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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 1512 (6 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
Nonparametric Estimation of StatePrice Densities Implicit In Financial Asset Prices
 JOURNAL OF FINANCE
, 1997
"... Implicit in the prices of traded financial assets are ArrowDebreu prices or, with continuous states, the stateprice density (SPD). We construct a nonparametric estimator for the SPD implicit in option prices and derive its asymptotic sampling theory. This estimator provides an arbitragefree metho ..."
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Cited by 339 (6 self)
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Implicit in the prices of traded financial assets are ArrowDebreu prices or, with continuous states, the stateprice density (SPD). We construct a nonparametric estimator for the SPD implicit in option prices and derive its asymptotic sampling theory. This estimator provides an arbitragefree method of pricing new, complex, or illiquid securities while capturing those features of the data that are most relevant from an assetpricing perspective, e.g., negative skewness and excess kurtosis for asset returns, volatility "smiles" for option prices. We perform Monte Carlo experiments and extract the SPD from actual S&P 500 option prices.
An empirical investigation of continuoustime equity return models
 Journal of Finance
, 2002
"... This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of timevarying intensity. We find that any reasonably descriptive continuoustime model for equityindex returns must allow for discrete jumps as well as stochastic volatility with a pronou ..."
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Cited by 240 (12 self)
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This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of timevarying intensity. We find that any reasonably descriptive continuoustime model for equityindex returns must allow for discrete jumps as well as stochastic volatility with a pronounced negative relationship between return and volatility innovations. We also find that the dominant empirical characteristics of the return process appear to be priced by the option market. Our analysis indicates a general correspondence between the evidence extracted from daily equityindex returns and the stylized features of the corresponding options market prices. MUCH ASSET AND DERIVATIVE PRICING THEORY is based on diffusion models for primary securities. However, prescriptions for practical applications derived from these models typically produce disappointing results. A possible explanation could be that analytic formulas for pricing and hedging are available for only a limited set of continuoustime representations for asset returns
The Dynamics of Stochastic Volatility: Evidence from Underlying and Option Markets
, 2000
"... This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultane ..."
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Cited by 159 (3 self)
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This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a timevarying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for shortdated options.
Of Smiles and Smirks: A TermStructure Perspective
 JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS
, 1998
"... An extensive empirical literature in finance has documented not only the presence of anamolies in the BlackScholes model, but also the "termstructures" of these anamolies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities). Theoretical ..."
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Cited by 140 (5 self)
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An extensive empirical literature in finance has documented not only the presence of anamolies in the BlackScholes model, but also the "termstructures" of these anamolies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities). Theoretical efforts in the literature at addressing these anamolies have largely focussed on two extensions of the BlackScholes model: introducing jumps into the return process, and allowing volatility to be stochastic. This paper employs commonlyused versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anamolies. We find that each model exhibits some "termstructure" patterns that are fundamentally inconsistent with those observed in the data. As a consequence, neither class of models constitutes an adequate explanation of the empirical evidence, although stochastic volatility models fare better than jumps in this regard.
Telling From Discrete Data Whether the Underlying ContinuousTime Model is a Diffusion
, 2001
"... ..."
Pricing and Hedging Spread Options
 SIAM Review
, 2003
"... Abstract. We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present ..."
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Cited by 59 (8 self)
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Abstract. We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.
Implied Volatility Skews and Stock Index Skewness and Kurtosis Implied by S&P 500 Index Option Prices
 Journal of Derivatives
, 1997
"... The BlackScholes (1973) option pricing model is used to value a wide range of option contracts. However, the model often inconsistently prices deep inthemoney and deep outofthemoney options. Options ’ professionals refer to this phenomenon as a volatility ‘skew ’ or ‘smile. ’ In this paper, we ..."
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Cited by 29 (0 self)
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The BlackScholes (1973) option pricing model is used to value a wide range of option contracts. However, the model often inconsistently prices deep inthemoney and deep outofthemoney options. Options ’ professionals refer to this phenomenon as a volatility ‘skew ’ or ‘smile. ’ In this paper, we apply an extension of the BlackScholes model developed by Jarrow and Rudd (1982) to an investigation of S&P 500 index option prices. We find that nonnormal skewness and kurtosis in optionimplied distributions of index returns contribute significantly to the phenomenon of volatility skews.