + 1 − ρ γ2 (t) .

(3.19)

Heston (1993, [16]) proposed the following additional restriction on the market-priceof-volatility-risk process, assuming it to be linear in local volatility,p
γ1 (t) = λ Vt ,where λ ∈ R is a real-valued constant.

(3.20)

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 21

With this, we know that the variance process under Q∗ has the dynamicspdVt = κθdt − (κ + λσV ) Vt dt + σV Vt dW̃t1 , V0 > 0,p
= κ∗ (θ∗ − Vt ) dt + σV Vt dW̃t1 ,(3.21)
V0 > 0,(3.22)

with W̃ 1 being a Q∗ -Wiener process.
Hence we obtain the same functional form as under P , but with transformed parameters for the speed of mean reversion and long-term average variance,κ∗ = κ + λσV
κθ
θ∗ =
.
κ + λσV(3.23)(3.24)For the second market-price-of-risk process γ2 , we obtain by substituting 3.20 into
equation (3.19) and under the aforementioned parameter restriction 2κθ ≥ σV2 , theequation1

γ2 (t) = p
1 − ρ2p µ−r√ − λρ Vt ,Vt

(3.25)and thus, if λ is given, both market-price-of-risk processes are fully defined.
Wong and Heyde (2006, [31]) also show that the equivalent local martingale measure√that corresponds to the market price of volatility risk λ Vt , exists if the inequality
−

κ
≤λ<∞
σV

(3.26)

is fulfilled. They further show that, if an equivalent local martingale measure Q∗
exists andκ + λσV ≥ σV ρ holds, the discounted stock price S̃ = S̃t(3.27)0≤t≤T, S̃t := St /Bt is a Q∗ -martingale.

Finally, provided both measures, P and Q∗ , exist, the Q∗ -dynamics of S and V ,again under the assumption that no dividends are paid, are given by (cf. Wong andHeyde, 2006, [31])pp
Vt St ρdW̃t1 + 1 − ρ2 dW̃t2 ,p
dVt = κ∗ (θ∗ − Vt ) dt + σV Vt dW̃t1 ,dSt = rSt dt +S0 ≥ 0(3.28)

V0 ≥ 0,(3.29)

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 22

where W̃ 1/2 are two independent Q∗ -Wiener processes and where κ∗ and θ∗ are
the risk-neutral counterparts to κ and θ as defined in (3.23) and (3.24) respectively.

3.3

Numerical analysis framework

3.3.1

Monte-Carlo simulations

As, to our knowledge, there exist no closed form solutions for the in section 2.3 defined value of the GLWB rider option V0G , we have to rely on numerical methods
to determine the option’s value, i.e. the value of the difference between expecteddiscounted future guarantee payments made by the insurer and the expected discounted future guarantee fees deducted from the policyholder’s fund assets. We
call the contract fair, if this difference is zero at inception of the contract. To thisend, for both equity models, we use Monte-Carlo simulations to compute the valueof the GLWB option as defined in section 2.3, that is to compute the expectation
V0G = EQ∗ GP0 − GF0 we use the Monte-Carlo approximation / estimator for V0G(cf. Glasserman, 2003, [14]),N

V̂0G = 
1 X  P  ˆk 
G0 S − GF0 Sˆk
N k=1(3.30) ˆkwhere each S = Ŝik

is one path of the fund’s underlying price process atthe policy calculation dates t̃i 0≤i≤M .
0≤i≤M

a)

Path generation

As there is a closed form solution for S available under the Black-Scholes-Merton
model, simulation of the financial market, i.e. path generation of S is rather simple:
Let zik , 0 ≤ i ≤ M, 0 ≤ k ≤ N be a realization of independent standard-normalkdistributed random variables Z
N (0, 1), 0 ≤ i ≤ M, 0 ≤ k ≤ N . Then the path
i ∼
k
ˆ
k
generation for one path S = Ŝin the Black-Scholes-Merton case works
i

0≤i≤M

as follows (cf. Glasserman, 2003, [14]),
Ŝ0k = S0(3.31)

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 23

and
k
Ŝi+1
= Ŝik exp




q

1 2
k
µ − σBS t̃i+1 − t̃i + σBS t̃i+1 − t̃i Zi .
2

(3.32)In the Heston model however, as there is no closed-form solution for S , numericalmethods are needed for the path generation of Ŝ k . We follow Lord, Koekkoek andvan Dijk (2008, [25]) in the remainder of this section.
A method to simulate paths for the Heston model without bias, using non-central
chi-squared random variables and a numerical Fourier inversion of the characteristic
function, has recently been derived by Broadie and Kaya (2004, 2006). However, thepart of the numerical Fourier inversion makes the algorithm highly time-consumingin the case if many observation dates of the underlying’s price are needed, i.e. if many
intermediate steps need to be computed, as, for instance, it is the case with highly
path-dependent products. Since there are indeed many intermediate steps neededfor our simulations, for instance when we analyze and compare different hedging
strategies, and as performance is an important issue with this kind of simulations,
we do not use this unbiased algorithm for our simulations, but, instead, use methods
that allow for a (comparably) fast, however biased generation of simulation paths ofS.
To this end, we use Euler discretizations of the instantaneous variance process V ,i.e. we use (very) small (typically equidistant) steps in time to approximate thedynamics of V given in its corresponding SDE. Let ∆t be the chosen step size of
our Euler discretization and let (tj )0≤j≤Ñ be the resulting time grid andV̂j0≤j≤Ñ

the corresponding simulated path2 of V , then a naı̈ve Euler discretization for V
would readV̂0 = V0

(3.33)andq √V̂j ∆t zj ,V̂j+1 = V̂j + κ θ − V̂j ∆t + σV

0 ≤ j ≤ Ñ − 1 ,(3.34)

where the (zj )0≤j≤Ñ again are realizations of independent standard-normal distributed random variables.
The problem with the simulation scheme in (3.34) is, that, given Ṽj > 0 , the proba2k

For ease of notation and better readability, we omitted the indication of the path number, () .

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 24

bility that Ṽj+1 will be negative is given by (cf. Lord, Koekkoek and van Dijk, 2008,[25]):− (1 − κ∆t) Ṽj − κθ∆t 
q √,P Ṽj+1 < 0 = N 
σV Ṽj ∆t

(3.35)

where N is the cumulative distribution function of the standard normal distribution.
That is, the probability that we end up with a negative value for the local variancewithin our simulations is strictly positive within the proposed scheme in (3.34).
Lord, Koekkoek and van Dijk (2008, [25]) introduce a general framework for all of thecommon fixes for this problem. All of the considered
Euler schemes can be unified 
within the scheme (with the auxiliary values

Ṽj)0≤j≤Ñ

V̂0 = V0

(3.36)

Ṽ0 = V0

(3.37)

and
r  
 
  

√
∆t zj
Ṽj+1 = f1 Ṽj − κ∆t · f2 Ṽj − θ + σV f3 Ṽj
0 ≤ j ≤ Ñ − 1 .
V̂j+1 = f3 Ṽj+1 ,(3.38)(3.39)

Now, the considered schemes can be summarized as presented in table 3.1, where|x| is the absolute value and x+ = max(x, 0) . For details and references, see Lord,
Koekkoek and van Dijk (2008, [25]).
Schemef1 (x) f2 (x) f3 (x)Absorption
Reflection
Higham and Mao
Partial truncation
Full truncation

x+
|x|
x
x
x

x+
|x|
x
x
x+

x+
|x|
|x|
x+
x+

Table 3.1: Overview of Euler schemes for simulation of the local volatility process
under the Heston model.
Lord, Koekkoek and van Dijk (2008, [25]) demonstrate that using the correct fix atthe boundary is important, and significantly impacts the magnitude of the bias. In
their examples, the numerical results indicate that the full truncation scheme pro-CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 25

duces the smallest bias, being far superior to the (in practice widely used) absorption scheme. The most biased scheme according to their comparison is the reflection
scheme with almost double the bias of the absorption scheme and about 100 times
the bias of the full truncation scheme. Therefore, we use the proposed full truncation
scheme in our simulations.
For the simulation of the underlying’s price

 
Ŝj

along the chosen time grid0≤j≤Ñ

(tj )0≤j≤Ñ , we switch to logarithms, resulting in
Ŝ0 = S0(3.40)and (with 0 ≤ j ≤ Ñ − 1 ),


q √


p
1
2
ln Ŝj+1 = ln Ŝj + µ − V̂j ∆t + V̂j ∆t ρzj + 1 − ρ z̃j ,2(3.41)

where zj are the values used in the simulation of the local variance, and z̃j areanother set of realizations of independent standard-normal distributed random variables.
We refer to Andersen (2007, [1]) and Van Haastrecht et. al. (2008, [29]) for moreelaborate simulation schemes of the Heston model, that aim at finding the best mix
between exact simulation and computational efficiency.

b)

Variance reduction techniques

In order to reduce the variance of the Monte-Carlo estimator of an option value, one
could use a higher number N of paths in the simulation, but this comes at the cost ofa higher computational effort and, as the central feature of the Monte Carlo method√
is a standard deviation of the form σM C / N (cf. Glasserman, 2003, [14]), wherefour times the number of paths only result in a bisection of the standard deviation,
this may prove inefficient. There are several alternative methods known how thevariance of the Monte-Carlo estimator can be reduced, well known is for instancethe use of the control variate technique, where (additionally to the option value ofinterest) the value of an option for which a closed-form solution exists is also priced
within the Monte-Carlo simulation. The computed Monte-Carlo price of this option
is then compared to its real value, given by the closed-form solution. The according
deviation is then used to improve the estimator for the value of the option which isof interest.

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 26

Another technique is the use of so-called antithetic variates (cf. Glasserman, 2003,[14]), where the simulation of the paths is done in form of pairs in case of the BlackScholes-Merton model and in form of quadruples in case of the Heston model. Thismethod uses the fact, that a Wiener process W = (Wt )0≤t≤T multiplied by −1 (i.e.
reflected with respect to the time axis) again is a Wiener process, and therefore, if
we have simulated one path of the Wiener process, we can use it in fact twice as
the pair (W, −W ) , resulting in a symmetrical distribution of the simulated paths,
as the outcome of the simulation generated by the first path will be balanced by theoutcome of the simulation result of the second path.
When the financial market in our model is driven by two Wiener processes, as itis the case with the Heston model, we can use the same approach to generate four
scenarios out of one realization of each Wiener process, resulting in the quadruple({W 1 , W 2 } , {W 1 , −W 2 } , {−W 1 , W 2 } , {−W 1 , −W 2 }) .
An additional benefit of the antithetic variates method is a reduced computational
effort, as in order to perform a Monte Carlo simulation with the same number ofpaths, the computational costs to produce realizations of the standard-normal distribution is reduced, because flipping the sign normally needs considerably less effort
than calculating an “original” draw from the standard-normal distribution.

3.3.2

Computation of sensitivities (Greeks)

Where no analytical solutions for the sensitivity of the option’s or guarantee’s value
to changes in model parameters (the so-called Greeks, see for instance Joshi (2003,[19]) for a detailed explanation) exist, we use Monte-Carlo methods to compute therespective sensitivities numerically. We use finite differences (cf. Glasserman, 2003,[14]) as approximations of the partial derivatives, where the direction of the shift ischosen accordingly to the direction of the risk, i.e. for the delta we shift the stock
downwards in order to compute the backward finite difference, and for the vega weshift the current volatility upwards, this time to compute a forward finite difference.

3.3.3

Valuation of European standard options via Fourier
inversion

In some of the hedging strategies considered in section 4, European standard or“plain vanilla” options are used. Under the Black-Scholes-Merton model, there are

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 27

closed form solutions for the price of European call and put options (cf. e.g. Bingham
and Kiesel, 2004, [5]). For strike price K and maturity T , the fair value of a call
option at time t is given by the Black (1976) formula
C BS (St , K, σBS , t, T ) = P (t, T ) [F (t, T ) · N (d1 ) − K · N (d2 )] ,(3.42)

where N (·) denotes the cumulative distribution function of the standard normal
distribution and d1 and d2 are given byd1 :=2ln(F (t, T )/K) + (σBS/2)(T − t)√
σBS T − t

(3.43)

and√d2 := d1 − σBS T − t .

(3.44)

Further, F (t, T ) denotes the forward price and P (t, T ) denotes the discount factor
to the option expiry date. With µ being the underlying’s drift under the risk-neutral
measure, and r the risk-free rate of return, we have
F (t, T ) := St exp (µ(T − t))(3.45)

P (t, T ) := exp (−r(T − t)) .

(3.46)

andSimilarly, the price of a European put option is given by
P BS (St , K, σBS , t, T ) = P (t, T ) [K · N (−d2 ) − F (t, T ) · N (−d1 )] .

(3.47)

Pricing via Fourier inversion under the Heston model
For the Heston stochastic volatility model, Heston (1993, [16]) found a semi-analyticalsolution for pricing European call and put options using Fourier inversion techniques.
The time- t prices for European call and put options with strike price K and maturity T can be expressed very similar to the Black-Scholes ones, namely (for thefollowing cf. Kahl and Jackel, 2005, [21])C Heston (St , K, Vt , t, T ) = P (t, T ) [F (t, T ) · P1 − K · P2 ](3.48)

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 28

and
P Heston (St , K, Vt , t, T ) = P (t, T ) [F (t, T ) · (P1 − 1) − K · (P2 − 1)] ,(3.49)

with F (t, T ) and P (t, T ) as defined in (3.45) and (3.46) respectively, and with
1 1
P1/2 := +2 πZ ∞f1/2 (u)du .

