Deriving Bond Price Dynamics from a Short-Rate Model

Summary

The document explains how to obtain the dynamics of a zero-coupon bond price from a stochastic model for the short rate. Under the risk-neutral measure, the bond price is expressed as the expected discount factor from the current time to maturity. Assuming the short rate is Markovian, the price can be written as a function of the current rate, time, and maturity.

Applying Itô’s lemma gives the price differential in terms of the rate process and derivatives of the bond price function. The response then uses the risk-neutral requirement that the bond’s proportional drift equals the short rate to simplify the dynamics: the drift is the short rate times price, and the diffusion depends on the rate volatility and the bond’s sensitivity to the rate. An explicit bond price still requires a solvable model; Hull–White is given as an example. The result assumes a valid solution and risk-neutral dynamics.

Key ideas

• Under the risk-neutral measure, a zero-coupon bond price is the expected discount factor to maturity.
• With a Markov short rate, the bond price depends on the current rate, time, and maturity.
• Itô’s lemma converts the short-rate process into a bond-price differential.
• The risk-neutral proportional drift of the bond price equals the short rate.
• An explicit price formula depends on choosing a tractable short-rate model.

Tags

# Spot Interest Rate at time $t$


# Spot Interest Rate at time $t$












I know that the general model for the dynamics of the spot interest rate is

$$dr(t)=\mu(r,t)dt+\sigma(r,t)dB(t)$$

My question is, if $P(t,T)$ is the bond value at time $t$, how would I derive $dP$?

## Answer by Antoine Conze (score 2, accepted)

https://quant.stackexchange.com/a/37362

Assume the SDE $dr(t)=\mu(r(t),t)dt + \sigma(r(t),t) dB(t)$ is under the risk neutral measure and that is has a solution.

By construction $P(t,T) = E[e^{-\int_t^T r(u) du}]$ under the risk neutral measure.

Because of the model Markov property you know $P(t,T) = P(r(t), t, T)$ where $P(r, t, T)$ is a function of current short rate, time and maturity, therefore $$ dP(t,T) = \frac{\partial P}{\partial r} dr(t) + \frac{\partial P}{\partial t} dt + \frac{1}{2}\frac{\partial^2 P}{\partial r^2} \sigma(r(t), t)^2 dt $$ Also under the risk neutral measure the drift for $dP(t,T)/P(t,T)$ is $r(t) dt$, hence the above equation simplifies to $$ dP(t,T) = P(t,T) r(t) dt + \frac{\partial P}{\partial r} \sigma(r(t), t) dB(t) $$ If you want to go further you need to be able to compute $E[e^{-\int_t^T r(u) du}]$ explicitly, which is possible for some models such as Hull & White.

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