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76
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
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 ...
Estimation of stochastic volatility models via Monte Carlo Maximum Likelihood
, 1998
"... This paper discusses the Monte Carlo maximum likelihood method of estimating stochastic volatility (SV) models. The basic SV model can be expressed as a linear state space model with log chisquare disturbances. The likelihood function can be approximated arbitrarily accurately by decomposing it int ..."
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Cited by 64 (6 self)
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This paper discusses the Monte Carlo maximum likelihood method of estimating stochastic volatility (SV) models. The basic SV model can be expressed as a linear state space model with log chisquare disturbances. The likelihood function can be approximated arbitrarily accurately by decomposing it into a Gaussian part, constructed by the Kalman filter, and a remainder function, whose expectation is evaluated by simulation. No modifications of this estimation procedure are required when the basic SV model is extended in a number of directions likely to arise in applied empirical research. This compares favorably with alternative approaches. The finite sample performance of the new estimator is shown to be comparable to the Monte Carlo Markov chain (MCMC) method.
Prediction in dynamic models with time dependent conditional heteroskedasticity, Working paper no
, 1990
"... This paper considers forecasting the conditional mean and variance from a singleequation dynamic model with autocorrelated disturbances following an ARMA process, and innovations with timedependent conditional heteroskedasticity as represented by a linear GARCH process. Expressions for the minimum ..."
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Cited by 43 (7 self)
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This paper considers forecasting the conditional mean and variance from a singleequation dynamic model with autocorrelated disturbances following an ARMA process, and innovations with timedependent conditional heteroskedasticity as represented by a linear GARCH process. Expressions for the minimum MSE predictor and the conditional MSE are presented. We also derive the formula for all the theoretical moments of the prediction error distribution from a general dynamic model with GARCHtl, 1) innovations. These results are then used in the construction of ex ante prediction confidence intervals by means of the CornishFisher asymptotic expansion. An empirical example relating to the uncertainty of the expected depreciation of foreign exchange rates illustrates the usefulness of the results. 1.
Markov Switching in GARCH Processes and Mean Reverting Stock Market Volatility
 Journal of Business and Economic Statistics
, 1997
"... This paper introduces four models of conditional heteroscedasticity that contain markov switching parameters to examine their multiperiod stockmarket volatility forecasts as predictions of optionsimplied volatilities. The volatility model that best predicts the behavior of the optionsimplied vol ..."
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Cited by 18 (2 self)
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This paper introduces four models of conditional heteroscedasticity that contain markov switching parameters to examine their multiperiod stockmarket volatility forecasts as predictions of optionsimplied volatilities. The volatility model that best predicts the behavior of the optionsimplied volatilities allows the studentt degreesoffreedom parameter to switch such that the conditional variance and kurtosis are subject to discrete shifts. The halflife of the most leptokurtic state is estimated to be a week, so expected market volatility reverts to nearnormal levels fairly quickly following a spike. keywords: conditional heteroscedasticity; asset price volatility; kurtosis; markov switching 1 1. Introduction Volatility clustering is a welldocumented feature of financial rates of return: Price changes that are large in magnitude tend to occur in bunches rather than with equal spacing. A natural question is how long financial markets will remain volatile, because volatility...
Which GARCH Model for Option Valuation
 Management Science
, 2004
"... Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a timeseries of asset ..."
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Cited by 17 (4 self)
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Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a timeseries of asset returns. This paper compares a range of GARCH models along a different dimension, using option prices and returns under the riskneutral as well as the physical probability measure. We judge the relative performance of various models by evaluating an objective function based on option prices. In contrast with returnsbased inference, we find that our optionbased objective function favors a relatively parsimonious model. Specifically, when evaluated outofsample, our analysis favors a model that besides volatility clustering only allows for a standard leverage effect. JEL Classification: G12
The Risk Premium of Volatility Implicit in Currency Options
 Journal of Business and Economic Statistics
, 1998
"... This paper provides an empirical investigation of the risk neutral variance process and the market price of variance risk implied in the foreign currency options market. There are three principal contributions. First, the parameters of Heston's (1993) meanreverting square root stochastic volatili ..."
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Cited by 17 (1 self)
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This paper provides an empirical investigation of the risk neutral variance process and the market price of variance risk implied in the foreign currency options market. There are three principal contributions. First, the parameters of Heston's (1993) meanreverting square root stochastic volatility model are estimated using dollar/mark option prices from 1987 to 1992. Second, it is shown that these implied parameters can be combined with historical moments of the dollar/mark exchange rate to deduce an estimate of the market price of variance risk. These estimates are found to be nonzero, time varying, and of sufficient magnitude to imply that the compensation for variance risk is a significant component of the risk premia in the currency market. Finally, the outofsample test suggests that the historical variance and the Hull and White implied variance contain no additional information than those imbedded in the Heston implied variance. KEY WORDS: Market price of variance ...