bluuebunny/crossref_metadata_2025 huggingface dataset:
Structural Default Risk Models
Authors: Kiesel Rüdiger , Scherer Matthias A.
Published: May 2010

Structural‐default models rely on the appealing interpretation of corporate default as a consequence of insufficient asset values to cover contractual obligations of the corporate. Multivariate extensions typically assume correlated firm‐value processes to introduce dependence between individual default events. On a theoretical level, the resulting default correlation and portfolio‐loss distribution depend on the choice of the (multivariate) asset‐value process and the definition of the default event. In most modeling frameworks, the explicit computation of default correlations and the portfolio‐loss distribution is not possible. Therefore, balancing realistic assumptions with computational tractability is a key challenge for all multivariate structural‐default models. Especially models for large portfolios, which are required for the pricing of portfolio derivatives and risk management, use simplifying assumptions concerning the firms involved, numerical approximations, and Monte Carlo simulations for the computation of the relevant quantities. An important example is Vasicek's asymptotic single‐factor model, which under several simplifying assumptions allows the derivation of a closed‐form expression for the portfolio‐loss distribution. This formula is used to compute regulatory capital for loan portfolios under the Basel II framework.
Portfolio Credit Risk: Statistical Methods
Authors: Rösch Daniel
Published: May 2010

The article gives an overview over method‐of‐moment and maximum‐likelihood approaches for joint estimation of default probabilities and correlations in credit portfolios from time series of default data.
Internal‐Ratings‐Based Approach
Authors: Pluto Katja , Tasche Dirk
Published: May 2010

Within the new Basel capital rules for banks (Basel II), the internal‐ratings‐based approach (IRBA) represents perhaps the most important innovation for regulatory minimum capital requirements. For the first time, subject to supervisory approval, banks are allowed to use their own risk assessments of credit exposures in order to determine the capital to be held against them. Within the IRBA, banks estimate the riskiness of each exposure on a stand‐alone basis. The risk estimates serve as input for a supervisory credit risk model (implicitly given by risk weight functions) that provides a value for capital that is deemed sufficient to cover against the credit risk of the exposure, given the assumed portfolio diversification. In order to obtain supervisory approval for the IRBA, banks must apply for IRBA and fulfill a set of minimum requirements aimed at ensuring the integrity of the rating model, rating process, and thus of the risk parameters and capital charges. Although risk quantification
via
IRBA can be of great use for the bank internal credit risk measurement and management, there are some limitations. The most important of these is surely that the IRBA provides no measure of risk concentrations. If banks are concerned about concentration effects or want to measure the diversification benefits of their portfolio, they need to develop their own, full‐fledged credit risk models.
Loan Valuation
Authors: Benzschawel Terry , Dagraca Julio , Fok Henry
Published: May 2010

We describe the pricing of corporate term loans and revolving credit lines within a risk‐neutral framework. The method accounts for credit‐ and market‐related term loan prepayments and for drawdown and repayment of revolving credit lines. Loans are priced using backward induction on a risk‐neutral credit‐transition lattice, testing for the prepayment criterion and default at each node. The risk‐neutral credit transitions are determined from historical data and modified to be consistent with market prices of credit default swaps. The model provides estimates of the average lives of loans and loan price and prepayments sensitivities to relevant market movements.
Credit Default Swaps
Authors: Cont Rama
Published: May 2010

A credit default swap (CDS) is a bilateral contract in which a protection seller compensates a protection buyer for the loss in the value of a reference security in the event of default of a reference entity. CDSs are sometimes viewed as insurance against default, but have important difference with insurance contracts. We describe the cash‐flow structure of a credit default swap, the principle behind the valuation of CDS positions, and the risk factors affecting a CDS.
Constant Maturity Credit Default Swap
Authors: Misirpashaev Timur S.
Published: May 2010

Constant maturity credit default swap (CM CDS) is a credit derivative with payments linked to periodic fixings of a standard CDS rate with a fixed tenor on a particular credit entity. We give references to sources describing trading conventions and strategies and summarize approaches to quantitative modeling of CM CDS, including the nonstochastic approximation, convexity adjustments, instantaneous hazard rate models, and stochastic forward spread models.
Credit Default Swap (
CDS
) Indices
Authors: Couderc Fabien , Finger Christopher C.
Published: May 2010

Credit default swap indices provide exposure to what is essentially a portfolio of credit default swaps, but in the form of a single standard contract. As a single contract, the index has some characteristics that distinguish it from standard CDS.
Basket Default Swaps
Authors: Overbeck Ludger
Published: May 2010

