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81
A JumpDiffusion Model for Option Pricing
 Management Science
, 2002
"... Brownian motion and normal distribution have been widely used in the Black–Scholes optionpricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (as ..."
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Cited by 114 (3 self)
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Brownian motion and normal distribution have been widely used in the Black–Scholes optionpricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called “volatility smile ” in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jumpdiffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of optionpricing problems, including call and put options, interest rate derivatives, and pathdependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.
Recovering Risk Aversion from Option Prices and Realized Returns. Manuscript
, 1998
"... A relationship exists between aggregate riskneutral and subjective probability distributions and risk aversion functions. Using a variation of the method developed by Jackwerth and Rubinstein (1996), we estimate riskneutral probabilities reliably from option prices. Subjective probabilities are es ..."
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Cited by 104 (3 self)
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A relationship exists between aggregate riskneutral and subjective probability distributions and risk aversion functions. Using a variation of the method developed by Jackwerth and Rubinstein (1996), we estimate riskneutral probabilities reliably from option prices. Subjective probabilities are estimated from realized returns. This paper then introduces a technique to empirically derive risk aversion functions implied by option prices and realized returns simultaneously. These risk aversion functions dramatically change shapes around the 1987 crash: Precrash, they are positive and decreasing in wealth and thus consistent with standard economic theory. Postcrash, they are partially negative and increasing and irreconcilable with the theory. Overpricing of outofthemoney puts is the most likely cause. A simulated trading strategy exploiting this overpricing shows excess returns even after accounting for the possibility of further crashes and transaction costs. * Jens Carsten Jackwerth is a visiting assistant professor at the London Business School. For helpful discussions I
Calibrating Volatility Surfaces Via RelativeEntropy Minimization
 Applied Mathematical Finance
, 1997
"... We present a framework for calibrating a pricing model to a prescribed set of option prices quoted... ..."
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Cited by 42 (2 self)
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We present a framework for calibrating a pricing model to a prescribed set of option prices quoted...
Derivative asset analysis in models with leveldependent and stochastic volatility
 CWI QUARTERLY
, 1996
"... In this survey we discuss models with leveldependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical BlackScholes model; they have been developed in an effort to build models that are flexible enough to cope wit ..."
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Cited by 38 (1 self)
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In this survey we discuss models with leveldependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical BlackScholes model; they have been developed in an effort to build models that are flexible enough to cope with the known deficits of the classical BlackScholes model. We start by briefly recalling the standard theory for pricing and hedging derivatives in complete frictionless markets and the classical BlackScholes model. After a review of the known empirical contradictions to the classical BlackScholes model we consider models with leveldependent volatility. Most of this survey is devoted to derivative asset analysis in stochastic volatility models. We discuss several recent developments in the theory of derivative pricing under incompleteness in the context of stochastic volatility models and review analytical and numerical approaches to the actual computation of option values.
Reconstructing The Unknown Local Volatility Function
 Journal of Computational Finance
, 1998
"... Using market European option prices, a method for computing a smooth local volatility function in a 1factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphas ..."
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Cited by 35 (5 self)
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Using market European option prices, a method for computing a smooth local volatility function in a 1factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.
A Market Model For Stochastic Implied Volatility
, 1998
"... In this paper a stochastic volatility model is presented that directly prescribes the stochastic development of the implied BlackScholes volatilities of a set of given standard options. Thus the model is able to capture the stochastic movements of a full term structure of implied volatilities. T ..."
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Cited by 27 (1 self)
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In this paper a stochastic volatility model is presented that directly prescribes the stochastic development of the implied BlackScholes volatilities of a set of given standard options. Thus the model is able to capture the stochastic movements of a full term structure of implied volatilities. The conditions are derived that have to be satisfied to ensure absence of arbitrage in the model and its numerical implementation is discussed.
