Changing to a Bond Numeraire with Deterministic Interest Rates

# Change of numeraire from bank account to Zcb


# Change of numeraire from bank account to Zcb












Why is there no drift adjustment when numeraire is changed from bank account (risk neutral measure) to zero coupon bond who matures at time of payoff (fwd risk neutral measure) ?

## Answer by Gordon (score 3)

https://quant.stackexchange.com/a/36375

Let $B_t= e^{\int_0^t r_sds}$ be the money-market account value at time $t$, and $P(t, T)$ be the value of the zero-coupon bond with maturity $T$ and unit face amount. Moreover, let $Q$ be the risk-neutral measure and $Q_T$ be the $T$-forward measure. If the interest rate $r_t$ is deterministic, then \begin{align*} P(t, T) &= E\left(e^{-\int_t^T r_s ds} \,|\, \mathcal{F}_t\right)\\ &= e^{-\int_t^T r_s ds} = \frac{B_t}{B_T}, \end{align*} where $E$ is the expectation under the risk-neutral measure $Q$. Moreover, for $0 \le t \le T$, \begin{align*} \frac{dQ}{dQ_T}\big|_t &= \frac{B_t P(0, T)}{P(t, T)}\\ &=\frac{B_t \frac{1}{B_T}}{\frac{B_t}{B_T}}=1. \end{align*} In particular, for any $A\in \mathcal{F}_T$, \begin{align*} Q(A) &= \int_{\Omega} \pmb{1}_A dQ \\ &= \int_{\Omega} \pmb{1}_A \frac{dQ}{dQ_T}\big|_T dQ_T \\ &= \int_{\Omega} \pmb{1}_A dQ_T \\ &=Q_T(A). \end{align*} That is, $Q=Q_T$.