(3.50)

0

The functions f1 and f2 are −iu ln Keϕ(u − i)f1 (u) := Re
iuF (t, T )(3.51)

 −iu ln Keϕ(u)f2 (u) := Re,iu(3.52)

andwith ϕ(·) being defined as the log-characteristic function of the underlying’s priceST at expiry under the risk-neutral measure Q∗ , that is
ϕ(u) := EQ∗ eiu ln ST St .

(3.53)The equations for the price of a call or put option given the log-characteristic function
ϕ(·) of the underlying’s price at expiry are generic and apply to any model.
Specifically for the Heston model, with τ = T − t , we have the log-characteristic
functionϕ(u) = eC(τ,u)+D(τ,u)Vt +iu ln F (t,T ) ,(3.54)

where the coefficients C and D are solutions of a two-dimensional system of ordinary
differential equations (ODE) of Riccati-type, and are given byκθC(τ, u) = 2
σV




c(u)ed(u)τ − 1(κ − ρσV ui + d(u)) τ − 2 ln,c(u) − 1(3.55)

ed(u)τ − 1c(u)ed(u)τ − 1(3.56)andκ − ρσV ui + d(u)D(τ, u) =
σV2




,

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 29

with the auxiliary functionsc(u) =

κ − ρσV ui + d(u)κ − ρσV ui − d(u)(3.57)

andqd(u) = (ρσV ui − κ)2 + iuσV2 + σV2 u2 .

(3.58)As Kahl and Jackel (2005, [21]) point out, the computation of the terms P1/2 in(3.50) includes the evaluation of the logarithm with complex arguments in (3.55),
which may lead to numerical instabilities for certain sets of parameters and/or longdated options. For that reason, we use the scheme proposed in their paper, which is
supposed to allow for a robust computation of the fair values of European call andput options for (practically) all levels of parameters and – as they state – even formaturities of many decades.
Kahl and Jackel also point out in their paper that the choice of the right integration
scheme is another crucial point for a robust implementation of the semi-analytical
Heston solution, since the integrands f1/2 can vary in their shape from simply exponentially decaying to highly oscillatory depending on the choice of parameters. As
in the proposed scheme, we use the adaptive Gauss-Lobatto quadrature method as
described in Gander and Gautschi (2000, [13]) for the numerical integration of P1
and P2 .

3.4

Black-Scholes-Merton Implied Volatility surface

The Black-Scholes-Merton Implied Volatility of an European standard option is the
specific volatility σBS for which the result of the Black-Scholes-Merton formulasin (3.42) for a call option or (3.47) for a put option respectively coincide with theobserved market prices of these options, i.e. the volatility implied by the market
price of the option. For instance in the case of a call option with maturity T , strike
set at K and a current level of the underlying St , this means that for a given timet market price C market (St , K, t, T ) of this option, the Black-Scholes-Merton implied
volatility σBS (St , K, t, T, r) , with the (assumed to be observable) risk-free rate ofCHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 30

return r , is defined as the unique3 value for which the following equation holds:
C market (St , K, t, T ) = C BS (St , K, σBS , t, T ).

(3.59)

If we have a matrix of option prices of call and put options on the same underlying but
for different maturities and different strikes, we call the corresponding set of BlackScholes-Merton (BSM) implied volatilities, the implied volatility surface, i.e. thesurface of the BSM volatilities implied by the observed market prices. For instance, if
the market prices are computed with the Black-Scholes-Merton model using only one
constant volatility σBS for all strikes and maturities, the resulting implied volatility
surface would be flat.
In reality however, the implied surfaces observed in the market are not flat but show
different kinds of dynamic “deformations” (cf. e.g Cont and Da Fonseca, 2002, [9],and Joshi, 2003, [19]) with the most prominent being the following:• skewness or smile: typically put options with a low moneyness (i.e. the strike
is significantly below the current level of the underlying) have a (much) higher
implied volatility than options struck at-the-money, i.e. the strike price coincides with the current spot price of the underlying, or options with a strikeabove the current level of the underlying (in-the-money puts or out-of-themoney calls). Depending on the underlying, the implied volatilities may riseagain with an increasing strike level. The emerging pattern is called the smileof the implied volatility surface.
• term structure: the implied volatilities observed in the market also usuallychange with the time to maturity of the options. Typically, the smile of thesurface flattens and the volatilies around the current level of the underlying (orthe forward value F (t, T ) = EQ∗ [ST |Ft ] hereof) rise or fall.
It is of interest to know the characteristics and dynamics of the implied volatility
surface that a certain financial market model generates, i.e. the volatility surface
which is implied by the prices of a given set of standard options computed within themodel. Ideally, if the model is used for market-consistent pricing, these computed
prices should coincide (within an error margin) with the observed market prices.
BS

In the Black-Scholes-Merton model, the Vega of a call option is given by ∂C=∂σBS0
S T − tN (d1 ) , which - by means of put-call parity and because the vega of a forward is zero- coincides with the Vega of the put option. Note that the Vega is always positive and therefore
the map from volatilities to prices is injective (cf. Joshi, 2003, [19]).
3√

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 31

Therefore, when confronted with the task of calibrating an equity model for pricing
purposes, there are usually two main approaches on how to determine the parameters(cf. Wilmott, 2006, [30]): The statistical approach first looks at the statistics of theunderlying and estimates the real-world model parameters from historical data of
the underlying, then tries to estimate the market price of risk processes (in caseof the Heston model, the parameter λ ) from option price data in order to get therisk-neutral parameters. The second approach aims at the market consistency of themodel and directly calibrates the risk-neutral parameters to option prices observablein the market, such that the model replicates these prices as accurately as possible.
The model is then used to determine the values of options that are not observablein the market.
Figure 3.1 shows the implied volatility surface that is implied by the option prices
generated by the Heston model with parameters as given in table 6.4 that we use
later on for our analyses in chapters 6 and 7.
The two implied volatility surfaces pictured in figure 3.2 illustrate the dynamics of
the implied volatility surface generated by the Heston model, as they show how thegenerated surface changes if the start value for the local variance, V0 , is changed toa higher or lower value than the long-term variance θ respectively.
The option prices were calculated via the Fourier inversion technique presented in
section 3.3.3 and the Newton-Raphson iteration procedure (cf. Bingham and Kiesel,
2004, [5]) was used for calculation of the implied volatilities.

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 32

Figure 3.1: Implied volatility surface generated by the Heston model with the option’s
strike stated as percentage of the underlying’s current spot price, and with model
parameters r = 0.04 , θ = 0.2202 , κ = 4.75 , σV = 0.55 , ρ = −0.569 and V0 = θ .

CHAPTER 3. FINANCIAL MARKET AND NUMERICAL ANALYSIS FRAMEWORK 33(i) Lower start value for the local variance: V0 = 0.102(ii) Higher start value for the local variance: V0 = 0.352

Figure 3.2: Implied volatility surfaces generated by the Heston model with the option’s strike stated as percentage of the underlying’s current spot price, with model
parameters r = 0.04 , θ = 0.2202 , κ = 4.75 , σV = 0.55 , ρ = −0.569 , and with
V0 = 0.102 used in (i) and V0 = 0.352 used in (ii) respectively.

Chapter 4
Hedging
In this Section, we describe the different (dynamic) hedging strategies that we use
for the hedging simulations whose results are presented in chapter 7. The presented
strategies may be applied by the insurer in order to reduce the financial risk of
the guarantees (and thereby reducing the required economic risk capital). We firstdescribe the assumed structure of the (hedge) portfolio of the insurer before we describe the considered hedging strategies for both equity models presented in chapter
3, Black-Scholes-Merton and Heston, and for different assumptions regarding theused hedge instruments.

4.1

Hedge portfolio

We assume that the insurer has sold a pool of policies with GLWB guarantees. We
denote by Ψt the cumulative (fair) option value for that pool at time t , i.e. Ψt is
the sum of the individual (fair) option values at time t of all policies in that pool,where the (fair) option value is defined and to be calculated as presented in section
2.3.
Additionally, we assume that the insurer cannot influence the value Ψt of the policy pool by changes in the underlying fund, like, for instance, changing the fund’s
exposure to risky assets or forcing the policyholder to switch to a different, e.g. lessvolatile, fund1 .
1

In practice, however, this is a frequently seen feature in products, which allows the insurer tohedge against volatility risk by means of product design, and effectively transfer volatility risk backto the policyholder. In our analyses, however, we concentrate on hedging strategies that aim atcompensating the (externally generated) changes of an option’s value.

34

CHAPTER 4. HEDGING

35

We further assume that the insurer invests the guarantee fees in a hedge portfolio
ΠHedgeand implements certain hedging strategies within this hedging portfolio. Also,t
if the guarantee of a contract is triggered, all of the subsequent guaranteed payments
are made from this hedge portfolio. Thus, the insurer’s cumulative profit/loss Πt (inwhat follows sometimes just denoted as the insurer’s profit) at time t that originatesfrom the pool of policies and the implemented hedging strategy is given byΠt = −Ψt + ΠHedge
,
t

(4.1)

where Ψt is the the time- t value of the implied guarantees of the pool of policies
and ΠHedge
is the time- t value of the corresponding hedge portfolio implemented byt
the insurer.
The following hedging strategies aim at reducing the insurer’s risk by implementing
certain investment strategies within the hedge portfolio ΠHedge. Note that the valuetΨt of the pool of policies at time t does not only depend on the number of the written
contracts and their respective face value, but also on retrospective and prospective
factors, such as - due to the path-dependent nature of the GLWB contracts - historicalprices of the fund at previous withdrawal dates, and several model assumptions
concerning mortality, lapsation, and the financial market.
The insurer’s choice of the financial market model and the corresponding choice ofparameters has a significant impact on the hedging strategies. Therefore, we will
differentiate in the following between the hedging model that is chosen and used by
the insurer, and the data-generating model that we use to simulate the developmentof the underlying and the market prices of European call and put options. This allows us, e.g., to analyze the impact on the insurer’s risk situation if the insurer bases
pricing and hedging on the Black-Scholes-Merton model (hedging model) with deterministic equity volatility, whereas in reality (data-generating model) equity volatility
is stochastic. We assume the value of the guarantee to be marked-to-model, where
the same model is used for valuation as the insurer uses for hedging. All other assets
in the insurer’s portfolio are marked-to-market, i.e. their prices are determined bythe (external) data-generating model.
We assume that, additional to the fund’s underlying and the money-market account,a market for European “plain vanilla” options on the underlying (i.e. simple European put and call options) exists. However, we assume that only options withlimited time to maturity are liquidly traded. As well as the underlying and themoney-market account, we assume the option prices (i.e. the implied volatilities ofCHAPTER 4. HEDGING

36

the options) to be driven by the data-generating model, and presume risk-neutrality
with respect to volatility risk, i.e. the market price of volatility is set to zero in casethe Heston model is used as data-generating model. Additionally, we assume thespread between bid and ask prices/volatilities to be zero.
For all considered hedging strategies we assume the hedging portfolio to consist of
positions in three assets, whose quantities are rebalanced at the beginning of eachhedging period: a position of quantity ∆St in the underlying, a position of quantity
X∆B
t in the money-market account and a position of ∆t in a 1-year ATMF straddle

option (that is, an option consisting of long positions in one call and one put, bothwith one year to maturity and struck at-the-money with respect to the maturity’s
forward, hereafter referred to by the acronym “ATMF”). We assume the insurer tohold the position in the straddle for one hedging period, then sell the options at thethen-current price, and set up a new position in a then 1-year ATMF straddle. For
each hedging period, the price of the currently used straddle option is denoted by
Xt . We assume that the portion of the hedge portfolio that is not invested in either
the underlying or the straddle options is invested in (or borrowed from) the money
market. Thus, the hedge portfolio at time t has the form
X
ΠHedge= ∆St St + ∆Btt Bt + ∆t Xt ,

(4.2)

where
ΠHedge− ∆St St − ∆X
t
t Xt
B
.
∆t =
Bt

4.2

(4.3)

Dynamic hedging strategies

For both considered hedging models, Black-Scholes-Merton and Heston, we analyze
three different types of (dynamic) hedging strategies.

4.2.1

No active hedging

The first strategy simply invests all guarantee fees in the money-market account anddoes no further hedging, i.e. ∆St = 0 and ∆X
t = 0 ∀ t . The strategy is obviously
identical for both models.

CHAPTER 4. HEDGING

4.2.2

37

Delta hedging

The second type of hedging strategy uses a position in the underlying in order tohedge the portfolio against small changes in the underlying’s price, i.e. to achieveso-called delta-neutrality of the portfolio.
Using the Black-Scholes-Merton model as hedging model, the position ∆St is chosen
as the delta of Ψt , i.e. the partial derivative of Ψt with respect to the underlying’s
spot price, because of the following reasoning: The portfolio of the insurer at time
t consists of the option value Ψt of the pool of policies and the hedge portfolio
ΠHedge. The hedge portfolio only consists of positions in the underlying and thetmoney-market account. Therefore we haveΠt = −Ψt + ∆St St + ∆Bt Bt .

(4.4)

A natural question to ask is how the value of the portfolio Πt changes from time t
to time t + dt . The change in the portfolio value originates in changes in all three
asset positions:dΠt = −dΨt + ∆St dSt + ∆Bt dBt .

