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Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes

Graduate School of Business, Columbia University, 415 Uris Hall, 3022 Broadway, New York, New York 10027-6902
Lehman Brothers, 745 Seventh Avenue, New York, New York 10019
mnb2{at}columbia.edu
okaya{at}lehman.com

The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results, and a large number of time steps may be needed to reduce the discretization bias to an acceptable level. This paper suggests a method for the exact simulation of the stock price and variance under Heston’s stochastic volatility model and other affine jump diffusion processes. The sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security. We compare our method with the more conventional Euler discretization method and demonstrate the faster convergence rate of the error in our method. Specifically, our method achieves an O(s^-1/2) convergence rate, where s is the total computational budget. The convergence rate for the Euler discretization method is O(s^-1/3) or slower, depending on the model coefficients and option payoff function.

Subject classifications: simulation; efficiency; exact methods; finance; asset pricing; computational methods.
History: Received July 2003; revision received October 2004; accepted January 2005.

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