An Iterative Approach for Obtaining a Closed-form Expansion for the Conditional Expectations of the Extended Cox-Ingersoll-Ross Process

Nopporn Thamrongrat, Sanae Rujivan

Abstract

In this paper, we develop an iterative approach for obtaining a closed-form expansion for the conditional expectation of the valuation process, defined by \[V_{t,T}:={e^{ - \int_t^T {g({v_s})ds} }}f({v_T}) + \int_t^T {h({v_s}){e^{-\int_t^s {g({v_u})du} }}} ds\]for $0\leq t\leq T$, where $v_t$ is assumed to follow the extended Cox-Ingersoll-Ross process, for any smooth real-valued functions $f, g$, and $h$. The novel analytical approach presented here at least serves for two major purposes: (i) to avoid the requirement of numerical integration or Monte Carlo (MC) simulations to compute the conditional expectation, which can substantially reduce the computational burden; (ii) to provide a simple closed-form expansion for the conditional expectation, which can be easily used by market practitioners. Furthermore, a multi-step closed-form expansion is constructed in order to improve the accuracy of our approach. The performance of the current approach is demonstrated by comparing our numerical results with some exact solutions and MC simulations from several examples.