(4.5)

Assuming the underlying follows a geometric Brownian motion as described in section
3.1.1 with the SDE
dSt = µSt dt + σBS St dWt ,S0 ≥ 0 ,(4.6)

then, from Itō we have (cf. Wilmott, 2006, [30])
dΨt =

∂Ψt
∂Ψt
1 2 2 ∂ 2 Ψtdt +dSt + σBS
St
dt .
∂t
∂St
2
∂St2

(4.7)

Thus the portfolio changes bydΠt = −

∂Ψt
∂Ψt
1 2 2 ∂ 2 Ψtdt −dSt − σBSSt
dt + ∆St dSt + ∆Bt rBt dt.
∂t
∂St
2
∂St2

(4.8)

The random parts in (4.8) are


∂Ψt
S
−
+ ∆t dSt ,
∂St

(4.9)

CHAPTER 4. HEDGING

38

and therefore, if we choose
∆St =

∂Ψt
∂St

(4.10)

the randomness is reduced to zero.
In the Black-Scholes-Merton framework with time-continuous trading and with notransaction costs, such a position is sufficient to perform a perfect hedge. In reality
however, time-discrete trading and transaction costs cause imperfections.
While delta hedging under the Black-Scholes-Merton model (given the typical assumptions), constitutes a theoretically perfect hedge, it does not under the Hestont
could still be used, arguing that this strategy
model. Delta hedging with ∆St = ∂Ψ∂Stconstitutes a first-order approximation (via Taylor’s theorem) to the real dynamicsof Ψt . However, as it is clear that in most cases the hedge cannot be perfect, thequestion arises if this imperfection can be effectively minimized.
This question leads to (locally) risk-minimizing strategies, that aim at minimizing
the variance of the instantaneous change of the portfolio. Under the Heston model2 ,the problem
Var (dΠt ) → min,∆St ∈ R,∆Xt ≡ 0

(4.11)

has the solution (cf. e.g. Ewald et al., 2007, [12])∆St =ρσV ∂ΨHeston∂ΨHeston
t
t
+
,
∂St
St
∂Vt

(4.12)

where the superscript “Heston” indicates that the time- t value of the pool of policies,
Ψt , is calculated within the Heston model.
To keep notation simple, this (locally) risk-minimizing strategy under the Heston
model is sometimes also referred to as “delta” hedge.

4.2.3

Delta and Vega hedging

The third type of hedging strategies incorporates the use of the straddle option Xt ,exploiting its sensitivity to changes in volatility for the purpose of neutralizing (or2Note that a (time-continuously) delta-hedged portfolio under the Black-Scholes-Merton modelis already risk-free. Therefore for the Black-Scholes-Merton model, the delta-hedging strategy
coincides with the locally risk minimizing strategy.

CHAPTER 4. HEDGING

39

at least limiting) the portfolio’s exposure to changes in volatility.
Under the Black-Scholes-Merton model, volatility is assumed to be constant; therefore using the model to hedge against a changing volatility may appear counterintuitive. Nevertheless, following Taleb (1997, [28]), we analyze some kind of ad-hoc vega
hedge in our simulations, that aims at compensating the deficiencies of the BlackScholes-Merton model: In order to perform the vega hedge, we do not compute the∂ΨBStBlack-Scholes-Merton vega ∂σBSof the guarantee’s value Ψt and compare it to the

∂X BS

t
corresponding Black-Scholes-Merton vega ∂σBSof the straddle option’s value Xt ,but, instead, we use the so-called modified vega of Ψt for comparison.
Since the expected future cash flows of all maturities of the pool of policies cannot
be expected to react the same way to changes in today’s volatility, the modified vega
applies a different weighting to the respective vega of each maturity. We use theinverse of the square root of time as simple weighting method and use the maturityof the hedging instrument Xt , i.e. one year, as the benchmark maturity (cf. Taleb
1997, [28]).
The modified vega of Ψt at a policy calculation date t̃i then has the form
ModVega(t̃i ) =MX1υt̃k (t̃i ) pt̃k − t̃ik=i+1(4.13)

where the υt̃k (t̃i ) denote the respective Black-Scholes-Merton vega, computed attime t̃i , of the expected discounted cash flow at time t̃k of the pool of policies.
The ratio between the modified vega and the vega of the straddle then is used todetermine the portfolio’s position in the straddles (i.e. the quantity of the straddles
bought).
Under the Heston model, we compare the two derivatives of Ψt and Xt with respect
to the current local variance Vt in the “vega” sense and then analogously determine
the option position required to offset the changes of the portfolio with respect tochanges in the local variance in first order, while ignoring terms of higher-order andCross-Greek 3 terms.
Of course, under both hedging models, the position in the underlying must be adjusted for the delta of the option position ∆Xt Xt .
3

Second-order and higher partial derivatives of the option price with respect to several different
model parameters, for instance the sensitivity of the option delta with respect to change in volatility,
the so-called “Vanna” (also known as “DvegaDspot” or “DdeltaDvol”), which is defined as thesecond-order derivative of the option value, once to the underlying spot price and once to the2ΠtBlack-Scholes volatility, in formulas: Vanna = ∂S∂t ∂σ.
BSCHAPTER 4. HEDGING

40

The hedge ratios for all three strategies used in our simulations are summarized intable 4.1 for the Black-Scholes-Merton model, and in table 4.2 for the Heston model,
where the supercripts “BS” and “Heston” of the option prices indicate which model
was used for computation of the option prices.
When using the Black-Scholes-Merton model for hedging, we use the assumed longterm volatility σBS for computation of the Greeks of the portfolio value ΨBS
t . Forthe computation of the sensitivities of the option price XtBS , however, we use theXBlack-Scholes-Merton implied volatility σBS(t) implied by the market price at time

t of the straddle option. This means for the simulation, that, at a rebalancing dateof the hedge portfolio, we first retrieve the market price of the straddle option fromthe (unknown) data-generating model, use this price to calculate the Black-ScholesMerton implied volatility of the option via a root finding algorithm (see section 3.4)
and then put this implied volatility in the Black-Scholes-Merton model in order tocalculate the sensitivities (delta and vega) of the straddle option.

∆S

∆X

(NH)

−

−

(D-BS)

∂ΨBS
t
∂St

−

(DV-BS)

BS
∂ΨBS
X ∂Xt
t
−
∆
t
∂St
∂St

. BS
∂X
ModVega(t) ∂σXt
BS

Table 4.1: Hedge ratios for the different strategies if the Black-Scholes-Merton modelis used as hedging model.

∆S

∆X

(NH)

−

−

(D-H)

∂ΨHeston∂ΨHestontt+ ρσStV ∂V
∂St
t

−

(DV-H)∂ΨHeston∂XtHeston
t
− ∆X
t
∂St
∂St

+ ρσStV
Heston∂ΨHeston
∂Xt
t
− ∆X
t
∂Vt
∂Vt

∂ΨHestont∂Vt.

∂XtHeston∂Vt

Table 4.2: Hedge ratios for the different strategies if the Heston model is used as
hedging model.

CHAPTER 4. HEDGING

4.2.4

41

Additional semi-static hedge

Additionally, for all “active” dynamic hedging strategies (Delta and Delta-Vega), we
assume that the hedger buys European put options at each policy calculation datesuch that the possible guarantee payments for the next calculation date are fully
hedged by the put options (assuming surrender and mortality rates are deterministic
and known by the insurer).
This strategy aims at avoiding having to hedge an option with short time to maturity
and hence having to deal with a potentially rapidly alternating delta (due to a highgamma of the option) if the option is near the strike (cf. Taleb, 1997, [28]).
This is possible for all four considered ratchet mechanisms, since the minimum guaranteed withdrawal amount at a policy calculation date is always known at the previous calculation date 4 .

4Actually, for the ratchet mechanisms“Noi Ratchet”, “Lookback Ratchet” and “Remainingh
WBB Ratchet” it holds that E Wt̃guar W Gt̃i = Wt̃guar , i.e. at a policy calculation date, thei+1iminimum guaranteed withdrawal amount of the next calculation date is the exact value of the current guaranteed withdrawal amount, due to the monotonically increasing property of these ratchetmechanisms. For the “Performance Bonus” product design, the minimum guaranteed withdrawal
amount is always equal to the initial guaranteed withdrawal amount, i.e. the guaranteed withdrawal
amount without any performance bonus.

Chapter 5
Software Solution
In this chapter we present the main aspects of the design and the architecture of
the software solution that we used to conduct the different numerical analyses of
the GLWB option presented in the forthcoming chapters 6 and 7. The two main
requirements on the design and the implementation of the software solution were
flexibility and reusability. That is, a single implementation of an insurance productor of a financial instrument should be sufficient for all of the desired simulations
and calculations. Similarly, once implemented numerical algorithms (like the Monte
Carlo routine for instance) should be working for different products, even those thatwill be implemented at a later point in time. Generally, in order to allow for apotentially growth and extension of the software solution for forthcoming analyses,
it should be easy to add new instruments and new models of the financial market
without needing to change much of the already existing code. All of this also aims
at keeping redundancy at a minimum, and can therefore also be expected to bebeneficial for software quality assurance.
First, we give a list of the main specifications of the software solution that we defined,
discuss the issues and questions arising from the implementation and present oursolution afterwards.

5.1

Functional specification

In this section, we specify the core functionality and the requested behavior of the
final software solution and the requested properties of the input and output data.
• core functionality
42

CHAPTER 5. SOFTWARE SOLUTION

43– The software solution should allow the user to conduct Monte-Carlo analyses of various financial instruments (including the GLWB products presented in chapter 2) and different models of the financial market (at least
the Black-Scholes-Merton and the Heston model must be implemented).
– The result of the Monte-Carlo simulations should be generic in the sensethat while the structure of the output may be pre-specified, its content
may change depending on the analyzed instrument.
– A toolkit for the (statistical) analysis of the Monte-Carlo results should be
provided to the user, including tools and functions to compute percentiles,
compute simple statistics (empirical moments like mean, standard deviation, skew and kurtosis) and risk measures (value at risk, conditional tailexpectation, lower moments, . . . ).
– The software solution should also provide functions for the pricing of European standard options, i.e. functions should be provided that implement
the closed-form formulas in case of the Black-Scholes-Merton model andthe inverse Fourier transformation in the case of the Heston model.
• input / output– The required input data should be easily transferable out of spreadsheet
files, i.e. file types that are used within spreadsheet calculation programs(.xls files for instance).
– The result of the analyses should also be easily transferable to spreadsheet
files or should be spreadsheet files in the first place.
• user interface– The user interface should be easy to use, should allow for a clear but
flexible input structure and should assist the user in avoiding mistakes.
– The user interface should also be supportive of task automatization, i.e.
it should allow for a kind of scripting language (e.g. a simple batch file,
Visual Basic for Applications, Perl or Python) in order to automate theuse of the core functionality of the software solution via the use of macros.
• general requirements– As the numerical analyses are expected to be rather time-consuming, thesoftware solution should aim at maximizing efficiency and speed of computation.

CHAPTER 5. SOFTWARE SOLUTION

44– The software solution should be reusable for subsequent analyses and tasks
and should therefore provide a flexible structure in both, the constructionof the code itself and the user interface.
– The implementation should allow for an easy way to add new financial instruments for analysis without having to change (too much) of the alreadyexisting code.
– The programming language used should be widely-used amongst academics and practitioners, and should also have a strong (online) community supporting it.
– The implementation should be easy to test and debug, i.e. the programmer should be able to locate and correct mistakes in the code with aslittle effort as possible.
– It must be possible for the programmer to expose functions directly tothe user (in the sense that the user is able to use the function directly)that have standard signatures, i.e. functions whose input and output
parameters are of standard types such as double, integer and string,as well as arrays and matrices of the aforementioned types.
– It must be clearly definable which functions are visible to and usable by
the user and which are not. The changes in the code that are needed for
exposing a function to the user should be kept minimal.

5.2

Architecture and implementation

The software solution that we build for our analyses is designed and implemented
for the use under Microsoft Windows. All of the core functionality of our software
solution was implemented using the C++ programming language and compiled with
the Visual C++ 9.0 compiler provided by Microsoft.
We used an object oriented approach for the software solution and hence also makeuse of the accompanying terminology and concepts, particularly the concept of (abstract) classes as representation of an object type and as container for the features
and traits of the object. We also use the concept of information hiding in this context, which says that the (implementational) details of the methods of an object
as well as the data structure should be hidden from the user of this object and be
encapsulated within the object. The concept also says that it is only of interest

CHAPTER 5. SOFTWARE SOLUTION

45

what is implemented by the object and not how, i.e. for the user is only of interest
which interface (as a collection of provided functionality) is implemented by the object. This concept is called polymorphism, as the same interface can be implemented
by several object types (classes) and therefore, if a function requires an object asargument that implements a specific interface, objects of all of these implementingclasses may be passed to the function, since only the provided functionality mattersand not the true type of the object.
In our software solution there are two main interfaces (or as they are known in C++:
abstract classes, i.e. classes that have so-called pure virtual methods, which declare
only the method’s signature but do not define the method’s body), which are theparent classes of all models and instruments that we implemented:• cProduct: This is the main interface for all financial products and instruments.
cMCProduct is derived from this class and represents all financial products andinstruments that support the simulation within a Monte Carlo routine, andhence implement the needed functionality. Every derived class of cProductand cMCProduct also contains the calculation dates of the product, i.e. for
instance for an European call option the start and end date of the option andfor a GLWB-type product all of the policy calculation dates.
• cModel: This is the abstract class which all of the financial market model classes
are derived from. Every class that represents a model of the financial market
also contains all of the corresponding model parameters and additional dataneeded within this model. For each model, there has to be a corresponding
scenario generator, i.e. a class that produces an arbitrary number of path
realizations for all of the considered assets in this market (typically at least
one stock and a money-market account).
Our software solution includes a central storage facility implemented in the classcStorage, in which objects of the types cProduct, cModel, as well as matrices ofdoubles and models for lapsation (cLapsation) and mortality (cMortality) can
be stored and retrieved from within all parts of the code. For storage and retrievalof the objects an identifier in form of an arbitrary string is used, i.e. when storingan object, apart from the object itself, a string has to be delivered by the user.
The string must be unique within the category (model, product, . . . ), otherwise thealready stored object with the same identifier is replaced by the new object.
This storage class is designed to allow only one single instance of itself, i.e. there
may never exist two parallel instances (objects) of this class. There are at least two

