Paper: arXiv 2307.12843

Abstract

We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine expansion (COS) method. The classical COS method is numerically very efficient in one-dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a cosine series. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs dense mathematical analysis including multivariate characteristic functions, convergence proofs, and the damped COS method, scoring high on math complexity. It includes numerical experiments up to five dimensions and provides explicit/implicit formulas for parameters, indicating strong empirical validation, warranting a high rigor score.

Research Flowchart

  flowchart TD
  A["Research Goal:<br/>Develop unified framework for<br/>multivariate density & option pricing<br/>from characteristic function"] --> B{"Key Methodology<br/>Damped COS Method"}
  B --> C["Inputs:<br/>Characteristic Function<br/>Truncation Range L<br/>Number of Terms N"]
  C --> D["Computation:<br/>Fourier-Cosine Expansion<br/>& Numerical Integration"]
  D --> E{"Convergence Analysis"}
  E -->|Exponential Decay| F["Outcome:<br/>Exponential Convergence"]
  E -->|General Integrands| G["Outcome:<br/>Efficient Multivariate<br/>Density Estimation<br/>up to 5D"]
  F --> H["Final Outcomes:<br/>European Option Prices<br/>Absolute Moments<br/>Distribution Functions"]
  G --> H