The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

CHAPTER 8

Interest Rate Derivatives: HJM Models

8.1. HULL-WHITE MODEL DERIVATION

8.1.1. Process and Pricing Equation

Stochastic interest rates add the extra dimension of trying to model a forward rate curve stochastically instead of just a stock price. The model is complicated by the fact that the first point of this curve is the risk-free (overnight, short, or spot) rate r that shows up in general risk-neutral pricing derivations.

We proceed with a simple arbitrage-free pricing model called the Hull-White model (following Cheyette 1992). Default-free zero bond prices B_T(t) of maturity T at time t are written as

which defines the instantaneous forward curve r_T(t),

representing the cost of borrow from time T to T + dT, that may be locked in at time t. It is defined only for T > t. In other words, r_T(t) is the forward curve observed at time t. The process for r_T(t) is what is to be modeled and the initial value is the current forward curve (i.e., current zero-coupon bond prices),

r_T(0) = f(T).

Now, writing a generalized Brownian motion process for B_T(t),

dB_T(t) = B_T(t)[µ_T(t) dt + σ_T(t) dz(t)]

using a Wiener process dz(t) together with arbitrary drift and volatility functions µ_T(t), σ_T(t), the risk premium argument still applies (see section 4.1 Risk Premium Derivation) and determines ...