# Malliavin Calculus

From a quant point of view, how would you explain Malliavin calculus in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing.

What would they bring to someone who has already learned stochastic Calculus with Ito's integral?

Would they be more useful for front office or middle office?

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I would suggest splitting this into two separate questions: one on Malliavin Calculus and one on Multi-Fractals Models. I could then maybe provide some comments on Malliavin Calculus: looking at the "proxy simulation scheme" technique can be useful, since it is a "discrete analog" (on the level of the discretization scheme, like Eurler scheme) of what Malliavin calculus does in the continuous setup. –  Christian Fries Jun 17 '13 at 13:46
here you go, thx for your comment. –  lmorin Jun 17 '13 at 14:57
In my mostly uneducated view, one purpose is to get quantitative formulations of the martingale representation theorem, i.e. go beyond existence to actually constructing what the integrand should be in the representation. –  quasi Jun 18 '13 at 15:49
I've struggled to find an easily readable introduction to the subject. Have you found anything that fits the bill? Especially one that comes at the problem from a probabilistic point of view, instead of analytic. –  quasi Jun 18 '13 at 15:50

I think this question has no easy answer but I'll give it a shot anyway (beware: oversimplification ahead!).

The main idea of the Malliavin calculus is to be able to differentiate stochastic processes like Brownian motion (or more general martingales with bounded quadratic variation), which are not differentiable in the traditional sense (because of their infinite variation).

Insofar the Malliavin calculus is the natural counterpart for stochastic differentiation to what the Ito calculus is for stochastic integration.

One of the practical application of Malliavin calculus is in the area of calculating option Greeks which makes sense since you would suspect that you needed derivatives to calculate these.

The main problem with the traditional approach is that the derivative needs to be approximated using the ﬁnite diﬀerence method and such approximations can become very rough. The integration by parts formula obtained from Malliavin calculus can transform a derivative into a weighted integral of random variables. This gives a more accurate and fast converging numerical solution than obtained by the classical method.

Some parts of the following thesis (on which parts of this answer are based too) might be helpful to dive deeper into the matter: The Malliavin calculus by Han Zhang.

To dive deeper into the practical applications (plus a primer on Malliavin calculus at the end!) can be found here: Smart Monte Carlo: Various Tricks Using Malliavin Calculus by Eric Benhamou.

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Thanks, that is exactly what I was looking for. –  lmorin Jun 19 '13 at 11:58
If you have some experience in multi fractals models, can you take a look here: quant.stackexchange.com/questions/8242/multi-fractals-models –  lmorin Jun 19 '13 at 12:41

If you want a really intuitive answer, I thought of two things to explain the key idea:

From a Masters student point of view:

Assume $\xi \sim N(0, \sigma^2)$, prove that:

$\mathbb{E}[f(\xi)\xi] = \sigma^2 \mathbb{E}[f'(\xi)] = \mathbb{E}[\xi^2]\mathbb{E}[f'(\xi)]$

A more intuitive financial explanation may go like this:

Consider you have a binary option, with payout as shown below:

Essentially taking a finite difference (red) ends up creating massive problems at the discontinuity, which may make perturbing the underlying density of the stock price the correct idea.

Its simple but may provide basic intuition.

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