The Pricing Problem Of Stock Index Futures With Stochastic Interest Rates

In this paper, we discuss the price interval of stock index futures with stochastic interest rates. We introduce a new Brownian motion to the model of stock index as the stochastic factor which influences interest rates , and thus the number of assets in the market is less than the number of stochastic factors, which means the financial market is incomplete. To make this market a complete one, we introduce the forward rate, which is driven by this Brownian motion, as a new risk factor, and under HJM model we have the new asset - zero-coupon bond. Then we can get the price of stock index futures by the martingale method. With all kinds of forward rate available, we obtain buying price, selling price and arbitrage-free interval of stock index futures, and furthermore, we simplify them under Vasicek model. Finally, we conduct an empirical study of S&P 500 index futures. The result turns out that this arbitrage-free interval includes the most of actual futures prices with a downward bias, and this bias decreases as the volatility of risk free rate increases.