bluuebunny/crossref_metadata_2025 huggingface dataset:
Filtering
Authors: Runggaldier Wolfgang
Published: May 2010

Stochastic filtering concerns the estimation of the state of a randomly evolving system that is only indirectly observed and the observations are, furthermore, affected by noise. The primary examples in finance concern factor models, where some factors are not directly observable. We review stochastic filtering in discrete and continuous time in linear and nonlinear models and describe some applications to pricing and portfolio management.
Equivalence of Probability Measures
Authors: Rheinländer Thorsten
Published: May 2010

Two probability measures are said to be equivalent if they define the same null sets. The Girsanov theorem characterizes the transformation of semimartingales under equivalent changes of measure.
Skorokhod Embedding
Authors: Obłój Jan
Published: May 2010

The Skorokhod embedding problem, in its essence, amounts to designing a stopping time such that, at that time, a given stochastic process has a prescribed distribution. It has been an active field of research in probability for over 40 years now. We discuss some of the key results in the domain, along with their applications in financial mathematics, namely, in robust pricing and hedging of exotic derivatives. More precisely, we establish a correspondence between solutions to the Skorokhod embedding problem and possible market models matching given prices of vanilla options for a fixed maturity. No‐arbitrage bounds on a price of an exotic derivative are then obtained through an optimization over solutions to the embedding problem. Pathwise understanding of the embedding often allows us to derive the associated robust (often model‐free) superhedges. We illustrate these techniques with an example of a one‐touch option. No‐arbitrage bounds on its price are induced by the Azéma–Yor and Perkins solutions to the Skorokhod embedding; the associated superhedging strategy was deduced by Hobson.
Markov Processes
Authors: Bossy Mireille , Champagnat Nicolas
Published: May 2010

We present the main concepts of the theory of Markov processes: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov's backward and forward equations, and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward stochastic differential equations (BSDEs) and describe the links between Markov processes and parabolic partial differential equations (PDEs). In particular, we state the Feynman–Kac formula for linear PDEs and BSDEs, and we give some examples of the correspondence between stochastic control problems and Hamilton–Jacobi–Bellman (HJB) equations and between optimal stopping problems and variational inequalities. Several examples of financial applications are given to illustrate each of these results, including European options, Asian options, and American put options.
Backward Stochastic Differential Equations: Numerical Methods
Authors: Martín Jaime San , Torres Soledad Torres
Published: May 2010

We review numerical methods for solving backward stochastic differential equations (BSDEs), forward–backward systems of stochastic differential equations (FBSDEs), and reflected backward stochastic differential equations (RBSDEs). Application to the pricing of European and American options are given.
Martingales
Authors: Yan Liqing
Published: May 2010

We give a brief introduction to martingales and related stochastic processes, and present some important theorems and their applications in quantitative finance.
Itô’s Formulas
Authors: Eisenbaum Nathalie
Published: May 2010

The Itô formula is introduced for 𝒞
2
‐functions of semimartingales. Various extensions are then presented for more general functions.
LéVy Copulas
Authors: Tankov Peter
Published: May 2010

Lévy copulas characterize the dependence among components of multidimensional Lévy processes. They are similar to copulas of probability distributions, but are defined at the level of Lévy measures. Lévy copulas separate the dependence structure of a Lévy measure from the one‐dimensional marginal measures, enabling one to construct parametric multidimensional Lévy models by combining arbitrary one‐dimensional Lévy processes with a Lévy copula from a parametric family. In finance, Lévy copulas are useful to model joint moves of several assets in various settings, including portfolio risk management, option pricing, insurance, and operational risk.
Convex Duality
Authors: Scandolo Giacomo
Published: May 2010

In this article, we provide a brief account of convex duality, a set of tools for the analysis of convex optimization problems. The unifying principles underlying these methods can be traced back to the basic duality that exists between a convex set of points in the plane and the set of supporting lines (hyperplanes.) First, we describe, in a finite‐dimensional setting, Lagrange duality, maybe the simplest form of duality. It is usually known as the
Lagrange multiplier method
. Then we introduce Fenchel duality; this is presented as a by‐product of Lagrange duality, even though these two forms of duality are formally equivalent. Finally, we discuss Lagrange and Fenchel duality in an infinite‐dimensional setting.
Squared Bessel Processes
Authors: Yor Marc J.
Published: May 2010

Squares of Bessel processes enjoy both an additivity property and a scaling property, which are, arguably, the main reasons why these processes occur naturally in a number of Brownian, or linear diffusions, studies.
Arbitrage Pricing Theory
Authors: Reisman Haim
Published: May 2010

The arbitrage pricing theory (APT) explains expected returns of portfolios in a given sequence of assets. The model identifies sets of reference variables with respect to which expected returns of portfolios have a multibeta representation. This is done under the assumption that the assets in the sequence have a factor structure and that there are no portfolios with large expected returns and unit variance. The model predictions may have approximation errors that disappear when the model is applied to diversified portfolios.
Efficient Market Hypothesis
Authors: LeRoy Stephen F.
Published: May 2010