CHAPTER 5. SOFTWARE SOLUTION

46

ways of implementing such a behavior, with one being the so-called Singleton pattern(cf. for instance Joshi, 2004, [20]), where the user has to address all of their calls
to the static method usually called instance() or getInstance() which returns areference or a pointer to the single object of the class, which was created at the timeof the first call of instance(). The other way of implementing this “single object
pattern” makes use of static methods and variables, i.e. variables whose lifetime
extend across the entire run of the program and methods that can be called without
an existing instance of the class. We chose the latter for our implementation, mainly
because of its superior simplicity.
The core design principle for the implementation of the Monte-Carlo functionality ofthe software solution is as follows: there is the function MonteCarlo, which expects
information about the financial market model to be used in the simulation andabout the product that is to be simulated. The information about the model ispassed in form of an object of a class derived from the abstract base class called
cModel. Every class that represents a model of the financial market used within theMonte Carlo simulation must be derived from this class. Each derived model classthen is supposed to contain the model type (in form of an enumeration) and thecorresponding parameters of the model. To each model, there is a corresponding
scenario generator, which will generate random scenarios according to the given
financial market model and for a given set of calculation dates. The base class of all
scenario generators is called cScenarioGenerator and has the following structure:
1class cScenarioGenerator {2public :3v i r t u a l void i n i t S c e n a r i o ( c S c e n a r i o& S c e n a r i o ) = 0 ;4v i r t u a l void g e n e r a t e S c e n a r i o ( c S c e n a r i o& S c e n a r i o ) = 0 ;5};where the class cScenario is a container for the financial market data (especiallyprices of the stocks and the money-market account at the calculation dates) in one
specific scenario. The function MonteCarlo creates an instance of cScenario andpasses it to the scenario generator for initialization and for each new scenario. The
purpose of the function initScenario is that in some financial market models not
all of the simulated market data is stochastic, but deterministic, i.e. the values
remain the same in each scenario. In order to achieve computational savings, in thiscase initScenario writes these deterministic values in the scenario object at the
beginning of the simulation and later on generateScenario only updates the values
for the assets that are (pseudo-) randomly generated.

CHAPTER 5. SOFTWARE SOLUTION

5.3

47

User interface

For the user interface we decided to follow a direct approach and integrate the solution into the spreadsheet environment provided by the Microsoft Office Excel software. The library containing the C++ implementation of our software solution iscompiled as a native Dynamic-link library 1 and marked via the file extension .xll
for the use within Microsoft Office Excel as an add-in. To achieve the interoperability of C++ functions and Microsoft Office Excel, we made use of version 3.0.0
of the “xlw C++ wrapper for Excel” library2 which makes use of the C application
programming interface (API) of Excel3 , and allows the programmer to easily export
C++ functions to Excel, where these “hand-made” functions are embedded just likethe built-in functions as, for instance, =SUM(...) are. To give an example: If a function is to be exported to Excel, which is called MyTest, takes a double as argument
and returns also a double, then the following code must be added to the project
files:
1#include <xlw / xlw . h>2using namespace xlw ;3

extern ”C” {4

LPXLFOPER EXCEL EXPORT xlMyTest ( XlfOper x l V a l u e ){5

EXCEL BEGIN ;

6

...

7

double i n p u t V a l u e = x l V a l u e . AsDouble ( ) ;8double r e t = Test ( i n p u t V a l u e ) ;9return XlfOper ( r e t ) ;10

EXCEL END;

11

}

12

X L R e g i s t r a t i o n : : Arg MyTestArgs [ ] = {{ ” Value ” , ”The i n p u t v a l u e ” , ”XLF OPER” }

13
14

};

15

XLRegistration : : XLFunctionRegistrationHelper registerMyTest (16” xlMyTest ” , ”MyTest” ,17” T e s t s t he xlw C++ wrapper f o r E x c e l . ” ,
1

See
for
instance
Microsoft’s
documentationofDynamic-link
http://msdn.microsoft.com/en-us/library/ms682589(VS.85).aspx.
2
3

librariesatProject website to be found at http://xlw.sourceforge.net.

This C API of Excel is documented in: Baarns Consulting Group. Microsoft Excel 97 Developer’s kit, Microsoft Press, 1997.

CHAPTER 5. SOFTWARE SOLUTION

MyTestArgs , 1 ) ;

18
19

48

}
In order to automate this procedure of adding the xlw-specific code to the project,
we implemented a script using the Ruby programming language4 which parses allof the header files of the project (i.e. all files in the project directory ending on.h) and searches for specific annotations to function declarations and produces theexport code for the matched function automatically. The annotation to the function
declaration which marks the function to be exported to Excel has the following
pattern:1//# EXPTOXLL( MyTest , T e s t s t h e x l w C++ wrapper f o r E x c e l . )2double MyTest

3(
const double Value // The i n p u t v a l u e

4
5);
The first argument after EXPTOXLL indicates the definable name of the function within
Excel (in the sense of using it in a spreadsheet cell via =MYTEST(...)), and the second
argument, as well as the comment part after the input argument (after the doubleslash), are used for the description of the fields of the Excel Function Wizard, as canbe seen in the screenshot shown in figure 5.1 (note that the automatization script
always adds the prefix FR to the function name in order to make the function names
distinctable from the built-in functions).

Figure 5.1: Screenshot showing the Microsoft Office Excel Function Wizard with the
exemplary C++ function that was exposed to Excel via the use of xlw.

4

See http://www.ruby-lang.org.

CHAPTER 5. SOFTWARE SOLUTION

5.4

49

Used libraries

Our software solution makes use of several open-source libraries for C++, for thereason that we wanted to avoid writing code that has already been written by others
and – more importantly – has been tested and used by many programmers. The
following list gives an overview of the used libraries and their purpose (cf. Joshi,
2004, [20]):• Boost: The boost project is an open source library designed to extend theC++ standard library (stl). It can be found at http://www.boost.org. The
intention is that the libraries incorporated in boost will become part of the
C++ Standard in the future, and hence, the license is very unrestrictive andallows the user basically to do whatever they want with the code (as opposed
to projects using the GNU license5 ). This is important if it is intended to use
the library within a product that is used or sold commercially. From the boost
website6 some of the requirements met by the license:– Must grant permission without fee to copy, use and modify the software
for any use (commercial and non-commercial).
– Must require that the license appear with all copies [including redistributions] of the software source code.
– Must not require that the license appear with executables or other binary
uses of the library.
– Must not require that the source code be available for execution or otherbinary uses of the library.
Of the many libraries of the boost project we use the following within oursoftware solution (in alphabetical order, with descriptions taken from the boost
website):– Bind: boost::bind is a generalization of the standard functions std::bind1st
and std::bind2nd. It supports arbitrary function objects, functions, function pointers, and member function pointers, and is able to bind anyargument to a specific value or route input arguments into arbitrary positions. We use this in the context of numerical quadrature, where arbitrary
5

See the website of the Free Software Foundation, http://www.fsf.org/licensing/, for details.

6

http://www.boost.org/users/license.html, retrieved August 20, 2009.

CHAPTER 5. SOFTWARE SOLUTION

50

unary functions (i.e. functions that take a double as argument and returning a double) can be passed to the quadrature function in order tohave them numerically integrated.
– Conversion: Polymorphic and lexical casts. Used in our software solution
for lexical casts from strings to doubles.
– Date Time: A set of date-time libraries based on generic programming
concepts. We use this for the calculation dates of the analyzed financial
instruments.
– Math/Statistical Distributions: A wide selection of univariate statistical distributions and functions that operate on them.
– Random: A complete system for random number generation. We use thisfor (pseudo-) random number generation, especially the implementationof the Mersenne Twister 19937 algorithm7 .
– Smart Ptr: Smart pointer class templates. We use smart pointers tomanage all of the cModel and cProduct objects, i.e. when an objectof this type is to be passed to a function, a smart pointer referencing
to the object is passed instead. This way we avoid potential memory
leakage when working with many dynamically generated objects. Thisis especially important because leaked memory would accumulate withevery function call from Excel and would not be freed again until the
Excel application is closed.
– uBLAS: uBLAS provides matrix and vector classes as well as basic linear algebra routines. Several dense, packed and sparse storage schemes
are supported. We use this for all numerical calculations and in highperformance areas of the code, where fast parsing of arrays is needed. In
those parts of the code where performance is not an issue, we often use
the std::vector template provided by the standard library instead.
• QuantLib: The QuantLib project is aimed at “providing a comprehensive
software framework for quantitative finance. QuantLib is a free / open-source
library for modeling, trading, and risk management in real-life.” The project
website can be found at http://www.quantlib.org. Similarly to boost, thelicense is very unrestrictive, allowing free use in commercial software. We tookparts of this library and used it for several numerical algorithms in our software
7

cf. Matsumoto, M., Nishimura, T. (1998), “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator”.

CHAPTER 5. SOFTWARE SOLUTION

51

solution, including the Gauss-Lobatto quadrature, which is used in the context
of the Heston model for the pricing of European “plain vanilla” options via
Fourier inversion (cf. section 3.3.3).
• xlw: We already mentioned the use of this library for generating Excel plug-ins,
known as xlls. The code for xlw can be obtained from http://xlw.sourceforge.net.

5.5

Usage example

In this section, we give a short example to illustrate the usage of our software solution
via the Microsoft Office Excel spreadsheet application. In this example we will
perform a 10-year Monte Carlo simulation of a portfolio of identical GLWB contracts
under the Heston stochastic volatility model and we will compute simple statisticsof the simulation result.
(i) =FRSchedule(Start; Step; NbSteps)
In order to generate a financial instrument, we need calculation dates, which
are often of the form “each month for x years”. For this purpose =FRSchedule
provides the functionality to define a start date, a step size (e.g. 1w for a
step size of one week, or 2m for the step size of two months), and the total
number of steps. The function will return a vector of dates according to thespecification. In our example we have annual calculation dates for a period of
10 years, therefore the function call is =FRSchedule(TODAY(); ‘‘1y’’; 10).
(ii) =FRSetPGMxBProduct(ID; LumpSum; GWRate; DeathBenefitMethod;
StepUpMethod;[...]; AcquisitionFee ;[...]; CalcDates)
This function generates an object of the type cGMxBProduct, the first argument,
ID, will be the string identifier of the object that is needed to pass it to otherfunctions (e.g. LookbackRatchet#01 would be a possible choice). The lastargument is a vector of the calculation dates, i.e. here we would pass the dates
that we created in the first step. The rest of the arguments are the parametersof the product (note that we omitted some of the parameters for the sake ofsimplicity of presentation). All of the functions that begin with FRSet... will
first check the parameters for correctness, will then create an object of their
specific type, will store the created object via the cStorage class, and will
finally return the identifier of the object back to the cell in Excel from where
the function was called.

CHAPTER 5. SOFTWARE SOLUTION

52(iii) =FRSetMortality(ID; Ages; DeathProbabilities)
For the portfolio of GLWB contracts, we need an underlying model of themortality of the insured. In this case, we only use a simple mortality table,
matching an annualized probability to each age. The first argument is theidentifier of the created mortality table, e.g. Mort#male#1965 would be a possible choice, whereas Ages and DeathProbabilities are vectors representing
the mortality probability for each age of the insured.
(iv) =FRSetLapsation(ID; ElapsedTime; LapsationProbabilities)
For the portfolio of GLWB contracts, we also need an underlying model of thelapsation of the insured. Again, we use a simple model where in each contract
year there is a certain annualized probability that the insured will surrender.
The first argument is the identifier of the created surrender probabilities, forinstance Lapse#01.
(v) =FRSetPGMxBPortfolio(ID; GMxBProductID; MortalityID; LapsationID; Age)In order to generate the product portfolio we have to pass to the function=FRSetPGMxBPortfolio its chosen identifier, the identifier of the underlying
product, the identifier of the mortality object, the identifier of the lapsation
object, and the age of the insured at outset of the contract, i.e. in our example
the call for a 65-year old would look like this:=FRSetPGMxBPortfolio(‘‘Portfolio#01’’; ‘‘LookbackRatchet#01’’;‘‘ Mort#male#1965’’; ‘‘Lapse#01’’; 65).
(vi) =FRSetMHeston(ID; Spot; ShortVol; LongVol; MeanReversion; VolOfVol;
Correlation; UndYC; CurrYC)
Now we have to create an object for the model of the financial market. The
function =FRSetMHeston will create an object of the model class cHestonModel,
which is derived of the abstract class cModel. Spot is the spot price of thestock at inception, the following parameters are model-specific (cf. section3.1), UndYC is the yield curve of the drift of the stock, and CurrYC is the yield
curve of the currency in which the money-market account is denoted.
(vii) =FRMonteCarlo(ProductID; ModelID; NbPaths; Seed)
To perform the actual Monte-Carlo simulation, we have to pass the identifierof the product that is to be analyzed and the identifier of the model that themodel that is to be used for scenario generation. Additionally, the number ofpaths and the seed for the random number generation is needed as argument.
The function will then try to fetch the model and the product from cStorage,

CHAPTER 5. SOFTWARE SOLUTION

53

and will then perform the Monte-Carlo simulation. All single path results arethen stored in cStorage, and the mean of the results is returned as a cellmatrix to Excel.
(viii) =FRStatistics(Statistic ; Percentile ; Dimension; RowNbr)
This function is used to calculate statistics from a simulation result which isstored in cStorage. The following statistics are supported: minimum, maximum, mean, standard deviation, variance, skewness, kurtosis, value at risk,
expected shortfall, and arbitrary percentiles between 0 and 1. The arguments
Dimension and RowNbr are used to select the value of the result matrix fromwhich the statistic is to be computed.

