A closed-form expansion for the conditional expectations of the extended CIR process

Abstract

This paper derives a closed-form expansion for the conditional expectation of a continuous-time stochastic process, given by Vt,T:=e−∫tTg(vs)dsf(vT) for 0≤t≤T, where vt evolves according to the extended Cox-Ingersoll-Ross process, for any C∞ functions f and g. We apply the Feynman-Kac theorem to state a Cauchy problem associated with Vt,T and solve the problem by using the reduction method. Furthermore, we extend our method to any piecewise C∞ function f; demonstrating our method can be applied to price options in financial derivative markets. In numerical study, we employ Monte Carlo simulations to demonstrate the performance of the current method.

Keywords

• ECIR process
• Conditional expectation
• Closed-form expansion
• Piecewise smooth function