Despite the fact that the topic of capital market efficiency plays a central role in introductory instruction in finance, it is difficult to determine exactly what it means. Fama's well‐known statement that market efficiency can be tested only jointly with an assumed returns model implies that market efficiency generates restrictions on the data that are distinct from those implied by particular models. It is difficult to find a clear characterization of what these additional restrictions are. One possibility is that the proponents of this theory identified market efficiency with rational expectations. Another possibility (suggested by empirical tests of efficiency) is that an efficient market is one in which expected returns are constant. A third possibility is that an efficient market is one in which asset prices are similar to those that would occur in the absence of frictions. All of these interpretations have difficulties. It is concluded that distinguishing market efficiency from particular returns models is a questionable research strategy.
Ambiguity
Authors: Ghirardato Paolo
Published: May 2010

In this article, the concepts of ambiguity and ambiguity aversion are introduced, and the decision models developed to incorporate them are briefly surveyed. Some of the consequences for financial modeling are also discussed.
Employee Stock Options
Authors: Henderson Vicky , Sun Jia
Published: May 2010

This article gives a brief introduction to employee stock options. We describe the common features of such options, give statistics on their usage around the world, describe observed exercise patterns, and discuss the literature on modeling exercise behavior, the valuation of employee options, and their cost to the company.
Arbitrage Strategy
Authors: Kardaras Constantinos
Published: May 2010

A (riskless)
arbitrage strategy
allows a financial agent to make certain profit out of nothing, that is, out of zero initial investment. This has to be disallowed on economic basis if the market is in equilibrium state, as opportunities for riskless profit would result in an instantaneous movement of prices of certain financial instruments. The principle of not allowing for arbitrage opportunities in financial markets has far‐reaching consequences, most notably the option‐pricing and hedging formulas in complete markets.
Fundamental Theorem of Asset Pricing
Authors: Schachermayer Walter
Published: May 2010

The fundamental theorem of asset pricing states that the absence of arbitrage in a market (model) is essentially equivalent to the existence of an equivalent probability measureQunder which the prices of tradable securities have the martingale property. We give a precise formulation of this theorem and discuss the origins and history of this theorem, which is one of the pillars of mathematical finance.
Risk‐Neutral Pricing
Authors: Schachermayer Walter
Published: May 2010

We review the valuation of uncertain future cash flows. This article has a history of several centuries in the actuarial literature as well as in Bachlier's “fundamental principle” for the valuation of options. The technique, which is a basic tool in mathematical finance, consists in calculating the expected value of the discounted cash flows. The crucial point is that we calculate this expectation under a so‐called “risk neutral” probability measure. In the context of the Black–Scholes model, the risk neutral measure is the unique probability measure which is equivalent to the given “physical” or “historic” probability measure and under which the discounted stock price process is a martingale.
Equivalent Martingale Measures
Authors: Schachermayer Walter
Published: May 2010

We survey the notion of equivalent martingale measure, which plays a central role in Mathematical Finance. We also focus on the ramifications, such as local and sigma‐martingales.
Expected Utility Maximization: Duality Methods
Authors: Biagini Sara
Published: May 2010

Expected utility maximization is a well‐established criterion for portfolio selection. The increasing need for complex probabilistic market models has led to the development of the convex duality technique for optimization. After presenting a variety of portfolio selection problems, we illustrate the dual approach as a resolution method, highlighting the most recent results in the field.
Superhedging
Authors: Campi Luciano
Published: May 2010

In incomplete financial markets, the lack of replicability of contingent claims justifies the introduction of superhedging. Given a contingent claim
H
, a superhedging strategy is an admissible strategy that super‐replicates
H
at maturity. The initial endowment that allows to pursue that strategy is called the
superhedging price
. This article is a short survey of the main results on this topic in the mathematical finance literature. We focus especially on continuous‐time hedging of European‐type options.
Free Lunch
Authors: Kardaras Constantinos
Published: May 2010

The concept of
absence of opportunities for free lunches
is one of the pillars in the economic theory of financial markets. This natural assumption has proved very fruitful and has lead to great mathematical, as well as economical, insights in quantitative finance. Formulating rigorously, the exact definition of absence of opportunities for riskless profit turned out to be a highly nontrivial fact that troubled mathematicians and economists for at least two decades. The purpose of this article is to give a quick (and, hence necessarily, incomplete) account of the recent work aimed at providing a simple and intuitive no‐free‐lunch assumption that would suffice in formulating a version of the celebrated fundamental theorem of asset pricing.
Minimal Entropy Martingale Measure
Authors: Schweizer Martin
Published: May 2010