Chapter 6
Contract Analysis
In this chapter, we analyze the GLWB rider options that we presented and definedin chapter 2. We first compare the pricing results for each product design under
different model assumptions for the financial market and the policyholder behavior.
Afterwards we analyze the distribution of the guaranteed withdrawal amount generated by the different product designs, and also analyze and compare the different
distributions of the point in time when the guarantee of the GLWB option triggers.
We conclude the chapter with an analysis of the deltas (i.e. the sensitivity of theoption’s value to changes in the underlying’s spot price) of the considered GLWB
product designs.

6.1

Determination of the fair guaranteed withdrawal rate

In this section, we first calculate the guaranteed withdrawal rate xW L that makes acontract fair (in the sense discussed in section 2.3), assumed all other policy parameters are given. In order to calculate xW L , we perform a root search with xW L asargument and the value of the GLWB option V0G (as defined in section 2.3) as function value. For all of our analyses we assume a fee structure of the GLWB contract
as given in Table 6.1.
Policy calculation dates are assumed to be annually, i.e. withdrawals and stepups are made only once per year at a prespecified point in time. The number ofpolicyholders in the pool is assumed to be large enough to allow for the use of the
Law of Large Numbers with respect to mortality and lapsation within the pool, and54

CHAPTER 6. CONTRACT ANALYSIS

55

Acquisition fee
4.00% of lump sum
Management fees 1.50% p.a. of NAV
Guarantee fees
1.50% p.a. of NAV
Withdrawal fees 0.00% of withdrawal amount
Table 6.1: Assumed fee structure for all considered contracts.
thus deterministic mortality and surrender rates may be applied for projection of
the cohort of policyholders. We further assume the policyholder to be a 65 years old
male. For pricing purposes, we use best-estimate mortality probabilities given in the
DAV 2004R table published by the German Actuarial Society (DAV).
Regarding the policyholder behavior, we assume that policyholders - as long as theircontracts are still in force - only chose between two options at a policy calculation
date: either to withdraw exactly the guaranteed annual withdrawal amount or toperform a full surrender and withdraw all of the remaining assets1 .
All results in this chapter have been calculated assuming an annual risk-free rate ofinterest of r = 4% .

6.1.1

Results for the Black-Scholes-Merton model

Table 6.3 displays the results for the fair guaranteed withdrawal rate xW L for all
four ratchet mechanisms that were defined in chapter 2 and for different assumptions
regarding the policyholder behavior, where the pricing was done using the BlackScholes-Merton model with different assumed values σBS for the equity volatility.
In regard to the policyholder behavior, we consider three different scenarios: nopolicyholder ever surrenders (no surr), surrender happens according to Table 6.2(surr 1) and surrender with twice the probabilities given in Table 6.2 (surr 2).
A comparison of the different product designs shows that, obviously, the highest
annual guarantee can be provided if no ratchet or performance bonus is provided
at all. If no surrender and a volatility of σBS = 20% is assumed, the guarantee issimilar to a “5 for life” product (4.98%). Including a Lookback Ratchet would needa reduction of the initial annual guarantee by 66 basis points to 4.32%. If a richer
ratchet mechanism is provided such as the Remaining WBB Ratchet, the guarantee
1

Please note that if the guaranteed withdrawal amount coincides with the remaining fund assets,
the withdrawal of all remaining assets is not considered to be a full surrender and the guarantee
does not end.

CHAPTER 6. CONTRACT ANALYSIS

56

Year

Surrender rate

1
2
3
4
5
≥6

6%
5%
4%
3%
2%
1%

Table 6.2: Assumed deterministic surrender rates.

needs to be reduced to 4.01%. About the same annual guarantee (4.00%) can beprovided if no ratchet is provided but a Performance Bonus is paid out annually.

Throughout our analyses, the Remaining WBB Ratchet and the Performance Bonus
allow for a similar annual guarantee. However, for lower volatilities, the Remaining
WBB Ratchet seems to be less valuable than the Performance Bonus and therefore
allows for a higher withdrawal rate while for higher volatilities the Performance Bonus
allows for higher withdrawal rates. Thus, the relative impact of volatility on the priceof a GLWB depends on the chosen product design and appears to be particularly highfor ratchet type products (II and III). This can also be observed when comparing
the No Ratchet case with the Lookback Ratchet. While - when volatility is increased
from 15% to 25% - for the No Ratchet case, the fair withdrawal rate decreases byjust over half a percentage point from 5.26% to 4.70%, it decreases by almost a fullpercentage point from 4.80% to 3.85% in the Lookback Ratchet case (all values forthe no surrender case). The reason for this is that for the products with ratchet, highvolatility leads to a possible lock in of high positive returns of the underlying fundin some years (as some kind of option on an option) and thus is a rather valuable
feature if volatilities are high.

If the insurance company assumes the deterministic surrender probabilities given
in table 6.2 when pricing the GLWB option, the fair withdrawal rates increase forall considered model points. The extent of the increase of the withdrawal rate israther similar over all product types and assumed values of the Black-Scholes-Merton
volatility σBS . The annual guarantee increases by around 15-20 basis points if thesurrender assumption from table 6.2 is made and increases by another 20 basis points
if this surrender assumption is doubled.

CHAPTER 6. CONTRACT ANALYSIS

57

Product
No Ratchet
Volatility

Surrender

I

Lookback
RatchetII

Rem. WBB Performance
Ratchet
Bonus
III
IV

σBS = 15%

No surr
Surr 1
Surr 2

5.26
5.45
5.66

4.80
5.00
5.22

4.43
4.62
4.83

4.37
4.57
4.79

σBS = 20%

No surr
Surr 1
Surr 2

4.98
5.16
5.35

4.32
4.50
4.71

4.01
4.18
4.38

4.00
4.19
4.40

σBS = 22%

No surr
Surr 1
Surr 2

4.87
5.04
5.23

4.13
4.30
4.50

3.85
4.01
4.20

3.85
4.03
4.24

σBS = 25%

No surr
Surr 1
Surr 2

4.70
4.86
5.04

3.85
4.01
4.20

3.61
3.76
3.94

3.62
3.81
4.01

Table 6.3: Fair guaranteed withdrawal rates xW L in % for different ratchet mechanisms, different policyholder behavior assumptions and for different values of theBlack-Scholes-Merton volatility σBS .

6.1.2

Results for the Heston model

For the pricing of the GLWB rider option under the Heston model we use the calibration given in table 6.4, where the Heston parameters (long-term local variance θ ,speed of mean reversion κ , “vol of vol” σV , equity correlation ρ , and starting value
for the local variance process V0 ) are those derived by Eraker (2004, [11]), whichhave also been stated in annualized form for instance by Ewald et al. (2007, [12]).
Parameter

Value

r
θ
κ
σV
ρ
V0

0.04
(0.220)2
4.75
0.55−0.569θ

Table 6.4: Benchmark parameters for the Heston model.
One of the key parameters in the Heston model is the market price of volatility riskλ (cf. section 3.2). Since absolute λ -values are hard to interpret, table 6.5 shows

CHAPTER 6. CONTRACT ANALYSIS

58

the corresponding values for the long-term local variance θ and the speed of mean
reversion κ for different choices of λ .
Market priceof volatility
risk

Speed ofmean
reversion κLong-run
local variance
θ

λ= 3
λ= 2
λ= 1
λ= 0
λ = −1
λ = −2λ = −3

6.40
5.85
5.30
4.75
4.20
3.65
3.10

(0.190)2(0.198)2(0.208)2(0.220)2(0.234)2(0.251)2(0.272)2

Table 6.5: Q -parameters for different choices of the market price of volatility risk
factor λ .

Higher values of λ correspond to a lower long-term variance and a higher speed ofmean reversion, while negative values of λ correspond to a higher long-term varianceand a lower speed of mean reversion. Thus, a negative value of λ leads to a higher
variation and a higher long-term mean of the local variance process, meaning that
if the insurer is “short volatility” (i.e. the insurer potentially suffers from volatility
increases) as it is the case if the insurer has a portfolio of sold GLWB rider options,then presumably a negative value of λ would be chosen for pricing purposes. λ = 2
implies a long-term volatility of θ = (19.8%)2 and λ = −2 implies a long-term
variance of θ = (25.1%)2 .
In table 6.6, we show the pricing results for the fair annual withdrawal rates under the
Heston model for all different product designs, the same scenarios for the policyholder
behavior as in the Black-Scholes-Merton case, and integer values of λ between −2
and 2 .
Under the Heston model, the fair annual guaranteed withdrawal appears to be thesame as under the Black-Scholes model with a comparable constant volatility. For
instance for λ = 0 , which corresponds to a long-term volatility of 22%, the fairannual guaranteed withdrawal rate for a contract without ratchet is given by 4.87%,
exactly the same number as under the Black-Scholes-Merton model. In the Lookback
Ratchet case, the Heston model leads to a fair guaranteed withdrawal rate of 4.17%,
the Black-Scholes-Merton model of 4.13%. For the other two product designs, again,
both asset models almost exactly lead to the same withdrawal rates. Also the impact

CHAPTER 6. CONTRACT ANALYSIS

59

Product
Market priceof volatility
risk factor
Surrender

No Ratchet
I

Lookback
RatchetII

Rem. WBB Performance
Ratchet
Bonus
III
IV

λ= 2

No surr
Surr 1
Surr 2

4.99
5.18
5.38

4.36
4.56
4.76

4.03
4.21
4.40

4.00
4.22
4.43

λ= 1

No surr
Surr 1
Surr 2

4.93
5.12
5.31

4.27
4.46
4.66

3.95
4.13
4.32

3.93
4.14
4.35

λ= 0

No surr
Surr 1
Surr 2

4.87
5.05
5.24

4.17
4.35
4.55

3.86
4.03
4.22

3.84
4.06
4.27

λ = −1

No surr
Surr 1
Surr 2

4.79
4.97
5.16

4.05
4.23
4.42

3.75
3.92
4.10

3.74
3.95
4.16

λ = −2

No surr
Surr 1
Surr 2

4.70
4.87
5.05

3.90
4.08
4.26

3.62
3.79
3.97

3.62
3.82
4.04

Table 6.6: Fair guaranteed withdrawal rates xW L in % under the Heston modelfor different ratchet mechanisms, different assumptions regarding the policyholder
behavior and for different values of the market price of volatility risk parameter λ .
that the different assumed policyholder behavior scenarios have on the pricing outcome seem to be of the same magnitude as in the Black-Scholes-Merton case, withan increase of around 20 basis points between the scenarios.
Thus, for the pricing (as opposed to hedging, see chapter 7) of GLWB rider options,the long-term volatility assumption is much more crucial than the question whether
volatility should be modeled stochastic or deterministic.

6.2

Distribution of withdrawals

In this section, we compare the distributions of the guaranteed withdrawal benefits(given the policyholder is still alive and did not surrender the contract) for each policy year and for all four different ratchet mechanisms that were presented in chapter
2. We use the Black-Scholes-Merton model for all simulations in this chapter andassume a risk-free rate of interest r = 4% , a drift of the underlying of µ = 7% andCHAPTER 6. CONTRACT ANALYSIS

60(i) No Ratchet(ii) Lookback Ratchet(iii) Remaining WBB Ratchet(iv) Performance Bonus

Figure 6.1: Development of empirical percentiles (25th - 75th dark blue area, 10th 90th light blue area), median (yellow points) and mean (red line) of the guaranteed
withdrawal amount over policy years 0 to 30 for each ratchet type and a single
premium of 10,000. The Black-Scholes-Merton model with parameters µ = 7% andσBS = 22% was used for data generation of the simulation.
a constant volatility of σBS = 22% .
For all four ratchet types, we use the guaranteed withdrawal rates derived in section
6.1 (without surrender). In figure 6.1, for each product design we show the development of arithmetic average, median, 10th - 90th percentiles, and 25th - 75th percentilesof the guaranteed annual withdrawal amount over time, assuming the policyholder
paid a single premium of 10,000.
Obviously, the different considered product designs lead to significantly different
risk/return profiles for the policyholder. While the No Ratchet case provides deterministic cash flows over time, the other product designs differ quite considerably. Both ratchet products have potentially increasing benefits. For the Lookback
Ratchet, however, the 25th percentile remains constant at the level of the first withdrawal amount. Thus, the probability that a ratchet never happens is higher than

CHAPTER 6. CONTRACT ANALYSIS

61

25%. The median increases for the first 10 years and then reaches some constantlevel implying that with a probability of more than 50% no withdrawal increments
will take place thereafter.
Product III (Remaining WBB Ratchet) provides more potential for increasing withdrawals: For this product, the 25th percentile increases over the first few years andthe median is increasing for around 20 years. In the 90th percentile, the guaranteedannual withdrawal amount reaches 1, 500 after slightly more than 25 years while
the Lookback Ratchet hardly reaches 1, 200 . On average, the annual guaranteed
withdrawal amount more than doubles over time while the Lookback Ratchet does
not, of course this is only possible since the guaranteed withdrawal at t = 0 is lower.
A completely different profile is achieved by the fourth product design, the product with Performance Bonus. Here, annual withdrawal amounts are rather highin the first years and are falling later. After 15 years, with a 75% probability no
more performance bonus is paid, after 25 years, with a probability of 90% no more
performance bonus is paid.
For all three product designs with some kind of bonus, the probability distributionof the annual withdrawal amount is rather skewed: the arithmetic average is significantly above the median. For the product with Performance Bonus, the medianexceeds the guarantee only in the first year. Thus, the probability of receiving aperformance bonus in later years is less than 50%. The expected value, however, ismore than twice as high.

