Cliquet option based contracts constitute a customized subclass of equity indexed annuities. The underlying options commonly are of monthly sum cap style paying a credited yield based on the sum of monthly-capped rates associated with some reference stock index. In this regard, cliquet type investments belong to the class of path-dependent exotic options. In [15] 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 [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 globally-floored 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 self-contained) paper to [8], as it presents a specific application of the results derived in [8] to the class of Meixner–Lévy processes.

The paper is organized as follows: In Section 2 we compile facts on the Meixner distribution and the related class of stochastic Meixner–Lévy processes. In Section 3 we introduce a geometric pure-jump stock price model driven by a Meixner–Lévy process. In Section 3.1 we establish a customized structure preserving measure change from the risk-neutral to the physical probability measure. Section 4 is dedicated to the pricing of cliquet options. We obtain semi-analytic expressions for the cliquet option price by using the probability distribution function of the driving Meixner–Lévy process in Section 4.1 and by an application of Fourier transform techniques in Section 4.2. In Section 5 we draw the conclusions.

Let (Ω,F,(Ft)t∈[0,T],Q) be a filtered probability space satisfying the usual hypotheses, i.e. Ft=Ft+:=∩s>tFs t}\mathcal{F}_{s}$]]> constitutes a right-continuous filtration and F denotes the sigma-algebra augmented by all Q -null sets (cf. p. 3 in [10]). Here, Q is a risk-neutral probability measure and 0<T<∞ denotes a finite time horizon. In the following, we compile various facts on the Meixner distribution and Meixner–Lévy processes from [1, 6, 12, 13] and [14].

A real-valued, càdlàg, pure-jump, time-homogeneous Lévy process M=(Mt)t∈[0,T] (with independent and stationary increments) satisfying M0=0 is called Meixner (-Lévy) process with scaling parameter α>0 0$]]>, shape/skewness parameter β∈(−π,π) , peakedness parameter δ>0 0$]]> and location parameter μ∈R , if Mt possesses the Lévy–Itô decomposition (2.1) Mt=θt+∫0t∫R0zdN˜Q(s,z) where R0:=R∖{0} , the drift parameter (2.2) θ:=μ+δαtan(β/2) is a real-valued constant and the Q -compensated Poisson random measure (PRM) is given by (2.3) dN˜Q(s,z):=dN(s,z)−dνQ(z)ds with positive and finite Meixner-type Lévy measure (2.4) dνQ(z):=δeβz/αzsinh(πz/α)dz (cf. [12, 13], Eq. (3) in [6]) satisfying νQ({0})=0 and ∫R0(1∧z2)dνQ(z)<∞. We denote the Lévy triplet of Mt by (θ,0,νQ) . (Note that this notation is not entirely consistent with [6, 8].) We recall that Mt possesses moments of all orders (cf. Section 5.3.10 in [13]). Evidently, Mt has no Brownian motion part. Since ∫R0|z|dνQ(z)=∞ the process Mt possesses infinite variation (cf. Section 5.3.10 in [13]). We write for any fixed t∈[0,T] Mt∼M(α,β,δt,μt) (cf. Section 3.6 in [1]) and say that M is Meixner distributed under Q with parameters α, β, δ and μ. From (2.1) and (2.2) we instantly receive the mean value (2.5) EQ[Mt]=θt=μt+δtαtan(β/2) standing in accordance with Eq. (11) in [6]. The variance, skewness and kurtosis of Mt are respectively given by VarQ[Mt]=δt2α2cos2(β/2),SQ[Mt]=2/(δt)sin(β/2),KQ[Mt]=3+2−cos(β)δt (cf. Table 6 in [1]). Furthermore, for all x∈R and t∈[0,T] the real-valued probability density function (pdf) of Mt under Q reads as (2.6) fMt(x):=(2cos(β/2))2δt2παΓ(2δt)eβ(x−μt)/α|Γ(δt+ix−μtα)|2 (cf. [1, 12], Eq. (4) in [6]) wherein Γ(ζ):=∫0∞uζ−1e−udu denotes the gamma function which is defined for all ζ∈C with Re(ζ)>0 0$]]>. Taking the definition of the gamma function and Euler’s formula into account, we get Γ(δt+ix−μtα)=∫0+∞uδt−1e−ucos(x−μtαlnu)du+i∫0+∞uδt−1e−usin(x−μtαlnu)du which implies |Γ(δt+ix−μtα)|2=(∫0+∞uδt−1e−ucos(x−μtαlnu)du)2+(∫0+∞uδt−1e−usin(x−μtαlnu)du)2. Note that the latter object appears in (2.6). The cumulative distribution function (cdf) of Mt does not possess a closed form representation but it can be computed numerically. Further on, the characteristic function of Mt can be computed by the Lévy–Khinchin formula (see e.g. [4, 5, 11, 13]) due to (2.7) ϕMt(u):=EQ[eiuMt]=eψ(u)t with i2=−1 , u∈R , t∈[0,T] and a characteristic exponent (2.8) ψ(u):=iu[μ+δαtan(β2)]+δ∫R0eiuz−1−iuzzeβz/αsinh(πz/α)dz. Moreover, let us define the Fourier transform, respectively inverse Fourier transform, of a deterministic function q∈L1(R) via qˆ(y):=∫Rq(x)eiyxdx,q(x)=12π∫Rqˆ(y)e−iyxdy. Then for all u∈R and t∈[0,T] we receive the well-known relationship ϕMt(u)=fˆMt(u) where fˆMt denotes the Fourier transform of the density function fMt defined in (2.6). An application of the inverse Fourier transform yields fMt(x)=12π∫Reψ(u)t−iuxdu thanks to (2.7). On the other hand, from Eq. (1) in [6] we know that (2.9) ϕMt(u)=eiuμt(cos(β/2)cosh((αu−iβ)/2))2δt where u∈R and t∈[0,T] . Taking the logarithm in (2.7) and (2.9), we finally deduce (2.10) ψ(u)=iuμ+2δ[lncos(β2)−lncosh(αu−iβ2)]. Further on, for the Meixner distribution the following properties are well-known (cf. [14], Section 5.3.10 in [13], Section 3.6 in [1], Corollary 1 in [6]). Lemma 2.1.

