Greeks Formulas for an Asset Price Model With Gamma Processes

Abstract

Greeks formulas of Delta, Rho, Vega, and Gamma are derived in closed form for asset price dynamics described by gamma processes and Brownian motions time-changed by a gamma process. The model considered here includes many well-known models of practical interest, such as the variance gamma model and the Black-Scholes model. Our approach is based upon the Malliavin calculus for jump processes by making full use of a scaling property of gamma processes with respect to the Girsanov transform. The existence of their variance is investigated. Numerical results are provided to illustrate that the derived Greeks formulas have faster rate of convergence relative to the finite difference method.

Sensitivity analysis for averaged asset price dynamics with gamma processes

“…First, setting µ(dt) = T δ T (dt) reduces to the European framework. With this setting, our results indeed recover the European formulas derived in Kawai and Takeuchi [14]. As a matter of course, µ(dt) = dt corresponds to the Asian option (also called average price option), while in real financial trading, the continuous average is simply impossible to compute and is thus not very interesting.…”

“…In addition, Asian options of geometric average type are also within our scope. This work can also be thought of as a considerable generalization of Kawai and Takeuchi [14], in the sense that some of the European Greeks formulas there can be recovered from our results. Finally, our approach taken in this paper can also be applied in various stochastic systems involving gamma processes.…”

“…In this paper, we take an approach again based on the scaling property, not of the gamma distribution as in Kawai and Takeuchi [14], but of the gamma Lévy process. This, rather simple, extension widens the applicability to the averaged asset price dynamics defined with the gamma process.…”

“…Takeuchi [17] studied the same problem for the stochastic differential equations with jumps via a martingale approach. 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. In Kawai and Takeuchi [14], various well known financial models, such as the BlackScholes model and the variance gamma model of Madan, Carr and Chang [15], are within its framework.…”

“…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. In Kawai and Takeuchi [14], various well known financial models, such as the BlackScholes model and the variance gamma model of Madan, Carr and Chang [15], are within its framework. The key idea there is to apply a scaling property of the gamma distribution with respect to the Esscher transform.…”