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Derivatives Pricing and Computational Finance / Monte Carlo Methods and Stochastic Volatility

Fourier Transform Pricing

For models beyond Black–Scholes the transition density is usually unknown — but its Fourier transform, the characteristic function, is often available in closed form (Heston, every Lévy/jump model). This note develops the technology that converts a characteristic function into option prices: the inversion lemma, the Lévy-process machinery of TCF 7.3.1 that supplies characteristic functions for jump models (Lévy–Khinchin, compound Poisson, Merton), the affine-jump-diffusion machinery that generates characteristic functions by solving Riccati ODEs, the Carr–Madan damped-FFT method with its complete derivation and implementation, and the COS method whose exponential convergence makes it the calibration engine of choice. The speed differential is not marginal: on the L8 benchmarks, COS is roughly a thousand times faster than Monte Carlo at a thousand times the accuracy — which is what makes daily recalibration of Heston/SLV (stochastic volatility and the Heston model) possible at all.
Learning objectives. By the end of this note you should be able to:
  • define the Fourier transform and the characteristic function, and prove the half-axis inversion lemma;
  • write the DFT as a matrix–vector product and explain what the FFT buys;
  • state the Lévy–Khinchin representation, interpret the characteristic triplet, derive the compound-Poisson characteristic function, and use the martingale criterion to find risk-neutral drifts (Merton's model);
  • explain the affine-jump-diffusion framework and derive the Black–Scholes characteristic function from the Riccati ODEs;
  • write the full Heston characteristic function and explain where it comes from;
  • derive the Carr–Madan formula end-to-end (damping, Fubini, the complex shift) and lay out its FFT implementation including the grid-reciprocity constraint;
  • derive the COS pricing formula including the analytic payoff coefficients χk,ψk\chi_k, \psi_k, and explain why convergence is exponential;
  • rank MC, Carr–Madan and COS by speed/accuracy and say when each is appropriate.

In this lesson

  1. 7.1 Motivation
  2. 7.2 The Fourier transform and characteristic functions
  3. 7.3 The inversion theorem and the half-axis lemma
  4. 7.4 The discrete Fourier transform and the FFT
  5. 7.5 Characteristic functions of Lévy processes
  6. 7.6 Affine jump diffusions: a characteristic-function factory (Duffie–Pan–Singleton)
  7. 7.7 Worked example: the Black–Scholes characteristic function
  8. 7.8 The Heston characteristic function
  9. 7.9 The Carr–Madan pricing technique
  10. 7.10 Numerical implementation: quadrature, truncation, and the FFT
  11. 7.11 The COS method (Fang & Oosterlee 2008)
  12. 7.12 Method comparison: the ranking

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

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