Lewis Model Revisited: Option Pricing with Lévy Processes

Abstract

This paper aims to discuss the mathematical details in Lewis’ model by considering the analyticity and integrability conditions of characteristic functions and payoff functions of contingent claims. In his seminal paper, Lewis shows that it is much easier to compute the option value in the Fourier space than computing in terminal security price space. He computes the option value as an integral in the Fourier space, the integrand being some elementary functions and the characteristic functions of a wide range of Lévy processes. The model also illustrates how the residue calculus leads to several variations of option formulas through the contour integrals. In this paper, we provide with, to a reasonable extent, some rigor into the mathematical background of Lewis’ model and validate his results for particular Lévy processes. We also simply give the analyticity conditions for the characteristic function of the Carr–Geman–Madan–Yor model and a simple derivation of the characteristic function of Kou’s double exponential model.

Appendices

Appendix A

A MATLAB code for variance gamma call option:

Appendix B

The characteristic function of the double exponential pure-jump process can be written as

$$\begin{aligned} {\phi }_t\left( z\right) =\exp \left( \lambda t\int Q_{\ln V}(\hbox {d}y)(\hbox {e}^{izy}-1)\right) \end{aligned}$$

where $Q_{\ln V}(\hbox {d}y)$ is the Lévy measure of the process and y is an (double) exponential variable with means $1/\eta _1$ (positive part) and $1/\eta _2$ (negative part).

Applying the integral for $y>0$ and $y<0$ parts separately

$$\begin{aligned} {\phi }_t\left( z\right) =\exp \left( \lambda t\left[ p\int ^{\infty }_0{\left( \hbox {e}^{izy}-1\right) {\eta }_1}\hbox {e}^{-{\eta }_1y}\hbox {d}y+q\int ^0_{-\infty }{\left( \hbox {e}^{izy}-1\right) {\eta }_2}\hbox {e}^{{\eta }_2y}\hbox {d}y\right] \right) \end{aligned}$$

and taking the integrals, we obtain

$$\begin{aligned}&\exp \left( \lambda t\left[ -p{\eta }_1\left( \frac{1}{iz-{\eta }_1}+\frac{1}{{\eta }_1}\right) +q{\eta }_2\left( \frac{1}{iz+{\eta }_2}-\frac{1}{{\eta }_2}\right) \right] \right) \\&\quad =\exp \left( \lambda t\left[ \frac{-piz}{iz-{\eta }_1}+\frac{-qiz}{iz+{\eta }_2}\right] \right) . \end{aligned}$$

With simple algebra, this will yield

$$\begin{aligned} {\phi }_t\left( z\right) =\exp \left( \lambda t\left[ \frac{z^2+iz(q{\eta }_1-p{\eta }_2)}{(iz+{\eta }_2)(iz-{\eta }_1)}\right] \right) . \end{aligned}$$

Recall that here $p,q>0$ and $p+q=1$ represent the probabilities of upward and downward jumps as in the Kou [29] model.