Convergence of Monte Carlo simulations involving the meanreverting square root process
Higham, D.J. and Mao, X. (2005) Convergence of Monte Carlo simulations involving the meanreverting square root process. Journal of Computational Finance, 8 (3). pp. 3561. ISSN 14601559

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Abstract
The meanreverting square root process is a stochastic differential equation (SDE) that has found considerable use as a model for volatility, interest rate, and other financial quantities. The equation has no general, explicit solution, although its transition density can be characterized. For valuing pathdependent options under this model, it is typically quicker and simpler to simulate the SDE directly than to compute with the exact transition density. Because the diffusion coefficient does not satisfy a global Lipschitz condition, there is currently a lack of theory to justify such simulations. We begin by showing that a natural EulerMaruyama discretization provides qualitatively correct approximations to the first and second moments. We then derive explicitly computable bounds on the strong (pathwise) error over finite time intervals. These bounds imply strong convergence in the limit of the timestep tending to zero. The strong convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. We spell this out for a bond with interest rate given by the meanreverting square root process, and for an upandout barrier option with asset price governed by the meanreverting square root process. We also prove convergence for European and upandout barrier options under Heston's stochastic volatility model  here the meanreverting square root process feeds into the asset price dynamics as the squared volatility.
Item type:  Article 

ID code:  160 
Keywords:  Monte Carlo simulations, stochastic differential equation, SDE, transition density, pathdependent options, stochastic volatility, Mathematics 
Subjects:  Science > Mathematics 
Department:  Faculty of Science > Mathematics and Statistics 
Depositing user:  Ms Sarah Scott 
Date deposited:  10 Feb 2006 
Last modified:  17 Oct 2017 13:20 
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URI:  https://strathprints.strath.ac.uk/id/eprint/160 
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