Suppose that discounted asset prices in a financial market are given by aP‐semimartingaleS. Among all probability measuresQthat turnSinto a localQ‐martingale, the minimal entropy martingale measure is characterized by the property that it minimizes the relative entropy with respect toP.Viaconvex duality, it is intimately linked to the problem of maximizing expected exponential utility from terminal wealth. It also appears as a limit ofp‐optimal martingale measures aspdecreases to 1. Like for most optimal martingale measures, finding its explicit form is easy ifSis an exponential Lévy process and quite difficult otherwise.
Minimal Martingale Measure
Authors: Föllmer Hans , Schweizer Martin
Published: May 2010

Suppose discounted asset prices in a financial market are given by aP‐semimartingale of the formS = S0 + M + A. The minimal martingale measure forSis characterized by the properties that it turnsSinto a local martingale and preserves the martingale property for any localP‐martingale stronglyP‐orthogonal toM. It plays a key role in finding locally risk‐minimizing strategies, and it comes up in various other contexts as well. Importantly, its density process can be written explicitly in terms ofMandA, so that one can use it very generally and broadly. In some specific settings, it also has other optimality properties.
Good‐Deal Bounds
Authors: Hodges Stewart D.
Published: May 2010

Good‐deal bounds provide a range of prices within which an instrument must trade if it is not to offer a surprisingly good reward‐for‐risk opportunity. This approach to valuation provides an important middle course between finding a unique value and finding probably weak bounds based on super‐replication. In a world where derivatives are difficult to hedge precisely, good‐deal bounds reflect this; they depend on model assumptions and on the prices of the securities that would be used to exploit mispricing.

Good‐deal bounds have two interpretations: we can think of them either as establishing normative bid and ask forward prices for a particular trader, or as predicting a range in which we expect the market price to lie.

We cover the main issues without going too deeply into mathematical technicalities, describing the nature of the construction and giving insights into the analysis, including the use of duality in the solutions. We also sketch the history of the topic, including that of the closely related generalized Sharpe ratio, and the link to coherent risk measures. There is also a discussion of the role of the utility function. The article concludes with a short account of the more recent and prospective literature.
Options: Basic Definitions
Authors: Davis Mark H.A.
Published: May 2010

This is an introductory article describing the features of option contracts and giving brief definitions of a number of specific contracts. The features are described by means of the following dichotonics: European/American, traded/OTC, physical/cash settlement, liquid/illiquid, vanilla/exotic and path‐dependent/path‐independent. The option classes defined are Asian, barrier, basket, Bermuda, chooser, digital, exchange, forward‐starting, cliquet, lookback, passport, quanto, and Russian options. Most of these are covered in greater detail elsewhere in the encyclopacdia.
Option Pricing: General Principles
Authors: Benth Fred E.
Published: May 2010

We explain the fundamental concepts of options pricing. In any reasonable market, arbitrage opportunities should not exist. This implies that any option that can be replicated (or perfectly hedged) must have a price coinciding with the cost of replication. It turns out that this cost can be represented in terms of the expected present value of the option's payoff, where the expectation is taken under an equivalent martingale measure (also called a
risk‐neutral probability
). The extensions to incomplete markets, where replication is not possible, are discussed.
Forwards and Futures
Authors: Vellekoop Michel
Published: May 2010

We discuss the existing pricing methodology for futures and forward contracts. Both the discrete‐time and the continuous‐time cases are treated, and we focus on complete and arbitrage‐free markets. Possible extensions and suggestions for further reading are provided at the end of the article.
Binomial Tree
Authors: Davis Mark H.A.
Published: May 2010

This article covers the theory of the Cox–Ross–Rubinstein binomial option pricing model. The single‐period model is discussed in detail, followed by its extension to multiple periods and convergence to the continuous‐time Black–Scholes model. The final two sections cover American options in the binomial model and computational aspects.
Arbitrage Bounds
Authors: Cox Alexander
Published: May 2010

A key question in option pricing concerns how to incorporate information about the prices of existing, liquidly traded options into the prices of exotic options. Rather than trying to calibrate market prices to models, an alternative approach is to construct model‐free bounds on the price of exotic options. These bounds are derived from arbitrages that may be constructed using relatively simple trading strategies in vanilla options, and their simplicity means that the bounds are generally valid for a wide class of models. This article discusses the construction of arbitrage bounds for a variety of classes of exotic options, and describes different techniques and approaches to the construction of such bounds. Particular examples include model‐free bounds on the price of one‐touch options, and pricing barrier options under the assumption of put–call symmetry.
Average Strike Options
Authors: Henderson Vicky
Published: May 2010

Average strike options or Asian options with floating strike are exotic, path‐dependent options with a payoff based on the difference between the terminal asset price and the average of an underlying asset price over a specified time period. This article gives an overview of the no‐arbitrage pricing methods for such options. Typically, contracts involve the arithmetic average of prices, and since the joint law of the average and the terminal stock price is needed, there is no closed‐form solution for the option price.
Foreign Exchange Options
Authors: Cekan Markus , Wendel Armin , Wystup Uwe
Published: May 2010

