Cliquet option pricing in a jump-diffusion Lévy model

Abstract

We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Lévy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density and distribution function of the emerging Lévy process. In this setting, we infer semi-analytic expressions for the cliquet option price by two different approaches. The first one involves the probability distribution function of the driving Lévy process whereas the second draws upon Fourier transform techniques. With view on sensitivity analysis and hedging purposes, we eventually deduce representations for several Greeks while putting emphasis on the Vega.

Cliquet option pricing with Meixner processes

“…The details are worked out in the remainder of the current section. Parallel to [8] and Eq. (1.1) in [3], we consider a monthly sum cap style cliquet option with payoff…”

“…By similar arguments as in the proof of Prop. 3.2 in [8], we obtain We next compute the emerging dw-integrals and finally substitute the resulting expression into (4.7) which yields (4.5).…”

“…There is an alternative method to derive expressions for E Q [Z k ], φ Z (x) and C 0 involving Fourier transforms and the Lévy-Khinchin formula. In the following, we present this method which has firstly been proposed in [8] in a cliquet option pricing context. Proposition 4.4.…”

“…In the literature, there are different pricing approaches for cliquet options involving e.g. partial differential equations (see [15]), Monte Carlo techniques (see [2]), numerical recursive algorithms related to inverse Laplace transforms (see [9]) and analytical computation methods (see [3,7,8]). The present article belongs to the last category.…”

“…The aim of the present paper is to provide analytical pricing formulas for globallyfloored locally-capped cliquet options with multiple resetting times where the underlying reference stock index is driven by a pure-jump time-homogeneous Meixner-Lévy process. In this setup, we derive cliquet option price formulas under two different approaches: once by using the distribution function of the driving Meixner-Lévy process and once by applying Fourier transform techniques (as proposed in [8]). All in all, the present article can be seen as an accompanying (but to a large degree selfcontained) paper to [8], as it presents a specific application of the results derived in [8] to the class of Meixner-Lévy processes.…”