Numerical computation of Theta in a jump-diffusion model by integration by parts

Abstract

Using the Malliavin calculus in time inhomogeneous jump-diffusion models, we obtain an expression for the sensitivity Theta of an option price (with respect to maturity) as the expectation of the option payoff multiplied by a stochastic weight. This expression is used to design efficient numerical algorithms that are compared with traditional finite-difference methods for the computation of Theta. Our proof can be viewed as a generalization of Dupire's integration by parts to arbitrary and possibly non-smooth payoff functions. In the time homogeneous case, Theta admits an expression from the Black-Scholes PDE in terms of Delta and Gamma but the representation formula obtained in this way is different from ours. Numerical simulations are presented in order to compare the efficiency of the finite-difference and Malliavin methods.Applied mathematical finance, European financial markets, Computational finance, Financial mathematics,

Optimal stopping and sensitivity analysis in regime switching models

“…In our model, there exists such a problem too. We can minimize the weight by choosing a optimal g ∈ G. We gain a similar conclusion to that in David and Privault (2009).…”

“…By Debelley and Privault (2004b), sensitivities in a jump diffusion model are computed by using the Malliavin calculus. By Davis and Johansson (2006) David and Privault (2009) obtain an expression for the sensitivity Theta of an option price as the expectation of the option payoff multiplied by a stochastic weight. Also, Bayazit and Nolder (2013) present sensitivities for options when the underlying dynamic follows an exponential Lévy process.…”

“…Integration by parts formulas can be based on multiple stochastic integral expansions as on the Wiener space, or on the absolute continuity of jump times, see e.g. Chapters 6 and 7 of Privault (2009). In the case of discrete systems driven by Markov chains, discrete multiple stochastic integral expansions for random variables have been constructed in e.g.…”

“…In this subsection, we aim at figuring out an optimal choice of g for efficient Monte Carlo simulation. Recall David and Privault (2009), based on the computation of T heta in a jump-diffusion model, Proposition 4.1 of David and Privault (2009) minimizes the variance of the weight of T heta. In our model, there exists such a problem too.…”

“…In the case of David and Privault (2009), it is proved that the best function chosen in the weight is the constant number. However, in our case, {g(i) = cσ(i) ; i ∈ M} obtains a faster convergence than {g(i) = 1 ; i ∈ M} does, as shown in the following graph (4.1).…”