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