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

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

Model

Model

Model

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

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

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

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

Weak Euler scheme

Estimator for the option price

Estimator for deltas

Estimators for deltas

Assumptions

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

Numerical integration error: proof

Monte Carlo error: price

Monte Carlo error: deltas If all the realizations are independent

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

Monte Carlo error: deltas

Main theorem

Higher-order integrators

Non-smooth payoff functions Bally, Talay (1996)

Non-smooth payoff functions

Other Greeks

Other Greeks: theta

Numerical tests: European call

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

Numerical tests: variance reduction

Numerical tests: binary option

Numerical tests: Heston stochastic volatility model

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