generalized black scholes

Question

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ where $d_{+}(\tau, x)$ = $\frac{1}{\sigma\sqrt{\tau}}$ * $[log\frac{x}{K} + (r + \frac{1}{2}\sigma^2)\tau]$, $d\_(\tau, x) = d_{+}(\tau, x) - \sigma \sqrt{\tau}$

However, I need to find the solution when, $dS_t = \mu_{t}S_tdt + \sigma_{t}S_tdW_t$ and $r_t$ are deterministic functions of t. I was asked to guess the solution, so it must be a very close analogue to the solution above. I thought about integrating over time, but I haven't been able to verify that this works, and I do need to verify the solution.

Any help in figuring out what the form and how to go about verifying that it is a solution would be appreciated.

Update: Someone asked to see some extra work, here is my guess of what the solution should be: c(t, x) = $xN(d_{+}(T - t), x))$ - $Ke^{-\int_0^{T - t}r_udu}$N(d_(T - t), x)) where $d_{+}(\tau, x) = \frac{1}{\int_0^\tau \sigma_udu}$ * $[log\frac{x}{K} + \int_0^\tau (r + \frac{1}{2}\sigma^2)]$, $d\_(\tau, x) = d_{+}(\tau, x) - \int_0^\sqrt{\tau} \sigma_udu$.

I don't know if this guess is even correct, and if it is I need to verify that it is a solution the Black-Scholes PDE.

Answer 1

What you need is to identify the distribution of the asset price $S_T$, conditional on the information set $\mathcal{F}_{t}$ at time $t$, for $0\leq t < T$. Note that \begin{align*} S_T &= S_t \exp\bigg(\int_{t}^T \Big(r_s-\frac{\sigma_s^2}{2}\Big)ds + \int_t^T\sigma_s dW_s \bigg). \end{align*} Let \begin{align*} P(t, T) = \exp\bigg(-\int_t^T r_s ds \bigg), \end{align*} and \begin{align*} \hat{\sigma} = \sqrt{\frac{1}{T-t}\int_t^T\sigma_s^2 ds}. \end{align*} Then \begin{align*} S_T &= F(t, T)e^{-\frac{\hat{\sigma}^2}{2}(T-t) + \hat{\sigma}\sqrt{T-t} Z}, \end{align*} where $F(t, T)=S_t/P(t, T)$ is the forward price, and $Z$ is a standard normal random variable independent of $\mathcal{F}_t$. Consequently, the value at time $t$ of the option payoff $[\psi(S_T-K)]^+$, where $\psi = \pm 1$, is given by \begin{align*} P(t, T) E\Big([\psi(S_T-K)]^+ \mid \mathcal{F}_t \Big) &= P(t, T)\psi\big[F(t, T) N(\psi d_1) - KN(\psi d_2) \big], \end{align*} where \begin{align*} d_{1} = \frac{\ln F(t, T)/K + \frac{\hat{\sigma}^2}{2}(T-t)}{\hat{\sigma}\sqrt{T-t}}, \end{align*} and \begin{align*} d_2 = d_1 - \hat{\sigma}\sqrt{T-t}. \end{align*}