ABSTRACT
This chapter is devoted to present some key Malliavin calculus tools that we will use throughout the book, together with some direct financial applications. More precisely, we introduce the Clark-Ocone formula, the general integration-by- parts formula, and the anticipating Itˆo's formula. The Clark-Ocone-Haussman formula leads to an explicit martingale representation of random variables and its better-known application is the construction of hedging strategies. The in- tegration by parts formula allows us to avoid the computation of the derivatives in the estimation of the Greeks. More precisely, it gives us a way to represent these derivatives as the expectation of the product of the payoff times some random weight. As particular examples, we study the estimation of the delta, gamma, and vega under different models, as well as the delta for an Asian option under the Black-Scholes model. The anticipating Itˆo formula extends the clas- sical Itˆo's formula to the case of non-adapted processes, as the future average volatility. As an application, we see that, in stochastic volatility models, the characteristic function of an asset price can be decomposed as the sum of the characteristic function of a mixed log-normal distribution, plus a second term due to the impact of correlation.