6.3

Distribution of the guarantee’s trigger time

In this section, we analyze the point in time τG , when the guarantee of the contract
triggers, i.e. we analyze the point in time when the insurer has to compensate for theguarantee payments to the policyholder for the first time since policy inception. For
this analysis, we assume that the policyholder does not die and does not surrenderduring the observation period, i.e. the guaranteed withdrawal amount is deducted
every year.
In figure 6.2, for each of the products, we show the simulation results for the probability distribution of the trigger time τG . Any probability mass at the end of
the simulation period refers to scenarios where the guarantee is not triggered. The
assumptions and parameters used for this simulation are the same as in section 6.2.

CHAPTER 6. CONTRACT ANALYSIS

62(i) No Ratchet(ii) Lookback Ratchet(iii) Remaining WBB Ratchet(iv) Performance Bonus

Figure 6.2: Distribution of guarantee’s trigger time for each of the product designs.
The Black-Scholes model with parameters µ = 7% and σBS = 22% was used forthe data generation of the simulation.

CHAPTER 6. CONTRACT ANALYSIS

63

For the No Ratchet product, trigger times vary from 7 to over 55 years. Witha probability of 17%, there is still some account value available at age 121 (thelimiting age of the mortality table). For this product, on the one hand, the insurancecompany’s uncertainty with respect to if and when guarantee payments have to be
paid is very high; on the other hand, there is a significant chance that the guarantee
is not triggered at all, meaning that the insurer never has to compensate for theguaranteed withdrawals of the insured, even if the insured survives until the limiting
age of the mortality table.

For the products with ratchet features, very late or even no triggers appear to beless likely. The more valuable a ratchet mechanism is for the client, the earlier theguarantee tends to trigger. While for the Lookback Ratchet still 2% of the contracts
do not trigger at all, the Remaining WBB Ratchet almost certainly triggers within
the first 40 years. However, on average the guarantee is triggered rather late, after
around 20 years.

The least uncertainty in the trigger time appears to be in the product with Performance Bonus. While the probability distribution looks very similar to that of theRemaining WBB Ratchet for the first 15 years, trigger probabilities then increase
rapidly and reach a maximum at t = 25 and 26 years. Later trigger times did
not occur at all within our simulation. The reason for this is quite obvious: The
Performance Bonus is calculated as 50% of the difference between the current account value and the remaining withdrawal benefit base. However, the remaining
withdrawal benefit base is annually reduced by the initially guaranteed withdrawal
amount and therefore reaches 0 after 26 years ( 1/3.85% ). Thus, after 20 years,
almost half of the account value is paid out as bonus every year.

This, of course, leads to a tremendously decreasing account value in later years.
Therefore, there is not much uncertainty with respect to the trigger time on the insurance company’s side. On the other hand, the complete longevity tail risk remains
with the insurer. Whenever the guarantee is triggered, the insurance company needs
to pay an annual lifelong annuity equal to the last guaranteed annual withdrawal
amount. This is the guarantee that needs to be hedged by the insurer. Thus, inthe following section, we have a closer look on the Greeks of the guarantees of thedifferent product designs.

CHAPTER 6. CONTRACT ANALYSIS

64(i) No Ratchet(ii) Lookback Ratchet(iii) Remaining WBB Ratchet(iv) Performance Bonus

Figure 6.3: Development over time of the empirical percentiles (dark blue line is themedian, blue lines are 25th - 75th , light blue lines 10th - 90th percentiles) and mean
(red dashed line) of the delta of a pool of policies, at each time multiplied with thethen current spot St of the underlying. The delta was computed within the BlackScholes model with parameters with r = 4% , µ = 7% and σBS = 22% , which was
also used for data generation of the simulation.

6.4

The Delta of the GLWB rider option

Within our Monte-Carlo simulation, for each scenario we can calculate different
sensitivities of the option value as defined in section 2.3, the so called Greeks. All
Greeks are calculated for a pool of identical policies with a total single premium
volume of US$ 100m under assumptions of future mortality and future surrender.
All the results shown in this section are calculated under standard mortality and nosurrender assumptions.
In figure 6.3, we chose to show different empirical percentiles as well as median andarithmetic average of the so-called delta, i.e. the sensitivity of the option value as
defined in section 2.3 with respect to changes in the price of the underlying.

CHAPTER 6. CONTRACT ANALYSIS

65

First of all, it is rather clear that all products throughout do have negative deltas
since the value of the guarantee increases with falling stock markets and vice versa.
Once the guarantee is triggered, no more account value is available and thus, fromthis point on, the delta is zero. Thus, in what follows, we call delta to be “high”
whenever its absolute amount is large.
At outset, the product without any ratchet or bonus does have the biggest delta andthus the highest sensitivity with respect to changes in the underlying’s price. The
reason for this mainly is that the guarantee is not adjusted when fund prices rise. In
this case, the value of the guarantee decreases much stronger than with any productwhere either ratchet lead to an increasing guarantee or a performance bonus leads toa reduction of the account value. On the other hand, if fund prices decrease, the first
product is deeper in-the-money since it does have the highest guarantee at outset.
Over time, all percentiles of the delta in the No Ratchet case are decreasing, and the
decay of the delta2 is quite steady in comparison to the other product designs.
For products II and III, the guarantee can never be far out-of-the-money due to theratchet feature. Thus delta increases in the first few years. All percentiles reach amaximum after ten years and tend to be decreasing from then on, i.e. there is achange of sign from plus to minus for the delta decay.
For the product with Performance Bonus, delta exposure is by far the lowest. Thisis consistent with our results of the previous section where we concluded that theuncertainty for the insurance company is the highest in the No Ratchet case and thelowest in the Performance Bonus case.

2

Delta decay is also known as the “Charm” or “DdeltaDtime” of an option, and is mathematicallydefined as the second derivative of the option’s value, once to the spot price of the underlying asset∂ 2 VtGand once to time, that is ∂St ∂t.

Chapter 7
Analysis of Hedge Efficiency
In this chapter we use the hedging strategies presented in chapter 4 to apply andanalyze them within Monte Carlo simulations, where we use different models for datagenerating, that is, we use different models for scenario generation of the underlying
asset’s spot price.

7.1

Objective and risk measures

We use the hedging strategies in order to hedge a pool of identical GLWB policies,
i.e. policies that have the same product design, the same age at inception of theinsured and the same product parameters like for instance the withdrawal rate xW L
and the fee structure. However, the insureds themselves are assumed to be separate
individuals with independent decisions to lapse the contract and with independent
time of deaths.
We assume the number of policyholders in the pool to be large enough to allow for
the use of the Law of Large Numbers with respect to mortality and lapsation within
the pool, and thus deterministic mortality and surrender rates may be applied for
projection of the cohort of policyholders.
In each scenario, we start with a pool of identical contracts that is then projected until
the limiting age of the mortality table is reached. We use the same denotations as inchapter 4, that is the time- t value of the guarantees embedded in the pool of policies
is denoted by Ψt , whereas the time- t value of the hedge portfolio implemented by
the insurer is denoted by ΠHedge, and the insurer’s portfolio balance Πt at time tt
for the pool of policies is given by the difference between the value of the embedded
66

CHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY

67

guarantees and the hedge portfolio, i.e. by
Πt = −Ψt + ΠHedge
,
t

(7.1)

where all of the cash-flows generated by the pool of policies are transacted throughthe hedge portfolio ΠHedge of the insurer, i.e. there are no transactions by the
insurance company to or from the portfolio Πt up until the end of the simulation(at time T ), where the insurer’s final profit/loss (short: P/L), ΠT , has to be settled.
We use the following three ratios to compare the performance of the different hedging
strategies, all in normalized form as a percentage of the premiums paid to the insurer
at t = 0 :• EP e−rT ΠT , the discounted expectation of the final value of the insurer’s
profit/loss under the real-world measure P , where T = ω − x0 is the last
policy calculation date. This is a measure for the insurer’s expected profit,
where a value of 1 means that, in expectation, for a single premium of 100 paidby the client, the insurance company’s expected profit is 1.
• CTE1−α (χ) = EP [−χ|−χ ≥ VaRα (χ)] , the conditional tail expectation of therandom variable χ , where χ is defined as the minimum of the discounted
valuesat all policy calculation dates, i.e. χ =n of the insurer’s profit/losso−rt̃i
min e Πt̃i i = 0, . . . , M , and VaR denotes the Value at Risk operator,that is VaRα (χ) = inf {l ∈ R|P(−χ > l) ≤ α} . This is a measure for the insurer’s downside risk given a certain hedging strategy: It can be interpreted
as the additional amount of money that would be necessary at outset suchthat the insurer’s profit/loss would never become negative over the life of thecontract, even if the financial market develops according to the average of theα (e.g. 10% ) worst scenarios in the stochastic model. In this sense, a valueof 1 means that, for a single premium of 100 paid by the client, the insurancecompany would need to hold 1 additional unit of capital upfront.
• CTE1−α (e−rT ΠT ) = EP −e−rT ΠT −e−rT ΠT ≥ VaRα (e−rT ΠT ) , the conditional
tail expectation of the discounted final profit/loss ΠT . This again is a risk
measure, however, with the difference to the previously defined CTE0.9 (χ) ,that it focuses on the value of the profit/loss at maturity T , i.e. after all
liabilities have been met (as in the model all of the insured have deceased),
and does therefore not factor in negative portfolio values that may have occurred over time. Thus a value of 1 in means that, in expectation over theCHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY

68

α (e.g. 10%) worst scenarios, for a premium of 100 paid by the client, theinsurance company’s expected discounted loss is 1. By definition, it holds
CTE1−α (χ) ≥ CTE1−α (e−rT ΠT ) .

7.2

Simulation results

For all of the considered hedging strategies we assume that the hedge portfolio isrebalanced on a monthly basis.
In the numerical analyses below, we set α = 10% for both risk measures and assume
a pool of identical policies with parameters as given in chapter 6, assuming that noneof the policyholders surrenders.
Our analysis focuses on model risk rather than parameter risk. Therefore, we use
the benchmark parameters for the capital market models presented in chapter 6 for
both, the hedging and the data-generating model, and we assume no risk regarding
mortality, thus using the same mortality tables1 for projection of the insured cohortand for hedging.

7.2.1

Comparison of hedge results

Table 7.1 gives the results for different hedging strategies and different data-generating
models as a percentage of the single premium paid by the client. In the table, thehedging strategies are abbreviated as follows: (NH) represents the “no active hedging” strategy presented in chapter 4, (D-BSM) represents the delta-hedging strategy
where the delta is computed using the Black-Scholes-Merton model. (D-H) stands
for the local risk-minimizing strategy performed using the Heston model, whereas(DV-BSM) indicates the delta-vega combination hedge using the additional straddle
options computed under the Black-Scholes-Merton model. (DV-H) finally represents
the extension of the local risk-minimizing strategy (D-H) for vega hedging also using
straddle options.
If no hedging (strategy (NH)) is in place, the insurance company’s portfolio has apositive delta due to the put character (and therefore negative delta, cf. section 6.4)of the sold GLWB guarantees. This effectively means the insurance company hasa (indirect) long position in the underlying and thus faces a rather high expected1Best-estimate mortality probabilities given in the DAV 2004R table published by the German
Actuarial Society (DAV).

CHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY

69

Data-generating model

Strategy

StatisticEP e−rT ΠT(NH)

CTE0.9 (χ)
CTE0.9 (e−rT ΠT )EP e−rT ΠT(D-BSM)

CTE0.9 (χ)
CTE0.9 (e−rT ΠT )Black-Scholes-Merton

HestonProduct

Product

I

0.48
1.71
1.44

CTE0.9 (χ)
CTE0.9 (e−rT ΠT )(DVBSM)
EP e−rT ΠT
CTE0.9 (χ)
CTE0.9 (e−rT ΠT )EP e−rT ΠT(DV-H)

III

IV

I

II

III

IV

10.43 07.77 06.67 3.88 10.36 7.97 6.82 04.13
25.29 20.07 17.54 15.12 25.76 20.97 18.54 15.97
23.41 18.27 15.90 13.35 22.93 18.55 16.25 13.51EP e−rT ΠT

(D-H)

II

CTE0.9 (χ)
CTE0.9 (e−rT ΠT )

0.27
3.25
2.74

0.21
3.12
2.71

0.17
2.02
1.78

0.57
2.77
2.44

0.29
4.76
4.14

0.17
4.51
3.99

0.13
3.35
3.02

0.52
2.63
2.28

0.42
4.59
4.03

0.34
4.44
3.98

0.21
3.36
2.95

0.82
1.75
1.35

0.81
2.41
1.80

0.75
3.01
2.40

0.47
1.88
1.53

0.49
1.40
1.15

0.41
1.99
1.58

0.33
1.95
1.60

0.19
1.49
1.21

Table 7.1: Results for different hedging strategies and different data-generating models as a percentage of the single premium paid by the client. The strategies are “noactive hedging” (NH), delta-hedging with the Black-Scholes-Merton model (D-BSM),
local risk-minimizing strategy using the Heston model (D-H), delta-vega hedge using
the Black-Scholes-Merton model (DV-BSM), and the extension for vega hedging of
the local risk-minimizing strategy under Heston (DV-H).

return combined with high risk. The results show that no active hedging effectively
means that the insurance company, on average over the worst 10% scenarios, would
need additional capital between 15% and 25%, depending on the respective productdesign, of the total premium volume paid by the clients in order to avoid a loss overtime.
The CTE0.9 (e−rT ΠT ) , i.e. the conditional tail expectation of the insurer’s discounted
final profit/loss ΠT , is around 23 for product type I (No Ratchet) and around 13 for
product IV (Performance Bonus) for both data-generating models, Black-ScholesMerton and Heston, which means that in expectation over the 10% worst scenarios,
for a premium of 100 paid by the client, the insurance company’s expected discounted
loss would be 23 or 13 respectively.
The corresponding values for the products with ratchet are in between. The differenceCHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY

70

in risk and expected return between the two data-generating models is rather small.
If the insurance company sets up a delta-hedging strategy based on the Black-ScholesMerton model, risk is significantly reduced for all products and under both datagenerating models. If the data-generating model is also the Black-Scholes-Merton
model, risk is reduced to less than 10% of its unhedged value for product I (NoRatchet). On the other side, the expected profit of the insurer is also reduced.
While without hedging, the No Ratchet product appeared to have the product designthat creates the most risk, with a delta-hedging strategy in place, the products witha ratchet feature (Lookback Ratchet and Remaining WBB Ratchet) now are theriskiest of the four considered designs.
The reason for this is that the delta is rather “volatile” for the products with ratchet,cf. figure 6.3 in section 6.4. Since fast changes in the delta lead to hedging errors due
to time-discrete readjusting of the hedge portfolio, this leads to an increase of therisk for the two ratchet type products. This result shows the impact of the gamma(the second order derivative of the option value with respect to the underlying price)on the hedgeability of an option. The larger the gamma, the higher discretization
errors and thus the higher the risk of a time-discrete delta-only hedge.
We now switch the data-generating model to the Heston model and analyze how theresults of the Black-Scholes-Merton delta-only hedge change. By sole introductionof stochastic equity volatility into the capital market, the riskiness of the GLWB
products, throughout all designs, is increased by roughly 50%. This is an indicationof the magnitude of the model risk that the insurance company bears. While therisk significantly increases, the insurance company’s expected return hardly changes,
meaning that the risk-return profile of the products worsens considerably.
In the last set of simulations of the delta-only strategies, we analyze what happens
if the calculation of the hedge ratios is performed within the Heston model (strategy(D-H)), i.e. we analyze what happens if the hedging is done with the “correct”
model. However, the risk is only reduced by a small amount in comparison to
the Black-Scholes-Merton delta-only hedge, which, in a reverse conclusion, can beinterpreted as the Black-Scholes-Merton not being such a poor choice in comparison
to more sophisticated models, even if it is known that the model is only a crude
approximation to the real dynamics.
Nevertheless, for both products with a ratchet mechanism and the Performance
Bonus product, the insurer’s expected profit is significantly increased if the Heston
model is used. Thus, in conclusion, by adapting the hedging model to the datagenerating model, the insurance company’s profit increases while its risk is only

CHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY

71

slightly reduced.
In the last two sets of simulations, we analyze the two hedging strategies wherevolatility risk is also tackled. The (DV-BSM) hedge further reduces risk significantly
compared to the two delta-only hedges, even though the hedge is set up with a modelthat assumes equity volatility to be deterministic, even constant.
Risk is further reduced by almost 50% and the results are even better than a (DBSM) hedge with the Black-Scholes-Merton model as scenario generator, which iscertainly not surprising, considering the hedge instrument used for vega hedging(a one-year ATMF straddle) also is sensitive to changes in delta and therefore (asvega and gamma tend to bear resemblance to each other) introduces a partial hedge
against the gamma of the insurer’s liability.
If the vega hedge is set up using the Heston model (DV-H), results improve evenfurther. The market risk within our simulation model now is below 2% of the initial
single premium paid by the client for all four product designs. Also, the differences
between the riskiness of the different product designs seems more evenly, as producttypes II and III benefit the most in terms of reducing risk if stategy (DV-H) is usedinstead of (DV-BSM).

7.2.2

Distribution of hedge portfolio’s value and insurer’sbalance

Figure 7.1 shows for each product type the empirical percentiles of the value of theinsurer’s hedge portfolio ΠHedge
over time, assumed a Black-Scholes-Merton deltat
only hedging strategy is in place and with the data-generating model also being
the Black-Scholes model with the same parameters that were used in chapter 6. If
the implemented delta hedge resulted in a perfect hedge, the value of the hedge
portfolio at time t , ΠHedge, should coincide with the value of the pool of policies,tΨt , for any t . Thus, if Ψt is negative, also ΠHedge
should be negative. The valuetof a GLWB contract becomes negative, if, for instance, the underlying fund has
performed well, the policyholder’s fund assets therefore are still substantial and thus
the guarantee fees (as percentage of the fund assets) paid by the policyholder arecomparably high, whereas the present value of the future guarantee payments of the
GLWB rider decreases as ruin is less likely and the guarantee therefore is farther
out-of-the-money.
As the guaranteed withdrawal amount is never increased within product design I(No Ratchet), the scenario of a negative option value – and thus a negative hedge

CHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY

72

portfolio – is quite often seen in the first 25 years after inception of the contract. If
the policyholder decides to surrender the contract in such a scenario, the insurancecompany has to close the hedge position, and thus realizes a loss. Hence, surrenderrisk is high in this case.
For product designs II (Lookback Ratchet) and III (Remaining WBB Ratchet), due
to their ratchet mechanisms that cause the present value of the future guarantee payments of the GLWB rider to decrease less (or even to increase) if the underlying fundhas performed well, the decline in the contract’s value is mitigated, and therefore,
while the hedge portfolio still may become negative, it is less likely to happen thanwith the No Ratchet product design. This effect is more distinctive with productdesign III, since – as seen in chapter 6 – the Remaining WBB Ratchet increases theguaranteed withdrawal amount faster than the Lookback Ratchet design.
In the fourth product design, the Performance Bonus design, the value of the contract
does not vary as much as in the other products, especially the downwards variation islimited, because any surplus is reduced quickly by the performance bonus payments
to the policyholder.
Although the results in the previous section indicate that product design I is theeasiest to hedge (which coincides with the finding of a in comparison rather steadydelta decay, cf. section 6.4), the variation of the value of the hedge portfolio during
the first 25 years of the contract is the highest for this product design, which is, on
the one hand due to the high initial guaranteed withdrawal amount, and on the other
hand due to the missing upside potential of the guarantee. This greater variation
may make the insurer more vulnerable to policyholder behavior.
After the guarantee has triggered, the active hedging ends, and the hedge portfolio’s
assets are completely transfered into the money-market account, where they are compounded at the risk-free rate r and the guarantee payments are deducted annually.
This proceeding is responsible for the steady development of the percentiles in thelater policy years (starting between year 20 to 30, depending on the product design).
In the very end, the value of the hedge portfolio coincides with the final profit/lossof the insurer, i.e. it holds ΠT = ΠHedge.
T
Finally, table 7.2 shows the empirical percentiles of the balance (or cumulative profit/loss) Πt of the insurer. Product designs II and III show almost no differences,apart from the higher upside potential of design II manifesting in a slightly skewed
distribution where the mean is higher than the median. However, product I seems
to have a better combination of upside potential and downside risk, leading to theCHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY(i) No Ratchet(ii) Lookback Ratchet(iii) Remaining WBB Ratchet(iv) Performance Bonus

73

Figure 7.1: For each product design, development over time of the empirical percentiles (dark blue line represents the median, blue lines are 25th - 75th , light bluelines are 10th - 90th percentiles) and mean (red dashed line) of the hedge portfolio
ΠHedgefor a pool of identical policies with a total single premium volume of US$t
100m, using the Black-Scholes model for both, delta hedging and as data-generating
model.
highest mean of the final profit/loss amongst all product designs.
In comparison, product design IV leads to a more symmetrical distribution of the
profit/loss of the insurer, with reduced risk in comparison to product design II and
III, and the lowest upside potential of all products.

CHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY(i) No Ratchet(ii) Lookback Ratchet(iii) Remaining WBB Ratchet(iv) Performance Bonus

74

Figure 7.2: For each product design, development over time of the empirical percentiles (dark blue line represents the median, blue lines are 25th - 75th , light bluelines are 10th - 90th percentiles) and mean (red dashed line) of the insurer’s portfolio
balance Πt for a pool of identical policies with a total single premium volume of US$
100m, using the Black-Scholes model for both, delta hedging and as data-generating
model.

CHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY

7.2.3

75

Unmodified Vega hedge

In this last section of the chapter, we would like to add some comments about vega
hedging. First of all, there is not one equity volatility, but quite a few different typesof equity volatility (e.g. actual volatility, realized/historical volatility and (BlackScholes-Merton) implied volatilities) that all can change in different ways, like forexample in their level, skew, slope, or convexity. On the other side, there is a great
variety of instruments that are liquidly traded in the market or that are commonly
available over-the-counter (OTC) from investment banks that exhibit different andoften complex sensitivities to changes in volatilities. Therefore, a single, uniquevega-hedging strategy that is generally applicable is hard to imagine to exist.
The second comment is in regard to the shortcomings of a somewhat intuitive andstraightforward way of setting up a delta-vega hedge within the Black-Scholes model:
One could simply calculate the vega of the portfolio’s option value Ψt , defined as
the first-order derivate with respect to the Black-Scholes volatility parameter σBS ,and then use the ratio between this vega of Ψt and the vega of hedge instrument,in our setting the vega of Xt , the straddle option’s value, in order to set up a vega
hedge consisting of a position of straddle options.
This strategy, however, results in a rather bad hedge performance (at least within
our analyses, with decade-long cash-flows) due to the following reason: A change
in current equity volatility under the Heston model would mean a change in shortterm volatility and a much smaller change in long-term volatility, due to the meanreversion property of the Heston model. This effect can also be seen in figure 3.2,
where different implied volatility surfaces are shown that are all generated by the
Heston model, but using different start parameters for the local volatility process.
Since volatility in the Black-Scholes model is assumed to be constant over time, andthus also the change in volatility is assumed to be permanent and constant over time,
the change in value of a long-dated option would be significantly overestimated (atleast if the Heston model is considered the “correct” model), and thus lead to a toolarge position in the straddle options.
The resulting hedge portfolio would lead to an increase of risk, because the vega hedge
is off target and may even introduce additional sensitivity to volatility risks into theinsurer’s portfolio, and because the insurer additionally needs to hedge the delta of
the straddle options, which leads to further sources of hedging errors. Therefore,
instead of reducing the insurer’s risk, this additional vega hedge would increase theinsurer’s risk, foiling the very idea of hedging.

CHAPTER 7. ANALYSIS OF HEDGE EFFICIENCY76

To illustrate this effect, we calculated the risk measures introduced in the beginningof this chapter for the unmodified vega hedge, using the Heston model for data
generation. The results are shown in table 7.2.
Product

Strategy(D-BSM)

Statistic

I

II

III

IV



EP e−rT ΠT

0.57
2.77
2.44

0.29
4.76
4.14

0.17
4.51
3.99

0.13
3.35
3.02

0.82
1.75
1.35

0.81
2.41
1.80

0.75
3.01
2.40

0.47
1.88
1.53

CTE0.9 (χ)
CTE0.9 (e−rT ΠT )(DV-BSM)mod. vega(DV-BSM)
unmod. vegaEP e−rT ΠT
CTE0.9 (χ)
CTE0.9 (e−rT ΠT )EP e−rT ΠT
CTE0.9 (χ)
CTE0.9 (e−rT ΠT )10.43 7.77 6.67 3.88
25.29 20.07 17.54 15.12
23.41 18.27 15.90 13.35

Table 7.2: Results under the Heston model as data-generating model for the BSM
delta hedge, the BSM delta-vega hedge using a modified version of the vega forsetting up the position in the straddle options, and the BSM delta-vega hedge using
the unmodified vega.

Chapter 8
Conclusion
8.1

Summary

Our main concern in the present thesis was the analysis of the properties of different types of Guaranteed Living Withdrawal Benefits (GLWB) riders, the latest
guarantee feature within Variable Annuities, from both, a client’s perspective andan insurer’s perspective, in particular in regard to pricing and hedging of this new
type of insurance contract.
In chapter 2, we first gave a high-level description of the considered Guaranteed
Living Withdrawal Benefits rider options and explained their general functionality.
Subsequently, we presented different step-up and ratchet mechanisms in the productdesigns of the GLWB options, that allow for potential (permanent) increases of
the guaranteed withdrawal amount during the lifetime of the contract, given theunderlying fund of the variable annuity performed sufficiently well.
In the last part of chapter 2, we described the general pricing framework that isused for evaluation of the GLWB rider option and for the purpose of finding the“fair” guaranteed withdrawal rate. We also described and discussed the assumptions
that were made regarding the policyholder behavior (the policyholder’s surrender
behavior to be precise) and regarding the mortality of the insured.
In chapter 3, we presented the models of the financial market that we used within
our analyses. The models are used for different purposes: first, as the financial
market model that is assumed to be used by the insurance company for pricing andhedging of the GLWB options, in particular for determing the positions in the hedge
instruments within the insurer’s hedge portfolio. Second, as the (external) datagenerating model in our analyses of the GLWB contracts in chapter 6, as well as in77