Basket credit derivatives are financial instruments whose payout depends on the credit behavior of a small number of credits. In this sense, they are between single‐name credit default swaps (CDS) and collateralized debt obligations (CDO)‐like structures. The standard products are
n
th‐to‐default basket in which the contingent payment is made when the
n
th default in a basket of at least
n
credits happens. Among these, first‐ or second‐to‐default swaps (FtD and StD) on a portfolio of three to five credits are most popular. As for CDOs, the benchmark valuation models are based on a copula for default times. However, some approaches based on spread models have also been developed.
Collateralized Debt Obligations (CDO)
Authors: Bruyere Richard , Jaeck Christophe
Published: May 2010

In the limited timeframe of a few years (2001–2007), CDOs and structured credit arguably became the hottest areas in capital markets and a crucible of financial innovation, until the 2007 subprime crisis marked the (temporary?) end of the party. This article provides a factual analysis of these developments starting from a typology of CDOs and a review of modeling techniques to an analysis of the impact of the subprime crisis on the CDO market.
CDO Square
Authors: Brasch Hans‐Jürgen
Published: May 2010

A CDO‐of‐CDO, or CDO square (CDO
2
), is a type of collateralized debt obligation (CDO) that has CDO tranches as reference assets. The CDO
2
market has been used for rating and capital arbitrage, repackaging, and diversification of loan and debt portfolios.
Credit Default Swap Index Options
Authors: Pedersen Claus M.
Published: May 2010

The basic conventions and terminology for portfolio swaptions are outlined and the standard model used by most market participants is explained. There is a brief discussion of other models and approaches.
Hazard Rate
Authors: Kim June Ho
Published: May 2010

The concept of hazard rates is introduced in the context of reduced‐form credit risk modeling. Starting from an intuitive overview, a concise and rigorous treatment of this essential concept is provided, with a focus on the underlying information filtration. An application to the pricing of credit default swaps (CDS) is given.
Duffie–Singleton Model
Authors: Schlögl Erik , Schlögl Lutz
Published: May 2010

The Duffie–Singleton model is a reduced‐form credit risk model in which instantaneous credit spreads can be identified with a “risk‐neutral mean‐loss rate due to default.” The model is driven by a set of state variables following a Markov process, and defaultable zero‐coupon bond prices are exponentially affine functions of these variables. A key assumption is the modeling of recovery in default as an exogenously given fraction of the market value of the defaultable claim. Unlike other exponentially affine model specifications, the one proposed by Duffie and Singleton is sufficiently flexible to allow default intensities (and thus credit spreads) to be negatively correlated with default‐free interest rates. The model is very tractable computationally and lends itself to econometric estimation or, alternatively, to calibration to observed market prices for purposes of relative valuation of credit derivatives.
Jarrow–Lando–Turnbull Model
Authors: Zagst Rudi , Scherer Matthias A.
Published: May 2010

The credit‐risk model of Jarrow, Lando, and Turnbull identifies the evolution of a firm's credit rating over time with some Markov chain. On the basis of this appealing economic interpretation, it is possible to valuate defaultable bonds and credit derivatives. The resulting prices explicitly depend on the initial rating and possible rating transitions of the reference firm in the future. The required martingale probabilities are obtained from empirical transition probabilities, which are adjusted by some risk premium.
Default Barrier Models
Authors: Gregory Jon
Published: May 2010

This article outlines the use of default barrier models in both single name and portfolio credit risk modeling. The economic and mathematic rationale for the modeling of default as the hitting time of a barrier is first described, followed by a description of the birth of such approaches from the classic Merton framework for pricing risky debt. The practical extensions of the Merton framework that have been used for default probability modeling and trading strategies such as capital structure arbitrage are considered. Finally the use of default barrier models in credit portfolio risk applications that require a description of credit portfolio losses is described.
Multiname Reduced Form Models
Authors: Lipton Alexander
Published: May 2010

In this article, we discuss single and multiname random‐intensity models. After providing a brief overview of single‐name models, we describe in some detail three popular multiname models, namely, the Duffie–Garleanu–Mortensen, Chapovsky–Rennie–Tavares, and Inglis–Lipton models. The main conclusion is that these models describe the market well, but, since they require large jumps in intensity, further work is needed if they are to be used for hedging purposes.
Gaussian Copula Model
Authors: Xianglin Li David
Published: May 2010

We provide a summary of the copula function approach to credit portfolio modeling as introduced by Li in 2000. We discuss the Gaussian copula model, the common specification of correlation matrix, its implementation and also the general shortcomings of the current copula function framework. Risk measures and hedging issues are also discussed.
Base Correlation
Authors: Xianglin Li David
Published: May 2010

Base correlations are default correlation parameters implied from market spreads of collateralized debt obligation tranches. They are analogous to the notion of implied volatility used for equity options. We also present three commonly used mapping methods from the base correlation curve for the standard index tranches to the base correlation curve for bespoke portfolios.
Random Factor Loading Model (for Portfolio Credit)
Authors: Andersen Leif B.G.
Published: May 2010

