In Shreve II, on p. 265 he states the Hull-White interest rate model as $$ dR(u) = \left( a(u) - b(u)R(u)\right) dt + \sigma(u)d\tilde{W}(u), $$ and then mentions "...$\tilde{W}(u)$ is a Brownian motion under a risk-neutral measure $\tilde{\mathbb{P}}$." However, when he defines a risk-neutral measure on p. 228, he states that $\tilde{\mathbb{P}}$ is a measure under which the discounted stock price is a martingale.
This definition doesn't really apply here, so what is meant by a "risk-neutral measure" when modelling interest rates? Also, why do interest rate models always seem to be stated under these risk-neutral probabilities?
Answer
It is a very interesting question. There is a brief explanation in the book Martingale methods in financial modelling. Basically, it says that, the interest short rate $r_t$ can be modeled in any martingale measure $Q$, however, as long as the zero-coupon bond price $P(t, T)$ is defined by \begin{align*} P(t, T) = E^{Q}\Big(e^{-\int_t^T r_s ds} \mid \mathcal{F}_t\Big) \end{align*} then the discounted bond price $$\frac{P(t, T)}{B(t)},$$ is a $Q-$martingale, and is arbitrage free. Here $B(t)= e^{\int_0^tr_sds}$ is the money market account value. This provides us the freedom to choose the martingale measure, and people always assume that the interest rate model is defined under the risk-neutral probability measure.
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