A Pathwise Approach To Sensitivity Estimation From Characteristic Functions

Sensitivities measure the change of assets price with respect to some parameter and when we try to hedge the financial risk, sensitivities reflect the relationship between assets price and the parameter representing the financial risk. Thus, sensitivities are very useful in financial risk management. There are two main methods to compute sensitivities:the likelihood ratio method and the pathwise derivative method. The likelihood ratio method is an important method to compute sensitivity, and it is widely used for its few conditions required. But the likelihood ratio method can cause an explosion in variance which leads to an inaccurate result. To more accurately estimate the sensitivity, this paper introduces another important method named pathwise derivative method.We know that Levy process is an important class of stochastic processes and it can be used to model the underlying asset price. But for most Levy processes, we only know their characteristic functions without knowing the clear expression of the density functions and distribution functions. Then, we firstly compute the approximated values of density functions and distribution functions at some points through the inverse transform method via the characteristic functions, then we can construct their cumulative distribution functions and the possibility density functions through the linear interpolation method by the approximated values. The main content of this paper are the following two points:1. The bias analysis of the estimator Previous studies concentrated on the estimating the sensitivities through the likelihood ratio method under the constructed density functions and distribution functions and they only gave out the convergence rate of the likelihood ratio estimator without a bound for the bias of the estimator. And under those Levy processes whose characteristic functions are only available, this paper focus on evaluating the sensitivities via the pathwise derivative method and deriving the bound for the bias of the pathwise derivative estimator.2. The comparison of the mean squared errors of the two estimators Previous researches only claimed that the pathwise derivative method has a smaller variance than the likelihood ratio method and in their numerical experiment we can find this property. But they did not give an clear proof to illustrate that property. This paper also compares the biases and the variances caused by the pathwise derivative method and the likelihood ratio method respectively when estimating the sensitivity by the two methods.Through the analysis of the above two points, we know how to better estimate the sensitivities by the pathwise derivative method. And in the numeric experiment, we can directly observe that the pathwise derivative method is more accurate than the likelihood ratio method.