We provide an overview of various flavors of the random factor loading (RFL) framework for credit portfolio loss modeling.
Local Correlation Model
Authors: Turc Julien , Herzog Benjamin
Published: May 2010

The local correlation model is a credit portfolio loss model that generalizes the Gaussian copula model and allows to account for the base correlation smile. The constant correlation assumption is initially relaxed within a large pool framework, which provides theoretical results used as a basis for the general model. The large pool assumption is then relaxed in order to take the portfolio's specificities into account.
Intensity Gamma Model
Authors: Joshi Mark , Stacey Alan
Published: May 2010

We review the intensity gamma model for pricing portfolio credit derivatives. This is a model for correlation of credit defaults based on a financially intuitive concept of business time similar to that in the variance gamma model for stock price evolution. Solving a simple equation calibrates each name to its credit spread curve. We show that the overall model can be calibrated to the market base correlation curve of a tranched collateralized debt obligation (CDO) index. Once this calibration is performed, obtaining consistent arbitrage‐free prices for nonstandard tranches, products based on different underlying names and even more exotic products such as CDO‐squared, is straightforward and rapid.
Special‐Purpose Vehicle (
SPV
)
Authors: Polizu Cristina , Jiang Jennifer
Published: May 2010

A special‐purpose vehicle (SPV) is a legal entity created by a firm for a specific business objective, usually in the context of securitization. SPVs are often used to finance a project or issue securities without putting at risk the original holder of the securitized portfolio. Derivative‐product companies (DPCs), structured‐investment vehicles (SIVs), and credit derivative‐product companies (CDPCs) are SPVs that operate in primarily one type of business: interest rate and foreign exchange derivatives, credit derivatives, and buy and hold assets or others. They might be exposed to credit risk, market risk, operational risk, and so on. Capital adequacy models are built to quantify the risks they are exposed to.
Structured Finance Rating Methodologies
Authors: Kocagil Ahmet E. , Papatheodorou Vasileios
Published: May 2010

Securitization is a mechanism by which a portfolio of financial assets such as loans, receivables, corporate debt, and other usually illiquid assets is used as a collateral for collateralized debt obligation (CDOs) or other structured finance (ABS) securities issued, typically by a special purpose vehicle (SPV), in the market. The rating agencies evaluate the different tranches of the issued securities and issue ratings in order to provide an independent opinion on the credit quality of these securities. The aim of this article is to review the ABS and CDO rating methodologies applied in the securitization market.
CDO Tranches: Impact on Economic Capital
Authors: Lee Yim T.
Published: May 2010

Collateralization debt obligations (CDOs) are financial securities backed by a pool of assets. Claims on the underlying assets of a CDO are divided into several tranches from the riskiest to the least risky. Many financial institutions including banks are holding CDO tranches of different transactions in their credit portfolios as a result of their recent activities in CDO investing and structuring. Modeling the impact of CDO tranches on the economic capital of credit portfolios is thus becoming increasingly important. This article presents the methodology to measure the impact of CDO tranches on the economic capital and capital allocation of credit portfolios. The methodology is illustrated by presenting an example of calculating capital allocation for a synthetic CDO tranche in a credit portfolio comprised of a tranche and other loans. It is demonstrated that adding a CDO tranche to a credit portfolio increases the systematic risk of the portfolio and, consequently, a CDO tranche could require much larger economic capital compared to a loan of similar credit quality and same maturity. An important implication of our finding is that by treating a CDO tranche as a loan equivalent, only a very poor approximation of economic capital is obtained.
Eurodollar Futures and Options
Authors: Yu Yuhua
Published: May 2010

Eurodollar futures and the options traded on those futures are among the most widely traded interest‐rate instruments worldwide. The futures contracts are based on the three‐month interest rates earned on US dollar deposits that are held in banks located outside the United States. This article explains the contract specifications and quoting conventions of the Eurodollar products listed at the Chicago Mercantile Exchange (CME). Further, it discusses the pricing methods of these contracts.
Bond Options
Authors: Piza Marcelo
Published: May 2010

This article describes various types of options on bonds and presents the main approaches used for their modeling and valuation. We start with a brief description of bond option contracts and the closely related options on bond futures as well as options embedded in the bond contract itself. We then present some standard approaches for valuation restricting ourselves to options on default‐free bonds with a focus on European bond options.

