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23
A finite difference scheme for option pricing in jump diffusion and exponential Lévy models
, 2003
"... We present a finite difference method for solving parabolic partial integrodierential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a timeinhomogeneous jumpdiffusio ..."
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Cited by 31 (1 self)
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We present a finite difference method for solving parabolic partial integrodierential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a timeinhomogeneous jumpdiffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicitimplicit finite dierence scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Our convergence analysis requires neither the smoothness of the solution nor the nondegeneracy of coefficients and applies to European and barrier options in jumpdiffusion and pure jump models used in the literature. Numerical tests are performed with smooth and nonsmooth initial conditions.
2004): “Model specification and risk premiums: Evidence from futures options,” Working paper, Columbia University, forthcoming in Journal of Finance
"... There are two central issues in option pricing: selecting an appropriate model and quantifying the risk premiums of the various underlying factors. In this paper, we use the information in the crosssection of S&P futures options from 1987 to 2003 to examine these issues. We first test for the prese ..."
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Cited by 17 (2 self)
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There are two central issues in option pricing: selecting an appropriate model and quantifying the risk premiums of the various underlying factors. In this paper, we use the information in the crosssection of S&P futures options from 1987 to 2003 to examine these issues. We first test for the presence of jumps in volatility by analyzing the higher moment behavior of option implied variance. The option data provides strong evidence supporting the presence of jumps in volatility. In conjunction with previous results, this implies that stochastic volatility, jumps in returns and jumps in volatility are all important components. Next, we find strong crosssectional evidence in support of jumps in returns, and modest evidence for jumps in volatility. We find evidence for reasonable jump risk premiums, but do not find any evidence for a diffusive volatility risk premium. We also find strong evidence for time variation in thejumpriskpremiums.
Recovering Volatility from Option Prices by Evolutionary Optimization
, 2004
"... We propose a probabilistic approach for estimating parameters of an option pricing model from a set of observed option prices. Our approach is based on a stochastic optimization algorithm which generates a random sample from the set of global minima of the insample pricing error and allows for the ..."
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Cited by 14 (4 self)
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We propose a probabilistic approach for estimating parameters of an option pricing model from a set of observed option prices. Our approach is based on a stochastic optimization algorithm which generates a random sample from the set of global minima of the insample pricing error and allows for the existence of multiple global minima. Starting from an IID population of candidate solutions drawn from a prior distribution of the set of model parameters, the population of parameters is updated through cycles of independent random moves followed by "selection" according to pricing performance. We examine conditions under which such an evolving population converges to a sample of calibrated models. The heterogeneity...
Retrieving Lévy processes from option prices: Regularization of an illposed inverse problem
, 2000
"... We propose a stable nonparametric method for constructing an option pricing model of exponential Lévy type, consistent with a given data set of option prices. After demonstrating the illposedness of the usual and least squares version of this inverse problem, we suggest to regularize the calibratio ..."
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Cited by 10 (2 self)
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We propose a stable nonparametric method for constructing an option pricing model of exponential Lévy type, consistent with a given data set of option prices. After demonstrating the illposedness of the usual and least squares version of this inverse problem, we suggest to regularize the calibration problem by reformulating it as the problem of finding an exponential Lévy model that minimizes the sum of the pricing error and the relative entropy with respect to a prior exponential Lévy model. We prove the existence of solutions for the regularized problem and show that it yields solutions which are continuous with respect to the data, stable with respect to the choice of prior and converge to the minimumentropy least square solution of the initial problem.
Spectral calibration of exponential Lévy models
 Finance and Stochastics
"... This research was supported by the Deutsche ..."
A jumpdiffusion Libor model and its robust calibration
 Preprint 1113, Weierstraß Institute (WIAS
, 2006
"... In this paper we propose a jumpdiffusion Libor model with jumps in a highdimensional space (R m) and test a stable nonparametric calibration algorithm which takes into account a given local covariance structure. The algorithm returns smooth and simply structured Lévy densities, and penalizes the ..."
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Cited by 6 (5 self)
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In this paper we propose a jumpdiffusion Libor model with jumps in a highdimensional space (R m) and test a stable nonparametric calibration algorithm which takes into account a given local covariance structure. The algorithm returns smooth and simply structured Lévy densities, and penalizes the deviation from the Libor market model. In practice, the procedure is FFT based, thus fast, easy to implement, and yields good results, particularly in view of the severe illposedness of the underlying inverse problem. 1
Hedging with options in models with jumps
 in "Proceedings of the II Abel Symposium 2005 on Stochastic analysis and applications
, 2005
"... in honor of Kiyosi Ito’s 90th birthday. We consider the problem of hedging a contingent claim, in a market where prices of traded assets can undergo jumps, by trading in the underlying asset and a set of traded options. We give a general expression for the hedging strategy which minimizes the varian ..."
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Cited by 6 (3 self)
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in honor of Kiyosi Ito’s 90th birthday. We consider the problem of hedging a contingent claim, in a market where prices of traded assets can undergo jumps, by trading in the underlying asset and a set of traded options. We give a general expression for the hedging strategy which minimizes the variance of the hedging error, in terms of integral representations of the options involved. This formula is then applied to compute hedge ratios for common options in various models with jumps, leading to easily computable expressions. The performance of these hedging strategies is assessed through numerical experiments.
ADAPTIVE WEAK APPROXIMATION OF DIFFUSIONS WITH JUMPS ∗
"... Abstract. This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler–Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretiz ..."
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Cited by 3 (0 self)
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Abstract. This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler–Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows and discrete dual backward functions. Combined with proper estimation of the statistical error, they lead to efficient and accurate computation of global error estimates, extending the results by A. Szepessy, R. Tempone, and G. E. Zouraris [Comm. Pure Appl. Math., 54 (2001), pp. 1169–1214]. Adaptive algorithms for either deterministic or trajectorydependent time stepping are proposed. Numerical examples show the performance of the proposed error approximations and the adaptive schemes. AMS subject classifications. diffusions with jumps, weak approximation, error control, Euler–Maruyama method, a posteriori error estimates, backward dual functions