Heath{Jarrow{Morton Model

For management of interest rate risk, we create an interest rate scenario by using an arbitrage-free model of the bond market, which describes the evolution of the forward rate. With this understanding, this chapter addresses the forward rate model introduced by Heath et al. (1992) (hereinafter, HJM). Additionally, we introduce the short rate model introduced by Hull and White (1990), which we treat as a special case in the HJM model. In interest rate models, the option price is typically valuated under the risk- neutral measure, and so these models have been developed as models specied under the risk-neutral measure. On the one hand, when we apply a model to risk management, we must use a model specied under the real-world measure. We consider this further by val- uating the VaR of a simple example. On the other hand, to construct an interest rate model under the real-world measure, it is necessary to estimate the market price of risk. We brie y summarize some approaches to estimation of that price in the short rate models.

Keywords: Arbitrage-free, Contribution rate, d-factor model, Drift coe- cient, Forward rate, HJM model, HullWhite model, Kalman lter, Market price of risk, Mean reversion rate, Principal component analysis (PCA), Real- world measure, Real-world model, Risk-neutral measure, Risk neutral valua- tion, Savings account, Short rate, Spot measure, State price de ator, Time- homogeneous, Volatility, Volatility component.