We start by describing the original approaches based on the Black–Scholes–Merton model where the quantity that is directly modeled is the bond's price or yield and discuss some of the inconsistencies related to this approach. This is followed by a presentation of the Black 1976 model. We also establish a relationship between the valuation of bond options and its relation to the standard valuation approach for vanilla interest rate derivatives such as caps, floors, and swaptions. We then turn to the valuation of bond options based on a full‐term structure model focusing on the special case of short term models. We present the Jamishidian decomposition, which allows for an explicit determination of the exercise boundary for European options on coupon bonds for a class of one‐factor short‐rate models satisfying a monotonicity condition. We conclude with a brief description of the approaches used for the pricing of Bermudan and American bond options.
Constant Maturity Swap
Authors: Veilex Laurent
Published: May 2010

This article describes interest rate derivatives that are simple extensions of vanilla interest rate swaps: in‐arrears swaps, averaging swaps, and constant maturity swaps (CMS). Unlike vanilla swaps, these securities are sensitive to volatilityviaso‐called convexity adjustments. We demonstrate two methods for calculating convexity adjustments: the standard Black– Scholes framework and the replication method.
B
ermudan Swaptions and Callable Libor Exotics
Authors: Andersen Leif B.G. , Piterbarg Vladimir V.
Published: May 2010

In this article we give an overview of callable interest rate products, a product class that includes Bermudan swaptions as well as callable Libor exotics. The main contractual features of these securities are discussed and an outline of valuation algorithms is provided.
Trigger Swaps
Authors: Pietersz Raoul
Published: May 2010

We discuss two classes of exotic interest rate derivatives: trigger swaps and targeted accrual redemption notes (TARNs). Subsequently, we outline possible valuation methodologies and assess the impact on the quality of risk estimates.
CMS
Spread Products
Authors: Galluccio Stefano , Scaillet Olivier
Published: May 2010

We discuss the pricing of constant maturity swap (CMS) spread options, first without taking into account the presence of volatility smiles and then after their inclusion. The notions of implied correlation and implied normal spread volatility are discussed.
Yield Curve Construction
Authors: Andersen Leif B.G. , Piterbarg Vladimir V.
Published: May 2010

Methods for construction of a continuous discount curve from a finite set of bonds, swaps, and forward rate agreements (FRAs) are discussed.
Term Structure Models
Authors: Vasicek Oldrich Alfons
Published: May 2010

Term structure models describe the behavior of interest rates of different maturities as a joint stochastic process. These models are a necessary tool for valuation and risk management of interest‐rate‐dependent securities.

A general theory of one‐factor term structure models was given by Vasicek. Invoking a no‐arbitrage argument, it was shown that the bond price satisfies a partial differential equation. Its solution, the bond pricing equation, is given as an expectation of the bond maturity value discounted at the short rate and adjusted for the price of risk.

A number of specific models have been developed. Vasicek gave an example of a term structure model in which the short rate follows a mean reverting random walk. Cox, Ingersoll, and Ross derived a model in which the volatility of the short rate is proportional to the square root of the short rate. Hull and White extended these two models by allowing the parameters to be time dependent, so that the model can be made consistent with initial data.

In one‐factor models, prices of interest‐rate‐contingent claims satisfy the same partial differential equation as bonds, with boundary conditions given by the claim payouts. More generally, derivatives can be valued by a change of probability measure.
Gaussian Interest‐Rate Models
Authors: Antonov Alexandre V.
Published: May 2010

We describe multifactor Gaussian interest‐rate models: their general features, key formulas, analytical caplet and swaption pricing, and numerical methods.
Affine Models
Authors: Cuchiero Christa , Teichmann Josef , Filipovic Damir
Published: May 2010

Affine term structure models have gained significant attention in the finance literature, mainly due to their analytical tractability and statistical flexibility. The aim of this article is to present both theoretical foundations and empirical aspects of the affine model class. Starting from the original one‐factor short‐rate models of Vasiček and Cox
et al.
, we provide an overview of the properties of regular affine processes and explain their relationship to affine term structure models. Methods for securities pricing and for parameter estimation are also discussed, demonstrating how the analytical tractability of affine models can be exploited for practical purposes.
Swap Market Models
Authors: Galluccio Stefano , Scaillet Olivier
Published: May 2010

We describe the basics and fundamentals of swap market models, that is, the continuous time modeling of a set of forward swap rates on a tenor structure under different probability measures. We review the three main classes known as the
co‐terminal, co‐initial
, and
co‐sliding models
, and describe their use in interest‐rate derivative pricing. Finally, we discuss numerical implementation
via
model calibration and approximation of pricing formulae.
Markov Functional Models
Authors: Kennedy Joanne
Published: May 2010

Markov‐functional models can fit the observed prices of liquid instruments in a similar fashion to market models, but also have the advantage that they can be implemented as efficiently as spot‐rate models. The defining property of Markov‐functional models is that pure discount bond prices are, at any time, a function of some low‐dimensional process that is Markovian in some martingale measure and this is the key to their efficient implementation.