(a)

If X∼M(α,β,δ,μ) , then cX+m∼M(cα,β,δ,cμ+m) with constants c>0 0$]]> and m∈R .

Let t∈[0,T] and define the stochastic stock price process St via (3.1) St:=S0eMt+bt with deterministic initial value S0 , a constant b∈R and a real-valued Meixner process Mt=θt+∫0t∫R0zdN˜Q(s,z) such as introduced in (2.1)–(2.4). Here, the constant b provides some additional degree of freedom which is introduced in order to ensure the arbitrage-freeness of the stock price model. More details on this topic will be given below. Verify that (3.1) belongs to the same model class (geometric Lévy models) as (2.2)–(2.3) in [8]. We next introduce the historical filtration Ft:=σ{Su:0≤u≤t}=σ{Mu:0≤u≤t}. Using Itô’s formula, we obtain the stochastic differential equation (SDE) dStSt−=(θ+b+∫R0[ez−1−z]dνQ(z))dt+∫R0[ez−1]dN˜Q(t,z) under Q . Let us further define the discounted stock price via Sˆt:=StBt where St is such as defined in (3.1) and Bt:=ert is the value of a bank account with normalized initial capital B0=1 and risk-less interest rate r>0 0$]]>. Due to (3.1) we find Sˆt=S0eMt+(b−r)t while Itô’s formula yields the following SDE under Q dSˆtSˆt−=(θ+b−r+∫R0[ez−1−z]dνQ(z))dt+∫R0[ez−1]dN˜Q(t,z). In accordance to no-arbitrage theory, the discounted stock price Sˆ must form a martingale under the risk-neutral probability measure Q . For this reason, we require the drift restriction (3.2) b=r−θ−∫R0[ez−1−z]dνQ(z). With this particular choice of the coefficient b, we deduce dStSt−=rdt+∫R0[ez−1]dN˜Q(t,z) under Q . Combining (3.1) and (3.2), we receive St=S0ertexp{∫0t∫R0zdN˜Q(s,z)−∫0t∫R0[ez−1−z]dνQ(z)ds} where the last factor on the right hand side constitutes a Doléans-Dade exponential which again shows the Q -martingale property of the discounted stock price process Sˆt=Ste−rt . Moreover, taking (2.2), (2.4), (2.8) and (2.10) into account, Eq. (3.2) can be expressed as (3.3) b=r−ψ(−i)=r−μ−2δln(cos(β/2)cos((α+β)/2)). Unless otherwise stated, from now on we assume that the constant b∈R appearing in (3.1) is such as given in (3.3). Though constituting an admissible choice, taking b=0 in (3.1)–(3.3) might be too restrictive in practical applications. In the following, we investigate the log-returns related to our model (3.1). For an arbitrary time step Δ>0 0$]]> and t≤T−Δ we obtain ln(St+ΔSt)≅MΔ+bΔ∼M(α,β,δΔ,(μ+b)Δ) by Lemma 2.1 (a). Here, the symbol ≅ denotes equality in distribution. Hence, in our stock price model (3.1) the log-returns are Meixner distributed. We stress that in [12] it was shown that the Meixner distribution fits empirical financial log-returns very well. Furthermore, for n∈N we introduce the time partition P:={0<t0<t1<⋯<tn≤T} and define the return/revenue process associated with the period [tk−1,tk] via Rk:=Stk−Stk−1Stk−1 where k∈{1,…,n} . A substitution of (3.1) into the latter equation yields (3.4) Rk=eYtk−Ytk−1−1 where Yt:=Mt+bt∼M(α,β,δt,(μ+b)t) is a Meixner–Lévy process. Taking (2.1)–(2.4) and (3.3) into