Stochastic Interest Models

The aim of this chapter is not to present a complete overview of stochastic interest rate models, but to show to which extend stochastic interest rates can be incorporated into a pricing formula for European-style stock options. To this end, we focus on only three typical one-factor short rate models, namely, the Vasicek model (1977), the CIR model (1985) and the Longstaff model (1989), which are again specified by a mean-reverting Ornstein-Uhlenbeck process, a mean-reverting square root process and a mean-reverting double square root process, respectively. In turn, these three processes correspond to these ones in stochastic volatility models discussed in Chapter 3. Since stochastic short rate appears in a risk-neutral stock process as drift, it becomes impossible for a square root process to incorporate a correlation between the short rate and the stock diffusion term with CFs. Therefore, we propose a modification of the stock price process so that the CIR model and the Longstaff model may be embedded into an option pricing formula. However, the mean-reverting Ornstein-Uhlenbeck process can be nested with the stock price process for a non-zero correlation without any modification. The extension of a one-factor stochastic interest rate models into a multi-factor case is straightforward, and some multi-factor models can be incorporated into the valuation of stock options in analogy to the one-factor model if the independence conditions are satisfied.