Abstract
The main purpose of this paper is to derive unbiased Monte Carlo estimators of various sensitivity indices for an averaged asset price dynamics governed by the gamma Lévy process. The key idea is to apply a scaling property of the gamma process with respect to the Esscher density transform parameter. Our framework covers not only the continuous Asian option, but also European, discrete Asian, average strike Asian, weighted average, spread options, and geometric average Asian options. Numerical results are provided to illustrate the effectiveness of our formulas in Monte Carlo simulations, relative to finite difference approximation.
Highlights
It has been widely known that the logarithmic derivatives of the density function of stochastic differential equations correspond essentially to the so-called Greeks, that is, sensitivity indices with respect to various model parameters of asset price dynamics
The pioneer work in this direction is of Fournieet al. [11], whose approach is based upon the integration-byparts formula developed in the Malliavin calculus on the Wiener space
Various types of Malliavin calculus and logarithmic derivatives have been studied on the Poisson space or on the Wiener-Poisson space
Summary
It has been widely known that the logarithmic derivatives of the density function of stochastic differential equations correspond essentially to the so-called Greeks, that is, sensitivity indices with respect to various model parameters of asset price dynamics. It is most important to have closed, but not necessarily unique, weights on hand, in order to design an efficient Monte Carlo evaluation In this direction, Greeks formulas are obtained in Davis and Johansson [9] and Cass and Friz [8] for jump diffusion processes. Kawai and Takeuchi [14] studied the computation of the Greeks of European payoffs for an asset price dynamics defined with gamma processes and with Brownian motions, possibly time-changed by one of the gamma processes. We close this study with some numerical results to illustrate remarkable improvements in Monte Carlo simulations in terms of estimator variance relative to the finite difference estimation
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