[Paper Review] Exponential stock models driven by tempered stable processes

Uwe Küchler, Stefan Tappe|arXiv (Cornell University)|Feb 26, 2014

Stochastic processes and financial applications38 references21 citations

TL;DR

This paper provides a systematic analysis of equivalent martingale measures for exponential stock models driven by tempered stable Lévy processes, focusing on preserving analytical tractability in option pricing. It establishes existence conditions for Esscher, minimal entropy, p-optimal, and Föllmer-Schweizer martingale measures, deriving explicit pricing formulae and demonstrating model stability via a DAX case study with implied volatility surfaces.

ABSTRACT

We investigate exponential stock models driven by tempered stable processes, which constitute a rich family of purely discontinuous L\'{e}vy processes. With a view of option pricing, we provide a systematic analysis of the existence of equivalent martingale measures, under which the model remains analytically tractable. This includes the existence of Esscher martingale measures and martingale measures having minimal distance to the physical probability measure. Moreover, we provide pricing formulae for European call options and perform a case study.

Motivation & Objective

• To establish existence conditions for equivalent martingale measures in exponential stock models driven by tempered stable processes.
• To ensure analytical tractability of option pricing under these measures by preserving the tempered stable nature of the driving process.
• To derive explicit pricing formulae for European call options using Fourier transform techniques under various martingale measures.
• To validate model stability and practical applicability through a case study using DAX historical data and implied volatility surface computation.

Proposed method

• Modeling stock prices via exponential Lévy processes driven by tempered stable processes with six parameters (α±, β±, λ±).
• Applying Esscher transforms to derive equivalent martingale measures under which the process remains tempered stable.
• Using bilateral Esscher transforms to construct minimal entropy and p-optimal martingale measures, with existence conditions derived via convex optimization and cumulant generating functions.
• Employing the Föllmer-Schweizer minimal martingale measure, which decomposes the process into two independent tempered stable components under the new measure.
• Deriving option pricing formulae via characteristic function inversion using Fourier transform methods.
• Validating results through a DAX-based case study, estimating parameters from historical data and computing option prices and implied volatility surfaces.

Experimental results

Research questions

• RQ1 Under what conditions does an Esscher martingale measure exist for a tempered stable-driven exponential stock model?
• RQ2 When does a minimal entropy martingale measure exist, and how does its existence depend on the parameters λ+ and λ−?
• RQ3 Can the Föllmer-Schweizer minimal martingale measure be constructed such that the driving process remains analytically tractable?
• RQ4 How do different martingale measures affect the implied volatility surface in practice?
• RQ5 How stable are the resulting option prices under small calibration errors in model parameters?

Key findings

• An Esscher martingale measure exists if and only if λ+ + λ− > 1 and r − q lies within the interval (f(−λ−), f(λ+ − 1)], with f(Θ) defined via the cumulant generating function.
• The minimal entropy martingale measure exists when λ+ < 1, or when λ+ ≥ 1 and the drift condition involving α±, β±, λ± is satisfied.
• The Föllmer-Schweizer minimal martingale measure exists if and only if the drift condition α+Γ(−β+) ((λ+ −1)β+ − λ+β+) + α−Γ(−β−) ((λ−+1)β− − λ−β−) ≥ r − q is met.
• Option pricing formulae are derived using Fourier transform techniques, enabling analytical computation of European call option prices under all studied martingale measures.
• The case study on the DAX index shows that option prices and implied volatility surfaces are locally flat under small perturbations of the mean and standard deviation, indicating robustness to calibration errors.
• Implied volatility surfaces computed under the minimal entropy measure exhibit a volatility smile for short maturities that flattens for longer maturities, consistent with results from [28] using a different calibration approach.