Finite Difference Formulas for Derivatives in Heat Equation Models
Summary
The document introduces finite difference methods as a way to approximate derivatives and prepare a heat equation for numerical solution. Taylor expansions around a point yield forward and backward first derivative estimates with first order accuracy, a centered first derivative estimate with second order accuracy, and a centered second derivative estimate with second order accuracy.
It then places these formulas on a discrete space and time grid, defining function values at grid locations and time steps. The resulting expressions correspond to forward, backward, and centered differences for the first derivative and a centered difference for the second derivative. The heat equation is relevant to quantitative finance because the Black–Scholes pricing equation can be transformed into this form, enabling numerical methods to be applied to pricing problems. The discussion distinguishes discretization error from floating point roundoff, but does not implement or compare complete solvers. It previews later coverage of stability conditions and accumulated error rather than establishing those results here.
Key ideas
- Taylor expansions provide finite difference approximations to derivatives.
- Forward and backward first derivative estimates have first order accuracy in the spacing.
- Centered approximations for the first and second derivatives have second order accuracy.
- A space and time grid converts the continuous heat equation into discrete values.
- Numerical solutions can accumulate discretization and floating point roundoff errors.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.