# Summary of Pricing Options of Log-Normal Claims Using Black's Formula

Cross posted from here.

Let $$B$$ be a $$Q$$-Brownian motion and $$X^{s,x}$$ given by

$$dX_t = X_t(\mu_t dt + \sigma_t dB_t),\quad X_s = x$$

for $$\mu, \sigma$$ deterministic. Let $$\mu_{s,t}=\int_s^t \mu_u du$$ and $$\sigma^2_{s,t} = \int_s^t \sigma^2_u du$$, i.e. $$X^{s,x}_t = x\exp(\mu_{s,t} - \sigma^2_{s,t}/2 + \int_s^t\sigma(u)dB_u)$$.

If $$\mu_t \equiv \mu$$, $$\sigma_t\equiv \sigma$$ are constant I know that

\begin{align} E_Q\left((X_t^{0,1} - K)^+\vert\mathcal{F}_s\right) & =E_Q\left((X_{t}^{s,x} - K)^+\right)\vert_{x = X_s^{0,1}}\\ & = x\exp(\mu_{s,t})\Phi(d_1(x,s,t)) - K\Phi(d_2(x,s,t))\vert_{x = X_s^{0,1}}\\ \end{align}

where $$d_1(x,s,t) = \frac{log(x/K) + \mu_{s,t} - \sigma_{s,t}^2/2}{\sqrt{\sigma^2_{s,t}}}$$ and $$d_2(x,s,t) = d_1(x,s,t) - \sqrt{\sigma^2_{s,t}}$$, using the Markov property and the standard computations from Black-Scholes pricing.

Question: The same formula should hold for $$\mu$$, $$\sigma$$ non-constant but how do I show that in a non-sketchy way? I tried to use markov kernels but...