The Heston characteristic function derived in the previous sub-note computes in closed form. From this, European option prices follow by Fourier inversion — the Gil-Pelaez formula. The result is a semi-analytic pricing formula: no closed form in the Black-Scholes sense (which gives an answer immediately), but one numerical integral per option price, evaluated by fast quadrature. On a modern CPU, pricing 500 strikes at 10 maturities from one characteristic function evaluation takes under 100ms — fast enough for production calibration.
This note derives the Gil-Pelaez formula, discusses the choice of integration contour, and presents the Carr-Madan FFT approach for simultaneous pricing across a grid of strikes.
Learning objectives. By the end of this note you should be able to:
- state the Gil-Pelaez formula for European option prices in terms of ;
- implement Heston European pricing via Gauss-Laguerre or trapezoidal quadrature;
- explain what the Carr-Madan FFT approach achieves and when to use it over direct integration;
- price a call using the numerical procedure and check against a BS formula at constant vol.
In this lesson
- 3.1 From characteristic function to probabilities
- 3.2 Numerical integration
- 3.3 The Carr-Madan FFT approach
- 3.4 Verification
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.