Option pricing in bilateral Gamma stock models
Uwe Küchler, Stefan Tappe
TL;DR
This paper develops a pricing framework for options in bilateral Gamma stock models by analyzing several martingale measures that preserve the bilateral Gamma structure. It derives existence conditions and explicit transformations for Esscher, minimal entropy, bilateral Esscher, and minimal martingale measures, and provides Fourier-based pricing formulas to compute option prices under these measures. It also compares the relative entropy and practical pricing implications, showing that MEMM and bilateral Esscher measures are close to each other and to the Esscher measure when market data are calibrated. A numerical illustration with DAX data demonstrates the behavior of the implied volatility surface and hedging considerations under these measures, and discusses hedging via Föllmer-Schweizer decomposition under the MMM. The work yields tractable, calibratable pricing and hedging tools for jump models with bilateral Gamma dynamics, offering insight into when different pricing measures are economically reasonable and computationally feasible.
Abstract
In the framework of bilateral Gamma stock models we seek for adequate option pricing measures, which have an economic interpretation and allow numerical calculations of option prices. Our investigations encompass Esscher transforms, minimal entropy martingale measures, $p$-optimal martingale measures, bilateral Esscher transforms and the minimal martingale measure. We illustrate our theory by a numerical example.