Semiparametric Pricing of Multivariate Contingent Claims
, 1999
"... This paper derives and implements a nonparametric, arbitragefree technique for multivariate contingent claims (MVCC) pricing. This technique is based on nonparametric estimation of a multivariate riskneutral density function using data from traded options markets and historical asset returns. “New ..."
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Cited by 27 (2 self)
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This paper derives and implements a nonparametric, arbitragefree technique for multivariate contingent claims (MVCC) pricing. This technique is based on nonparametric estimation of a multivariate riskneutral density function using data from traded options markets and historical asset returns. “New ” multivariate claims are priced using expectations under this measure. An appealing feature of nonparametric arbitragefree derivative pricing is that fitted prices are obtained that are consistent with traded option prices and are not based on specific restrictions on the underlying asset price process or the functional form of the riskneutral density. Nonparametric MVCC pricing utilizes the method of copulas to combine nonparametrically estimated marginal riskneutral densities (based on options data) into a joint density using a separately estimated nonparametric dependence function (based on historical returns data). This paper provides theory linking objective and riskneutral dependence functions, and empirically testable conditions that justify the use of historical data for estimation of the riskneutral dependence function. The nonparametric MVCC pricing technique is implemented for the valuation of bivariate underperformance and outperformance options on the S&P500 and DAX index. Price deviations are
Implied Trinomial Trees of the Volatility Smile
 Journal of Derivatives
, 1996
"... This material is for your private information, and we are not soliciting any action based upon it. This report is not to be construed as an offer to sell or the solicitation of an offer to buy any security in any jurisdiction where such an offer or solicitation would be illegal. Certain transactions ..."
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Cited by 25 (0 self)
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This material is for your private information, and we are not soliciting any action based upon it. This report is not to be construed as an offer to sell or the solicitation of an offer to buy any security in any jurisdiction where such an offer or solicitation would be illegal. Certain transactions, including those involving futures, options and high yield securities, give rise to substantial risk and are not suitable for all investors. Opinions expressed are our present opinions only. The material is based upon information that we consider reliable, but we do not represent that it is accurate or complete, and it should not be relied upon as such. We, our affiliates, or persons involved in the preparation or issuance of this material, may from time to time have long or short positions and buy or sell securities, futures or options identical with or related to those mentioned herein. This material has been issued by Goldman, Sachs & Co. and/or one of its affiliates and has been approved by Goldman Sachs International, regulated by The Securities and Futures Authority, in connection with its distribution in the United Kingdom and by Goldman Sachs Canada in connection with its distribution in Canada. This material is distributed in Hong Kong by Goldman Sachs (Asia) L.L.C., and in Japan by Goldman Sachs (Japan) Ltd. This material is not for distribution to private customers, as defined by the rules of The Securities and Futures Authority in the United Kingdom, and any investments including any convertible bonds or derivatives mentioned in this material will not be made available by us to any such private customer. Neither Goldman, Sachs & Co. nor its
Computation of Deterministic Volatility Surfaces
 Journal Comp. Finance
, 1998
"... The `volatility smile' is one of the wellknown biases of BlackScholes models for pricing options. In this paper, we introduce a robust method of reducing this bias by pricing subject to a deterministic functional volatility oe = oe(S; t). This instantaneous volatility is chosen as a spline whose w ..."
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Cited by 22 (0 self)
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The `volatility smile' is one of the wellknown biases of BlackScholes models for pricing options. In this paper, we introduce a robust method of reducing this bias by pricing subject to a deterministic functional volatility oe = oe(S; t). This instantaneous volatility is chosen as a spline whose weights are determined by a regularised numerical strategy that approximately minimises the difference between BlackScholes vanilla prices and known market vanilla prices over a range of strikes and maturities; these BlackScholes prices are calculated by solving the relevant partial differential equation numerically using finite element methods. The instantaneous volatility generated from vanilla options can be used to price exotic options where the skew and termstructure of volatility are important, and we illustrate the application to barrier options. Contents 1 Introduction 4 2 Spacetime spline representation of oe(S; t) 7 3 Regularised optimisation strategy 9 3.1 The costfunctiona...