The driving process (or in dimension
k
 > 1, a prior model based upon it) is chosen to capture the basic dynamics of the options market. The functional forms are derived numerically from market prices and the martingale properties necessary to make the model arbitrage‐free. We show how the LIBOR form of the model can be specified using the numeraire approach and the market prices of caplets. This model is qualitatively very similar to the LIBOR market model. The approach is particularly suited to formulating models appropriate for pricing callable LIBOR products such as Bermudan swaptions.
Hedging of Interest Rate Derivatives
Authors: Hyer Tom
Published: May 2010

Interest rate derivatives are claims based on the forward rate curve, so their hedging is best understood by studying their relation to that curve. Thus, interest rate risk is tantamount to forward rate risk; we make this definition precise and display the risk of an example swap. We explain curve build methods as a choice of mapping from observable market prices to forward rates, and define their
response functions
, which determine the instrument hedges computed with a given curve build method. We discuss real‐world methods for approximating second‐order risks. The language of response functions can be extended to volatilities in the Heath– Jarrow– Morton (HJM) picture; using this we analyze vega‐hedging methods. Finally, we discuss the complications introduced by credit and counterparty issues.
Inflation Derivatives
Authors: Jäckel Peter , Bonneton Jerome
Published: May 2010

We give an outline of the inflation market, the most common products, and the most frequently used models.
Swaps
Authors: Traven Sergei
Published: May 2010

The concept of interest rate swap is described. We discuss plain vanilla swaps, basis swaps, and cross‐currency swaps. Valuation formulas for those contracts are given.
Finite Difference Methods for Barrier Options
Authors: Randall Curt
Published: May 2010

This article describes the numerical solution of partial differential equations (PDEs) in the context of computational finance for the pricing of barrier options. Barrier options cease to exist (knockout barrier options) or only come into existence (knock‐in barrier options) when some observable market parameter crosses a designated level (the barrier). They have become ubiquitous in financial contracts for virtually all asset classes.

A brief discussion of continuously monitored barrier option pricing using PDEs describes the use of coordinate transformations to obtain smooth, rapid convergence, highly desirable properties that are difficult to obtain using traditional lattice methods.

In the majority of barrier option contracts, barrier monitoring frequency is at most daily, for example, at daily market close. For these options, pricing as if continuously monitored is almost always a gross approximation. Gradients of option value with respect to the underlying can become very strong near the barrier(s). The majority of the article is devoted to a more detailed discussion of PDE methods for the pricing of discretely monitored barrier options, including customization of the finite difference grid and time discretizations. Numerical examples illustrate how easy‐to‐implement grid structuring techniques can yield orders of magnitude reduction in finite difference truncation error and smooth, monotonic convergence.
Finite Difference Methods for Early Exercise Options
Authors: Toivanen Jari
Published: May 2010

Usually, options with early exercise possibility like American and Bermudan options have to be priced numerically. In this article, approaches based on partial differential operators and their finite difference discretization are considered. The early exercise possibility leads to an inequality constraint for the price of the option. Linear complementarity problem (LCP) and other formulations incorporating this constraint are discussed. Fairly standard finite difference discretizations can be used for the underlying partial differential operator. Several solution and approximation methods including the projected successive over relaxation (PSOR) method and penalty methods are described. As an example, an American put option is priced using different methods under the Black–Scholes model.
Partial Differential Equations
Authors: Pham Huyên
Published: May 2010

This survey article outlines the various contexts where partial differential equations (PDEs) arise in finance. We discuss, in particular, option pricing, portfolio optimization, and calibration from the PDE viewpoint, and indicate further issues for PDE in finance.
Finite Element Methods
Authors: Achdou Yves , Pironneau Olivier
Published: May 2010

The finite element method is well suited to the numerical solution of the partial differential equations arising in finance because they allow for
a posteriori
error estimates and mesh adaptivity. The method is described on three simple examples and its advantages are emphasized.
Fourier Transform
Authors: Černý AleŠ
Published: May 2010

This article is a concise introduction to applications of Fourier transform and fast Fourier transform (FFT) in option pricing. The first section defines the discrete Fourier transform (DFT) and states its most important properties. The second section explains how the binomial asset pricing model can be implemented by means of a circular convolution and how circular convolution can in turn be computed using three DFTs. This algorithm is generalized to the multinomial model. The third section discusses fast implementation of DFT and in particular it analyzes how the length of the input vector can (adversely) affect the speed of computation. We also discuss continuous Fourier transform and option pricing in continuous‐time affine models. We show how continuous‐time pricing formulae can be efficiently implemented for multiple strike values using FFT. Additional references and suggestions for further reading are provided.
Monotone Schemes
Authors: Jakobsen Espen R.
Published: May 2010

We discuss viscosity solutions for boundary value problems for linear and nonlinear partial differential equations (PDEs) and their relation to monotone numerical schemes for such problems. The main results that we present are a convergence theory for general problems and error estimates for convex ones.
Integral Equation Methods for Free Boundaries
Authors: Chadam John
Published: May 2010

