In quantitative finance, it is often necessary to analyze the distribution of the sum of specific functions of observed values at discrete points of an underlying process. Examples include the probability density function, the hedging error, the Asian option, and statistical hypothesis testing. We propose a method to calculate such a distribution, utilizing a recursive method, and examine it using various examples. The results of the numerical experiment show that our proposed method has high accuracy.
IntroductionThis paper introduces a recursive method to compute interesting quantities related to probability distributions in various financial applications. The method is versatile, and hence, with slight modifications, it is easy to apply the basic framework to various applications. More precisely, the method is based on a convolution-like formula, applied to compute the distribution of the sum of values of one-dimensional processes observed at discrete points. Financial applications include numerical densities of asset price or volatility models, hedging error distributions, arithmetic Asian option prices, and statistical hypothesis tests.Various kinds of stochastic processes are used in quantitative finance, such as the Cox-Ingersoll-Ross model (CIR), the constant elasticity of variance model (CEV), stochastic volatility, and GARCH models. The probability distributions of the processes in financial models can be used for risk management, asset pricing, hedging analysis, parameter estimation, and statistical hypothesis testing. In many cases, the closed form formulas for the density function of stochastic models are not known, and it is advantageous to develop a numerical procedure to compute the probability distributions or density functions.When trading a financial option, the investor usually performs a hedging procedure to reduce risk. In general, continuous models of asset price movements assume a continuous hedging process. However, in practice, because continuous trading is not applicable, a discrete