Diffusion processes and applications to financial time series

Abstract

Diffusion processes are effective tools for modeling financial and economic phenomena. Diffusion models have been implemented with great success in financial markets where stochastic calculus based on such models allow researchers to probe the dynamics of processes ranging from stock prices, yields and interest rates to volatility studies and exchange rates. These processes, according to (Pienaar, 2016), allow for the investigation and quantification of the dynamics of various real world financial models. The dynamics of diffusion processes are governed by stochastic differential equations (SDEs), which dictate how these processes evolve over time. A key component in the analysis of such systems is the transitional density, which allows one to make predictions about the state of the process, or functions of the state of the process, when its parameters are known/fixed, or perhaps more importantly, when the parameters are not known a transition density allows one to estimate parameters and subsequently perform inference. Unfortunately, with the exception of certain processes, many of these models' transition density cannot be expressed by an explicit analytical expression. Therefore, efficient and consistent approximation techniques, to obtain an analytical expression for the transition density function, is of paramount interest and importance. The Hermite expansion method, of (Sahalia, 1998), outlines one of the most effective methods of obtaining an approximation to the transition density. The Saddlepoint, or Cumulant Truncation approximation method, provides a strong and robust alternative approximation method, Varughese,2013) and (Pienaar, 2016). In the present paper, we explore how these techniques can be used to analyse popular non-linear diffusion models from the world of finance. In particular, we focus on the construction of the transition density approximations for the Ornstein-Uhlenbeck (OU) model, Cox-Ingersoll and Ross (CIR) model and the Heston model, and the application of these models to real-world datasets, such as the CBOE volatility/VIX index and the S&P 500 stock index. The Sapplepoint or Cumulant Truncated approximate transition density will be used to perform inference on the mentioned datasets.