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31
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 97 (2 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.
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 46 (2 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.
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 18 (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
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 10 (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 9 (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.
ACCURATE EVALUATION OF EUROPEAN AND AMERICAN OPTIONS UNDER THE CGMY PROCESS
"... A finitedifference method for integrodifferential equations arising from Lévy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be secondorder ..."
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Cited by 5 (0 self)
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A finitedifference method for integrodifferential equations arising from Lévy driven asset processes in finance is discussed. The equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula. The discretization is shown to be secondorder accurate independently of the degree of the singularity in the Lévy measure. The singularity is dealt with by means of an integration by parts technique. An application of the fast Fourier transform gives the overall amount of work O(MN log N), rendering the method fast.
Continuousdiscrete unscented kalman filtering
, 2005
"... The unscented Kalman filter (UKF) is formulated for the continuousdiscrete state space model. The exact moment equations are solved approximately by using the unscented transform (UT) and the measurement update is obtained by computing the normal correlation, again using the UT. In contrast to the ..."
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Cited by 5 (1 self)
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The unscented Kalman filter (UKF) is formulated for the continuousdiscrete state space model. The exact moment equations are solved approximately by using the unscented transform (UT) and the measurement update is obtained by computing the normal correlation, again using the UT. In contrast to the usual treatment, the system and measurement noise sequences are included from the start and are not treated later by extension of the state vector. The performance of the UKF is compared to Taylor expansions (extended Kalman filter EKF, second and higher order nonlinear filter SNF, HNF), the Gaussian filter, and simulated Monte Carlo filters using a bimodal GinzburgLandau model and the chaotic Lorenz model.
The Importance of the Loss Function in Option Pricing
, 2001
"... Which loss function should be used when estimating and evaluating option pricing models? Many different functions have been suggested, but no standard has emerged. We do not promote a particular function, but instead emphasize that consistency in the choice of loss functions is crucial. First, for a ..."
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Cited by 4 (0 self)
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Which loss function should be used when estimating and evaluating option pricing models? Many different functions have been suggested, but no standard has emerged. We do not promote a particular function, but instead emphasize that consistency in the choice of loss functions is crucial. First, for any given model, the loss function used in parameter estimation and model evaluation should be identical, otherwise suboptimal parameter estimates will be obtained. Second, when comparing models, the estimation loss function should be identical across models, otherwise unfair comparisons will be made. We illustrate the importance of these issues in an application of the socalled Practitioner BlackScholes (PBS) model to S&P500 index options. We find reductions of over 50 percent in the root mean squared error of the PBS model when the estimation and evaluation loss functions are aligned. We also find that the PBS model outperforms a benchmark structural model when the estimation loss functions are identical across models, but otherwise not. The new PBS model with aligned loss functions thus represents a much tougher benchmark against which future structural models can be compared.
Volatility Surface: Theory, Rules of Thumb, and Empirical Evidence
, 2003
"... Implied volatilities are frequently used to quote the prices of options. The implied volatility of a European option on a particular asset as a function of strike price and time to maturity is known as the asset’s volatility surface. Traders monitor movements in volatility surfaces closely. In this ..."
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Cited by 2 (0 self)
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Implied volatilities are frequently used to quote the prices of options. The implied volatility of a European option on a particular asset as a function of strike price and time to maturity is known as the asset’s volatility surface. Traders monitor movements in volatility surfaces closely. In this paper we develop a noarbitrage condition for the evolution of a volatility surface. We examine a number of rules of thumb used by traders to manage the volatility surface and test whether they are consistent with the noarbitrage condition and with data on the trading of options on the S&P 500 taken