Questions About Backward Finite Differences for Black–Scholes

Summary

The document presents a proposed backward-time finite-difference scheme for solving the Black–Scholes partial differential equation. Starting from the option payoff at expiration, it uses a Taylor approximation to step the value backward in time. It substitutes the Black–Scholes equation for the time derivative and proposes central differences for the spatial derivatives.

It raises two numerical-analysis questions: how to handle the payoff’s lack of smoothness at expiration when approximating first and second derivatives, and why the backward Taylor step appears to ignore that the underlying price also varies with time. The document contains no answer, computed example, convergence evidence, or discussion of boundary conditions and stability. It is therefore useful as a statement of implementation concerns rather than a complete finite-difference method or validated pricing result.

Key ideas

• The proposed scheme starts from the option payoff at expiration and steps backward in time.
• The Black–Scholes equation supplies the time derivative for the recurrence.
• Central differences are proposed to approximate spatial derivatives.
• The question highlights payoff nonsmoothness and the treatment of the underlying price during a time step.
• No solution, stability analysis, or numerical evidence is provided.

Tags

# Finite Difference Method for Black-Scholes differential equation


# 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

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