Two analytically tractable jump-diffusion models dominate the early literature: Merton (1976) with Gaussian log-jump sizes (giving a symmetric contribution to the smile) and Kou (2002) with double-exponential jump sizes (giving asymmetric, leptokurtic tails with analytic option prices). Both extend Black-Scholes by adding a compound Poisson component to GBM; both price European options in closed form (Merton via an infinite series, Kou via a recursion). They differ in how well they match empirical tails and in computational tractability.
Learning objectives. By the end of this note you should be able to:
- state the SDE and characteristic function for the Merton and Kou models;
- price a European call under Merton's model using the series formula;
- calibrate Merton or Kou to an observed smile cross-section;
- explain why the double-exponential is computationally attractive for barrier options.
In this lesson
- 3.1 The Merton jump-diffusion model
- 3.2 Merton European call price
- 3.3 The Kou double-exponential model
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Advanced Derivatives and Exotic Options syllabus.