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 Slide 1 Agenda I.Motivation for Fast Monte Carlo Engines II.Smart Computation of the Greeks III.Typology of Options and Practical Use IV.Other Developments: Smart Calibration, Conditional Expectations and Design of Efficient Monte Carlo Engines

3. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 2 I. Motivation for Fast Monte Carlo Engines

4. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 3 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 Slide 4 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 Slide 5 II. Smart Computation of the Greeks

7. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 6 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 Slide 7 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 Slide 8 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 Slide 9 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 Slide 10 Comparison of the Methods Continued 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 Slide 11 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 Slide 12 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 Slide 13 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 Slide 14 Necessary and Sufficient Conditions Condition Expressing the Malliavin derivative

16. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 15 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 Slide 16 For European Options, BS Type of Malliavin weighting functions:

18. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 17 II. Typology of Options and Practical Use

19. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 18 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 Slide 19 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 Slide 20 Comparison Call and Digital For a call For a Binary option

22. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 21 Simulations (Corridor Option)

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

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

25. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 24 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 Slide 25 II. Other Developments: Smart Calibration, Conditional Expectations and Design of Efficient Monte Carlo Engines

27. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 26 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 Slide 27 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 Slide 28 Conditional Volatility in Heston Model

30. Quantitative Finance, Risk Conference, NY, November 3-4, 2002 Slide 29 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 Slide 30 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