Applying the HJM-approach when volatility is stochastic

We propose a simple approach to extend the Heath, Jarrow and Morton (1992) model to a framework where the volatility of bond prices and forward rates is stochastic and where volatility-specific risk cannot necessarily be hedged in trading bonds. The arbitrage-free modeling of Heath, Jarrow and Morton cannot readily be transposed to such an environment. HJM’s major insight is that the drift of forward rates must be a function of the volatility structure of forward rates to preclude arbitrage opportunities. This implies that to price interest-rate dependent claims, one need only specify the volatility structure of forward rates. Given the initial term structure (observed in the market), any contingent claim can be priced. While the HJM-restriction still applies when the volatility of forward rates is stochastic, it is not sufficient to price interest-rate contingent claims. Indeed one needs to specify the arbitrage-free process of volatility under the risk-neutral measure. The traditional HJM restriction does not provide any guidance in that respect. We solve that problem by transposing HJM’s idea to the volatility term structure. We show that given an initial term structure of futures prices on interest-rate yields, one need only specify the volatility structure of the futures price on interest-rate yields to derive the arbitrage-free dynamics of the volatility of forward rates. Thus, taking as input the initial term structure of futures prices, the term strucuture of forward rates and specifying a volatility structure for the futures price on yields, one can price all interest rate contingent claims in our model. Our model thus requires as input two term structures, but only one volatlity structure need be specified. We provide a detailed analysis of a gaussian specification of the model, for which volatility exhibits mean reversion. It is parsimonious as it requires only three parameters to be estimated. We use that specification to draw comparisons with a simple extended-Vasicek model in order to investigate the effects of stochastic volatility on interest-rate options. The results support the widespread assertion that even though stochastic volatility might not be essential for the modeling of the term structure, it might have an important impact on interest-rate derivatives.