The article deals with pricing and hedging of foreign exchange options from a trader's perspective. We focus on first‐generation exotic options and give some examples on second‐generation exotic options. We split up some of these options into their individual components and point out the risk arising from this type of options, as well as show ways to minimize these risks. Market conventions in plain vanilla and exotic options are described and deviations as well as exceptional features in these markets are shown.
Currency Forward Contracts
Authors: Korchmáros Tamás
Published: May 2010

Companies can hedge their foreign exchange exposure with structured forwards, while leaving the possibility open for favorable market moves. Like forwards, they offer a fixed level at which the exposure is hedged, the worst‐case rate, and they are in most cases zero‐cost strategies. To offer this flexibility, the worst‐case rate is less favorable than the prevailing market rate. Structured forwards enable companies to construct a hedging solution that suits their risk–return profile and give the opportunity to incorporate a given market view with the aim of outperforming the market forward rate.
Pricing Formulae for Foreign Exchange Options
Authors: Weber Andreas , Wystup Uwe
Published: May 2010

We provide closed‐form solutions for the value of selected first‐generation exotic options in the Black–Scholes model as used frequently to quote the “theoretical value (TV)”. These formulae allow a fast computation; all involve the normal density and cumulative distribution function.
Vanna–Volga Pricing
Authors: Wystup Uwe
Published: May 2010

The vanna–volga method, also called the
traders' rule of thumb
, is an empirical procedure that can be used to infer an implied‐volatility smile from three available implied volatility quotes for a given maturity. It is based on the construction of locally replicating portfolios whose associated hedging costs are added to corresponding Black–Scholes prices to produce smile consistent values. Besides being intuitive and easy to implement, this procedure has a clear financial interpretation, which further supports its use in practice. The vanna–volga method is commonly used in foreign exchange options markets, where three main volatility quotes are typically available for a given market maturity: the delta‐neutral straddle, referred to as at‐the‐money (ATM); the risk reversal (RR) for 25 delta call and put; and the butterfly (BF) with 25 delta wings. The application of vanna‐volga pricing allows us to derive implied volatilities for any option's delta, in particular for those outside the basic range set by the 25 delta put and call quotes. It is also used to determine smile‐adjusted prices of exotic options.
Basket Options
Authors: Avellaneda Marco
Published: May 2010

Equity basket options are derivative contracts that have as underlying asset a basket of stocks. Their value is sensitive both to the volatility of the underlying stocks and to their correlation. We discuss some issues related to their pricing, including the notion of implied correlation.
Butterfly
Authors: Liverance Eric
Published: May 2010

A butterfly spread is an option strategy with limited upside and limited downside that uses call options of three different strikes but the same maturity on the same underlying. This produces a structure that at maturity pays off only in scenarios where the price of the underlying is between the highest and lowest strikes. One can think of this strategy as buying a low‐strike call option and financing part of the upfront cost with the rest of the structure. The effect is to limit the upside potential, but significantly reduce the cost of the structure. It is typically used for expressing a view that the underlying will trade within a restricted range or that implied volatility will decline. The prices of butterflies can be used to infer the market‐implied terminal probability distribution of the underlying.
Gamma Hedging
Authors: Roubinet Charles‐Henri
Published: May 2010

Gamma is defined as the second derivative of a derivative product with respect to the underlying price. If we consider the evolution through time of the value of a delta‐hedged self‐financing portfolio in the Black and Scholes framework, we can show that, locally, it is equal to

(known as the
breakeven
relation). Hence, neutralizing the gamma of a portfolio is a way of controlling this evolution. Because there is no pure gamma instrument, hedging gamma will also change the vega exposure of the portfolio. Nevertheless, it can be done using vanilla options of well‐chosen strikes and maturities (usually short ones), or with other instruments such as variance swaps. If one wishes to incorporate the dynamics of the market in the model by using a more complex setting, such as stochastic volatility models or Levy processes, the relation presented earlier is modified and other “gamma” terms that require a hedge appear. Finally, it can be shown that for multiunderlying options, the breakeven relation integrates the “cross gamma” terms for which a hedge can also be performed using basket trading strategies.
Correlation Swap
Authors: Bossu Sébastien
Published: May 2010

A correlation swap is a type of exotic derivative security that pays off the observed statistical correlation between the returns of several underlying assets, against a preagreed price. At the time of writing, it is traded over‐the‐counter (OTC) on equity and foreign exchange derivatives markets. This article focuses on equity correlation swaps, which appeared in the early 2000s, as a means to hedge the parametric risk exposure of exotic trading desks to changes in correlation.
Stock Pinning
Authors: Lipkin Mike
Published: May 2010

It had long been empirically observed that optionable stocks occasionally closed exactly at a strike price on option expiration dates. This phenomenon, known as
pinning
, or
stock pinning
, is keenly appreciated by practitioners but remained unexplored by theorists until quite recently. The experimental evidence for pinning is discussed, as well as a successful simple predictive model.
Variance Swap
Authors: Liverance Eric
Published: May 2010

