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A limit theorem for generalized tempered stable processes and their quadratic variations with stable index tending to two

Masaaki Fukasawa, Mikio Hirokane

TL;DR

This work analyzes multidimensional Generalized Tempered Stable (GTS) processes as the stability index $\alpha$ approaches 2, showing that, under a specific scaling of the Lévy measure, the process converges to Brownian motion while the normalized quadratic covariation converges to an independent stable process of index 1. The authors apply this limit to finance by constructing a pure-jump exponential Lévy model perturbing the Black–Scholes framework and derive explicit first-order asymptotics for call and digital option prices, as well as the at-the-money implied volatility skew using Fourier methods (Carr–Madan). A cumulant expansion is developed to support these asymptotics, and the results are complemented by numerical experiments illustrating the predicted $t^{-1/2}$ decay of the ATM skew. The findings provide a principled bridge between stable-type jump models and near-Black–Scholes pricing, with practical implications for calibration and risk-management of short-maturity options.

Abstract

We study the limit of the joint distribution of a multidimensional Generalized Tempered Stable (GTS) process and its quadratic covariation process when the stable index tends to two. Under a proper scaling, the GTS processes converges to a Brownian motion that is a stable process with stable index two. We renormalize their quadratic covariation processes so that they have a nondegenerate limit distribution. We show that the limit is a stable process with stable index one and is independent of the limit Brownian motion of the GTS processes. In addition, we apply this convergence result to finance. By using the scaled GTS process defined above, we construct a pure jump asset price model approaching to the Black-Scholes model. To evaluate how $α$-stable jumps affect the implied volatility, we obtain the asymptotic expansion of the at-the-money implied volatility skew when the model approaches to the Black-Scholes model.

A limit theorem for generalized tempered stable processes and their quadratic variations with stable index tending to two

TL;DR

This work analyzes multidimensional Generalized Tempered Stable (GTS) processes as the stability index approaches 2, showing that, under a specific scaling of the Lévy measure, the process converges to Brownian motion while the normalized quadratic covariation converges to an independent stable process of index 1. The authors apply this limit to finance by constructing a pure-jump exponential Lévy model perturbing the Black–Scholes framework and derive explicit first-order asymptotics for call and digital option prices, as well as the at-the-money implied volatility skew using Fourier methods (Carr–Madan). A cumulant expansion is developed to support these asymptotics, and the results are complemented by numerical experiments illustrating the predicted decay of the ATM skew. The findings provide a principled bridge between stable-type jump models and near-Black–Scholes pricing, with practical implications for calibration and risk-management of short-maturity options.

Abstract

We study the limit of the joint distribution of a multidimensional Generalized Tempered Stable (GTS) process and its quadratic covariation process when the stable index tends to two. Under a proper scaling, the GTS processes converges to a Brownian motion that is a stable process with stable index two. We renormalize their quadratic covariation processes so that they have a nondegenerate limit distribution. We show that the limit is a stable process with stable index one and is independent of the limit Brownian motion of the GTS processes. In addition, we apply this convergence result to finance. By using the scaled GTS process defined above, we construct a pure jump asset price model approaching to the Black-Scholes model. To evaluate how -stable jumps affect the implied volatility, we obtain the asymptotic expansion of the at-the-money implied volatility skew when the model approaches to the Black-Scholes model.
Paper Structure (11 sections, 8 theorems, 121 equations, 3 figures)

This paper contains 11 sections, 8 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalized Tempered Stable Processes
  3. Limit Theorem for the Scaled GTS Process
  4. Asymptotics of Implied Volatility and Option Prices
  5. A Perturbation Model with Stable-Type Jumps
  6. Results : Asymptotics of the implied volatility and Option Prices
  7. Asymptotic Expansion of the Cumulant
  8. Proof : Asymptotic Expansions of Call and Digital Option Prices
  9. Proof : Asymptotic Expansion of the ATM Volatility Skew
  10. Numerical Experiment
  11. Decay of the Characteristic Function of a GTS Process

Key Result

Lemma 2.1

Let $X$ be an $\mathbb{R}^d$-valued Lévy process with triplet $(m,V,\nu)$ and $\lbrack*\rbrack{X}$ be the optional quadratic covariation process of $X$. Then, $\mathbb{R}^d\times\mathbb{R}^{d\times d}$-valued Lévy process $(X,\lbrack*\rbrack{X})$ has the following characteristic function: where $z\in\mathbb{R}^d$ and $U\in\mathbb{R}^{d\times d}$.

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 4: For each $\alpha=1.2,1.7$ and $1.9$, the ATM implied volatility skew $\partial_K\hat{\sigma}(t,K,\alpha)|_{K=1}$ is calculated at each maturity $t$ and plotted with the solid lines. Then, for each $\alpha=1.2,1.7$ and $1.9$, $(2-\alpha)A_{\mathrm{ATM}}(t)$, the first-order term of the asymptotic expansion, is plotted with the dashed lines. We did this calculations and the plots with Python.

Theorems & Definitions (27)

  • Lemma 2.1: The joint distribution of a Lévy process and its quadratic covariation process
  • proof
  • Definition 2.2: Generalized tempered stable process
  • Example 2.3: the CGMY process
  • Definition 3.1: Scaled generalized tempered stable process
  • Definition 3.2: Normalized quadratic covariation
  • Theorem 3.3: Main result
  • proof
  • Definition 4.1: Scaled pure-jump exponential Lévy model
  • Remark 4.2
  • ...and 17 more