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.

During the last decades, cliquet option based contracts became a very popular and frequently sold investment product in the insurance industry. These contracts can be considered as a customized subclass of equity indexed annuities which combine savings and insurance benefits [

In the literature, there are different pricing approaches for cliquet options involving e.g. partial differential equations (see [

The aim of the present paper is to provide analytical pricing formulas for globally-floored locally-capped cliquet options with multiple resetting times where the underlying reference stock index is driven by a drifted time-homogeneous Lévy process with Brownian diffusion component and compound Poisson jumps. In our framework, jumps represent rare events such as crashes, large drawdowns or upward movements. The dates of e.g. market crashes are modeled as arrival times of a standard Poisson process while the jump amplitudes can be both positive and negative. With reference to Section 4.1.1 in [

The paper is organized as follows: In Section

Let

Furthermore, let us define the Fourier transform, respectively inverse Fourier transform, of a function

Note that the characteristic function (

In what follows, we investigate in detail the jump part of the Lévy process

If

If

Combine (

Let

Verify that the proof of Proposition

The density can directly be read off in (

This representation is an immediate consequence of Proposition

Recall that the stochastic process

Since

In quantitative risk management, it is often of interest to compute the probability of large drawdowns (shocks) in asset prices like e.g.

This section is dedicated to the pricing of cliquet options in the Lévy jump-diffusion stock price model presented in Section

More explicit expressions for

The proof essentially follows the same lines as the proof of Proposition 3.1 in [

The remaining challenge now consists in finding appropriate computation techniques for the entities

Let us first apply a method involving probability distribution functions (cf. [

By a case distinction, we find that the distribution function of

If we insert (

In accordance to Proposition 2.4 in [

There is an alternative method to derive expressions for

First of all, verify that

It is possible to derive an alternative representation for the characteristic function

Our argumentation in the proof of Proposition

By the definition of the characteristic function (recall (

There is an alternative method involving (

A substitution of (

Note that the expressions in (

Inspired by the Fourier transform techniques applied in the proof of Proposition

Suppose that the cliquet option price

Verify that the emerging integrand

We recall that Fourier transform techniques have also been applied in the context of cliquet option pricing in e.g. [

In this section, we are concerned with sensitivity analysis and the computation of Greeks in our cliquet option pricing context. Let us start with an investigation of the Greek

First of all, note that the only ingredient in (

In the distribution function context studied in Section

Taking (

In this paper, we investigated the pricing of a monthly sum cap style cliquet option with underlying stock price modeled by a jump-diffusion Lévy process with compound Poisson jumps. In Section

A future research topic might consist in a transformation of the presented techniques and results to a time-inhomogeneous Lévy process setup which, in particular, contains a time (and state) dependent volatility coefficient

To read more on cliquet option pricing in a pure-jump Meixner–Lévy process setup, the reader is referred to the accompanying article [