ABSTRACT
This chapter is a first introduction to the main tools of Malliavin calculus. We define the main properties of the Malliavin derivative operator and its adjoint, the divergence operator and we study their main properties, as the integration- by-parts formula. We see how to compute Malliavin derivatives for different types of processes. In particular, we see that the Malliavin derivative of a diffusion process can be computed as the solution of a stochastic differential equation (SDE) and that, alternatively, it can be represented as the solution of an ordinary differential equation (ODE). We detail the explicit computations in several classical asset models, like the Black-Scholes and the CEV. We also study the case of stochastic volatilities as the SABR, the Heston, or the 3/2 model. Moreover, we compare numerically the efficiency of the different pro- posed computational methods. We see that Malliavin derivatives are not just abstract random variables but they can be computed explicitly for most models in finance. As examples, we have studied the Black-Scholes and the CEV model, as well as some common volatility models as Heston, SABR, and the 3/2 model.
