Cliquet option pricing in a jump-diffusion L\'{e}vy model

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 L\'{e}vy 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 L\'{e}vy 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 L\'{e}vy 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.


Introduction
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 [3]. The underlying options usually are of monthly sum cap style paying a credited yield based on the sum of monthly-capped rates associated with some reference (stock) index. More precisely, the investor pays a contractually specified amount to the issuer of the option prior to its maturity date and, in turn, receives at maturity a payoff depending on the performance of some designated reference index. In this regard, cliquet type investments belong to the class of pathdependent exotic options. The most popular choice in the insurance branch are cliquet contracts with globally-floored and locally-capped payoffs. These products can be utilized to protect against downside risk while yielding significant upside potential, yet avoiding extreme payoffs due to their local capping (cf. [3,10,16]). In [16] cliquet options are regarded as "the height of fashion in the world of equity derivatives".
In the literature, there are different pricing approaches for cliquet options involving e.g. partial differential equations (see [16]), Monte Carlo techniques (see [1,2,5,9]), numerical recursive algorithms related to inverse Laplace transforms (see [10]) and analytical computation methods (see [1,3,5,8,9,11]). In [3] the authors provide semi-analytic pricing formulas for path-dependent equity-linked contracts. They distinguish between cliquet options and monthly sum cap contracts and derive expressions for various Greeks. In their approach, it is crucial to know the probability distribution of the returns of the underlying reference index. In [1] the author uses a Lévy process specification to model the evolution of the underlying reference portfolio and investigates the valuation of life insurance policies providing interest rate guarantees. The driving Lévy process is of jump-diffusion type with normally distributed jump amplitudes while a special focus in [1] is laid on valuation under different risk-neutral pricing measures. In [11] the valuation of insurance contracts is discussed while emphasis is put on the impact of different Lévy process model specifications. It is shown that changing the underlying asset model implies a significant change in the prices of guarantees, indicating a substantial model risk. In [8] the pricing of cliquet options in a geometric Meixner model is investigated. The considered option is of monthly sum cap style while the underlying stock price is driven by a pure-jump Meixner-Lévy process yielding Meixner distributed log-returns. The paper [8] provides a specific application of the results derived in the present article. In [10] cliquet option prices are computed numerically by a recursive algorithm involving inverse Laplace transforms. This method is applied to a lognormal and a jump-diffusion model with deterministic volatility as well as to the Heston stochastic volatility model. In addition, a sensitivity analysis in each model is presented. Moreover, in [7] cliquet option pricing in a jump-diffusion model with time-dependent coefficients is examined. The jumps in the stock price trajectory are interspersed by an increasing standard Poisson process and the time-dependent coefficients are approximated by piecewise constant functions. In [7] there are solely cliquet options with a single resetting time discussed. In [16] the author investigates cliquet option pricing with partial differential equations (PDEs) while putting a special focus on the important role of volatility surface modeling. Recently, there have been extensions beyond Lévy settings to regime switching Lévy models (see e.g. [5,9]). In [5] the authors investigate the pricing of equitylinked annuities with cliquet-style guarantees in regime-switching stochastic volatility models with jumps. They propose a transform-based pricing method involving density projections and continuous-time Markov chain approximations. The considered models include exponential and regime-switching Lévy processes as well as stochastic volatility models with general jump size distributions. In [9] the valuation of equity-linked life insurance contracts in a regime switching Lévy model is studied. The model parameters depend on a continuous-time finite-state Markov chain, and closed-form pricing formulas based on Fourier transform techniques are derived.
