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I am trying to write the SDE for a forward bond, $dP(t,T_1,T_2)$, under the $T_1$-Forward measure, $Q_{T_1}$. I can easily do this by:

  1. Writing the equation of $dP(t,T_1)$ and $dP(t,T_2)$ under the Risk-neutral measure ($Q$).
  2. Applying Ito's formula for ratios.
  3. Finally, changing the measure to $Q_{T_1}$.

I run into a problem when I try to write the two SDEs under the $T_1$-forward measure directly. What does $dP(t,T_1)$ look like under $T_1$-forward measure? Should it not be identity? Then how would one apply Ito's formula on the ratio?

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We consider a financial market with three assets: a zero-coupon bond of maturity $T_1$, a second one with maturity $T_2$ and the money market account $B_t$. Assuming the market's risk-free rate $r_t$ is normally-distributed, the spot dynamics of the assets under the risk-neutral measure $Q$ are given by:

$$\begin{align} \frac{\text{d}B_t}{B_t}&=r_t\text{d}t \\[6pt] \frac{\text{d}P(t,T_1)}{P(t,T_1)}&=r_t\text{d}t+\sigma(t,T_1)\text{d}W_t^{(1)} \\[6pt] \frac{\text{d}P(t,T_2)}{P(t,T_2)}&=r_t\text{d}t+\sigma(t,T_2)\text{d}W_t^{(2)} \\[6pt] \text{d}W_t^{(1)}\text{d}W_t^{(2)}&=\rho\text{d}t \end{align}$$

The $T_1$-forward measure $Q_{T_1}$ is defined such that all $T_1$-forward assets are martingales. Let us define:

$$\begin{align} \tilde{B}_t &\triangleq \frac{B_t}{P(t,T_1)} \\[6pt] \tilde{P}(t,T_2)&\triangleq \frac{P(t,T_2)}{P(t,T_1)} \end{align}$$

By Itô's Lemma, the forwards dynamics are:

$$\begin{align} \frac{\text{d}\tilde{B}_t}{\tilde{B}_t} &= \sigma^2(t,T_1)\text{d}t-\sigma(t,T_1)\text{d}W_t^{(1)} \\[6pt] \frac{\text{d}\tilde{P}(t,T_2)}{\tilde{P}(t,T_2)} &=\left(\sigma^2(t,T_1)-\rho\sigma(t,T_1)\sigma(t,T_2)\right)\text{d}t+\Sigma(t)\cdot\text{d}W_t\end{align}$$

where:

$$\begin{align} \Sigma(t)&\triangleq \bigg(-\sigma(t,T_1),\sigma(t,T_2)\bigg) \\[2pt] W_t&\triangleq \bigg(W_t^{(1)},W_t^{(2)}\bigg) \end{align}$$

Using Girsanov theorem, we define the $T_1$-forward measure such that the following processes are Brownian Motions under $Q_{T_1}$:

$$\begin{align} \tilde{W}_t^{(1)}&=W_t^{(1)}-\int_0^t\sigma(s,T_1)\text{d}s \\[6pt] \tilde{W}_t^{(2)}&=W_t^{(2)}-\rho\int_0^t\sigma(s,T_1)\text{d}s \end{align}$$

which turn into martingales the forward dynamics:

$$\begin{align} \frac{\text{d}\tilde{B}_t}{\tilde{B}_t} &= \sigma(t,T_1)\text{d}\tilde{W}_t^{(1)} \\[6pt] \frac{\text{d}\tilde{P}(t,T_2)}{\tilde{P}(t,T_2)} &=\Sigma(t)\cdot\text{d}\tilde{W}_t\end{align}$$

Therefore spot dynamics under the $T_1$-forward measure are:

$$\begin{align} \frac{\text{d}B_t}{B_t}&=r_t\text{d}t \\[6pt] \frac{\text{d}P(t,T_1)}{P(t,T_1)}&=\left(r_t+\sigma^2(t,T_1)\right)\text{d}t+\sigma(t,T_1)\text{d}\tilde{W}_t^{(1)} \\[6pt] \frac{\text{d}P(t,T_2)}{P(t,T_2)}&=\left(r_t+\rho\sigma(t,T_1)\sigma(t,T_2)\right)\text{d}t+\sigma(t,T_2)\text{d}\tilde{W}_t^{(2)} \end{align}$$

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