Supported by

Workshop on Stochastic Analysis and Computational Finance, November 2005Imperial College (London)

G.N. Milstein and M.V. Tretyakov

Numerical analysis of Monte Carlo Numerical analysis of Monte Carlo evaluation of Greeks by finite differencesevaluation of Greeks by finite differences

J. Comp. Fin. 8, No 3 (2005), 1-33

MC evaluation of Greeks by finite differencesMC evaluation of Greeks by finite differences

Plan ModelModel Other approachesOther approaches Finite difference approach Finite difference approach Numerical integration errorNumerical integration error Monte Carlo errorMonte Carlo error Other GreeksOther Greeks Numerical examplesNumerical examples ConclusionsConclusions

ModelModel

ModelModel

ModelModel

Other approachesOther approaches

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

Other approachesOther approaches

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

Finite difference approachFinite difference approach

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

Finite difference approachFinite difference approach

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

Weak Euler schemeWeak Euler scheme

Estimator for the option priceEstimator for the option price

Estimator for deltasEstimator for deltas

Estimators for deltasEstimators for deltas

AssumptionsAssumptions

Numerical integration errorNumerical integration error

Proof.

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

Numerical integration error: Numerical integration error: proofproof

Monte Carlo error: priceMonte Carlo error: price

Monte Carlo error: deltasMonte Carlo error: deltas

If all the realizations are independent

Monte Carlo error: deltasMonte Carlo error: deltas

Boyle (1997); Glasserman (2003), Glasserman, Yao (1992), Glynn (1989); L’Ecuyer, Perron (1994)

Monte Carlo error: deltasMonte Carlo error: deltas

Main theoremMain theorem

Higher-order integratorsHigher-order integrators

Non-smooth payoff functionsNon-smooth payoff functions

Bally, Talay (1996)

Non-smooth payoff functionsNon-smooth payoff functions

Non-smooth payoff functionsNon-smooth payoff functions

Other GreeksOther Greeks

Other Greeks: thetaOther Greeks: theta

Numerical tests: European callNumerical tests: European call

Numerical tests: variance reductionNumerical tests: variance reduction

Newton (1997); Milstein, Schoenmakers (2002); M&T (2004)

Numerical tests: variance reductionNumerical tests: variance reduction

Numerical tests: variance reductionNumerical tests: variance reduction

Numerical tests: binary optionNumerical tests: binary option

Numerical tests: binary optionNumerical tests: binary option

Numerical tests: Numerical tests: Heston stochastic volatility modelHeston stochastic volatility model

Numerical tests: Numerical tests: Heston stochastic volatility Heston stochastic volatility modelmodel

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

ConclusionsConclusions