# Finite Difference Method for Black-Scholes differential equation

I am using the backward finite difference Method to simulate the price.

The algorithm is the follows:

Suppose we know $V_T(S_T)=payoff$, we can use a backward recurrence:$$V_{T-\delta t}=V_T-\frac{\partial V}{\partial t}\cdot \delta t$$

This follows just from the Talyor approximation around time $T$.

$\frac{\partial V}{\partial t}$ can be obtained from Black-Scholes-Partial differential equation, i.e.$$\frac{\partial V}{\partial t}=rV-rS\frac{\partial V}{\partial S}-\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}$$, where the partial differential can be approximated by central difference.

Question: (1) At time $T$ my $V_T$ is equal to the payoff of the option, this is not really differentiable, how can I assume that the partial differential $\frac{\partial^2 V}{\partial S^2}$ and $\frac{\partial V}{\partial S}$ exists? If these doesn't exist, I am not really able to define the recurrence in the first formula.

(2) By the Taylor approximation around time $T$ in the first formula, $V_T$ depends also on $S_T$ which is again time dependend, why I can neglect the time dependence in the $S_T$ term and only approximate around $t=T$

Thanks

• Answer to (1): $V$ is differentiable in $S$ for $t < T$ and that is all you need. Lookup finite differences schemes, explicit, implicit or mixt for a proper setup of the scheme including boundary conditions. Answer to (2): do not confuse $S_T$ the time $T$ (random) value of a stochastic process with $S$ the state variable in the PDE. – Antoine Conze May 4 '17 at 7:32
• @AntoineConze Thank you for the clarification, I definitely misunderstood it. – quallenjäger May 4 '17 at 17:45