Variance swaps are derivatives used for speculating on the differential between implied option volatility and skew, on the one hand, and realized volatility, on the other hand. The variance swap market is very well developed in the equities and foreign exchange over‐the‐counter markets. A formula for the fair value involves both the level of volatility and the amount of market skew—change in option implied volatility with strike. Variance swaps can be perfectly statically hedged and replicated under the assumption that every option strike is tradable, there is no gapping in the underlying, and so on. However, the payoff is quadratic with respect to the implied/realized vol differential. For this reason, volatility swaps are sometimes preferred, which pay off linearly on the volatility differential rather than the variance differential. Recently, it has been shown, by Carr and Lee, that a model‐independent dynamic replication strategy exists for volatility swaps.
Volatility Swaps
Authors: Ren Yong
Published: May 2010

A volatility swap is very similar to a variance swap; it is a forward contract on realized volatility.
Gamma Swap
Authors: Lee Roger
Published: May 2010

A gamma swap is a weighted variance swap, with weight given by the underlying level (normalized by the initial underlying level). We discuss the gamma swap's motivations, model‐free replication and valuation, applications to dispersion trading, relations to variance swaps, and valuation via characteristic function.
Himalayan Option
Authors: Gharavi Reza K.
Published: May 2010

A Himalayan option is an option on a small basket where at end of predetermined time periods, the best performing assets are removed from the basket. The final payoff is the sum of the performance of the assets at the time of their removal. This article provides an overview of the product. It describes the motivation behind the product, discusses its valuation, and describes its risk profile.
Altiplano Option
Authors: Gharavi Reza K.
Published: May 2010

An Altiplano option is a barrier option on a small basket where the barriers are on individual assets rather than the whole basket. This article provides an overview of the product. It describes the motivation behind the product, discusses its valuation, and describes its risk profile.
Constant Proportion Portfolio Insurance
Authors: Hirsa Ali
Published: May 2010

This article describes constant proportion portfolio insurance, its features, and valuation. It explains gap risk and cites various works done in modeling gap risk
Exchange‐Traded Funds (ETFs)
Authors: Tari Michael J.
Published: May 2010

A number of financial entities are necessary to create and maintain exchange‐traded funds (ETFs), indexed products that trade like stocks on major bourses. ETFs differ from mutual funds in that they trade throughout the day, can be shorted, and generally carry lower fees. The close to 2000 ETFs trading today cover a plethora of different investment categories, offering investors an inexpensive means to construct well‐diversified portfolios.
Volume‐Weighted Average Price (VWAP)
Authors: Madhavan Ananth N.
Published: May 2010

The volume‐ (or time‐) weighted average price is a common benchmark used to evaluate performance in trading and execution.
Equity Swaps
Authors: Hinnerich Mia
Published: May 2010

In this article, we consider equity swaps. We provide product definitions and examples of several contracts. We also provide some illustrations of how these contracts can be used.
Volatility Index Options
Authors: Benhamou Eric , Ciuca Marian
Published: May 2010

Volatility index options are options on a volatility index. Volatility index options enable one to take a pure volatility exposure without the need to take positions in the index itself and without the need to delta‐hedge. We describe how volatility indices—such as the VIX—are defined and discuss pricing procedures for options on such indices.
Realized Volatility Options
Authors: Lee Roger
Published: May 2010

We survey four approaches to pricing options on quadratic variation (“realized variance”) and on its square root (“realized volatility”). The first approach uses Fourier analysis to price realized volatility/variance options, assuming the availability of the characteristic function of realized variance; it leads to explicit formulas in, for instance, the Heston model and lévy models. The second approach does not assume a model for the realized variance, but instead prices volatility/variance options in terms of European options, under an independence assumption. The third approach assumes the availability of volatility and/or variance swap quotes, and prices volatility/variance options in terms of the swaps. The fourth approach assumes only the continuity of the underlying returns process; it finds lower and upper bounds, as well as subreplicating and superreplicating hedges of variance options, by use of model‐free strategies involving European options.
Discretely Monitored Options
Authors: AitSahlia Farid
Published: May 2010

This article reviews pricing techniques for options with path‐dependent payoffs and which are monitored over a discrete set of dates.
Implied Volatility Surface
Authors: Kamal Michael , Gatheral Jim
Published: May 2010

We describe some empirical observations that are most relevant for the construction and validation of realistic models of the volatility surface for equity option indices. We consider both static and dynamic properties of the volatility surface. We discuss how a simple measure of the volatility skew shows power‐law decay with increasing term to expiration and discuss the theoretical implications of this result. We then point out that the dynamics of the volatility are surprisingly simple: several principal compenent analysis (PCAs) or closely related analyses confirm that a few simple modes explain most of the variation of the volatility surface. Furthermore, the mode associated with overall level of implied volatility is dominant and strongly and negatively correlated with the returns of the index itself. Finally, we show that the volatility skew plays a central role in linking the statics to the dynamics of the volatility surface. Connections to the theoretical literature are highlighted throughout our discussion.
Moment Explosions
Authors: Friz Peter K. , Ressel Martin Keller
Published: May 2010

