The Black–Scholes PDE has a closed-form solution only for the simplest contracts. For everything else — barriers, American exercise, dividends at discrete dates — we solve the PDE numerically. This note develops the workhorse: replace derivatives by difference quotients on a grid, then march the solution backwards from the payoff. The mathematical drama is stability: the cheapest scheme (explicit) blows up unless the time step is tiny, the implicit scheme is unconditionally stable but only first-order accurate in time, and Crank–Nicolson achieves second order and unconditional stability for the cost of one tridiagonal solve per step. Understanding why — via the eigenvalues of the iteration matrix — is the heart of the note and a standard exam question.
Learning objectives. By the end of this note you should be able to:
- set up the FDM grid in both the direct formulation and the heat-equation formulation, and explain the trade-off;
- derive the explicit, implicit and Crank–Nicolson schemes from Taylor-series difference quotients;
- carry out the stability analysis of the explicit scheme and state the condition ;
- interpret the explicit scheme as a trinomial tree and negative coefficients as a stability alarm;
- impose Dirichlet/Neumann boundary conditions and write the schemes in matrix form;
- compare FDM against Monte Carlo and state when each wins.
In this lesson
- 5.1 Two equivalent formulations
- 5.2 Difference approximations and the grid
- 5.3 Explicit method (forward difference in time)
- 5.4 Stability and convergence
- 5.5 Implicit method (backward difference in time, BTCS)
- 5.6 Crank–Nicolson method
- 5.7 Boundary conditions
- 5.8 Method comparison
The full lesson, with worked examples and exercise solutions, is for members.
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