We outline a unified approach for treating free boundary problems arising in finance using integral equation methods. Starting with the partial differential equations (PDE) formulations of the free boundary problems, we show how to derive nonlinear integral equations for the free boundaries in a variety of finance applications. Methods to treat theoretical (existence, uniqueness) questions and analytical and numerical approximations are sketched in this integral equation context.
Tikhonov Regularization
Authors: Crépey Stéphane
Published: May 2010

An important issue in quantitative finance is
model calibration
. The calibration problem is the
inverse
of the pricing problem. Instead of computing prices in a model with given values for its parameters, one wishes to compute the values of the model parameters that are consistent with observed prices. Now, it is well known to physicists that such inverse problems are typically
ill posed
. So, if one perturbs the data (e.g., if the observed prices change by some small amount between today and tomorrow it is quite typical that a numerically determined best fit solution of the calibration problem switches from one “basin of attraction” to the other; thus the numerically determined solution is
unstable
. To achieve a robust calibration, we need to introduce some
regularization
. The most widely known and applicable regularization method is
Tikhonov(–Phillips)
regularization method. In this paper we provide a survey of the Tikhonov regularization and illustrate it by applying the regularization to the problem of calibrating a local volatility model.
Quadrature Methods
Authors: Newton David P.
Published: May 2010

Quadrature applied in Finance is a relatively new technique. By its nature, it is extremely fast to converge and competes across the range of option types, even with multiple dimensions, heavy path‐dependency and early exercise.
Fourier Methods in Options Pricing
Authors: Lord Roger
Published: May 2010

We provide a short introduction to Fourier option pricing methods, for both European and Bermudan options. The first part considers the pricing of European options using both Lévy inversion methods and the Carr–Madan approach. The second part considers the application of these techniques to the pricing of Bermudan and early exercise options. The key element here is to recognize that the continuation value of a Bermudan option is the convolution of the transition density with the option value at the next exercise date.
Quantization Methods
Authors: Pagès Gilles
Published: May 2010

In this paper, we present an overview of the recent developments in the applications of optimal quantization to solve nonlinear problems arising in finance like the pricing and hedging of American‐style options, portfolio management, nonlinear filtering, and stochastic partial differential equations (SPDEs). It is is a spatial discretization based on the nearest neighbor projection on a finite codebook “optimally fitted” to a random vector
X
.
Monte Carlo Greeks
Authors: Curran Michael
Published: May 2010

This article provides a brief overview of methods for the estimation of hedge parameters (delta, gamma, vega, etc.) for financial derivatives within a Monte Carlo framework. Improvements on the basic resimulation method that are discussed include the pathwise method and the likelihood ratio method. We review the form these computations take and demonstrate the methods with a simple example.
Variance Reduction
Authors: Lapeyre Bernard
Published: May 2010

We have given here an introduction to variance reduction methods in the context of financial applications. Special emphasis is given to importance sampling and control variates, but other standard methods are also described. For each are these methods, financially relevant examples have been given.
Sensitivity Computations: Integration by Parts
Authors: Chen Nan
Published: May 2010

Derivative price sensitivities, or greeks, play an important role in the practice of risk management to quantify the potential effects of the changes of underlying market parameters on the values of derivatives. However, how to calculate them efficiently is a challenging problem for computational finance.

An obvious approach is to simulate replications of the model at perturbed parameters and then to use finite difference to form estimators. While this method has its own merits depending on the circumstances, it usually yields estimators with often unacceptably high variances, unless major computational efforts are made in terms of long calculation times. To obtain estimators with lower variance, traditional methods either differentiate the payoff functions of derivatives or differentiate the probability density of the underlying price. The former approach fails when the payoff functions are discontinuous while the latter meets difficulty if the explicit form of the density is not available.

The integration‐by‐parts method overcomes both shortcomings of the traditional methods. It shifts the differential operator from the payoffs to the underlying diffusions in order to remove the smoothness requirement on the payoff functions. This method can be traced back to the Malliavin calculus in the field of stochastic analysis.
Quasi‐MonteCarlo Methods
Authors: Niederreiter Harald
Published: May 2010

This article presents a survey of quasi‐Monte Carlo methods for multidimensional numerical integration and their applications to computational finance. Quasi‐Monte Carlo methods are deterministic versions of Monte Carlo methods, which outperform Monte Carlo methods for many types of integrals. First, a general background on quasi‐Monte Carlo methods is given. Then we describe principles for the construction of low‐discrepancy sequences, which are basic ingredients of any implementation of quasi‐Monte Carlo methods. Special emphasis is placed on the currently most powerful constructions based on the digital method and the theory of (T,s)‐sequences. Next, the important concept of effective dimension is discussed. One section is devoted to scrambling schemes for low‐discrepancy sequences. A review of applications to pricing problems, in particular for mortgage‐backed securities and options, concludes the article.
Rare‐Event Simulation
Authors: Asmussen Søren
Published: May 2010

