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Optimal Malliavin weighting functions for the simulations of the Greeks

Optimal Malliavin weighting functions for the simulations of the Greeks. MC 2000 (July 3-5 2000) Eric Ben-Hamou Financial Markets Group London School of Economics, UK benhamou_e@yahoo.com. Outline. Introduction & motivations Review of the literature Results on weighting functions

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Optimal Malliavin weighting functions for the simulations of the Greeks

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  1. Optimal Malliavin weighting functions for the simulations of the Greeks MC 2000 (July 3-5 2000) Eric Ben-Hamou Financial Markets Group London School of Economics, UK benhamou_e@yahoo.com

  2. Outline • Introduction & motivations • Review of the literature • Results on weighting functions • Numerical results • Conclusion MC 2000 Conference Slide N°2

  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. MC 2000 Conference Slide N°3

  4. Traditional method for the Greeks • 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 MC 2000 Conference Slide N°4

  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. MC 2000 Conference Slide N°5

  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 MC 2000 Conference Slide N°6

  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. MC 2000 Conference Slide N°7

  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) MC 2000 Conference Slide N°8

  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” MC 2000 Conference Slide N°9

  10. Integration by parts • Conditions…Notations • Chain rule • Leading to MC 2000 Conference Slide N°10

  11. Necessary and sufficient conditions • Condition • Expressing the Malliavin derivative MC 2000 Conference Slide N°11

  12. Minimal weighting function? • Minimum variance of • Solution: The conditional expectation with respect to • Result: The optimal weight does depend on the underlying(s) involved in the payoff MC 2000 Conference Slide N°12

  13. For European options, BS • Type of Malliavin weighting functions: MC 2000 Conference Slide N°13

  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) MC 2000 Conference Slide N°14

  15. Finite difference versus Malliavin method • Malliavin weighted scheme: not payoff sensitive • Not the case for “bump and re-price” • Call option MC 2000 Conference Slide N°15

  16. For a call • For a Binary option MC 2000 Conference Slide N°16

  17. Simulations (corridor option) MC 2000 Conference Slide N°17

  18. Simulations (corridor option) MC 2000 Conference Slide N°18

  19. Simulations (Binary option) MC 2000 Conference Slide N°19

  20. Simulations (Binary option) MC 2000 Conference Slide N°20

  21. Simulations (Call option) MC 2000 Conference Slide N°21

  22. Simulations (Call option) MC 2000 Conference Slide N°22

  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 MC 2000 Conference Slide N°23

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