1. Supported by Workshop on Stochastic Analysis and Computational Finance, November 2005 Imperial College (London) G.N. Milstein and M.V. Tretyakov Numerical analysis of Monte Carlo evaluation of Greeks by finite differences J. Comp. Fin. 8, No 3 (2005), 1-33

2. MC evaluation of Greeks by finite differences Plan Model Model Other approaches Other approaches Finite difference approach Finite difference approach Numerical integration error Numerical integration error Monte Carlo error Monte Carlo error Other Greeks Other Greeks Numerical examples Numerical examples Conclusions Conclusions

3. Model

4. Model

5. Model

6. Other approaches Broadie, Glasserman (1996); Milstein, Schoenmakers (2002)

7. Other approaches Fournie, Lasry, Lebuchoux, Lions, Touzi (1999, 2001); Benhamou (2000)

8. Finite difference approach Standard finite difference formulas Weak-sense numerical integration of SDEs Monte Carlo technique

9. Finite difference approach Newton (1997); Wagner (1998); Milstein, Schoenmakers (2002); M&T (2004)

10. Weak Euler scheme

11. Estimator for the option price

12. Estimator for deltas

13. Estimators for deltas

14. Assumptions

15. Numerical integration error Proof. It is based on the Talay-Tubaro error expansion (Talay, Tubaro (1990); M&T (2004))

16. Numerical integration error: proof

17. Monte Carlo error: price

18. Monte Carlo error: deltas If all the realizations are independent

19. Monte Carlo error: deltas Boyle (1997); Glasserman (2003), Glasserman, Yao (1992), Glynn (1989); L’Ecuyer, Perron (1994)

20. Monte Carlo error: deltas

21. Main theorem

22. Higher-order integrators

23. Non-smooth payoff functions Bally, Talay (1996)

24. Non-smooth payoff functions

26. Other Greeks

27. Other Greeks: theta

28. Numerical tests: European call

29. Numerical tests: variance reduction Newton (1997); Milstein, Schoenmakers (2002); M&T (2004)

30. Numerical tests: variance reduction

32. Numerical tests: binary option

34. Numerical tests: Heston stochastic volatility model

36. Supported by Approximate deltas by finite differences taking into account that the price is evaluated by weak-sense numerical integration of SDEs together with the MC technique Exploit the method of dependent realizations in the MC simulations Rigorous error analysis Conclusions