# Change of numeraire in exchange options with random interest rate

At time $$t$$, the market offers a (possibly random) bounded interest rate $$r_t$$ and two assets whose prices are given by

\begin{align*} {{\rm d S^{(1)}_t \over S^{(1)}_t}} &= b^{(1)}_t d t + \sigma_1 dB^{(1)}_t, \\ {{\rm d S^{(2)}_t \over S^{(2)}_t}} = b^{(2)}_t d t & + \sigma_2 (\rho dB^{(1)}_t + \sqrt{1-\rho^2} dB^{(2)}_t) \end{align*} where $$(B^{(1)}_t, B^{(2)}_t)$$ is a two dimensional standard Brownian motion under the historical probability measure $$\mathbb{P}$$, $$\rho \in (-1,1)$$, $$\sigma_1 , \sigma_2 >0$$, and $$(b^{(1)}, b^{(2)})$$ is a bounded $$\mathbb{R}^2$$-valued (possibly random) process.

I want to compute the value {at time $$t$$ }of the option with payoff $$\Psi = (S^{(2)}_T - KS^{(2)}_T)_{+}$$ at time $$0$$ for $$K \ge 0$$.But i don't know how to deal with the interest rate being random.