Malliavin Calculus for Monte Carlo Greeks on Exotic Derivatives
Summary
The document discusses whether Malliavin calculus has practical value in finance, especially for computing Greeks on exotic derivatives. Its main use case is a payoff with nonsmooth features, where finite-difference estimates can be difficult or unstable. Malliavin methods can express sensitivities as Monte Carlo expectations with alternative weighting schemes, but choosing a useful weight is a further problem. When the process distribution is tractable, likelihood ratio methods may provide a more direct or effective approach.
The answers emphasize that Malliavin calculus alone does not guarantee better estimates. Variance reduction and other techniques may improve results, and those improvements can sometimes make finite differences adequate. The document offers practitioner-oriented opinions rather than benchmark results or a worked comparison, and it notes a difference between academic interest and the practical goal of producing estimates useful to traders. Its advice is to assess methods on concrete examples.
Key ideas
- Malliavin calculus can help estimate Monte Carlo Greeks for exotic payoffs with nonsmooth features.
- The method yields alternative weighting schemes, but the weights still need to be chosen effectively.
- Likelihood ratio methods can be attractive when the dynamics have a tractable probability law.
- Variance reduction may matter as much as the choice of Greek estimation method.
- Practical value depends on whether the resulting estimates improve a specific application.
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# How do practitioners use the Malliavin calculus (if at all)? # How do practitioners use the Malliavin calculus (if at all)? This question is inspired by the remark due to Vladimir Piterbarg made in a related thread on Wilmott back in 2004: > 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. At the same time, it seems that recently there has been a rather noticeable flow of academic papers and books devoted to applications of the Malliavin calculus to finance (see, e.g., Malliavin Calculus for Lévy Processes with Applications to Finance by Di Nunno, Øksendal, and Proske, Stochastic Calculus of Variations in Mathematical Finance by Malliavin and Thalmaier, and the references therein). Question. So do practitioners actually use the Malliavin calculus to compute Greeks these days? Are there any other real-world financial applications of the Malliavin calculus? Or does Dr. Piterbarg's assessment of the practical potential of the theory remain to be essentially accurate? ## Answer by TheBridge (score 6, accepted) https://quant.stackexchange.com/a/703 Well the problems where Malliavin Calculus is applicable are mostly regarding greeks of exotic derivatives where some non smoothness in the payoff function creates trouble when trying to get this by finite difference methods. The thing is in my opinion that Malliavin Calculus is only an opening as it gives you basically an infinite number of ways to get those derivatives by Monte Carlo simulations. Then you have to determine an optimal weight, and it appears that when the probability law of the dynamic is tractable (maybe this is related to Piterbarg's comment)you can use Likelyhood Ratio Method to get those bad greeks and this method is optimal. Anyway Malliavin Calculus alone won't really improve the solution to the problem if you don't use some other techniques together with it(such variance reduction techniques), and it might be the case that those techniques provide sufficient improvements to make difference methods good enough and so to avoid to resort to Malliavin Calculus. But anyway it is ALWAYS a good thing to have as many as possible methods as you can when dealing with a problem. So Piterbarg comment is a little provocative and it should not prevent you from getting your own idea on the subject by implementing some Malliavin Calculus techniques on concrete examples. Here are a few references if you want to get acquainted with Malliavin Calculus and its application to Greek comptuations : Benhamou - Smart Monte Carlo, Various Trikcs using Malliavin Calculus Friz - An Introduction to Malliavin Calculus Glasserman - Malliavin Greeks without Malliavin Calculus (this one is fun) Prével - Greeks via Malliavin Regards. ## Answer by quant_dev (score 4) https://quant.stackexchange.com/a/771 Piterbarg (who's a very smart and no-nonsense guy) looks at the problem from the practitioner's perspective, i.e. "will this make the traders happy?". Academicians tend to answer the question "will this make the journal referees happy?". Two different goals.
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