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Optimal importance sampling with explicit formulas in continuous time. (English) Zbl 1150.91019

In the Black-Scholes model the problem arises how to select a change of drift which minimizes the variance of Monte Carlo estimators for prices of path-dependent options. The authors follow the approach to a continuous-time setting, where the optimal deterministic drift in the Black-Scholes model is identified as the solution of a one-dimensional variational problem. The main idea is briefly summarized with an heuristic argument, which involves derivatives of non-differential Brownian paths, applies Laplace asymptotic in infinite dimensions, and assumes the validity of a minimax result. In spite of these issues, this characterization of the optimal drift is essentially correct. In continuous time the variational problem reduces to the familiar Euler-Lagrange ordinary differential equation. In the case of Asian options the optimal change of drift admits closed-form solutions.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60F10 Large deviations
65C05 Monte Carlo methods
91G60 Numerical methods (including Monte Carlo methods)
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