Optimal Malliavin weighting functions for the simulations of the Greeks MC 2000 (July ) Eric Ben-Hamou Financial Markets Group London School of Economics, UK

3-5 July 2000MC 2000 ConferenceSlide N°2 Outline Introduction & motivations Review of the literature Results on weighting functions Numerical results Conclusion

3-5 July 2000MC 2000 ConferenceSlide N°3 Introduction When calculating numerically a quantity –Do we converge? to the right solution? –How fast is the convergence? Typically the case of MC/QMC simulations especially for the Greeks important measure of risks, emphasized by traditional option pricing theory.

3-5 July 2000MC 2000 ConferenceSlide N°4 –Finite difference approximations: “bump and re- compute” –Errors on differentiation as well as convergence! –Theoretical Results: Glynn (89) Glasserman and Yao (92) L’Ecuyer and Perron (94): –smooth function to estimate: - independent random numbers: non centered scheme: convergence rate of n -1/4 centered scheme n -1/3 - common random numbers: centered scheme n -1/2 –rates fall for discontinuous payoffs Traditional method for the Greeks

3-5 July 2000MC 2000 ConferenceSlide N°5 How to solve the poor convergence? Extensive litterature: –Broadie and Glasserman (93, 96) found, in simple cases, a convergence rate of n -1/2 by taking the derivative of the density function. Likelihood ratio method. –Curran (94): Take the derivative of the payoff function. –Fournié, Lasry, Lions, Lebuchoux, Touzi (97, 2000) Malliavin calculus reduces the variance leading to the same rate of convergence n -1/2 but in a more general framework. –Lions, Régnier (2000) American options and Greeks –Avellaneda Gamba (2000) Perturbation of the vector of probabilities. –Arturo Kohatsu-Higa (2000) study of variance reduction –Igor Pikovsky (2000): condition on the diffusion.

3-5 July 2000MC 2000 ConferenceSlide N°6 Common link: All these techniques try to avoid differentiating the payoff function: Broadie and Glasserman (93) –Weight = likelihood ratio –should know the exact form of the density function

3-5 July 2000MC 2000 ConferenceSlide N°7 Fournié, Lasry, Lions, Lebuchoux, Touzi (97, 2000) : “ Malliavin” method does not require to know the density but the diffusion. Weighting function independent of the payoff. Very general framework. infinity of weighting functions. Avellaneda Gamba (2000) other way of deriving the weighting function. inspired by Kullback Leibler relative entropy maximization.

3-5 July 2000MC 2000 ConferenceSlide N°8 Natural questions There is an infinity of weighting functions: –can we characterize all the weighting functions? –how do we describe all the weighting functions? How do we get the solution with minimal variance? –is there a closed form? –how easy is it to compute? Pratical point of view: –which option(s)/ Greek should be preferred? (importnace of maturity, volatility)

3-5 July 2000MC 2000 ConferenceSlide N°9 Weighting function description Notations (complete probability space, uniform ellipticity, Lipschitz conditions…) Contribution is to examine the weighting function as a Skorohod integral and to examine the “weighting function generator”

3-5 July 2000MC 2000 ConferenceSlide N°10 Integration by parts Conditions…Notations Chain rule Leading to

3-5 July 2000MC 2000 ConferenceSlide N°11 Necessary and sufficient conditions Condition Expressing the Malliavin derivative

3-5 July 2000MC 2000 ConferenceSlide N°12 Minimum variance of Solution: The conditional expectation with respect to Result: The optimal weight does depend on the underlying(s) involved in the payoff Minimal weighting function?

3-5 July 2000MC 2000 ConferenceSlide N°13 For European options, BS Type of Malliavin weighting functions:

3-5 July 2000MC 2000 ConferenceSlide N°14 Typology of options and remarks Remarks: –Works better on second order differentiation… Gamma, but as well vega. –Explode for short maturity. –Better with higher volatility, high initial level –Needs small values of the Brownian motion (so put call parity should be useful)

3-5 July 2000MC 2000 ConferenceSlide N°15 Finite difference versus Malliavin method Malliavin weighted scheme: not payoff sensitive Not the case for “bump and re-price” –Call option

3-5 July 2000MC 2000 ConferenceSlide N°16 For a call For a Binary option

3-5 July 2000MC 2000 ConferenceSlide N°17 Simulations (corridor option)

3-5 July 2000MC 2000 ConferenceSlide N°18 Simulations (corridor option)

3-5 July 2000MC 2000 ConferenceSlide N°19 Simulations (Binary option)

3-5 July 2000MC 2000 ConferenceSlide N°20 Simulations (Binary option)

3-5 July 2000MC 2000 ConferenceSlide N°21 Simulations (Call option)

3-5 July 2000MC 2000 ConferenceSlide N°22 Simulations (Call option)

3-5 July 2000MC 2000 ConferenceSlide N°23 Conclusion Gave elements for the question of the weighting function. Extensions: –Stronger results on Asian options –Lookback and barrier options –Local Malliavin –Vega-gamma parity