Outside GBM, SDEs have no closed-form solutions and paths must be generated step by step. This note covers the two workhorse schemes — Euler (simple, strong order ½) and Milstein (one extra term, strong order 1) — and, more importantly, the theory of what "accurate" means for a random object: the distinction between strong (pathwise) and weak (distributional) convergence, which determines both how schemes are graded and which accuracy you actually need for pricing. The note ends with the practical pathology of finance's favourite non-Lipschitz SDE: the CIR square-root process can be driven negative by Euler, and the fixes (full truncation et al.) are exam material. The derivation of Milstein's correction term from the stochastic Taylor expansion lives in the stochastic Taylor expansion derivation.
Learning objectives. By the end of this note you should be able to:
- write the Euler and Milstein schemes for a general scalar SDE and for GBM;
- define strong and weak convergence, state the orders for Euler/Milstein, and explain which mode option pricing needs;
- explain why the exact lognormal step beats Euler for GBM;
- describe the CIR positivity problem and the full-truncation fix;
- outline jump-diffusion simulation.
In this lesson
- 3.1 Euler discretisation
- 3.2 Strong and weak convergence
- 3.3 The Milstein scheme
- 3.4 Positivity and the CIR process
- 3.5 Jump diffusion (preview for L8)
The full lesson, with worked examples and exercise solutions, is for members.
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