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Smart Monte Carlo: Various Tricks Using Malliavin Calculus Quantitative Finance, NY, Nov 2002

Smart Monte Carlo: Various Tricks Using Malliavin Calculus Quantitative Finance, NY, Nov 2002 Eric Benhamou eric.benhamou@gs.com Goldman Sachs International. Agenda. Motivation for Fast Monte Carlo Engines Smart Computation of the Greeks Typology of Options and Practical Use

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Smart Monte Carlo: Various Tricks Using Malliavin Calculus Quantitative Finance, NY, Nov 2002

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  1. Smart Monte Carlo:Various Tricks Using Malliavin Calculus Quantitative Finance, NY, Nov 2002 Eric Benhamou eric.benhamou@gs.com Goldman Sachs International

  2. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Agenda • Motivation for Fast Monte Carlo Engines • Smart Computation of the Greeks • Typology of Options and Practical Use • Other Developments: Smart Calibration, Conditional Expectations and Design of Efficient Monte Carlo Engines

  3. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 I. Motivation for Fast Monte Carlo Engines

  4. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Multi-Asset Products • Growing demand of multi-asset products have urged to develop generic pricing engines (often using Monte Carlo): • Parser to enter tailor made complex payoffs • Ability to design easily multi-asset models • Modelling components easy and fast to calibrate • Powerful risk engine • Stability of prices and risks • Fast pricing and generation of risk reports

  5. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Computing Challenge of Monte Carlo Trading Book • The two most time-consuming steps are: • Calibration • Risk  How can we create generic smart Monte Carlo engines to speed up calibration and Greek computation?

  6. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 II. Smart Computation of the Greeks

  7. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 The Challenge of Fast Greeks • Price sensitivities required for: • Pricing (measure of the error and price charge) • Estimation of the risk of the book (hedging) • PNL explanation and back testing • Credit valuation adjustment and VAR

  8. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Traditional Method for the Greeks • Finite difference approximation: “bump and re-price” • Two types of errors: • Differentiation • Convergence • Obviously very inefficient for payoffs containing discontinuities like binary, corridor, range accrual, step-up, cliquet, ratchet, boost, scoop, altiplano, barrier and other types of digital options for example

  9. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 How to Avoid Poor Convergence?Avoid Differentiating • Take the derivative of the payoff function • Pathwise method (Broadie Glasserman (93)) • Take the derivative of the probability function • Likelihood ratio method (Broadie Glasserman (96)) • Do an integration by parts • Compute a weighting function using Malliavin calculus (Fournié et al. (97), Benhamou (00)) • Compute the Vector of perturbation numerically  Work of Avellaneda, Gamba (00)

  10. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Comparison of the Methods • All these techniques try to avoid differentiating the payoff function: • Likelihood ratio • Weight = likelihood ratio • Advantage: easy to use • Drawback: requires to know the exact form of the density function

  11. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Comparison of the MethodsContinued • Malliavin method: • Does not require knowing the density only the diffusion • Weighting function independent of the payoff • Very general framework • Infinity of weighting functions • Numerical estimation of the weighting function • Other way of deriving the weighting function • Inspired by Kullback Leibler relative entropy maximization • Spirit close to importance sampling

  12. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 The Best Weighting Function? • There is an infinity of weighting functions: • Can we characterize all the weighting functions? • Can 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? • Practical point of view: • Which option(s)/ Greek should be preferred? (importance of maturity, volatility)

  13. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 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” • Notations: general diffusion first variation process Malliavin derivative Skorohod integral

  14. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 How to Derive the Malliavin Weights? • Integration by parts: • Chain rule • Greeks is to compute

  15. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Necessary and Sufficient Conditions • Condition • Expressing the Malliavin derivative

  16. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 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

  17. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 For European Options, BS • Type of Malliavin weighting functions:

  18. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 II. Typology of Options and Practical Use

  19. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 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) • Use of localization formula to target the discontinuity point

  20. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Finite Difference Versus Malliavin Method • Malliavin weighted scheme: not payoff sensitive • Not the case for “bump and re-price” • Call option

  21. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Comparison Call and Digital • For a call • For a Binary option

  22. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Simulations (Corridor Option)

  23. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Simulations (Binary Option)

  24. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Simulations (Call Option)

  25. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Industrial Use • Fast Greeks formulae can be derived easily in the case of: • Market models (with payoff like Asian cap knock-out, Asian digital cap…etc) • Stochastic volatility models homogeneous (like Heston model) • Fast Greeks particularly useful for path-dependent payoffs

  26. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 II. Other Developments: Smart Calibration, Conditional Expectations and Design of Efficient Monte Carlo Engines

  27. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Smart Calibration • When using calibration algorithms, one needs to compute gradient with respect to various model parameters  One can use localization formula to isolate the discontinuity of the payoff function to get faster estimate of the gradient

  28. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Conditional Expectation • Conditional expectation can be seen as a Dirac function in one point. To smoothen payoff, one can do integration by parts like for the Greeks • Typical example is in Heston model, to compute the conditional volatility

  29. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Conditional Volatility in Heston Model

  30. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Design of a Generic Risk Engine for Monte Carlo Trades • According to the payoff profile, at parsing time, should branch or not on Malliavin calculus weighting formula and use a localization formula • When distributing the various trades across the different computers of the pool, should aggregate them according to trades requiring same Malliavin weighting

  31. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Conclusion • Malliavin weights enable to derive weights knowing only the diffusion coefficients • Combined with the localization of the discontinuity, method quite powerful • Extensions: • Use of vega-gamma parity in homogeneous models • Extension to jump diffusion models

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