A survey is given of how to efficiently compute small probabilities efficiently by Monte Carlo simulation. The methods include importance sampling and conditional Monte Carlo. Applications cover large exceedances of sums both with light and heavy tails, random walk maxima, and barrier options.
Exercise Boundary Optimization Methods
Authors: Jäckel Peter , Andersen Leif B.G.
Published: May 2010

The valuation of derivatives contracts with early exercise features in a Monte Carlo framework requires special techniques due to the inherent incompatibility of the framework's forward‐looking nature and the associated Hamilton–Jacobi–Bellman equation's backward‐looking nature. The approaches to handle this difficulty fall into two categories. The first one is founded on parametric representations of the valuation function (regression‐based techniques). The second approach is based on the optimization of a parametric representation of the exercise domain geometry (exercise boundary optimization methods). In this article, the mathematical foundation of exercise boundary optimization methods is explained, and a practical explanation of its algorithmic implementation is given.
Early Exercise Options: Upper Bounds
Authors: Andersen Leif B.G. , Broadie Mark
Published: May 2010

In this article, we discuss how to generate upper bounds for American or Bermudan securities by Monte Carlo methods. These techniques provide a useful supplement to strategies that provide lower bound estimates, allowing one both to generate valid confidence intervals for the true option price and to test the accuracy to any proposed approximation to the optimal exercise strategy.
Performance Measures
Authors: Bacmann Jean‐Francois , Scholz Stefan
Published: May 2010

The measurement of performance is the cornerstone of the evaluation of a fund. We start by examining the hypothesis underlying the most commonly used measure, namely the Sharpe ratio, in particular, the normal distribution of asset returns. As many asset classes, for example hedge funds, do not exhibit normally distributed returns, more recent measures have been considered. Some of them, the Sortino ratio and the Stutzer index, rely on a loss aversion principle. The more recent Omega function is a transformation of the cumulative density function that is also used.

Factor models offer the possibility to incorporate market conditions in the analysis. The performance is measured by the alpha, which corresponds to the remaining return once the influences of factors have been stripped out. A large part of the research focuses on the definition of the factors to be included in performance analysis. In the mutual fund context, four factors, including a market factor, a size factor, a book‐to‐market factor, and a momentum factor, are frequently chosen. Recent contributions are devoted to the understanding of the dynamic behavior of active managers, as well as the estimation of alpha, by trying to distinguish skill from luck. Finally, we show how performance can be measured with characteristic benchmarks when the weights of the underlying assets are known.
Stochastic Control
Authors: Elliott Robert J. , Siu Tak Kuen
Published: May 2010

Uncertainty is ubiquitous. It plays a primary role in economics, finance, and actuarial science, as many problems in these fields involve decisions under uncertainty. The modeling, analysis, and control of uncertainty are central issues in these fields. The theory of stochastic optimal control studies the control of (dynamical) systems involving uncertainty. It originates from the calculus of variations, which has a long history in mathematics of nearly 300 years or so. The theory of stochastic optimal control has expanded since the middle of the last century. These developments are partly due to the emergence of many powerful mathematical tools such as Ito's stochastic calculus. The expansion is also attributed partly to applications in fields such as engineering, aerospace, and manufacturing. Robert Merton pioneered the use of stochastic control in finance and applied it to obtain an elegant solution to the optimal portfolio problem in the early 1970s. His work stimulated the development of continuous‐time finance, and transformed the theory and practice of finance. In this article, we discuss applications of stochastic control to the optimal consumption–investment problem. We review three approaches namely, the martingale method, the dynamic programming approach, and the stochastic Pontryagin's maximum principle.
Transaction Costs
Authors: Bouchard Bruno , Jouini Elyes
Published: May 2010

Standard models for financial markets are based on the simplifying assumption that trading orders can be given and executed in continuous time with no friction. This assumption is clearly a strong idealization of the reality. In particular, securities should not be described by a single price but by a bid and ask curve. When bid and ask prices do not depend on the traded quantities, securities prices can be modeled in terms of proportional transaction costs. This article presents such transaction costs models as well as the associated duality theory that has been deveoped during the last decade and that has influenced the treatment of many other types of market frictions.
Behavioral Portfolio Selection
Authors: Shefrin Hersh
Published: May 2010

Portfolios satisfy both financial needs and psychological needs. Behavioral portfolio selection is a framework for understanding how psychology impacts investors' portfolios. Whereas risk and return are the central variables in neoclassical portfolio selection, the behavioral approach emphasizes the roles of emotions such as hope and fear, aspirations, the framing of outcomes as gains or losses relative to reference points, regret aversion, self‐control, and judgmental biases.
Drawdown Minimization
Authors: Browne Sid , Kosowski Robert
Published: May 2010

