(Solved): 9. Derive the pricing formula for a put option with payoff VT=(KST)+ under a stochastic i ...

9. Derive the pricing formula for a put option with payoff $$V_T={\left(K?S_T\right)}^{+}$$ under a stochastic interest rate. Assume that the volatility vector of the asset and the $$T$$-maturity zero-coupon bond are $${?}_s={?}_s\left(\begin{array}{c}?\\ \sqrt{1?{?}^2}\end{array}\right)\text{, and }{?}_P={?}_P\left(\begin{array}{l}1\\ 0\end{array}\right),$$ where both $${?}_s$$ and $${?}_P$$ are time-dependent scalars. a. (4) Show that the correlation between $$d\ln S_t$$ and $$d\ln P(t,T)$$ is $$?$$. b. (4) Express the option formula in terms of $${?}_s,{?}_P$$ and $$?$$.