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 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 [4], we state that jump-diffusion models are easy to simulate and efficient Monte Carlo methods for option pricing are available. Jumpdiffusion models also perform well when it comes to implied volatility smile interpolation (see Section 13 in [4]). In our setup, we derive cliquet option price formulas under two different approaches: once by using the distribution function of the driving Lévy process and once by applying Fourier transform techniques. In the context of sensitivity analysis, we eventually provide expressions for several Greeks related to our model. The paper is organized as follows: In Section 2 we introduce the jump-diffusion stock price model and derive representations for the probability density and distribution function of the driving Lévy process. In Section 3 we are concerned with cliquet option pricing under both a distribution function and a Fourier transform approach. Section 4 is dedicated to sensitivity analysis and the computation of different Greeks. In Section 5 we draw the conclusions and briefly mention some future research topics.

A Lévy stock price model and its distributional properties
Let (Ω, F, (F t ) t∈[0,T ] , Q) be a filtered probability space satisfying the usual hypotheses, i.e. F t = F t+ := ∩ s>t F s constitutes a right-continuous filtration and F denotes the sigma-algebra augmented by all Q-null sets (cf. p. 3 in [13]). Here, Q is a riskneutral probability measure and 0 < T < ∞ denotes a finite time horizon. In the sequel, we introduce a stochastic model for the stock price process S t . Let t ∈ [0, T ] and consider the stochastic differential equation (SDE) where η, σ and θ are deterministic functions, W constitutes an F -adapted standard Brownian motion under Q and N is a Poisson random measure (PRM). We further introduce the Q-compensated PRM (cf. Eq. (3.14) in [4]). In the above setup, we refer to η, σ and θ as the drift, volatility and jump function, respectively. We assume that W and N are Q-independent and set In the next step, we specify the emerging coefficients as follows η(t, S t ) := η(t)S t , σ(t, S t ) := σ(t)S t , θ(t, z, S t− ) := θ(t, z)S t− while assuming that θ(t, z) > −1 for all (t, z) ∈ [0, T ] × R and that (cf. Section 9.1 in [6]). Consequently, we obtain the geometric SDE which possesses the discontinuous Doléans-Dade solution for all t ∈ [0, T ]. From now on, we set η(t) ≡ η, σ(t) ≡ σ > 0 and θ(t, z) := e z − 1 in order to obtain a time-homogeneous Lévy process specification. If we do so, the latter equation can be written as with a real-valued Lévy process where γ := η − σ 2 /2 and t ∈ [0, T ]. Note that X 0 = 0 Q-a.s. We denote the characteristic triplet of X by (γ, σ, ν). Moreover, the first moment of X is given by whereas the characteristic function of X can be computed by the Lévy-Khinchin formula (see e.g. [4,6,14,15]) due to More details on Lévy processes can be found in e.g. [4,6,14,15]. In the next step, we define the discounted stock priceŜ where S t is such as defined in (2.2) and B t := e rt is the value of a bank account with normalized initial capital B 0 = 1 and risk-less interest rate r > 0. Due to (2.2), we findŜ while Itô's formula yields the following geometric SDE under Q dŜ t In accordance to no-arbitrage theory, the discounted stock price processŜ t must form a martingale under the risk-neutral probability measure Q. For this reason, we have to require the drift restriction With this particular choice of the drift coefficient η, we obtain under Q. Summing up, if we model the stock price process S t as in the latter equation, then the discounted stock priceŜ t constitutes a Q-martingale. Furthermore, let us define the Fourier transform, respectively inverse Fourier transform, of a function q ∈ L 1 (R) viâ Proposition 2.1 (density function). Suppose that the Lévy process X t is such as defined in (2.3). Then for all t ∈ [0, T ] and x ∈ R the probability density function Proof. Note that the characteristic function (2.4) is the Fourier transform of the density function f Xt (·), that is, We next apply the inverse Fourier transform and hereafter take (2.4) and (2.5) into account which yields the density function (2.6).
In what follows, we investigate in detail the jump part of the Lévy process X denoted by which constitutes a càdlàg, finite activity compound Poisson process (CPP) with finitely many jumps in each time interval. In the latter equation, N t constitutes a standard Poisson process under Q with deterministic jump intensity λ > 0, i.e.
