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Journal ArticleDOI

A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options

Steven L. Heston1
01 Apr 1993-Review of Financial Studies (Oxford University Press)-Vol. 6, Iss: 2, pp 327-343
TL;DR: In this paper, a closed-form solution for the price of a European call option on an asset with stochastic volatility is derived based on characteristi c functions and can be applied to other problems.
Abstract: I use a new technique to derive a closed-form 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 strike-price biases in the BlackScholes (1973) model. The solution technique is based on characteristi c functions and can be applied to other problems.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors examined the pricing of aggregate volatility risk in the cross-section of stock returns and found that stocks with high sensitivities to innovations in aggregate volatility have low average returns.
Abstract: We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, Journal of Financial Economics 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility. IT IS WELL KNOWN THAT THE VOLATILITY OF STOCK RETURNS varies over time. While considerable research has examined the time-series relation between the volatility of the market and the expected return on the market (see, among others, Campbell and Hentschel (1992) and Glosten, Jagannathan, and Runkle (1993)), the question of how aggregate volatility affects the cross-section of expected stock returns has received less attention. Time-varying market volatility induces changes in the investment opportunity set by changing the expectation of future market returns, or by changing the risk-return trade-off. If the volatility of the market return is a systematic risk factor, the arbitrage pricing theory or a factor model predicts that aggregate volatility should also be priced in the cross-section of stocks. Hence, stocks with different sensitivities to innovations in aggregate volatility should have different expected returns. The first goal of this paper is to provide a systematic investigation of how the stochastic volatility of the market is priced in the cross-section of expected stock returns. We want to both determine whether the volatility of the market

2,936 citations


Cites background from "A Closed-Form Solution for Options ..."

  • ...A two-factor pricing kernel with the market return and stochastic volatility as factors is also the standard setup commonly assumed by many stochastic option pricing studies (see, for example, Heston (1993))....

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Book
01 Jan 2000
TL;DR: Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.
Abstract: This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behaviour of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.

2,826 citations

Journal ArticleDOI
TL;DR: In this article, an option pricing model that allows volatility, interest rates and jumps to be stochastic is presented. But it is not known whether and by how much each generalization improves option pricing and hedging.
Abstract: Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant timeseries data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance. IN THE LAST TWO DECADES, option pricing has witnessed an explosion of new

2,777 citations


Cites background or methods from "A Closed-Form Solution for Options ..."

  • ...…V), (8) where the risk-neutral probabilities, 1I1 and 2, are recovered from inverting the respective characteristic functions (see Bates (1996a,c), Heston (1993), and Scott (1997) for similar treatments): HIj(t, T; S (t), R (t), V(t)) 1 1 f0 [e`On[K]fj(t, T, S(t) I tR (, (t); 4)1 2 + - j e R()d…...

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  • ...…model of Cox and Ross (1976); (iv) the Markovian models of Rubinstein (1994) and Alt-Sahalia and Lo (1996); (v) the stochastic-volatility models of Heston (1993), Hull and White (1987a), Melino and Turnbull (1990, 1995), Scott (1987), Stein and Stein (1991), and Wiggins (1987); (vi) the…...

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  • ...Second, this general model allows for a flexible correlation structure between the stock return and its volatility, as opposed to the perfect correlation assumed in, for instance, Heston (1993)....

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Journal ArticleDOI
TL;DR: In this paper, the fast Fourier transform is used to value options when the characteristic function of the return is known analytically, and it is shown how to use it for value selection.
Abstract: This paper shows how the fast Fourier Transform may be used to value options when the characteristic function of the return is known analytically.

2,306 citations


Cites methods from "A Closed-Form Solution for Options ..."

  • ...Characteristic functions have also been used in the pure diffusion context with stochastic volatility (Heston 1993) and with stochastic interest rates (Bakshi and Chen 1997)....

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Journal ArticleDOI
TL;DR: In this article, the authors provide an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical Treatment of a range of valuation and econometric problems.
Abstract: In the setting of ‘‘affine’’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

2,114 citations

References
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Journal ArticleDOI
TL;DR: In this paper, a theoretical valuation formula for options is derived, based on the assumption that options are correctly priced in the market and it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks.
Abstract: If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.

28,434 citations


"A Closed-Form Solution for Options ..." refers background in this paper

  • ...Many plaudits have been aptly used to describe Black and Scholes’ (1973) contribution to option pricing theory....

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  • ...Standard arbitrage arguments [Black and Scholes (1973), Merton (1973)] demonstrate that the value of any asset U(S, v, t) (including accrued payments) must satisfy the partial differential equation (PDE) (6) The unspecified term (S, v, t) represents the price of volatility risk, and must be…...

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Book
12 Sep 2011
TL;DR: In this paper, the authors deduced a set of restrictions on option pricing formulas from the assumption that investors prefer more to less, which are necessary conditions for a formula to be consistent with a rational pricing theory.
Abstract: The long history of the theory of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion with zero drift. Since that time, numerous researchers have contributed to the theory. The present paper begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new "down-and-out" option are derived. The effects of dividends and call provisions on the warrant price are examined. The possibilities for further extension of the theory to the pricing of corporate liabilities are discussed.

9,635 citations

Book
01 Jan 1994
TL;DR: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
Abstract: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes

7,270 citations

Journal ArticleDOI
TL;DR: In this paper, the authors use an intertemporal general equilibrium asset pricing model to study the term structure of interest rates and find that anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices.
Abstract: This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. 1. INTRODUCTION THE TERM STRUCTURE of interest rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relationship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market's anticipations of future events. An explanation of the term structure gives us a way to extract this information and to predict how changes in the underlying variables will affect the yield curve. In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, previous theories of the term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalizations of the certainty equilibrium relationships. The literature in the area is voluminous, and a comprehensive survey would warrant a paper in itself. It is common, however, to identify much of the previous work in the area as belonging to one of four strands of thought. First, there are various versions of the expectations hypothesis. These place predominant emphasis on the expected values of future spot rates or holdingperiod returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that the implied forward rates are equal to the expected spot rates. Generally, this approach is characterized by the following propositions: (a) the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities. The liquidity preference hypothesis, advanced by Hicks [16], concurs with the importance of expected future spot rates, but places more weight on the effects of the risk preferences of market participants. It asserts that risk aversion will cause forward rates to be systematically greater than expected spot rates, usually

7,014 citations