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 FORUMS > The Quantitative Finance FAQs Project < refresh >
 Topic Title: What is Malliavin Calculus? Created On Thu May 27, 04 11:10 PM Topic View: Branch View Threaded (All Messages) Threaded (Single Messages) Linear

Paul
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Posts: 5984
Joined: Jul 2001

Thu May 27, 04 11:10 PM

Thank you, sehrlich...

chiral3
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Posts: 1722
Joined: Nov 2002

Fri May 28, 04 03:29 PM

Oh, a challenge to describe this in FAQ/easy terms....

Malliavin calculus is sometime refered to as a variational calculus for stochastic calculus.

The idea is that you form shifts on Weiner spaces. The family of shifts, y, form a Cameron-Martin space. From this you can form the Malliavin derivative D (wrt e) of (x + ey). Let e go to zero. Since the derivative is defined, one can continue to define the calculus......

The two papers for learning this are Oksendals' and Peter Fritz (sp?)

PS - really it is just a caculus of functions on Wiener spaces. One of the powers of it is that, via integration by parts that naturally falls out of the calculus, when computing various greek sensitivities, you can move the derivative off the payoff functional. The payoff functional is a function of X_t's, given by the usual BS SDE, and its expectation defines the contigent claim on the security. Since this this SUCKS so bad, it can be 1) impossible or 2) computationally expensive to take derivatives. The Malliavsin integration by parts takes the derivative off the functional. This way Greeks can be computed. It extends to other, more, exotic things involving S.

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I am clearly an idiot.

Edited: Wed Jun 09, 04 at 06:40 PM by chiral3

SU2
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Posts: 32
Joined: Sep 2004

Sat Oct 02, 04 04:04 PM

I guess where Malliavin calculus really becomes powerful the ability to deal with densities induced by a fairly arbitrary stochatic differential equation, without assuming any parametric forms of it. The power Malliavin calculus in that aspect was first demonstrated via a probabilistic proof of Hormander's sum-of-squares theorem.

Now, if you recall the Black and Scholes option price; where the price of the option was expressed in terms of Phi, the cummulative distribution function of a normal random variable. Hence, any sensitivities of the option price would more or less be dependant on the existence and smoothness of the underlying density. This is precisely the point where Malliavin calculus takes over, and it works for asset dynamics governed by very general classes of stochastic differential equations.

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piterbarg
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Mon Nov 15, 04 10:27 PM

Not to be a party-pooper, but Malliavin calculus is essentially useless in finance. Any practical result ever obtained with Malliavin calculus can be obtained by much simpler methods by eg differentiating the density of the underlying process
-V

muniangel
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Posts: 110
Joined: May 2005

Mon May 23, 05 08:26 AM

chiral3,can you upload the two papers you recommanded?

TKS,Im interesting it malliavin calculus!

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What is essence of science?

muniangel
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Posts: 110
Joined: May 2005

Mon May 23, 05 08:27 AM

or email:munianguo@yahoo.com

TKs again

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What is essence of science?

Friesenstein
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Joined: Nov 2005

Thu Nov 03, 05 08:56 AM

The author of "Introduction to Malliavin Calculus" (mentioned in chiral3's reply) is Peter K. Friz. You may download the PDF from his website at cambridge university..

Regards
C

Senior Member

Posts: 11057
Joined: Feb 2005

Sun Nov 13, 05 12:38 AM

Beautiful.
Quote

Some Lecture Notes:

Heat Kernels, Parabolic PDEs and Diffusion Processes

An Introduction to Malliavin Calculus

Some Geometric Aspects in Diffusion Theory

This is it.

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That the ultimate felicity of man consists in the contemplation of God – St. Thomas Aquinas.

player
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Posts: 2284
Joined: Aug 2002

Mon Nov 14, 05 01:00 PM

I was asked to define this in layman terms in an interview recently.

This is what I said:

It is in effect the attempt to carry out standard calculus to BM. You take BM and bump it up at a specific point in time in a certain direction. And only bump it up at that time. At no other time is it bumped up.

I'll be interested to know what the experts in MC think of this defn... And remember this had to be defined in layman terms

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IF YOU CAN IMAGINE IT YOU CAN ACHIEVE IT: IF YOU CAN DREAM IT YOU CAN BECOME IT

Escher
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Joined: Jan 2008

Fri Jul 11, 08 11:53 PM

Quote

Originally posted by: piterbarg
Not to be a party-pooper, but Malliavin calculus is essentially useless in finance. Any practical result ever obtained with Malliavin calculus can be obtained by much simpler methods by eg differentiating the density of the underlying process
-V

This is correct, but the real strength of malliavin calculus is that it allows you to calculate weights which act as unbiased estimators for greeks without explicitly knowing the density of the underlying process. It is sufficient to know the dynamics of the process.

phdquant
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Fri Apr 24, 09 06:01 PM

If you allow me Malliavin calculus find a new application in the recent hot topic of asymptotic development of implied volatility of options priced in a general stochastic volatility model. This subject is currently actively looked by Osajima, Ustunel, Nourdin,Malliavin and many other stochastic analysists. It is quite extraordinary how this approach converge to the Henry Labordere methods (geometry on manifolds) and give similar results or on old perturbation approachs such as Hagan's.

If I am correct Labordere is a former employee of you Vladimir , think he actually works for Lorenzo Bergomi at SocGen.

I have some papers and can advise on theoritical side :
*Nualar (malliavin Calculus)
*Oksendal
*Fritz
*Baly

I will check for correct spelling of those great persons and articles in a few hours

jfuqua
Senior Member

Posts: 1255
Joined: Jul 2002

Sat Apr 25, 09 04:04 PM

See Technical Forum under: Malliavin Calculus with Stochastic Vol. models for a file with refernces.
For some reason this Forum does not allow attachments

crougeaux
Junior Member

Posts: 7
Joined: Jun 2011

Sun Jul 24, 11 03:28 PM

Hi

I programmed Malliavin calculus in order to approximate the delta and the gamma of an option.
You can test it here:

http://pricing-option.com/MonteCarlo.aspx

Christophe