Normalized price process $Z(t)=\frac{\Pi(t)}{B(t)}$

Question

If an interest rate model with the following $P$-dynamics for the short rate.

$$dr(t)=\mu(t,r(t))dt+\sigma(t,r(t))d\bar{W}(t)$$

Now consider a $T$-claim of the form $\chi = \Phi(r(T))$ with corresponding price process $Π(t)$.

Can anyone help me to find stochastic differential of $Π(t)$ ?

and show that the normalized price process

$$Z(t)=\frac{\Pi(t)}{B(t)}$$

is a $Q$-martingale.?

I appreciate any help.

Thanks.

Answer 1

You Know that $dB_t=r_tB(t)dt$ . Ito's formula give us \begin{align} dZ(t)=\frac{1}{B(t)}d\,\Pi(t)-\frac{\Pi(t)}{B\,^2(t)}dB(t)+0 \end{align} As your teacher mentioned, $d\Pi(t)=r(t)\Pi(t)dt+\sigma(\Pi(t),t)dW(t)$,Thus we have \begin{align} & dZ(t)=\frac{1}{B(t)}[r(t)\Pi(t)dt+\sigma(\Pi(t),t)dW(t)]-\frac{\Pi(t)}{B\,^2(t)}r(t)B(t)dt\\ & dZ(t)=\frac{1}{B(t)}r(t)\Pi(t)dt+\frac{1}{B(t)}\sigma(\Pi(t),t)dW(t)-\frac{1}{B(t)}r(t)\Pi(t)dt\\ \end{align} then \begin{align} dZ(t)=\frac{1}{B(t)}\sigma(\Pi(t),t)dW(t) \end{align} Martingale Representation Theorem shows that $Z(t)$ is a Martingale.