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49
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 144 (4 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.
All in the Family: Nesting Symmetric and Asymmetric GARCH Models
 Journal of Financial Economics
, 1995
"... This paper develops a parametric family of models of generalized autoregressive heteroskedasticity (GARCH). The family nests the most popular symmetric and asymmetric GARCH models, thereby highlighting the relation between the models and their treatment of asymmetry. Furthermore, the structure perm ..."
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Cited by 90 (0 self)
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This paper develops a parametric family of models of generalized autoregressive heteroskedasticity (GARCH). The family nests the most popular symmetric and asymmetric GARCH models, thereby highlighting the relation between the models and their treatment of asymmetry. Furthermore, the structure permits nested tests of different ypes of asymmetry and functional forms. Daily U.S. stock return data reject all standard GARCH models in favor of a model in which, roughly speaking, the conditional standard deviation depends on the shifted absolute value of the shocks raised to the power three halves and past standard deviations.
JumpDi®usion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing
 Review of Derivatives Research
, 2000
"... Abstract. This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (Parti ..."
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Cited by 76 (2 self)
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Abstract. This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (Partial IntegroDifferential Equation) and demonstrates how this equation can be used to fit the model to European option prices. For numerical pricing of general contingent claims, we develop an ADI finite difference method that is shown to be unconditionally stable and, if combined with Fast Fourier Transform methods, computationally efficient. The paper contains several detailed examples from the S&P500 market.
DeltaHedged Gains and the Negative Market Volatility Risk Premium
 The Review of Financial Studies
, 2001
"... We investigate whether the volatility risk premium is negative by examining the statistical properties of deltahedged option portfolios (buy the option and hedge with stock). Within a stochastic volatility framework, we demonstrate a correspondence between the sign and magnitude of the volatility r ..."
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Cited by 73 (3 self)
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We investigate whether the volatility risk premium is negative by examining the statistical properties of deltahedged option portfolios (buy the option and hedge with stock). Within a stochastic volatility framework, we demonstrate a correspondence between the sign and magnitude of the volatility risk premium and the mean deltahedged portfolio returns. Using a sample of S&P 500 index options, we provide empirical tests that have the following general results. First, the deltahedged strategy underperforms zero. Second, the documented underperformance is less for options away from the money. Third, the underperformance is greater at times of higher volatility.Fourth, the volatility risk premium significantly affects deltahedged gains even after accounting for jumpfears. Our evidence is supportive of a negative market volatility risk premium.
An Equilibrium Guide to Designing Affine Pricing Models, Mathematical Finance forthcoming
, 2007
"... We examine equilibriummodels based on EpsteinZin preferences in a framework where exogenous state variables which drive consumption and dividend dynamics follow affine jump diffusion processes. Equilibrium asset prices can be computed using a standard machinery of affine asset pricing theory by im ..."
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Cited by 26 (9 self)
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We examine equilibriummodels based on EpsteinZin preferences in a framework where exogenous state variables which drive consumption and dividend dynamics follow affine jump diffusion processes. Equilibrium asset prices can be computed using a standard machinery of affine asset pricing theory by imposing parametric restrictions on market prices of risk, determined by preference and model parameters. We present a detailed example where large shocks (jumps) in consumption volatility translate into negative jumps in equilibrium prices of the assets. This endogenous ”leverage effect ” leads to significant premiums for outofthemoney put options. Our model is thus able to produce an equilibrium ”volatility smirk ” which realistically mimics that observed for index options. KEY WORDS: EpsteinZin preferences, affine asset pricing model, general equilibrium, option pricing ∗We thank two anonymous referees and the associate editor for valuable comments. We have also
Applied Stochastic Processes and Control for JumpDiffusions: Modeling, Analysis and Computation
 Analysis and Computation, SIAM Books
, 2007
"... Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions usi ..."
