Finite Differences for Derivatives in Heat and Black-Scholes PDEs

Summary

This tutorial introduces finite difference methods for approximating derivatives in a parabolic partial differential equation, using the heat equation as its example. It motivates the approach for quantitative finance by noting that the Black-Scholes equation can be transformed into the heat equation. Taylor expansions around a point, taken in forward and backward directions, provide approximations for first and second derivatives.

The tutorial then discretizes space and time into a grid, representing the function at each grid point and time step. It names forward, backward, and central difference forms and distinguishes time indices from powers or derivative orders. It also describes truncation error from omitted higher-order terms and rounding error from finite-precision storage. The discussion is introductory: it does not present a complete numerical solution, stability conditions, boundary conditions, or option-pricing results. Those choices, along with step size and precision, affect whether a computed solution is reliable.

Key ideas

• Taylor expansions yield finite difference approximations for derivatives from nearby function values.
• Discretizing space and time turns the continuous heat equation into values on a grid.
• The heat equation provides a route to numerical methods for the transformed Black-Scholes equation.
• Finite difference calculations incur truncation error and rounding error.
• Grid spacing, discretization choice, and numerical precision affect solution accuracy.

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