Convergence of Monte Carlo Simulations involving the Mean-Reverting Square Root Process
Convergence of Monte Carlo Simulations involving the Mean-Reverting Square Root Process
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DOI：
10.21314/jcf.2005.136
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发表时间：
2005-04
影响因子：
0.9
通讯作者：
D. Higham;Xuerong Mao
中科院分区：
文献类型：
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作者：
D. Higham;Xuerong Mao
均值回复平方根过程是一个随机微分方程，它被广泛用作波动率、利率和其他金融量的模型。该方程没有一般的显式解，尽管其跃迁密度可以表征。在此模型下，为了评估路径依赖型期权，直接模拟期权定价通常比使用精确的转换密度进行计算更快、更简单。由于扩散系数不满足全局Lipschitz条件，目前缺乏理论来证明这种模拟。我们开始表明，一个自然的欧拉-丸山离散化提供了定性正确的近似的第一和第二时刻。然后，我们推导出明确的可计算的边界上的强（路径）错误在有限的时间间隔。这些界限意味着强收敛的时间步长趋于零的限制。强收敛的结果可以用来证明蒙特卡罗模拟计算金融产品的预期收益的方法。我们对利率服从均值回复平方根过程的债券和资产价格服从均值回复平方根过程的上下障碍期权进行了分析。我们还证明了收敛的欧式和向上和出障碍期权下赫斯顿的随机波动率模型-这里的均值回复平方根过程馈入资产价格动态的平方波动率。
The mean-reverting 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 path-dependent 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 Euler-Maruyama 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 mean-reverting square root process, and for an up-and-out barrier option with asset price governed by the mean-reverting square root process. We also prove convergence for European and up-and-out barrier options under Heston's stochastic volatility model - here the mean-reverting square root process feeds into the asset price dynamics as the squared volatility.