bluuebunny/crossref_metadata_2025_split huggingface dataset:
Options: Basic Definitions
Authors: Davis Mark H.A. 
Published: May 2010

Abstract
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

Abstract

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.
Arbitrage Bounds
Authors: Cox Alexander 
Published: May 2010

AbstractA 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

Abstract
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

Abstract
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

Abstract
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.
Vanna–Volga Pricing
Authors: Wystup Uwe 
Published: May 2010

Abstract

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.
Butterfly
Authors: Liverance Eric 
Published: May 2010

AbstractA 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

Abstract

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

Abstract
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

Abstract

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

Abstract
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.
Himalayan Option
Authors: Gharavi Reza K. 
Published: May 2010

Abstract
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.
Exchange‐Traded Funds (ETFs)
Authors: Tari Michael J. 
Published: May 2010

Abstract
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.
Volatility Index Options
Authors: Benhamou Eric , Ciuca Marian 
Published: May 2010

AbstractVolatility 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

AbstractWe 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.
Implied Volatility Surface
Authors: Kamal Michael , Gatheral Jim 
Published: May 2010

Abstract
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

Abstract

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

Abstract
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

Abstract

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

Abstract

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.
Normal Inverse
G
aussian Model
Authors: Jönsson Henrik , Masol Viktoriya , Schoutens Wim 
Published: May 2010

Abstract

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.
Creditrisk+
Authors: Wilde Tom 
Published: May 2010

Abstract
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.
Saddlepoint Approximation
Authors: Hurd Thomas R. 
Published: May 2010

Abstract
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

Abstract
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

AbstractA 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.
Exposure to Default and Loss Given Default
Authors: Resti Andrea 
Published: May 2010

AbstractTwo 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.
Counterparty Credit Risk
Authors: Pykhtin Michael 
Published: May 2010

Abstract

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.
Total Return Swap
Authors: Wehn Carsten S. 
Published: May 2010

Abstract
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.