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 BT(t) of maturity T at time t are written as

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which defines the instantaneous forward curve rT(t),

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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, rT(t) is the forward curve observed at time t. The process for rT(t) is what is to be modeled and the initial value is the current forward curve (i.e., current zero-coupon bond prices),

rT(0) = f(T).

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

dBT(t) = BT(t)[µT(t) dt + σT(t) dz(t)]

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

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