Skip to content
Advanced Derivatives and Exotic Options / Local Volatility

Pricing in the Local Vol Model — PDE and Monte Carlo

Once a local vol surface σL(S,t)\sigma_L(S, t) is calibrated, the practical question is how to price exotic options under it. Unlike Black-Scholes (constant σ\sigma → analytic formula for Europeans), local vol models have no closed-form solutions for path-dependent exotics. Two numerical approaches dominate: backward PDE methods (the natural extension of Black-Scholes finite differences) and Monte Carlo simulation of the SDE dSt=(r−q)St dt+σL(St,t)St dWtQdS_t = (r-q)S_t\,dt + \sigma_L(S_t, t) S_t\, dW_t^{\mathbb{Q}}.

Both routes start from the calibrated surface and price consistently with the market — this is the core advantage of local vol over Black-Scholes with a flat vol: every exotic is priced on a surface that reprices all vanilla instruments. The limitation is the surface dynamics (the forward smile problem, treated in the next note), not the numerical scheme.

Learning objectives. By the end of this note you should be able to:
  • write the backward PDE for a local vol model and identify how it differs from the Black-Scholes PDE;
  • implement a local vol Monte Carlo with Euler-Maruyama and assess its path-discretisation error;
  • price a knock-out barrier option under both PDE and Monte Carlo and explain why the two may disagree;
  • describe when PDE is preferable to Monte Carlo and vice versa for exotic payoffs.

In this lesson

  1. 3.1 Backward PDE under local vol
  2. 3.2 Monte Carlo under local vol
  3. 3.3 PDE vs Monte Carlo — when to use which

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or browse the Advanced Derivatives and Exotic Options syllabus.