Let (
S

t

)

t
 ≥ 0

be the discounted price process in a stochastic volatility model. A moment explosion takes place, if the moment

of some given order
u
 ∈ ℝ becomes infinite (“explodes”) after some finite time
T
*
(
u
). Moment explosions are closely related to the shape of the implied volatility surface, where they can be used to obtain approximations for deep in‐the‐money and out‐of‐the‐money strikes. Furthermore, moment explosions may lead to infinite prices of derivatives with superlinear payoff, and to the breakdown of error estimates for numerical approximation schemes. Comparison results for parabolic partial differential equations (PDEs) combined with an exponentially affine ansatz for the solution, allow to link explosion times to the blow‐up time of ordinary differential equations of the (generalized) Riccati type.
Implied Volatility: Volvol Expansion
Authors: Wang Zaizhi , Miri Mohammed
Published: May 2010

This article presents the volatility of volatility series expansion, a powerful technique for deriving fast‐computing analytical formulae for European options. The main idea is to apply perturbation method to the volvol parameter, calculating the first and second order of the difference between a stochastic volatility model and a Black–Scholes model. In the general case, we can reduce the integration of the exact formula to some simpler integration. The method can be applied successfully to several stochastic volatility models including Heston model, GARCH model, SABR model, and so on, to enhance the calibration and pricing routines.
SABR
Model
Authors: Henry‐Labordère Pierre
Published: May 2010

The short expiration limit of implied volatility for stochastic volatility models (SVMs) has been extensively analyzed with a view to developing arbitrage‐free parametrizations of market smiles with fast calibration. In particular, the SABR formula due to Hagan
et al.
is widely used in practice. In this article, we present the SABR formula and describe how geometrical methods may be used to derive accurate short‐time asymptotics of the implied volatility for more general time‐homogeneous SVMs. We conclude by presenting a more accurate version of the SABR formula derived using these methods.
Constant Elasticity of Variance (
CEV
) Diffusion Model
Authors: Linetsky Vadim , Mendoza Rafael
Published: May 2010

The
constant elasticity of variance
(CEV) spot price model is a one‐dimensional diffusion model with the instantaneous volatility specified to be a power function of the underlying spot price, σ(
S
) = 
aS
β
. The model has been introduced by Cox as one of the early alternatives to the geometric Brownian motion to model asset prices. The CEV process is closely related to Bessel processes and is analytically tractable, leading to closed‐form options pricing formulas. Options prices in the CEV model exhibit implied volatility skews and, for β < 0, there is a positive probability of hitting zero (bankruptcy). This article surveys the CEV model.
Tempered Stable Process
Authors: Kienitz Jörg
Published: May 2010

A tempered stable process is a Levy process whose small jumps behave like a stable processes, while the large jumps have been “tempered” so that the tail of the density decays exponentially. This article describes tempered stable processes, their distributional and path properties, and their use in option pricing.
Lognormal Mixture Diffusion Model
Authors: Mercurio Fabio
Published: May 2010

The lognormal mixture diffusion model is a diffusion model in which the asset price has a marginal density given by a known mixture of lognormal densities. Such a model is analytically tractable and can be easily used to price exotic derivatives under smile effects.
Normal Inverse
G
aussian Model
Authors: Jönsson Henrik , Masol Viktoriya , Schoutens Wim
Published: May 2010

The normal inverse Gaussian (NIG) process is a Lévy process with no Brownian component and NIG‐distributed increments. The NIG process can be constructed either as a process with NIG increments or, alternatively,
via
random time change of Brownian motion using the inverse Gaussian process to determine time. The article presents the normal inverse Gaussian distribution function and the corresponding characteristic function and Lévy measure. Further, the NIG process is used to construct a market model for financial assets. Option pricing can be done using the NIG density function, the NIG Lévy characteristics, or the NIG characteristic function.
Generalized Hyperbolic Models
Authors: Eberlein Ernst
Published: May 2010

Generalized hyperbolic (GH) Lévy motions constitute a subclass of Lévy processes that are generated by
GH distributions
. Analytical and probabilistic properties of GH processes are described and some applications in option pricing and risk management are presented.
Regime‐Switching Models
Authors: Chourdakis Kyriakos
Published: May 2010

Many financial time series exhibit sudden changes in the structure of the data‐generating process. Regime‐switching models attempt to capture this behavior by allowing the data‐generating process to change in time. We review examples of regime‐switching methods, estimation techniques, and filtering methods for these models and discuss their use in derivative pricing.
Exponential Lévy Models
Authors: Tankov Peter
Published: May 2010

Exponential Lévy models generalize the classical Black and Scholes setup by allowing the stock prices to jump while preserving the independence and stationarity of returns. This article introduces most common examples of such models and discusses the pricing and hedging of options in this framework.
Creditrisk+
Authors: Wilde Tom
Published: May 2010

