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.


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