CHAPTER 8. CONCLUSION

78

chapter 7, where the efficiency of the hedging strategies is analyzed.
We used two models of the financial market, one with deterministic (and constant)equity volatility, the Black-Scholes-Merton model, and a second one, the Heston
model, in which the instantaneous variance of the underlying asset’s spot price isassumed to evolve according to an one-factor mean-reverting square-root process,similar to the process that is used within the Cox-Ingersoll-Ross model. We further
described how an equivalent martingale measure can be derived for both models,
which then can be used for pricing purposes within the general framework that we
presented in the last section of chapter 2.
In section 3.3 of chapter 3 we described the numerical methods that we used within
our analyses. Besides pricing and calculation of partial derivatives (Greeks) via
Monte-Carlo techniques, we also described the valuation of European standard options under both financial market models that were presented in the previous chapter,in particular the time-efficient valuation via Fourier inversion of European standard
options under the Heston model.
In the last section of chapter 3, we introduced the notion of the Black-Scholes-Merton
Implied Volatility. Afterwards, we presented an illustration of the implied volatility
surface generated by the Heston model, using the parameters that we used for ouranalyses. Additionally, we showed how the shape of the implied volatility surface
generated by the Heston model changes if the starting value of the local variance
process is changed.
As mentioned, we analyzed the hedge efficiency of different hedging strategies that
may be applied by the insurance company in order to reduce the risk that originatesfrom selling GLWB options. The analyzed strategies were presented in chapter 4,
where we described different types of dynamic hedging strategies that differ mainlyin the type and number of the hedge instruments that are used within the respective
strategy, i.e. some strategies use the money-market account and the underlying only,
while others make additional use of standard options on the underlying. For bothfinancial market models, Black-Scholes-Merton and Heston, we specified the hedge
ratios that correspond to the different hedging strategies in table 4.1 and table 4.2.
Chapter 5 deals with the design, architecture and implementation of the software
solution that was used to conduct our analyses. We gave details on how we madeuse of Microsoft Excel as an user interface to our software solution, and on how weseamlessly embedded the functionality of the software solution into Excel by meansof the open-source library xlw. We also specified and described all of the open-source

CHAPTER 8. CONCLUSION

79

libraries that we used within our software solution and concluded the chapter witha detailed usage example of the final software solution.
In chapter 6, we presented the first set of analyses that dealt with the pricing of
the GLWB options and the distinctions in the characteristics of the product designs
that originate from the ratchet and bonus features that are implemented in therespective product types. We found that different ratchet and bonus features canlead to significantly different cash-flows to the insured. Similarly, the probability
that, at some point in time, the insurer has to compensate for guaranteed payments,
the amount of the guaranteed payments, and the distribution of the point in time
when the guarantee of the GLWB option triggers differ significantly for the different
product designs, even if they all come at the same guarantee fee. We also found thatthe development of the Greeks - that is the sensitivities of the GLWB option value
with respect to changes in certain market parameters - over time is also significantly
different, depending on the selected product features.
In the subsequent chapter 7 we analyzed different (dynamic) hedging strategies (noactive hedging, delta only, delta and vega) and analyzed the distribution of theinsurer’s cumulative profit/loss and certain risk measures hereof. We found thatthe insurer’s risk can be significantly reduced by suitable hedging strategies, butwith considerable differences between the considered hedging strategies. However,
we also found that – which is in line with our findings in chapter 6 – the productdesign has a significant impact on how risky a product is to the insurer and on how
well it is hedgeable. Thus both, the constitution of a hedging portfolio (following acertain hedging strategy) and the insurer’s risk after hedging, differ significantly forthe analyzed products.
We then quantified the model risk that is inherent in the use of the considered hedging
strategies by using different models of the financial market for data generation andcalculation of the hedge positions. The results hereof can be used as an indication
for the model risk that an insurer takes who utilizes a certain model for hedging,whilst in the real world, financial markets behave differently. Within this analysis,
we focused on the risk an insurer takes by assuming a constant equity volatility in thehedging model whereas in the real world equity volatility is stochastic, and showedthat this risk can be substantial.
With the last analysis in chapter 7, we were able to show that, while – using theBlack-Scholes-Merton model for hedging – a delta-vega hedging strategy based ona modified version of vega can lead to a significant reduction of volatility risk, eventhough the model assumes deterministic volatility, a somewhat more intuitive andCHAPTER 8. CONCLUSION

80

straightforward attempt to hedge against volatility risk based on the unmodified vega
may lead to results inferior to the case with no vega hedging at all, which would beclearly not the intention of implementing a vega hedge in the first place.
Our results - in particular with respect to model risk - should be of interest to both,
insurers and regulators, in particular in regard to the neglected model risk that ispresent if the analysis of the hedge efficiency of a certain strategy is done under thesame data-generating model that is used by the insurer as hedging model.

8.2

Outlook and future research

Further research could aim at extending our findings in several areas, starting withthe product design of the GLWB riders: For instance, we could add designs to ouranalyses that include a deferment period with a guaranteed compounding of the withdrawal benefit base during that period, or we could extend our analyses to productdesigns that allow for partial surrender of the policyholder or a withdrawal amount
below the guaranteed withdrawal amount.
Also, different forms of defining and deducting the guarantee fees would be an interesting feature to examine, e.g. guarantee fees that are not fixed as a percentage ofthe policyholder’s current fund assets, but as a percentage of the current withdrawal
benefit base, which would result in much more foreseeable guarantee payments received by the insurer.
Furthermore, the GLWB riders considered in this thesis all expect an initial single
premium payment of the policyholder. Therefore it would be interesting how GLWB
riders that allow for regular premium payments compare against these products interms of riskiness to the insurer.
Similarly, the models of the financial market that we used in this work could be
extended in various ways, for instance by incorporating jumps in the dynamics of
the underlying – since jumps can be expected to considerably influence the performance of the considered hedging strategies. The jumps would be added on top of
the modeled stochastic equity volatility, as it is the case e.g. in the Bates (1996,[4]) model, where the Heston stochastic volatility model and the Merton (1976, [23])jump-diffusion model were combined. Also, adding stochastic interest rates to theexisting model seems important, especially if product designs are considered thatallow for regular premiums.
Likewise, different approaches to how the stochasticity of the actual and/or the im-

CHAPTER 8. CONCLUSION

81

plied equity volatility are modeled are thinkable. For instance, using the SABR
model (Hagan, 2002, [15]) for data-generating, while hedging is done under the Heston model, or even considering separate models for implied and actual volatility.
In addition, a systematic analysis of parameter risk appears worthwhile, with respect to the financial market parameters as well as to mortality probabilities anddeterministic surrender rates.
In the present work policyholder behavior is modeled deterministic (probabilistic),
and the policyholder’s decision is furthermore restricted to just two options: fullsurrender or withdrawal of the guaranteed withdrawal amount. Therefore, it is certainly of interest how the results of our analyses change if the policyholder behavior
is modeled differently: First, under a framework that allows for arbitrary withdrawal
amounts between nil and all of the remaining fund assets, and, second, under optimal
and/or stochastic policyholder behavior.
For instance, we could follow Le Courtois and Nakagawa (2009, [22]), who used an
approach for the modeling of stochastic policyholder behavior via a mixed Poissonprocess (also known as doubly stochastic Poisson process or Cox process) with astochastic intensity process that is correlated to the underlying asset’s spot price.
This approach models stochastic surrender rates that are influenced by market movements and that allows for the modeling of external effects like e.g. higher surrender
rates due to a deteriorated reputation of the insurance company. However, within
this approach, the calibration of the Poisson process could constitute a problem.
Further, an approach where the moneyness or the value of the GLWB option’s value
directly affects the policyholder’s likelihood to surrender appears useful.
Regarding the computation of the optimal policyholder behavior, an approach similar to that used in Bacinello, Biffis, and Millossovich (2009, [3]), which is based on
the Least-Squares Monte-Carlo method, seems promising.
In general, an analysis of the robustness of the hedging strategies against policyholder
behavior appears worthwhile.
Also, within the scope of future research work, there is much room for refinement regarding the numerical methods used in our analyses: We could aim at improving thequality of the calculated Greeks, as well as decreasing the computational effort thatis necessary to calculate them. Possible starting points would be the use of different
techniques for valuating the GLWB rider options, and more efficient schemes forthe numerical calculation of the partial derivatives, like, for instance, the Likelihood
Ratio method (cf. Glasserman, 2003, [14]).
Hedging by product design: Likewise, it would be interesting to analyze how theCHAPTER 8. CONCLUSION

82

insurer can reduce risk by means of product design, e.g. by offering investments funds
within their products that are managed to keep the volatility of the fund’s returns
constant at some prespecified volatility target (so-called “VolTarget”1 funds), or byreserving the right to switch the policyholder’s assets to less risky funds (e.g. fundswith an higher portion in bonds or the money market), should market volatilities
increase. Ideally, such product designs would allow for pricing and hedging using amodel that assumes only constant equity volatility.

1

At the time of writing this thesis, such products were for instance offered by Société Générale,
like e.g. the “SGI Vol Target BRIC” index.

Bibliography[1] Andersen, L. (2007), Efficient Simulation of the Heston Stochastic Volatility
Model, Working paper, Banc of America Securities.
[2] Bauer, D., Kling, A., and Russ, J. (2008), A Universal Pricing Framework for
Guaranteed Minimum Benefits in Variable Annuities. ASTIN Bulletin, Volume:38 (2), 621–651, November 2008.
[3] Bacinello, A. R., Biffis, E., and Millossovich, P. (2009), Regression-Based Algorithms for Life Insurance Contracts with Surrender Guarantees, to appear in
Quantitative Finance, version as of April 7, 2009.
[4] Bates, D. (1996), Jump and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options, Review of Financial Studies 9: 69–107.
[5] Bingham, N. H., Kiesel, R. (2004), Risk-Neutral Valuation: Pricing and Hedging
of Financial Derivatives, Springer Verlag, Berlin.
[6] Black, F. and Scholes, M. (1973), The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81, 1973, 637–654.
[7] Coleman, T.F., Kim, Y., Li, Y., and Patron, M. (2005), Hedging Guarantees in
Variable Annuities Under Both Equity and Interest Rate Risks, Cornell University, New York.
[8] Coleman, T.F., Kim, Y., Li, Y., and Patron, M. (2007), Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks, The Journalof Risk and Insurance, 2007, Vol. 74, No.2, 347–376.
[9] Cont, R., and Da Fonseca, J. (2002), Dynamics Of Implied Volatility Surfaces,
Quantitative Finance, Vol. 2, No. 1, 2002.
[10] Cox, J.C., Ingersoll, J.E., and Ross, S.A. (1985), A Theory of the Term Structureof Interest Rates. Econometrica 53, 385-407.
83

BIBLIOGRAPHY

84[11] Eraker, B. (2004), Do stock prices and volatility jump? Reconciling evidence
from spot and option prices, Journal of Finance, Volume 59, Number 3, pp.
1367–1404, June 2004.
[12] Ewald, C.-O., Poulsen, R., Schenk-Hoppe, K. R. (2007), Risk Minimizationin Stochastic Volatility Models: Model Risk and Empirical Performance, Swiss
Finance Institute Research Paper Series N07 - 10.
[13] Gander, W., and Gautschi, W. (2000), Adaptive Quadrature - Revisited. BIT,
40(1): 84–101, March 2000.
[14] Glasserman, P. (2003), Monte Carlo Methods in Financial Engineering,
Springer, Berlin.
[15] Hagan, P., Kumar, D., Lesniewski, A., and Woodward, D. (2002), Managing
smile risk, Wilmott Magazine, September, 84-108, 2002.
[16] Heston, S. (1993), A closed form solution for options with stochastic volatility
with applications to bond and currency options, Review of Financial Studies 6,
no. 2, 327-344.
[17] Holz, D., Kling, A., and Ru, J. (2007), GMWB For Life An Analysis of Lifelong
Withdrawal Guarantees. Working Paper, Ulm University, 2007.
[18] Hull, J.C. (2008), Options, Futures and Other Derivatives, Prentice Hall.
[19] Joshi, M. S. (2003), The Concepts and Practice of Mathematical Finance, Cambridge University Press.
[20] Joshi, M. S. (2004), C++ Design Patterns and Derivatives Pricing, Cambridge
University Press.
[21] Kahl, C. and Jckel, P. (2005), Not-so-complex logarithms in the Heston model,
Wilmott Magazine, 94-103, September 2005.
[22] Le Courtois, O., and Nakagawa, H. (2009), Surrender Risk and the Default of
Insurance Companies, working paper, EM Lyon Business School, France, andHitotsubashi University, Japan.
[23] Merton, R. (1976), Option pricing when underlying stock return are discontinuous, Journal of Financial Economics 3: 125–144.

BIBLIOGRAPHY

85[24] Mikhailov, S. and Ngel, U. (2003), Hestons Stochastic Volatility Model Implementation,Calibration and Some Extensions, Wilmott Magazine (2003).
[25] Lord, R., Koekkoek, R., and van Dijk, D. (2008), A Comparison of biased
simulation schemes for stochastic volatility models, Working Paper, Tinbergen
Institute.
[26] Milevsky, M., and Posner, S.E. (2001), The Titanic Option: Valuation of the
Guaranteed Minimum Death Benefit in Variable Annuities and Mutual Funds.
The Journal of Risk and Insurance, Vol. 68, No. 1, 91-126, 2001.
[27] Milevsky, M. and Salisbury, T.S. (2006), Financial Valuation of Guaranteed
Minimum Withdrawal Benefits. Insurance: Mathematics and Economics: 38,
21-38, 2006.
[28] Taleb, N. (1997), Dynamic Hedging: Managing Vanilla and Exotic Options,Wiley.
[29] Van Haastrecht, A., Pelsser, A. (2008), Efficient, Almost Exact Simulation of the
Heston Stochastic Volatility Model, Working paper, Delta Lloyd and Universityof Amsterdam.
[30] Wilmott, P. (2006), Paul Wilmott on Quantitative Finance, Wiley & Sons.
[31] Wong, B., Heyde, C. C. (2006), On changes of measure in stochastic volatility
models, Journal of Applied Mathematics and Stochastic Analysis, Volume 2006,
Article ID 18130, 1-13.

Ehrenwörtliche Erklärung

Ich erkläre hiermit ehrenwörtlich, dass ich die vorliegende Arbeit selbstständig angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken
sind als solche kenntlich gemacht. Die Arbeit wurde bisher keiner anderen Prüfungsbehörde vorgelegt und auch noch nicht veröffentlicht.

Ich bin mir bewusst, dass eine unwahre Erklärung rechtliche Folgen haben wird.

Ulm, den 22. September 2009

(Unterschrift)