Black-Scholes equation

The Black-Scholes equation from finance theory is $$\frac{\partial V}{\partial t}+ rs\frac{\partial V}{\partial s} + \frac{\sigma^2 s^2}{2} \frac{\partial^2 V}{\partial s^2} - rV = 0 \qquad 0 < t<T$$subject to the final boundary condition $V (s, T) = F (s)$ .

We can multiply the equation by $e^{- r t}$ , and define $V_{1} = e^{- r t} V$ . Then the equation becomes $$\frac{\partial V_1}{\partial t}+ rs\frac{\partial V_1}{\partial s} + \frac{\sigma^2 s^2}{2} \frac{\partial^2 V_1}{\partial s^2} = 0$$
Let $t_{1} = - (σ^{2} / 2) t$ . Then $\frac{\partial V_{1}}{\partial t} = \frac{\partial V_{1}}{\partial t_{1}} (- σ^{2} / 2)$ , and upon dividing by $- σ^{2} / 2$ on both sides we get $$\frac{\partial V_1}{\partial t_1}= \frac{2r}{\sigma^2}s\frac{\partial V_1}{\partial s} + s^2\frac{\partial^2 V_1}{\partial s^2}$$
Then we can solve this using the same method as we did for a variant of the heat equation

An alternative way is to consider the change of variables and the substitution $V (s, t) = e^{a x + b τ} = U (x, τ)$ , with $x = \ln s$ , $τ = \frac{σ^{2}}{2} (T - t)$ , $a = \frac{1}{2} - \frac{r}{σ^{2}}$ , and $b = - {(\frac{1}{2} + \frac{r}{σ^{2}})}^{2}$ . Then this gets us that $$\frac{\partial U}{\partial \tau} = \frac{\partial^2 U}{\partial x^2}$$with an initial condition of the form $U (x, 0) = e^{- a x} F (e^{x})$ . We can solve the this equation getting $$U(x, \tau) = \frac1{\sqrt{4\pi \tau}}\int_{-\infty}^\infty e^{-ay}F(e^{y}) \exp\left(\frac{(x-y)^2}{4\tau}\right) , dy$$
Doing all the substitutions back we get that the solution is $$V(s, t) = \frac{e^{-r(T-t)}}{\sqrt{2\pi \sigma^2(T-t)}} \int_0^\infty F(s^) \exp\left(-\frac{(\ln(s/s^)+(r-\sigma^2/2)(T-t))^2}{2\sigma^2(T-t)}\right), ds^*$$