An application of Malliavin calculus is to calculate the sensitivity of financial Greeks.
However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to first specify a Hilbert space H, but I didn't see what it is for calculating the sensitivity of financial Greeks.
Can anyone give a guidance? Do we really need to firstly specify a Hilbert space H, and can we take derivative of ANY random variable?
Applying the Malliavin calculus in order to calculate sensitivies is a little bit difficult, since the Malliavin derivative is not as simple to use as the ordinary derivative operator.
Essentially (and a bit imprecisely), the Hilbert space you mention is a space of random variables which generate a $\sigma$-algebra with the property that you can take the "$\omega$-wise" derivative with respect to random variables measurable with respect to this $\sigma$-algebra. As a concrete example, if you consider a Brownian motion $W$, you will be able to take Malliavin derivatives of random variables such as $f(W_{t_1},\ldots,W_{t_n})$. Thus, the Hilbert space you mention is a way of specifying which random variables you can take the derivative with respect to.
However, taking derivatives of random variables is not really what you directly want when considering the calculation of sensitivities. In order to apply the Malliavin calculus for this, some more advanced results are necessary. Consider a $p$-dimensional SDE $$ dX^x_t = \mu(X^x_t) dt + \sigma(X^x_t)dW_t, X_0 = x $$ where $W$ is a Brownian motion. Thus, $X^x$ is the solution to the SDE starting in $x$. The solution process $X^x$ will be adapted to the filtration generated by the Brownian motion, and is therefore a candidate for the use of the Malliavin calculus. Under suitable regularity conditions, it holds that with $u(x) = Ef(X^x_T)$, where $$ \nabla u(x) = E\left(f(X^x_T)\frac{1}{T}\int_0^T (\sigma^{-1}(X^x_T)Y^x_T)^t dW_t\right) $$ where $Y^x$ is the Jacobian of $X^x$, that is, the derivative $\omega$-by-$\omega$ of the mapping $x\mapsto X^x$ from $\mathbb{R}^p$ to $\mathbb{R}^p$. This is a formula which proven using the Malliavin calculus, and it is useful for calculation of greeks: If $X$ were, say, the stock price process, then the above formula for the gradient would be the sensitivity of a contingent claim with expiry at time $T$ and payoff $u$, with respect to the initial stock price, in other words, the delta greek. The above formula and its proof can be found in E. Fournié et al.: Applications of Malliavin calculus to Monte Carlo methods in finance, Finance and Stochastics, 1999.
If you do not mind a bit of shameless self-promotion on my side, you can also read about the method and see some numerical examples in this master's thesis:
