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Fast and explicit European option pricing under tempered stable processes

Gaetano Agazzotti, Jean-Philippe Aguilar

TL;DR

This work develops Mellin-Barnes residue representations for tempered stable ($\mathrm{TS}$) densities and uses them to derive fast, hyperparameter-free series pricing for digital and European options in exponential $\mathrm{TS}$ models. It unifies one-sided and two-sided TS frameworks, introduces new multi-dimensional MB representations, and delivers compact, tractable pricing formulas for KoBoL, CGMY, BG, and VG special cases. The methods achieve high accuracy with modest summation orders and compare favorably with state-of-the-art Fourier methods, while avoiding parameter tuning inherent to damping or truncation schemes. The results offer a practical, robust alternative for pricing under $\mathrm{TS}$ dynamics, with public code available in TS-Pricing for reproducibility and further extensions to broader $\beta$-ranges and additive processes.

Abstract

We provide series expansions for the tempered stable densities and for the price of European-style contracts in the exponential Lévy model driven by the tempered stable process. These formulas recover several popular option pricing models, and become particularly simple in some specific cases such as bilateral Gamma process and one-sided TS process. When compared to traditional Fourier pricing, our method has the advantage of being hyperparameter free. We also provide a detailed numerical analysis and show that our technique is competitive with state-of-the-art pricing methods.

Fast and explicit European option pricing under tempered stable processes

TL;DR

This work develops Mellin-Barnes residue representations for tempered stable () densities and uses them to derive fast, hyperparameter-free series pricing for digital and European options in exponential models. It unifies one-sided and two-sided TS frameworks, introduces new multi-dimensional MB representations, and delivers compact, tractable pricing formulas for KoBoL, CGMY, BG, and VG special cases. The methods achieve high accuracy with modest summation orders and compare favorably with state-of-the-art Fourier methods, while avoiding parameter tuning inherent to damping or truncation schemes. The results offer a practical, robust alternative for pricing under dynamics, with public code available in TS-Pricing for reproducibility and further extensions to broader -ranges and additive processes.

Abstract

We provide series expansions for the tempered stable densities and for the price of European-style contracts in the exponential Lévy model driven by the tempered stable process. These formulas recover several popular option pricing models, and become particularly simple in some specific cases such as bilateral Gamma process and one-sided TS process. When compared to traditional Fourier pricing, our method has the advantage of being hyperparameter free. We also provide a detailed numerical analysis and show that our technique is competitive with state-of-the-art pricing methods.

Paper Structure

This paper contains 37 sections, 14 theorems, 123 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. TS distributions
  3. Notations
  4. One-sided and double-sided tempered stable distributions
  5. One-sided TS distributions
  6. Double-sided TS distributions
  7. A continuity property
  8. Particular cases
  9. Series expansion of TS distributions
  10. Notations
  11. Mellin-Barnes representation of TS densities
  12. Series expansion of TS densities
  13. Option pricing under TS processes
  14. Option pricing framework
  15. Digital options
  16. ...and 22 more sections

Key Result

Proposition 2.1

Let $(\alpha_+,\lambda_+,\alpha_-,\lambda_-)\in (0,+\infty)^4$ and $(\beta_n^+)_{n\in\mathbb{N}}\in (0,1)^\mathbb{N}$ and $(\beta_n^+)_{n\in\mathbb{N}}\in (0,1)^\mathbb{N}$ be two sequences such that $\beta_n^\pm \underset{n\to\infty}{\longrightarrow}\beta_\pm\in(0,1)^2$ with $(\beta_n^+)_{n\in\math

Figures (3)

  • Figure 1: Series expansion \ref{['eq:serie-gts-density']} of the double-sided TS density truncated at $\bm{n}=(60,60,60)$vs. numerical Fourier inversion of the characteristic function \ref{['eq:char_TS']} for $t=1$. TS parameters from \ref{['eq:numerical-params']}.
  • Figure 2: Comparison between the call pricing formula in the TS model (proposition \ref{['prop:eur-series-gts']}) and the PROJ method. We let $S_0$ vary in $[0.2,1.9]$, and we choose various maturities ($T\in\{0.05,0.1,0.2,0.5\}$).
  • Figure 3: Relative error (dotted lines) and computation time (continuous lines) under one and double-sided tempered stable as well as bilateral Gamma process. We use $S_0=1$, $K=1.5$ and $T=1.2$.

Theorems & Definitions (35)

  • Remark 1
  • Remark 2
  • Proposition 2.1
  • proof
  • Example 1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 25 more