account, we get (3.5) Yt=γt+∫0t∫R0zdN(s,z) where γ:=r+δ[αtan(β2)−2ln(cos(β/2)cos((α+β)/2))−∫R0eβz/αsinh(πz/α)dz] is a real-valued constant. Recall that the Meixner–Lévy process Y given in (3.5) above just is a special case of the more general Lévy process X defined in Eq. (2.3) in [8]. For this reason, the cliquet option pricing results derived in [8] simultaneously apply in our current Meixner modeling case. More details on this topic are given in Section 4 below. Also note that R1,…,Rn are Q -independent random variables and that Rk>−1 -1$]]> Q -almost surely for all k. Since Y is a Lévy process under Q , we observe Ytk−Ytk−1≅Yτ (stationary increments) where τ:=tk−tk−1 (equidistant partition). Here, the symbol ≅ denotes equality in distribution. For the sake of notational simplicity, we always work under the assumption of equidistant time points in the following, unless otherwise stated. Taking (3.4) into account, we obtain the subsequent relationship between the cumulative distribution functions of Rk and Yτ (3.6) Q(Rk≤ξ)=Q(Yτ≤ln(1+ξ)) where ξ>−1 -1$]]> is an arbitrary real-valued constant.

This section is devoted to the pricing of cliquet options in the Meixner stock price model presented in Chapter 3. Since the Meixner process Y in (3.5) above just is a special case of the more general Lévy process X defined in Eq. (2.3) in [8], the cliquet option pricing results derived in [8] simultaneously apply to our present Meixner–Lévy modeling case. 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 HT=K+Kmax{g, ∑k=1nmin{c,Rk}} where T is the maturity time, K denotes the notional (i.e. the initial investment), g is the guaranteed rate at maturity, c≥0 is the local cap and Rk is the return process given in (3.4). Recall that the payoff HT is globally-floored by the constant K(1+g) and locally-capped by c. By a case distinction, we get HT=Kmax{1+g,1+ ∑k=1nmin{c,Rk}}=K(1+g+max{0, ∑k=1nZk}) where for all k∈{1,…,n} the appearing objects (4.1) Zk:=min{c,Rk}−g/n are independent and identically distributed (i.i.d.) random variables. Note that Rk is Ftk -measurable such that HT is Ftn -measurable. Since tn≤T , it holds Ftn⊆FT such that HT constitutes an FT -measurable claim. As before, let us denote the constant interest rate by r>0 0$]]>. Then the price at time t≤T of a cliquet option with payoff HT at maturity T is given by the discounted risk-neutral conditional expectation of the payoff, i.e. Ct=e−r(T−t)EQ(HT∣Ft). Combining the latter equations, we obtain (4.2) C0=Ke−rT(1+g+EQ[max{0, ∑k=1nZk}]) which shows that the considered cliquet option with payoff HT essentially is a plain-vanilla call option with strike zero written on the basket-style underlying ∑k=1nZk .

In the subsequent sections, we derive explicit expressions for ϕZ(x) and EQ[Z1] appearing in the pricing formula (4.3). As before, we stick to the presumption of equidistant resetting times and set τ=tk−tk−1 for all k∈{1,…,n} in the following.