Optimal Malliavin weighting functions for the simulations of the Greeks

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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_at_yahoo.com

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Outline

• Introduction motivations
• Review of the literature
• Results on weighting functions
• Numerical results
• Conclusion

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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.

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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) LEcuyer 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

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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.

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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

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• 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.

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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)

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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

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Integration by parts

• ConditionsNotations
• Chain rule
• Leading to

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Necessary and sufficient conditions

• Condition
• Expressing the Malliavin derivative

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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

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For European options, BS

• Type of Malliavin weighting functions

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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)

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Finite difference versus Malliavin method

• Malliavin weighted scheme not payoff sensitive
• Not the case for bump and re-price
• Call option

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• For a call
• For a Binary option

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Simulations (corridor option)
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Simulations (corridor option)
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Simulations (Binary option)
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Simulations (Binary option)
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Simulations (Call option)
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Simulations (Call option)
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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