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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
Modeling Term Structures of Defaultable Bonds
, 1999
"... This article presents convenient reducedform models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a creditspread option ..."
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Cited by 423 (23 self)
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This article presents convenient reducedform models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a creditspread option
Transform Analysis and Asset Pricing for Affine JumpDiffusions
 Econometrica
, 2000
"... In the setting of ‘‘affine’ ’ jumpdiffusion 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 applicat ..."
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Cited by 384 (32 self)
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In the setting of ‘‘affine’ ’ jumpdiffusion 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 fixedincome pricing models, with a role for intensitybased models of default, as well as a wide range of optionpricing 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.
2000): “Specification Analysis of Affine Term Structure Models
 Journal of Finance
"... This paper explores the structural differences and relative goodnessoffits of affine term structure models ~ATSMs!. Within the family of ATSMs there is a tradeoff between flexibility in modeling the conditional correlations and volatilities of the risk factors. This tradeoff is formalized by our ..."
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Cited by 336 (30 self)
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This paper explores the structural differences and relative goodnessoffits of affine term structure models ~ATSMs!. Within the family of ATSMs there is a tradeoff between flexibility in modeling the conditional correlations and volatilities of the risk factors. This tradeoff is formalized by our classification of Nfactor affine family into N � 1 nonnested subfamilies of models. Specializing to threefactor ATSMs, our analysis suggests, based on theoretical considerations and empirical evidence, that some subfamilies of ATSMs are better suited than others to explaining historical interest rate behavior. IN SPECIFYING A DYNAMIC TERM STRUCTURE MODEL—one that describes the comovement over time of short and longterm bond yields—researchers are inevitably confronted with tradeoffs between the richness of econometric representations of the state variables and the computational burdens of pricing and estimation. It is perhaps not surprising then that virtually all of the empirical implementations of multifactor term structure models that use time series data on long and shortterm bond yields simultaneously have focused on special cases of “affine ” term structure models ~ATSMs!.AnATSM accommodates timevarying means and volatilities of the state variables through affine specifications of the riskneutral drift and volatility coefficients. At the same time, ATSMs yield essentially closedform expressions for zerocouponbond prices ~Duffie and Kan ~1996!!, which greatly facilitates pricing and econometric implementation. The focus on ATSMs extends back at least to the pathbreaking studies by Vasicek ~1977! and Cox, Ingersoll, and Ross ~1985!, who presumed that the instantaneous short rate r~t! was an affine function of an Ndimensional state vector Y~t!, r~t! � d 0 � d y Y~t!, and that Y~t! followed Gaussian and squareroot diffusions, respectively. More recently, researchers have explored formulations of ATSMs that extend the onefactor Markov represen
Term Premia and Interest Rate Forecasts in Affine Models
, 2001
"... I find that the standard class of a#ne models produces poor forecasts of future changes in Treasury yields. Better forecasts are generated by assuming that yields follow random walks. The failure of these models is driven by one of their key features: The compensation that investors receive for faci ..."
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Cited by 250 (8 self)
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I find that the standard class of a#ne models produces poor forecasts of future changes in Treasury yields. Better forecasts are generated by assuming that yields follow random walks. The failure of these models is driven by one of their key features: The compensation that investors receive for facing risk is a multiple of the variance of the risk. This means that risk compensation cannot vary independently of interest rate volatility. I also describe and empirically estimate a class of models that is broader than the standard a#ne class. These "essentially a#ne" models retain the tractability of the usual models, but allow the compensation for interest rate risk to vary independently of interest rate volatility. This additional flexibility proves useful in forming accurate forecasts of future yields. Address correspondence to the University of California, Haas School of Business, 545 Student Services Building #1900, Berkeley, CA 94720. Phone: 5106421435. Email address: du#ee@haas.b...
A Markov Model for the Term Structure of Credit Risk Spreads
 Review of Financial Studies
, 1997
"... This article provides a Markov model for the term structure of credit risk spreads. The model is based on Jarrow and Turnbull (1995), with the bankruptcy process following a discrete state space Markov chain in credit ratings. The parameters of this process are easily estimated using observable data ..."
