Modeling Two Stock Prices using Diffusion Process

Abstract

In real life applications, the parameters of a stochastic differential equation (SDE) are unknown and need
to be estimated. In most cases what is available is only sampled data of the process at discrete times. It is a common practice to use the discretization of the original continuous time process for the modeling. SDEs have solutions in continuous-time called diffusion process. The methodology as applied to stock price involves a discrete time stochastic model for the dynamical system under study. For a small time interval, ∆t, the possible changes with their corresponding transition probabilities are determined. The expected change and the covariance matrix for the change are determined for the discrete time stochastic process. The system of SDEs is obtained by letting the expected change divided by ∆t,- be the drift coefficient and the square root of the covariance matrix divided by ∆t,- be the diffusion coefficient. The SDE model is inferred by similarities in the forward Kolmogorov equations between the discrete and the continuous stochastic processes. The resulting SDE model with estimated drift and volatility parameters is solved using the multi-dimensional Euler-Maruyama scheme for SDEs as implemented using the R packages ’Sim.DiffProc’ and ’yuima’.