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117
A closedform solution for options with stochastic volatility with applications to bond and currency options
 Review of Financial Studies
, 1993
"... I use a new technique to derive a closedform solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond option ..."
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Cited by 704 (4 self)
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I use a new technique to derive a closedform solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strikeprice biases in the BlackScholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems. Many plaudits have been aptly used to describe Black and Scholes ’ (1973) contribution to option pricing theory. Despite subsequent development of option theory, the original BlackScholes formula for a European call option remains the most successful and widely used application. This formula is particularly useful because it relates the distribution of spot returns I thank Hans Knoch for computational assistance. I am grateful for the suggestions of Hyeng Keun (the referee) and for comments by participants
The Variance Gamma Process and Option Pricing.
 European Finance Review
, 1998
"... : A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional par ..."
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Cited by 197 (26 self)
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: A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model d...
Do stock prices and volatility jump? Reconciling evidence from spot and option prices
, 2001
"... This paper studies the empirical performance of jumpdiffusion models that allow for stochastic volatility and correlated jumps affecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the insample estimation ..."
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Cited by 97 (2 self)
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This paper studies the empirical performance of jumpdiffusion models that allow for stochastic volatility and correlated jumps affecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the insample estimation period. This contrasts previous findings where stochastic volatility paths are found to be too smooth relative to the option implied dynamics. While the models perform well during the high volatility estimation period, they tend to overprice long dated contracts outofsample. This evidence points towards a too simplistic specification of the mean dynamics of volatility.
New Insights Into Smile, Mispricing and Value At Risk: The Hyperbolic Model
 Journal of Business
, 1998
"... We investigate a new basic model for asset pricing, the hyperbolic model, which allows an almost perfect statistical fit of stock return data. After a brief introduction into the theory supported by an appendix we use also secondary market data to compare the hyperbolic model to the classical Black ..."
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Cited by 80 (7 self)
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We investigate a new basic model for asset pricing, the hyperbolic model, which allows an almost perfect statistical fit of stock return data. After a brief introduction into the theory supported by an appendix we use also secondary market data to compare the hyperbolic model to the classical BlackScholes model. We study implicit volatilities, the smile effect and the pricing performance. Exploiting the full power of the hyperbolic model, we construct an option value process from a statistical point of view by estimating the implicit riskneutral density function from option data. Finally we present some new valueat risk calculations leading to new perspectives to cope with model risk. I Introduction There is little doubt that the BlackScholes model has become the standard in the finance industry and is applied on a large scale in everyday trading operations. On the other side its deficiencies have become a standard topic in research. Given the vast literature where refinements a...
A Study towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options Valuation
, 1999
"... The purpose of this paper is to bridge two strands of the literature, one pertaining to the objectiveorphysical measure used to model the underlying asset and the other pertaining to the riskneutral measure used to price derivatives. We propose a generic procedure using simultaneously the fundame ..."
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Cited by 74 (4 self)
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The purpose of this paper is to bridge two strands of the literature, one pertaining to the objectiveorphysical measure used to model the underlying asset and the other pertaining to the riskneutral measure used to price derivatives. We propose a generic procedure using simultaneously the fundamental price S t and a set of option contracts ### I it # i=1;m # where m # 1 and # I it is the BlackScholes implied volatility.We use Heston's #1993# model as an example and appraise univariate and multivariate estimation of the model in terms of pricing and hedging performance. Our results, based on the S&P 500 index contract, show that the univariate approach only involving options by and large dominates. Abyproduct of this #nding is that we uncover a remarkably simple volatility extraction #lter based on a polynomial lag structure of implied volatilities. The bivariate approachinvolving both the fundamental and an option appears useful when the information from the cash market ...
Maximum likelihood estimation for stochastic volatility models
 JOURNAL OF FINANCIAL ECONOMICS
, 2007
"... We develop and implement a method for maximum likelihood estimation in closedform of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure ..."
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Cited by 48 (3 self)
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We develop and implement a method for maximum likelihood estimation in closedform of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by proxies based on the implied volatility of a shortdated atthemoney option. The approximation results in a small loss of accuracy relative to the standard errors due to sampling noise. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine Heston model and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.
Complete Models with Stochastic Volatility
, 1996
"... The paper proposes an original class of models for the continuous time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentiallyweighted moments of historic logprice. The instantaneous volatility is therefore driven ..."
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Cited by 42 (3 self)
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The paper proposes an original class of models for the continuous time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentiallyweighted moments of historic logprice. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that unlike many other models of nonconstant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preferenceindependent options prices. We find a partial differential equation for the price of a European Call Option. Smiles and skews are found in the resulting plots of implied volatility. Keywords: Option pricing, stochastic volatility, complete markets, smiles. Acknowledgement. It is a pleasure to thank the referees of an earlier draft of this paper whose perceptive comments have resulted in many improvements. 1 Research supported in part by Record Treasu...
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.