Vasicek Short-Rate Dynamics Under the T-Forward Measure

Summary

The document explains how the Vasicek short-rate process changes when moving from the risk-neutral measure to the measure associated with a zero-coupon bond maturing at time T. The key step is a change of Brownian motion: the risk-neutral Brownian increment differs from the T-forward Brownian increment by a drift adjustment tied to the bond’s volatility. Since the Vasicek bond volatility is expressed using the model’s B(t,T) function, this shifts the drift of the short rate by a term involving volatility squared and B(t,T).

The answer gives the measure-change relationship and the resulting drift, but the question’s surrounding derivation and definitions are incomplete in the provided text. It therefore offers a concise formula-level explanation rather than a full derivation of the Vasicek bond price or a discussion of parameter estimation. The adjustment applies in the stated Vasicek framework and depends on the chosen bond numeraire and volatility convention.

Key ideas

• Changing to the T-forward measure changes the Brownian motion by a term linked to the zero-coupon bond’s volatility.
• Under the Vasicek model, the bond volatility is expressed through the function B(t,T).
• The short-rate drift under the forward measure gains an adjustment involving volatility squared and B(t,T).
• The provided explanation states the result but does not show the full derivation.

Tags

# Vasicek interest rate of T-forward measure


# Vasicek interest rate of T-forward measure












I know dr of risk-neutrual measure is

There is a price of a pure-discount bond can be derived by computing the expectation, I get:

where A and B are:

why dr becomes to:

under T-forward measure?

## Answer by jaehyukchoi49 (score 0)

https://quant.stackexchange.com/a/71140

The two BMs in the risk-neutral and $T$-forward measures are related by $$ dW(t) = dW^T(t) + \sigma_P dt, $$ where $\sigma_P$ is the volatility of the zero-coupon bond $P(t,T)$. This is because $P(t,T)$ is the numeraire of the $T$-forward measure.

Because the volatility of $P(t,T)$ is $\sigma_P = -\sigma B(t,T)$, you get the result: $$ dr(t) = \big(k\theta - \sigma^2 B(t,T) - k r(t)\big)dt + \sigma dW^T(t). $$

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