Quantitative Finance

Chapter 5

Finite difference methods for partial differential equations

Option pricing problems can typically be represented as a partial differential equation (PDE) subject to boundary conditions, see for example the Black/Scholes PDE in Section 4.2.1 or the option pricing PDE in the presence of stochastic volatility in Section 6.3.. The idea behind finite difference methods is to approximate the partial derivatives in the PDE by a difference quotient, e.g.

$\begin{array}{cc}{\partial }_{1}C(t,S(t))\approx \frac{C(t+\text{Δ}t,S(t))-C(t,S(t))}{\text{Δ}t}& (5.1)\end{array}$

$\begin{array}{cc}{\partial }_{2}C(t,S(t))\approx \frac{C(t,S(t)+\text{Δ}S)-C(t,S(t))}{\text{Δ}S}& (5.2)\end{array}$

Thus the PDE is discretised in both the time and the space dimension and the resulting solution converges to the continuous (true) solution as the discretisation grid is made finer and finer. Potentially there ...