CreditRisk+ is a portfolio credit risk model developed and published by the bank Credit Suisse in 1997. CreditRisk+ offers an approach for calculating the aggregate risk of a portfolio of credit‐risky instruments such as loans or bonds, based on individual borrower characteristics including credit ratings and estimated recovery amounts. Portfolio credit risk models such as CreditRisk+ and Gaussian copula models are widely used by financial institutions in the assessment and allocation of economic capital and related measures such as return on capital. CreditRisk+ has the distinguishing features that it avoids Monte Carlo simulation, using instead a fast numerical algorithm to calculate aggregate risk, and that it was published as an open methodology document, with the intention of encouraging financial institutions and risk practitioners to develop their own implementations of the model. Since its introduction, CreditRisk+ has consistently attracted the interest of practitioners, financial regulators, and academics, resulting in a significant body of literature on the model, including many enhancements and alternative approaches.
Large Pool Approximations
Authors: Iscoe Ian , Kreinin Alex
Published: May 2010

We consider approximations for the portfolio loss distribution based on limit theorems of probability theory. The applications of the law of large numbers, the compound Poisson approximation, and the central limit theorem are considered in the conditional independence framework suitable for financial risk management applications.
Saddlepoint Approximation
Authors: Hurd Thomas R.
Published: May 2010

The saddlepoint method is a classic technique of applied mathematics for developing asymptotic expansions of integrals that typically arise in Fourier analysis. Such integrals are important in finance too, particularly in portfolio credit risk with a large number of obligors. This article discusses the general method, and explores how it can be used for computing credit risk measures and pricing collateralized debt obligations.
Credit Scoring
Authors: Sabato Gabriele
Published: May 2010

Credit scoring models play a fundamental role in the risk management practice at most banks. They are used to quantify credit risk at counterparty or transaction level in the different phases of the credit cycle (e.g., application, behavioral, collection models). The credit score empowers users to make quick decisions or even to automate decisions and this is extremely desirable when banks are dealing with large volumes of clients and relatively small margin of profits at individual transaction level (i.e., consumer lending, but increasingly also small business lending). In this article, we analyze the history and new developments related to credit scoring models. We find that with the new Basel Capital Accord, credit scoring models have been remotivated and given unprecedented significance. Banks, in particular, and most financial institutions worldwide, have either recently developed or modified existing internal credit risk models to conform with the new rules and best practices recently updated in the market. Moreover, we analyze the key steps of the credit scoring model's lifecycle (i.e., assessment, implementation, validation), highlighting the main requirement imposed by Basel II. We conclude that banks that are going to implement the most advanced approach to calculate their capital requirements under Basel II will need to increase their attention and consideration of credit scoring models in the near future.
Credit Rating
Authors: Ashcraft Adam , Schuermann Til
Published: May 2010

A credit rating represents an overall assessment of the creditworthiness of a borrower. In this way the rating is a forecast, and like all forecasts it is noisy. For that reason, credit rating agencies make use of discrete ratings. Such a rating may also be generated by a model internal to a financial institution like a bank for the assessment of credit risk, specifically for the probability of default. Ratings typically depend on firm (or household) financial information such as indebtedness (leverage), revenue, profitability, and so on. Because defaults are rare, it is very difficult to assess the accuracy of a given rating model. Although the rating scale for structured credit products is the same as for single obligor ratings, risk assessment is quite different. Corporate bond (obligor) ratings are largely based on firm‐specific risk characteristics. Since asset‐backed structures represent claims on cash flows from aportfolioof underlying assets, the rating of a structured credit product must take into account systematic risk.
Recovery Rate
Authors: Das Sanjiv R. , Hanouna Paul
Published: May 2010

We provide an overview of the definitions of recovery rates, and some empirical statistics. We summarize the relation of recovery rates to default rates. We describe some economic implications and also present approaches to implying recovery rates from models of default‐related securities.
Exposure to Default and Loss Given Default
Authors: Resti Andrea
Published: May 2010

Two important risk drivers in credit risk are exposure risk (measured by exposure at default (EAD) and loss given default (LGD) or recovery rate (RR)). The former is related to the fact that the amount due at default could not be known with certainty. This is the case, for example, of revolving credit lines, where the borrower could increase his/her credit usage before default. Several measures can be used to assess exposure risk, with CCF (credit conversion factor) being the most widely used one. Empirical analyses have shown that CCFs are mainly impacted by the borrower's rating, the time‐to‐default, and the state of the economic cycle. LGDs and recovery rates can be measured in several ways: whereasmarket LGDsare mostly based on postdefault prices (and therefore require the exposures to be traded on a secondary market), inworkout LGDsit is required to record and discount all actual recovery flows occurring after the default. The main drivers of LGD are the debt seniority, the borrower's industry and liabilities mix, and the presence of collateral. Recently, LGDs have been found to increase together with PDs: measures of expected and unexpected loss can prove to be severely underestimated if such a correlation is ignored.
Credit Portfolio Simulation
Authors: Kalkbrener Michael
Published: May 2010

