Exponential stock models driven by tempered stable processes

TL;DR

This work investigates exponential stock models driven by tempered stable processes, with the stock price given by $S_t = S_0 e^{X_t} B_t$ and $B_t = e^{rt}$, where $X$ is a tempered stable process. It develops a systematic framework for the existence of equivalent martingale measures (EMMs) that preserve tractability, including Esscher and bilateral Esscher transforms, minimal-entropy and $p$-optimal measures, and the Föllmer–Schweizer minimal martingale measure (FSMMM), and derives option-pricing formulas under these measures. The paper establishes precise existence criteria (e.g., conditions on $\\eta^\\\pm$, $\\lambda^\\\pm$, and $r-q$) and shows that under FSMMM the driving process becomes the sum of two independent tempered-stable processes, enabling analytic pricing via Fourier methods. A case study calibrates the model to DAX data, illustrating the existence of multiple EMMs, the behavior of the implied volatility surface, and the conditions under which FSMMM exists, thereby demonstrating practical applicability and the nuanced landscape of martingale measures in tempered-stable settings.

Abstract

We investigate exponential stock models driven by tempered stable processes, which constitute a rich family of purely discontinuous Lévy 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.

Figures (8)

• Figure 1: The left plot shows the values of the German stock index DAX from January 3, 2011 until December 28, 2012. The right plot shows the corresponding log returns.
• Figure 2: Histogram for the log returns together with the fitted normal distribution in the left plot and the fitted tempered stable distribution in the right plot.
• Figure 3: The left plot shows the function $f$ from Section \ref{['sec-Esscher']} together with the interest rate $r$ as dashed line. The right plot shows the graph of $\Phi$ together with the graph of $\Theta \mapsto -\Theta$ as dashed line.
• Figure 4: The left plot shows the relative entropies on the interval $[-2,0]$, where the $x$-axis is $\theta$ and the $y$-axis is $\mathbb{H}(\mathbb{P}^{(\theta,\Phi(\theta))} \,|\, \mathbb{P})$. The right plot shows the $2$-distances on the interval $[-2,0]$, where the $x$-axis is $\theta$ and the $y$-axis is $\mathbb{H}_2(\mathbb{P}^{(\theta,\Phi(\theta))} \,|\, \mathbb{P})$.
• Figure 5: The function $r_a \mapsto c(r_a)$ defined according to (\ref{['fct-const']}) on the interval $[0,0.2]$, where the $x$-axis is the annualized interest rate $r_a$. The shaded area indicates the values of $r_a$ for which the FS minimal martingale measure exists.

Theorems & Definitions (31)

• Remark 2.1
• Lemma 2.2
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Remark 3.4
• Theorem 4.1