Abstract
The Malliavin–Thalmaier formula was introduced in [P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance, Springer-Verlag, Berlin, 2006] as an alternative expression for the density of a multivariate smooth random variable in Wiener space. In comparison with classical integration by parts formulae, this alternative formulation requires the application of the integration by parts formula only once to obtain an expression that can be simulated. Therefore, this expression is free from the curse of dimensionality. Unfortunately, when this formula is applied directly in computer simulation, it exhibits unstable behavior. We propose an approximation to the Malliavin–Thalmaier formula in the spirit of the theory of kernel density estimation to solve this problem. In the first part of this paper, we obtain a central limit theorem for the estimation error. And in the latter part, we apply the Malliavin–Thalmaier formula for the calculation of Greeks in finance.
Highlights
Let (Q = C([O, T]; Rd), 'F, P) denote the canonical Wiener probability space whose canonical process denoted by W is a Wiener process
Field on n, 'F, is given by the smallest fF-field generated by the sets of the type {x E Q; x(t¡) E A¡, ... ,x(tn) E An} forn E N, t¡, ...,tn E [O,T] and A 1, . , An are Borel sets in Rd
The measure Pis such that the canonical process W: Qx[O, T] ---t Rd is a measurable map from the product fF-field 'F023([0, T])
Summary
The goal ofthe present article is to estirnate through sirnulations the probability density function ofF using Malliavin Calculus and discuss sorne of its applications, in Finance. The result applied to estirnate a density is the classical integration by parts formula of Malliavin Calculus that can be stated as follows. Malliavin and Thalmaier [6] (Section 4.5.) gave a new integration by parts formula that seems to alleviate the computational burden for simulation of densities in high dimension. The simulation results for our proposed approximation are as follows After obtaining these error estimations and the corresponding optimal parameter h, we app1y the Ma11iavin-Thalmaier formula to finance, especially to the calculation of Greeks. To avoid introducing further terminology, we will keep referring to pp(x) as the "density"
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