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Cited by 19 (7 self)
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Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theory of simple jumpdiffusions. Next, markedjumpdiffusions are treated using compound Poisson processes to include random marked jumpamplitudes in parallel with the equivalent Poisson random measure formulation. Otherwise, the approach is quite applied, using basic principles with no abstractions beyond Poisson random measure. This treatment is suitable for those in classical applied mathematics, physical sciences, quantitative finance and engineering, but have trouble getting started with the abstract measuretheoretic literature. The approach here builds upon the treatment of continuous functions in the regular calculus and associated ordinary differential equations by adding nonsmooth and jump discontinuities to the model. Finally, the stochastic optimal control of markedjumpdiffusions is developed, emphasizing the underlying assumptions. The survey concludes with applications in biology and finance, some of which are canonical, dimension reducible problems and others are genuine nonlinear problems. Key words. Jumpdiffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic differential equations, stochastic chain rules, stochastic optimal control AMS subject classifications. 60G20, 93E20, 93E03 1. Introduction. There
Numerical valuation of options with jumps in the underlying
, 2005
"... A jumpdiffusion model for a singleasset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integrodifferential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in tim ..."
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Cited by 15 (3 self)
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A jumpdiffusion model for a singleasset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integrodifferential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in time by the second order backward differentiation formula (BDF2). The resulting system is solved by an iterative method based on a simple splitting of the matrix. Using the fast Fourier transform, the amount of work per iteration may be reduced to O(n log 2 n) and only O(n) entries need to be stored for each time level. Numerical results showing the quadratic convergence of the methods are given for Merton’s model and Kou’s model.
ImplicitExplicit Numerical Schemes for JumpDiffusion Processes
 Calcolo
, 2004
"... We study the numerical approximation of viscosity solutions for Parabolic IntegroDifferential Equations (PIDE). Similar models arise in option pricing, to generalize the BlackScholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jum ..."
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Cited by 11 (3 self)
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We study the numerical approximation of viscosity solutions for Parabolic IntegroDifferential Equations (PIDE). Similar models arise in option pricing, to generalize the BlackScholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the nonlocal nature of the integral term, unconditionally stable implicit difference scheme are not practically feasible. Here we propose to use ImplicitExplicit (IMEX) RungeKutta methods for the time integration to solve the integral term explicitly, giving higher order accuracy schemes under weak stability timestep restrictions. Numerical tests are presented to show the computational efficiency of the approximation.
Bayesian Option Pricing Using Asymmetric Garch
 CORE DP 9759, LouvainlaNeuve
, 1997
"... This paper shows how one can compute option prices from a Bayesian inference viewpoint, using an econometric model for the dynamics of the return and of the volatility of the underlying asset. The proposed evaluation of an option is the predictive expectation of its payoff function. The predictive d ..."
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Cited by 10 (1 self)
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This paper shows how one can compute option prices from a Bayesian inference viewpoint, using an econometric model for the dynamics of the return and of the volatility of the underlying asset. The proposed evaluation of an option is the predictive expectation of its payoff function. The predictive distribution of this function provides a natural metric with respect to which the predictive option price, or other option evaluations, can be gauged. The proposed method is compared to the Black and Scholes evaluation, in which a predictive mean volatility is plugged, but which does not provide a natural metric. The methods are illustrated using an asymmetric GARCH model with a data set on a stock index in Brussels. The persistence of the volatility process is linked to the prediction horizon and to the option maturity.
Option Pricing for a StochasticVolatility JumpDiffusion Model with LogUniform JumpAmplitudes
, 2006
"... An alternative option pricing model is proposed, in which the stock prices follow a diffusion model with square root stochastic volatility and a jump model with loguniformly distributed jump amplitudes in the stock price process. The stochasticvolatility follows a squareroot and meanreverting d ..."
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Cited by 7 (2 self)
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An alternative option pricing model is proposed, in which the stock prices follow a diffusion model with square root stochastic volatility and a jump model with loguniformly distributed jump amplitudes in the stock price process. The stochasticvolatility follows a squareroot and meanreverting diffusion process. Fourier transforms are applied to solve the problem for riskneutral European option pricing under this compound stochasticvolatility jumpdiffusion (SVJD) process. Characteristic formulas and their inverses simplified by integration along better equivalent contours are given. The numerical implementation of pricing formulas is accomplished by both fast Fourier transforms (FFTs) and more highly accurate discrete Fourier transforms (DFTs) for verifying results and for different output.