Recall that the compensated compound Poisson process thanks to (2.1). Note that N t shall not be mixed up with the Poisson random measure dN (s, z). Obviously, we may write X t = γt + σW t + L t . We assume that the stochastic processes W t , N t and the random variables Y 1 , Y 2 , . . . altogether are Q-independent.
Example 2.2. If Y j is normally distributed with mean µ and variance δ 2 under Q for all j, then the Lévy measure possesses the Lebesgue density and z ∈ R. Here, µ and δ 2 model the mean respectively variance of the jump sizes.
In this setup, we receive β = µ, E Q [X t ] = t(γ + λµ) and Var Q [X t ] = t(σ 2 + λδ 2 + λµ 2 ). Evidently, choosing a negative µ makes the occurrence of downward jumps more likely than upward jumps and vice versa. We remark that a similar model specification with normally distributed jump sizes has firstly been proposed in [12].
If Y j is exponentially distributed with parameter α > 0 under Q for all j, then the Lévy measure possesses the Lebesgue density dν(z) = λp α (z) dz where p α (z) := αe −αz and z ∈ [0, ∞[. We presently find β = 1/α. [12]) with mean µ and variance δ 2 under Q for all j. Then for all t ∈ [0, T ] and x ∈ R the probability density function of X t under Q takes the form Then for all t ∈ [0, T ] and x ∈ R the probability density function of X t under Q takes the form Proof. Combine (2.6) with Example 2.2 and Example 2.3.

Proposition 2.5 (distribution function)
. Let X t = γt + σW t + Nt j=1 Y j and assume that the standard Poisson process N t jumps m times in the time interval [0, t], that is, N t = m with m ∈ N 0 . As in [12], suppose that Y j is normally distributed with mean µ and variance δ 2 . Then for any Borel set A ⊂ R and t ∈ [0, T ] the cumulative probability distribution function of X t under Q possesses the representation In accordance to Section 4.3 in [4] and the properties of conditional probabilities, the following ("quickly converging" [4]) series representation for the distribution function of X t under Q holds Since Y j ∼ N (µ, δ 2 ) for all j, we find that the stochastic process (γt + σW t + m j=1 Y j ) t also is normally distributed under Q with mean γt + mµ and variance σ 2 t + mδ 2 . Thus, by the definition of the cumulative distribution function, we get where ϕ denotes the probability density function of the normal distribution (see [12] and Example 2.2 above). Putting the latter equations together, we end up with (2.7).
Remark 2.6. Verify that the proof of Proposition 2.5 only works, if the random variables Y j are normally distributed for every j. If Y j is e.g. exponentially distributed for all j = 1, . . . , m (as proposed in Example 2.3), then it is unclear how to compute the probability emerging in the sequel of (2.8).
Corollary 2.7 (Eq. (4.12) in [4]). Under the assumptions of Proposition 2.5, for all t ∈ [0, T ] and x ∈ R the probability density function of X t under Q is given by Proof. The density can directly be read off in (2.7). Also see [12].
Corollary 2.8. If the Borel set A = ] − ∞, a] ⊆ R is an interval, then for any a ∈ R and t ∈ [0, T ] the distribution function in (2.7) takes the form where Φ denotes the standard normal cumulative distribution function.
Proof. This representation is an immediate consequence of Proposition 2.5.
Recall that the stochastic process S t will serve as our stock price model when it comes to cliquet option pricing in the subsequent section. In this context, for n ∈ N we introduce the time partition P := {0 < t 0 < t 1 < · · · < t n ≤ T } and define the return/revenue process associated with the period [t k−1 , t k ] via where k ∈ {1, . . . , n} and X is the Lévy process defined in (2.3). Note that R 1 , . . . , R n are Q-independent and that R k > −1 Q-almost sure for all k. For the sake of notational simplicity, we always work under the assumption of equidistant time points in the following and define τ := t k − t k−1 . If we want to refrain from this assumption again, then τ simply has to be replaced by the difference t k − t k−1 in all subsequent equations -with (3.17) as an exception.
