Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations

The e¢ cient and accurate calculation of sensitivities of the price of …nancial derivatives with respect to perturbations of the parameters in the underlying model, the so called ‘Greeks’, remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial di¤erential equations or stochastic di¤erential equations (Monte Carlo techniques) are being used. The computation of the ‘Greeks’is essential to risk management and to the hedging of …nancial derivatives and typically require substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic di¤erential equations produces a time-discretization error and a statistical error in the process of pricing …nancial derivatives and calculating the associated ‘Greeks’. In this article we show how a posteriori error estimates and adaptive methods for stochastic di¤erential equations can be used to control both these errors in the context of pricing and hedging of …nancial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated ‘Greeks’. These expansions allow the user to simultaneously …rst control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance. 2000 Mathematics Subject Classi…cation. 49Q12, 62P05, 65C. Keywords and phrases. Sensitivity analysis, parabolic partial di¤erential equations, stochastic di¤erential equations, Euler scheme, a posteriori error estimate, adaptive algorithms, hedging, …nancial derivatives. email: [email protected], Nyström was partially supported by grant VR-70629701 from the Swedish research council VR. yemail: [email protected], Önskog was …nancially supported by the Swedish Defense Research Agency, FOI.