Risk-Neutral Measures for Interest Rate and Bond Models
Summary
The document explains what a risk-neutral measure means in an interest rate model. It frames the key condition in terms of bond prices: when a zero-coupon bond is valued as the conditional expectation of its future payoff discounted by the short rate, its price divided by the money market account is a martingale under the chosen measure. This is the no-arbitrage pricing condition that gives the measure its relevance, even when there is no stock price in the model.
A second explanation starts with dynamics under the real-world probability, then introduces a market price of risk and uses a change of measure to make discounted bond prices martingales. The short rate’s specific stochastic differential equation does not by itself define the risk-neutral measure; the pricing condition does. The discussion is conceptual and focuses on a simplified one-factor setting with a bond and money market account. It does not address measure selection, model calibration, or the complications of multiple traded bonds and factors.
Key ideas
- Risk-neutrality in an interest rate model is expressed through discounted tradable bond prices being martingales.
- A zero-coupon bond price can be represented as the expected discounted payoff under a martingale measure.
- The money market account provides the discounting numeraire for the bond pricing condition.
- A change of measure can adjust bond dynamics using the market price of risk.
- The explanation uses a simplified setting and does not cover calibration or richer rate models.
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Full text
# What's Risk-Neutral in an Interest Rate Model?
# What's Risk-Neutral in an Interest Rate Model?
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 by Gordon (score 10, accepted)
https://quant.stackexchange.com/a/18186
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.
## Answer by zer0hedge (score 2)
https://quant.stackexchange.com/a/32605
The definition of a risk-neutral probability measure depends on the model. The (one factor) Interest Rate Model in Shreve II consists of a single zero-coupon bond $B(t,T)$ with maturity $T$ and of a money market account. So we want discounted bond price to be a martingale under risk-neutral probability measure. We define it as usual (i.e. Shreve II, 5.2.2.):
Assume that the interest rate $R(t)$ and the bond $B(t,T)$ processes satisfy their respective stochastic differential equations under the actual probability: $$ dR(t) = \xi(t,R(t))dt + \phi(t, R(t))dW(t)$$ $$ dB(t,T) = \mu(t,T)B(t,T)dt + \sigma(t,T)B(t,T)dW(t)$$ where $W(t)$ is a Brownian motion.
The discount process $D(t) = e^{-\int_0^t R(s)ds}$ so as usual $ dD(t) = -R(t)D(t)dt$
We want the discounted bond price to be a martingale: $$ d(D(t)B(t,T)) = D(dB(t,T) - R(t)B(t,T)dt) = D(t)B(t,T)\sigma(t,T)\Big(\frac{\mu(t,T) -R(t)}{\sigma(t,T)}dt + dW(t)\Big) = D(t)B(t,T)\sigma(t,T)\Big(\theta(t)dt + dW(t)\Big)$$
where we defined the market price of risk $\theta(t) = \frac{\mu(t,T) -R(t)}{\sigma(t,T)}$.
We introduce risk-neutral probability measure $\tilde{\mathbb{P}}$ using Girsanov's theorem as usual.
The above considerations do not depend on the form of SDE for the interest rate process $R(t)$ so it is ok to start right from the riks-neutral probability measure as it is done in Shreve's book.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.