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Cited by 237 (12 self)
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This article provides a Markov model for the term structure of credit risk spreads. The model is based on Jarrow and Turnbull (1995), with the bankruptcy process following a discrete state space Markov chain in credit ratings. The parameters of this process are easily estimated using observable data. This model is useful for pricing and hedging corporate debt with imbedded options, for pricing and hedging OTC derivatives with counterparty risk, for pricing and hedging (foreign) government bonds subject to default risk (e.g., municipal bonds), for pricing and hedging credit derivatives, and for risk management. This article presents a simple model for valuing risky debt that explicitly incorporates a firm's credit rating as an indicator of the likelihood of default. As such, this article presents an arbitragefree model for the term structure of credit risk spreads and their evolution through time. This model will prove useful for the pricing and hedging of corporate debt with We would like to thank John Tierney of Lehman Brothers for providing the bond index price data, and Tal Schwartz for computational assistance. We would also like to acknowledge helpful comments received from an anonymous referee. Send all correspondence to Robert A. Jarrow, Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853. The Review of Financial Studies Summer 1997 Vol. 10, No. 2, pp. 481523 1997 The Review of Financial Studies 08939454/97/$1.50 imbedded options, for the pricing and hedging of OTC derivatives with counterparty risk, for the pricing and hedging of (foreign) government bonds subject to default risk (e.g., municipal bonds), and for the pricing and hedging of credit derivatives (e.g. credit sensitive notes and spread adjusted notes). This model can also...
The JumpRisk Premia Implicit in Options: Evidence from an Integrated TimeSeries Study
 Journal of Financial Economics
"... Abstract: This paper examines the joint time series of the S&P 500 index and nearthemoney shortdated option prices with an arbitragefree model, capturing both stochastic volatility and jumps. Jumprisk premia uncovered from the joint data respond quickly to market volatility, becoming more promi ..."
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Cited by 210 (1 self)
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Abstract: This paper examines the joint time series of the S&P 500 index and nearthemoney shortdated option prices with an arbitragefree model, capturing both stochastic volatility and jumps. Jumprisk premia uncovered from the joint data respond quickly to market volatility, becoming more prominent during volatile markets. This form of jumprisk premia is important not only in reconciling the dynamics implied by the joint data, but also in explaining the volatility “smirks” of crosssectional options data.
Explaining the rate spread on corporate bonds
 Journal of Finance
, 2001
"... The purpose of this article is to explain the spread between spot rates on corporate and government bonds. We find that the spread can be explained in terms of three elements: (1) compensation for expected default of corporate bonds (2) compensation for state taxes since holders of corporate bonds p ..."
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Cited by 207 (3 self)
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The purpose of this article is to explain the spread between spot rates on corporate and government bonds. We find that the spread can be explained in terms of three elements: (1) compensation for expected default of corporate bonds (2) compensation for state taxes since holders of corporate bonds pay state taxes while holders of government bonds do not, and (3) compensation for the additional systematic risk in corporate bond returns relative to government bond returns. The systematic nature of corporate bond return is shown by relating that part of the spread which is not due to expected default or taxes to a set of variables which have been shown to effect risk premiums in stock markets Empirical estimates of the size of each of these three components are provided in the paper. We stress the tax effects because it has been ignored in all previous studies of corporate bonds. 1
Testing ContinuousTime Models of the Spot Interest Rate
 Review of Financial Studies
, 1996
"... Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are rec ..."
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Cited by 194 (7 self)
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Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then meanreverts strongly when far away from the mean. The volatility is higher when away from the mean. The continuoustime financial theory has developed extensive tools to price derivative securities when the underlying traded asset(s) or nontraded factor(s) follow stochastic differential equations [see Merton (1990) for examples]. However, as a practical matter, how to specify an appropriate stochastic differential equation is for the most part an unanswered question. For example, many different continuoustime The comments and suggestions of Kerry Back (the editor) and an anonymous referee were very helpful. I am also grateful to George Constantinides,
Post'87 Crash Fears in the S&P 500 Futures Option Market
, 1998
"... Postcrash distributions inferred from S ..."