Proposition 2.9. Let P = {0 < t 0 < t 1 < · · · < t n ≤ T } and put τ = t k − t k−1 for k ∈ {1, . . . , n} (equidistant time points). Define the return process R k as in (2.10). Then for any fixed real-valued ξ > −1 the distribution function of R k under Q admits the series representation where Φ denotes the standard normal cumulative distribution function.
Proof. Since X is a Lévy process under Q, we observe X t k − X t k−1 ∼ = X τ (stationary increments) where τ = t k − t k−1 and the symbol ∼ = denotes equality in distribution. Taking (2.10) and (2.9) into account, we obtain where ϕ denotes the density function of the normal distribution (recall Example 2.2). We finally perform the integration and end up with (2.11).
In quantitative risk management, it is often of interest to compute the probability of large drawdowns (shocks) in asset prices like e.g. Q(S u ≤ κS t ), 0 ≤ t < u ≤ T , where κ constitutes some stress scenario percentile like 60%, for instance. Due to (2.2), we find Q(S u ≤ κS t ) = Q(X u−t ≤ ln κ) which can easily be computed by Corollary 2.8.

Cliquet option pricing
This section is dedicated to the pricing of cliquet options in the Lévy jump-diffusion stock price model presented in Section 2. In accordance to (1.1) in [3], we consider a cliquet option with payoff where T is the maturity time, K denotes the notional (the initial investment), g is the guaranteed rate at maturity, c ≥ 0 is the local cap and R k is the return process defined in (2.10). This option is of monthly sum cap style with credited rate based on the sum of the monthly-capped rates [3]. Verify that the payoff H T is globally-floored and locally-capped. A popular choice in the insurance industry is to take g = 0 (globally-floored by zero) and n = 12 (monthly-capped by c). Further, note that the payoff H T actually is a function of multiple random variables, i.e. H T = h(R 1 , . . . , R n ) = h(S t0 , . . . , S tn ) wherein h and h are appropriately defined functions while the resetting times of the cliquet option are ordered as follows 0 < t 0 < t 1 < · · · < t n ≤ T . In this regard, a notation like H t0,...,tn (T ) might be more intuitive than simply writing H T . However, by a case distinction we observe where Z k := min{c, R k } − g/n denote i.i.d. random variables. Moreover, we introduce a bank account dB t = rB t dt with constant interest rate r > 0 and initial capital B 0 = 1, i.e. B t = e rt . Then the price at time t ≤ T of a cliquet option with payoff H T at maturity T is the discounted risk-neutral conditional expectation of the payoff, i.e.
Combining the latter equations, we obtain which shows that the considered cliquet option with payoff H T essentially is a plainvanilla call option with strike zero written on the basket-style underlying n k=1 Z k . Proposition 3.1 (Cliquet option price). Let k ∈ {1, . . . , n} and consider the independent and identically distributed random variables Z k = min{c, R k } − g/n where c ≥ 0 is the local cap, R k is the return process defined in (2.10) and g is the guaranteed rate at maturity. Denote the maturity time by T , the notional by K and the risk-less interest rate by r. Then the price at time zero of a cliquet option with payoff H T can be represented as where Re denotes the real part and the characteristic function φ Z (x) is defined via More explicit expressions for φ Z (x) and E Q [Z 1 ] are derived in several propositions below.
Proof. The proof essentially follows the same lines as the proof of Proposition 3.1 in [3] whereas our proof does not make use of the Rademacher random variable introduced in [3]. To begin with, we recall that similar to (3.2) and (3.3) in [3]. As a consequence, we deduce where the characteristic function φ Z (x) is such as defined in (3.3). In the derivation of the latter equation, we used the fact that Z 1 , . . . , Z n are i.i.d. random variables under Q. Since we get Substituting this into (3.1) leads us to (3.2).
