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European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis

A. H. Nzokem

TL;DR

The paper addresses European option pricing when log-returns follow a Generalized Tempered Stable distribution, calibrated to S&P 500 data. It constructs an arbitrage-free market via an Esscher transform to obtain a risk-neutral GST process and implements two pricing routes: an Extended Black-Scholes framework using FRFT-based GST densities and a Generalized Black-Scholes approach using a 12-point Newton–Cotes quadrature. The results show that Black-Scholes underprices near-the-money and in-the-money options under GTS dynamics, while prices for deeply out-of-the-money and deeply in-the-money options converge with the GST-based prices, illustrating the impact of heavy tails and skewness. The work contributes a data-driven, numerically robust methodology for option pricing under a flexible Levy-based model and demonstrates practical implications for pricing decisions in markets exhibiting tail risk and asymmetry.

Abstract

The paper investigates the performance of the European option price when the log asset price follows a rich class of Generalized Tempered Stable (GTS) distribution. The GTS distribution is an alternative to Normal distribution and $α$-stable distribution for modeling asset return and many physical and economic systems. The data used in the option pricing computation comes from fitting the GTS distribution to the underlying S\&P 500 Index return distribution. The Esscher transform method shows that the GTS distribution preserves its structure. The extended Black-Scholes formula and the Generalized Black-Scholes Formula are applied in the study. The 12-point rule Composite Newton-Cotes Quadrature and the Fractional Fast Fourier (FRFT) algorithms were implemented, and they yield the same European option price at two decimal places. Compared to the option price under the GTS distribution, the Black-Scholes (BS) model is underpriced for the Near-The-Money (NTM) and the in-the-money (ITM) options. However, the BS model and GTS European options yield the same option price for the deep out-of-the-money (OTM) and the deep-in-the-money (ITM) options.

European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis

TL;DR

The paper addresses European option pricing when log-returns follow a Generalized Tempered Stable distribution, calibrated to S&P 500 data. It constructs an arbitrage-free market via an Esscher transform to obtain a risk-neutral GST process and implements two pricing routes: an Extended Black-Scholes framework using FRFT-based GST densities and a Generalized Black-Scholes approach using a 12-point Newton–Cotes quadrature. The results show that Black-Scholes underprices near-the-money and in-the-money options under GTS dynamics, while prices for deeply out-of-the-money and deeply in-the-money options converge with the GST-based prices, illustrating the impact of heavy tails and skewness. The work contributes a data-driven, numerically robust methodology for option pricing under a flexible Levy-based model and demonstrates practical implications for pricing decisions in markets exhibiting tail risk and asymmetry.

Abstract

The paper investigates the performance of the European option price when the log asset price follows a rich class of Generalized Tempered Stable (GTS) distribution. The GTS distribution is an alternative to Normal distribution and -stable distribution for modeling asset return and many physical and economic systems. The data used in the option pricing computation comes from fitting the GTS distribution to the underlying S\&P 500 Index return distribution. The Esscher transform method shows that the GTS distribution preserves its structure. The extended Black-Scholes formula and the Generalized Black-Scholes Formula are applied in the study. The 12-point rule Composite Newton-Cotes Quadrature and the Fractional Fast Fourier (FRFT) algorithms were implemented, and they yield the same European option price at two decimal places. Compared to the option price under the GTS distribution, the Black-Scholes (BS) model is underpriced for the Near-The-Money (NTM) and the in-the-money (ITM) options. However, the BS model and GTS European options yield the same option price for the deep out-of-the-money (OTM) and the deep-in-the-money (ITM) options.
Paper Structure (10 sections, 5 theorems, 27 equations, 6 figures, 3 tables)

This paper contains 10 sections, 5 theorems, 27 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Generalized Tempered Stable (GTS) Process
  3. Parameter Estimation based on daily S&P 500 price
  4. Pricing European Options under GTS Process
  5. GTS Process: Risk Neutral Esscher Measure
  6. Extended Black-Scholes Formula
  7. Generalized Black-Scholes Formula
  8. Parameter $q$ Evaluation
  9. Empirical Analysis: A case of S&P 500 return
  10. Conclusion

Key Result

Theorem 1

Consider a variable $Y \sim GTS(\textbf{$\mu$}, \textbf{$\beta_{+}$}, \textbf{$\beta_{-}$}, \textbf{$\alpha_{+}$},\textbf{$\alpha_{-}$}, \textbf{$\lambda_{+}$}, \textbf{$\lambda_{-}$})$, the characteristic exponent can be written

Figures (6)

  • Figure 1: S&P 500 Index Historical Data
  • Figure 2: $\mu=-0.693477$, $\beta_{+}=0.682290$, $\beta_{-}=0.242579$, $\alpha_{+}=0.458582$, $\alpha_{-}=0.414443$, $\lambda_{+}=0.822222$, $\lambda_{-}=0.727607$
  • Figure 3: Estimations: $f(\xi,\tau, h^{*})$ versus $f(\xi, \tau, h^{*} + 1)$
  • Figure 4: Estimations: $F(\xi,\tau, h^{*})$ versus $F(\xi, \tau, h^{*} + 1)$
  • Figure 5: Optimal parameter ($q$) and minimum Error value ($ER(k,q)$)
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • proof
  • Corollary 4
  • Theorem 5