Back to Sensitivity Computation

We present and analyze three types of sensitivity computation for a general diffusion process with a focus on the sensitivity with respect to the initial value. First, an in-depth analysis of the (elementary but popular) finite difference method—also known as the shock or bump method—is carried out with both a constant and decreasing step settings. Then the tangent process method, based on pathwise differentiation, is described for smooth enough payoffs. As a third method, we revisit the log-likelihood method applied to the Euler scheme of the diffusion. As a conclusion, we provide some first stakes on the way to Malliavin calculus by proving the formulas of both Haussmann-Clark-Ocone and Bismut. We show how to use them for the computations of various sensitivities. Numerical experiments illustrate the typical behavior of these methods depending on the regularity of the payoffs under consideration.