The remaining challenge now consists in finding appropriate computation techniques for the entities E Q [Z 1 ] and φ Z (x) emerging in (3.2). In the subsequent sections, we present different methods to derive expressions for the mentioned entities. Similar to the notation introduced in Proposition 2.9, for arbitrary k ∈ {1, . . . , n} we set τ = t k − t k−1 in the following. We also assume that the jump amplitudes are normally distributed, as pointed out in Example 2.2.

Cliquet option pricing with distribution functions
Let us first apply a method involving probability distribution functions (cf. [3]). We initially investigate the treatment of φ Z (x) as defined in (3.3).
Then the characteristic function of Z k under Q can be represented as where Φ denotes the standard normal cumulative distribution function.
Proof. By a case distinction, we find that the distribution function of Z k is given by if R k ≤ c and ξ ≤ c − g/n, whereas Q(Z k > ξ) = 0 otherwise (cf. (3.15) in [3]). Since R k > −1 Q-a.s. for all k, we deduce Z k > −1 − g/n Q-a.s. for all k. Thus, Z k + 1 + g/n > 0 Q-a.s. for all k. With respect to (3.5), we obtain if R k ≤ c and w ≤ 1 + c, whereas Q(Z k + 1 + g/n > w) = 0 otherwise. Further on, verify that for the characteristic function the following relation holds Moreover, we recall that for any random variable Λ ≥ 0 with finite first moment, its characteristic function can be represented as (This equality follows from integration by parts; cf. Eq. (3.14) in [3].) Combining the latter equation with (3.6) and (3.5a), we deduce which can be rewritten as Merging (2.11) into the latter equation while noting that in (2.11) it holds ξ > −1, we finally end up with (3.4).

If we insert (3.4) into (3.3)
, we eventually get a representation for the characteristic function φ Z (x). Let us proceed with the computation of E Q [Z k ]. Proposition 3.3. Suppose that Z k = min{c, R k } − g/n where k ∈ {1, . . . , n}. Then the first moment of Z k under Q is given by where Φ denotes the standard normal distribution function.
Proof. In accordance to Proposition 2.4 in [4], we have .

Cliquet option pricing with Fourier transform techniques
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.
Proposition 3.4. Suppose that Z k = min{c, R k } − g/n where k ∈ {1, . . . , n} and let a > 0 be a finite real-valued dampening parameter. Then the first moment of Z k under Q can be represented as where the characteristic function φ Xτ is given by Proof. First of all, verify that which implies Hence, the evaluation of E Q [Z k ] is equivalent to the evaluation of a put option with underlying R k and strike c ≥ 0. Taking (2.10) into account, we receive where τ = t k − t k−1 and X is the real-valued Lévy process introduced in (2.3). Furthermore, we define the function ζ(u) := e au c + 1 − e u + with a finite real-valued dampening parameter a > 0. Since ζ ∈ L 1 (R), its Fourier transform exists and reads aŝ ζ(y) = (c + 1) 1+a+iy (a + iy)(1 + a + iy) .
Proposition 3.5. Suppose that Z k = min{c, R k } − g/n with k ∈ {1, . . . , n} and c ≥ 0. Then the characteristic function of Z k under Q reads as where the density f Xτ of X τ under Q is such as given in (2.9).
Proof. By the definition of the characteristic function (recall (2.4)), we get Taking (3.12) and (2.10) into account, the latter equation can be expressed as where the density f Xτ is such as given in (2.9). Next, verify that 1 + c − e u + = 1 + c − e u 1 u≤ln(1+c) (where 1 denotes the indicator function) which implies Since the last integral can be rewritten as we eventually obtain (3.14).
There is an alternative method involving (3.9) to derive an expression for E Q [Z k ] which is presented in the following. Corollary 3.6. In the setup of Proposition 3.5, we receive the representation wherein Φ denotes the standard normal distribution function and Proof. A substitution of (3.14) into (3.9) yields Taking (2.13) into account, we obtain the equalities where the arguments κ m 1 and κ m 2 are such as defined in the sequel of (3.15). Putting the latter equations into (3.16), we ultimately get (3.15).
