标题： 离散采样的扩散过程的似然函数的闭式近似：指数展开法 显示英文标题

标题： A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion

摘要： 本文讨论了任意扩散过程似然函数的闭式近似。 该近似基于有限时间步长 $\Delta t$ 的转移概率指数假设，以及对其对数与高斯分布对数偏差的级数展开。 通过这种方法（称为 {\em 指数展开}），转移概率被表示为 $\Delta t$ 的幂级数。 如果包含越来越多的项，这会渐近精确，并且即使截断到前几项（比如 3 项）也能提供非常准确的结果。 这种展开的系数可以通过递归直接确定，且涉及简单的单维积分。 我们给出了几个金融相关的例子，并将我们的结果与离散采样扩散的最新近似方法 [Aït-Sahalia, {\it 金融学杂志} {\bf 54} , 1361 (1999)] 进行了比较。 我们发现，指数展开在大多数情况下提供了类似的准确性，但在低波动率情形下表现更好。 此外，目前方法的实现相对更简单。 在泛函积分框架下，指数展开能够显著获得金融衍生品定价核的良好近似。 这一点通过应用于简单的路径依赖利率衍生品得到了说明。 最后，我们讨论了这些结果如何用于提高数值（无论是确定性还是随机性）方法在衍生品定价中的效率。 显示英文摘要

摘要： In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotically exact if an increasing number of terms is included, and provides remarkably accurate results even when truncated to the first few (say 3) terms. The coefficients of such expansion can be determined straightforwardly through a recursion, and involve simple one-dimensional integrals. We present several examples of financial interest, and we compare our results with the state-of-the-art approximation of discretely sampled diffusions [A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that the exponent expansion provides a similar accuracy in most of the cases, but a better behavior in the low-volatility regime. Furthermore the implementation of the present approach turns out to be simpler. Within the functional integration framework the exponent expansion allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. This is illustrated with the application to simple path-dependent interest rate derivatives. Finally we discuss how these results can also be used to increase the efficiency of numerical (both deterministic and stochastic) approaches to derivative pricing.