One measure of riskiness of an investment is “drawdown”, defined, most often in the asset management space, as the decline in net asset value from a historic high point. Another measure of riskiness that is oftentimes confused terminologically with drawdown is that of “shortfall”, which is simply the gap, or loss level of the current value from the initial or some other given value. In this article, we review the multiperiod continuous time literature on optimal portfolio choice and fund manager incentives in the presence on drawdown and shortfall minimization. The models differ in their assumptions regarding investment horizons (finite or infinite), constraints (fixed or stochastic benchmark), stochastic processes (diffusion with and without jumps) as well as objective function (general goal problem or expected utility maximization). In general, without transactions costs, the incorporation of drawdown constraints induces a portfolio insurance strategy, which, in the stationary stochastic model case, is that of a constant proportions portfolio insurance (CPPI) with different “floor” levels determined by the horizon and the objective.
Robust Portfolio Optimization
Authors: Iyengar Garud
Published: May 2010

We show how to formulate and solve robust portfolio selection problems. The objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. We introduce “uncertainty structures” for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second‐order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs. We also survey recent literature on robust portfolio selection.
Diversification
Authors: Fernholz Robert
Published: May 2010

Diversification involves spreading investments among various assets in order to improve portfolio performance in some manner. Mathematical analysis of portfolio diversification was introduced in 1952 by Markowitz with his concept of mean/variance portfolio efficiency. A portfolio was called
efficient
if among all portfolios of the same assets, the portfolio had minimum variance for a given expected rate of return. Hence, in the setting of expected portfolio return and variance, portfolio diversification served to control the risk of a portfolio. In 1982, Fernholz and Shay showed that if the expected compound growth rate of a portfolio is considered rather than the expected rate of return, then portfolio diversification can increase the expected growth rate as well as control risk.
Risk Exposures
Authors: Crouhy Michel , Galai Dan , Mark Robert
Published: May 2010

Risk factors can be broadly grouped together into the following categories: market risk, credit risk, liquidity risk, operational risk, legal and regulatory risk, business risk, strategic risk, and reputation risk. These categories can then be further decomposed into more specific categories. This article reviews the definitions of the various risk categories and their main drivers.
Risk Management: Historical Perspectives
Authors: Jorion Philippe
Published: May 2010

Financial risk management builds on the work of pioneers in financial theory. Spurred by the rapid expansion of derivatives markets, new risk‐management tools were developed in response to the need to manage financial risks better. This led to the development of position‐based risk measures such as Value at Risk (VAR), which are now widely used. Later, these methods were extended to other types of risk such as credit and operational risk. In spite of these technical advances, however, risk managers need to be keenly aware of the limitations of these models. Even so, risk management has now become an essential function for financial institutions.
Economic Capital Allocation
Authors: Kalkbrener Michael
Published: May 2010

Capital allocation techniques are of central importance in portfolio management and risk‐based performance measurement. After a short review of the capital allocation problem and the existing literature, we present an axiomatic characterization of capital allocation. We analyze the relation between these axioms and properties of the associated risk measure and show that the allocation by the gradient (or Euler principle) is an immediate consequence of the proposed axioms. The axiomatization is applied to the most widely used risk measures in the finance industry to derive explicit capital allocation formulae for these measures.
Spectral Measures of Risk
Authors: Acerbi Carlo
Published: May 2010

Spectral measures of risk are the subclass of coherent measures of risk characterized by the additional properties of law invariance and comonotonic additivity. We provide a formal definition of spectral measures of risk and their connection with deviation measures known in actuarial mathematics. We discuss why this class is particularly important from the point of view of financial risk management. We finally provide consistent estimators.
Operational Risk
Authors: Chavez‐Demoulin Valérie , Embrechts Paul
Published: May 2010

Under the capital requirements of the Basel II regime, banks have to provide estimates of their operational risk on a yearly basis and, for larger international banks, use a Value‐at‐ Risk estimate at the confidence level of 99.9%. The so‐called loss distribution approach (LDA) allows for broad methodological flexibility for the estimation of this risk capital. In this article, we discuss some of the underlying statistical issues.
Regulatory Capital
Authors: Duellmann Klaus
Published: May 2010

The 2004 Basel II framework marks a milestone in the transition from the rules‐based 1988 Basel I Accord to a more risk‐sensitive and principles‐based approach to bank regulation. It significantly expands the regulatory scope by a new capital charge for operational risk and by its new three‐pillar structure consisting of minimum capital requirements, supervisory review process, and market discipline. Depending on the risk category, the minimum capital requirements of its most advanced approaches are either built on banks' internal models or on regulatory models requiring banks' own estimates of risk components as input. The evolutionary nature of the Basel II framework will encourage banks to migrate to more risk‐sensitive approaches in the future, depending on the complexity of their business and their risk management expertise.