Pricing European Calls with the Explicit Euler Finite Difference Method

Summary

The document explains how to approximate European vanilla option prices by solving the Black–Scholes partial differential equation with an explicit Euler finite difference scheme. It lays out the PDE domain, expiry payoff, and call boundary conditions, then describes discretizing time and spot derivatives to calculate each time layer from the previous one. The solution is marched backward from expiry to the present, producing a price surface over spot and time.

A C++ design separates payoff and option data, PDE coefficients and conditions, grid storage, and the explicit solver. The article illustrates the method with a call option and a plotted output surface: the expiry slice matches the piecewise linear payoff, while earlier prices become smoother around the strike. The example demonstrates implementation structure and qualitative behavior rather than a comparison against an analytic price. It flags stability, consistency, and convergence as important topics left untreated, and notes that more advanced schemes such as Crank–Nicolson are covered separately.

Key ideas

• The explicit scheme computes a new time layer directly from values at the prior layer.
• For option pricing, the calculation starts with the expiry payoff and proceeds backward to the present.
• The finite difference grid approximates the Black–Scholes equation over discrete spot and time points.
• Call boundary conditions set the value at low spot to zero and adjust the high-spot value for discounting.
• The method’s reliability depends on stability and convergence conditions that the article does not analyze.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.