Note that the expressions in (3.4), (3.8), (3.10), (3.14) and (3.15) altogether are independent of k. This is not a surprising observation, since Z 1 , . . . , Z n are i.i.d. random variables and we have chosen equidistant resetting times with τ = t k − t k−1 .
Inspired by the Fourier transform techniques applied in the proof of Proposition 3.4, we now focus on the derivation of an alternative representation for the cliquet option price C 0 given in Eq. (3.1).
Theorem 3.7 (Fourier transform cliquet option price). Let k ∈ {1, . . . , n} and consider the independent and identically distributed random variables Z k = min{c, R k }− g/n where c ≥ 0 is the local cap, g is the guaranteed rate at maturity and R k is the return process defined in (2.10). For arbitrary n ∈ N we set ̺ := nc − g and denote the maturity time by T , the notional by K and the risk-less interest rate by r. Then the price at time zero of a cliquet option paying at maturity can be represented as where f Xτ (u) constitutes the probability density function claimed in (2.9).
Proof. Suppose that the cliquet option price C 0 is such as given in (3.1). We only need to evaluate the expectation Z k + appearing in (3.1). Taking the definition of Z k and (3.12) into account, we obtain An application of the inverse Fourier transform yields constitutes the Fourier transform of ϑ. In the next step, we compute the characteristic function of D. Taking the definition of D and the Q-independence of the random variables R 1 , . . . , R n into account, we deduce With respect to (2.10), we obtain where τ = t k − t k−1 and f Xτ (u) is such as given in (2.9). By a case distinction, we get Verify that the emerging integrand [e iy(e u −1−c) − 1]f Xτ (u) is finite for all u ∈ [−∞, ln(1 + c)]. Also note that the appearing factors altogether are independent of k.
Combining the latter equations with (3.1), we finally get the asserted cliquet option price formula (3.17).
We recall that Fourier transform techniques have also been applied in the context of cliquet option pricing in e.g. [9] and [11].

Hedging and Greeks
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 Rho which is defined as the derivative of the cliquet option price C 0 with respect to the interest rate r. Due to (3.2), respectively (3.17), we find where T denotes the maturity time of the option. Further on, both the Delta and Gamma of the cliquet option vanish, i.e.
since we assumed t 0 = 0 in (2.10) such that neither R 1 nor Z 1 contains S 0 . In accordance to Section 3.2 in [3], we claim that in any cliquet option pricing context the most important Greek to study is the Vega which is defined as where σ > 0 denotes the volatility parameter of the Lévy process X defined in (2.3).
In the Fourier transform framework studied in Section 3.2, we get the following result.
Putting the latter equations together, we obtain (4.1) which completes the proof.
Proof. Taking (3.2) into account, we get Using (3.8) and the ordinary chain rule, we obtain where ϕ 0,1 = Φ ′ denotes the probability density function of the standard normal distribution. On the other hand, by (3.3) and (3.4) we deduce Putting the latter equations together, we end up with the asserted representation (4.2).

Conclusions
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 2, we derived representations for the probability density and distribution function of the involved Lévy process. Moreover, we obtained semi-analytic expressions for the cliquet option price by using the probability distribution function of the driving Lévy process in Section 3.1 and by an application of Fourier transform techniques in Section 3.2. With view on existing literature, the main contribution of the paper consists of the Fourier transform cliquet option price formula provided in Theorem 3.7. In Section 4, we concentrated on sensitivity analysis and computed the Greeks Rho, Delta, Gamma and Vega. 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 σ(t), respectively σ(t, X t ), in order to obtain more realistic (implied) volatility surfaces. In this context, we refer to Section 4 in [16] as well as Sections 1.2.1 and 11.1.2 in [4].
To read more on cliquet option pricing in a pure-jump Meixner-Lévy process setup, the reader is referred to the accompanying article [8].