Solve Black scholes PDE without using any transformation

Question

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$

is to transform it into the heat equation and then using the explicit euler FTCS scheme.

I was wondering if you could directly use the explicit euler scheme on the Black scholes PDE without using any transofrmation, just substitute the approximations of the derivatives.

In that case i realized that i have no initial condition $V(S,0)$ to start solving. And one of the boundary condition works for $S \rightarrow \infty$, will it work if i restrict $S=S_{max}$.

Is it possible to solve the Black scholes PDE this way, if it is what is the initial condition and the boundary condition i should take .

Answer 1

Yes it can be done. However, bear in mind that a naive explicit FD scheme is not unconditionally stable (see CFL stability condition).

As far as your initial/boundary conditions issue is concerned:

[Time domain]

Use terminal condition $V(S,T)=h(T)$ where $h(T)$ figures the payoff of the target derivative claim at maturity $T$ (e.g. $h(T)=(S-K)^+$ for a European call option struck at $K$ expiring at $T$). This is simply an expression of the absence of arbitrage opportunity. Note that this condition can become an initial condition if you change variables i.e. replace the variable $t$ (current time) by $\tau = T-t$ (remaining time to expiry).

[Space domain]

Either use claim-dependent Dirichlet conditions. For instance for a European call, $\forall t \in [0,T] $

$$V(S=0,t)=0$$ $$\lim_{S \rightarrow \infty} V(S,t) = P(t,T) (F(t,T)-K)$$

Or use more general boundary conditions on the derivatives of the option price, typically like no Gamma when sufficiently OTM, $\forall t \in [0,T] $

$$ \lim_{S\rightarrow 0}\frac {\partial^2 V}{\partial S^2} (S,t) = 0$$ $$ \lim_{S\rightarrow \infty}\frac {\partial^2 V}{\partial S^2} (S,t) = 0$$

Because you will truncate the spot price domain to $[S_{min},S_{max}]$ with $0 < S_{min} < S_{max} < \infty$ you obviously need to be careful here: applying a boundary condition such as $V(S_{min},t)=0$ will only be a good approximation of the true boundary condition $V(S=0,t)=0$ if $S_{min}$ is sufficiently small.