Chapter 9: Option Pricing Using Monte Carlo Methods

Monte Carlo simulation is a very general technique with wide applications. In this chapter, we briefly review the technique in the context of option pricing.

Recall that the Wiener process is a family (W_t)_t≥0, where W₀ = 0, W_t ∼ N(0, t) for all t ≥ 0, ΔW_t = W_t+Δt − W_t on non-overlapping time intervals are independent, and W_t depend continuously on t. Here N(μ, σ²) denotes the normal distribution with mean μ and variance σ². As a consequence of the definition, W_t − W_s ∼ N(0, t − s) and so E[W_t − W_s] = 0, Var(W_t − W_s) = t − s implying E[(ΔW_t)²] = Δt. So, ΔW are independent and normally distributed with mean 0 and variance Δt.