This article provides a short introduction to Monte Carlo simulation techniques in structural credit portfolio models. The main focus is on importance sampling, a particularly efficient variance reduction technique for this class of models.
Counterparty Credit Risk
Authors: Pykhtin Michael
Published: May 2010

Counterparty credit risk (CCR) is the risk that a counterparty in a financial contract will default prior to the expiration of the contract and will fail to make all the payments required by the contract. In this article, we briefly describe approaches to managing CCR and review different aspects of modeling CCR. We define counterparty credit exposure for a single derivative contract and then extend this definition to a portfolio of contracts. We show how netting and margin agreements affect the exposure. For margin agreements, we describe the standard approach to modeling available collateral. The article also covers pricing CCR
via
credit valuation adjustment (CVA) and modeling portfolio loss and economic capital.
Credit Risk
Authors: Crouhy Michel
Published: May 2010

We define the four main types of credit risk: default risk, bankruptcy risk, downgrading risk, and settlement risk. Credit risk at the portfolio level is also discussed with its main component, concentration risk. Credit events are also defined in the context of the International Swaps and Derivatives Association documentation for the fast growing credit default swaps market.
Total Return Swap
Authors: Wehn Carsten S.
Published: May 2010

A total return swap (TRS) allows synthetic investment in a reference asset without the asset being legally owned. For the TRS payer, the situation is similar to selling the market and credit risk of the reference asset; the TRS receiver's investment is comparable to a direct investment in the reference asset. The valuation of a TRS is quite similar to other credit derivatives; hence, the valuation methods are alike.
Recovery Swap
Authors: Das Sanjiv R. , Hanouna Paul
Published: May 2010

We present a brief introduction to recovery swaps, their replication using no‐arbitrage arguments, and their applications. We also provide a brief survey of the literature on recovery rates.
Credit Default Swaption
Authors: Laurent Jean‐Paul
Published: May 2010

Credit default swap (CDS) options, also called
credit swaptions
, allow an investor to buy protection on a reference name at a specified spread. We define the cash flows associated with the option and describe a method for obtaining a Black formula for CDS options. One could think of other specifications, for instance associated with a shifted square root process for the short‐term credit spread of the reference entity, which can be put within a convenient Cox process framework. We discuss the merits of the two approaches.
Forward‐Starting
CDO
Tranche
Authors: Baheti Prasun , Mashal Roy , Naldi Marco
Published: May 2010

Reset tranches are a class of default‐path‐dependent structures that include forward‐starting contracts and subordination/leverage step‐ups. The attachment and/or width of the tranche are reset at a predetermined time (the reset date) as deterministic functions of the random amount of losses incurred by the reference portfolio up to that time. We discuss a simple algorithm for fast quasi‐analytical valuation of these products.
Leveraged Super‐Senior Tranche
Authors: Arnsdorf Matthias
Published: May 2010

A leveraged super‐senior (LSS) note is a structure that allows investors to take a leveraged exposure to the super‐senior (SS) part of a collateralized debt obligation (CDO). This provides an enhanced level of return while typically maintaining a AAA rating. Leverage is achieved by posting an initial collateral amount that is less than the notional of the underlying SS tranche. All credit losses to the investor are capped at the collateral amount, but the coupon is paid on the full notional. Early unwind clauses are typically included in the trade to mitigate the risk to the issuer that losses exceed the collateral amount. Compared to a standard SS tranche the investor is exposed to mark‐to‐market (MTM) risk as well as credit risk, that is, for certain market moves his/her principal could be reduced even if no credit losses have occurred owing to a forced unwind. The issuer faces a so‐called gap risk, the risk that the MTM of the tranche will fall below the initially posted collateral amount before a trade unwind can take place.
Managed CDO
Authors: Jobst Andreas A.
Published: May 2010

Managed collateralized debt obligations (CDOs) have become the most prevalent form of large‐scale securitization. As opposed to static structures, these CDOs refinance the market value of a dynamic reference portfolio (“blind pool”) that is administered like a managed fund subject to prespecified investment limits and guidelines.
Collateralized Debt Obligation (CDO) Options
Authors: Toutain Olivier
Published: May 2010

This article is a primer on the different types of derivative contracts based on synthetic CDO tranches. It describes the mechanisms of those contracts and the potential modeling solutions for their valuations. We briefly explain the advent of those contracts and their use during the boom years of 2001– 2008.
Intensity‐Based Credit Risk Models
Authors: Gregory Jon
Published: May 2010

This article outlines the use of reduced‐form models for pricing credit derivatives structures. The basis of reduced‐form models and their use in pricing credit default swaps (CDS) and bonds are described. The extension of reduced‐form models to more sophisticated credit derivatives structures such as options, hybrids, and portfolio products is also commented on.