Pricing formulae for derivatives in insurance using the Malliavin calculus

In this paper we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process, by using the Malliavin calculus. In analogy with the celebrated Black-Scholes formula, we aim at expressing the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of Stop-Loss contracts the building block is given by the distribution function of the terminal cumulated loss, taken at the Value at Risk when computing the Expected Shortfall risk measure.


Introduction
Risk analysis in the context of insurance or reinsurance is often based on the study of properties of a so-called cumulative loss process L := (L t ) t∈[0,T ] over a period of time [0, T ] where T > 0 denotes the maturity of a contract. Usually, L takes the form where N := (N t ) t∈[0,T ] is a counting process, and the random variables (X i ) i∈N * represent the amount of the claims. A typical contract in reinsurance is the Stop-Loss contract that offers protection against an increase in either (or both) severity and frequency of a company's loss experience. More precisely, Stop-loss contracts provide to its buyer (another insurance company) the protection against losses which are larger than a given level K and its payoff function is given by a "call" function. In some cases, there is also an upper limit given by some real number M , which specifies the maximal reimbursement amount. Thus the payoff of such a contract is given by (1.1) In full generality the risk carried out by the claims is neither hedgeable nor related to a financial market, hence the premium of the Stop-Loss is equal to E[Φ(L T )] which immediately re-writes as There is a large number of papers describing how to approximate the compound distribution function of the cumulated loss L T , and to compute the Stop-Loss premium. The aggregate claims distribution function can in some cases be calculated recursively, using, for example, the Panjer recursion formula, see Panjer [12] and Gerber [11]. Various approximations of Stop-Loss reinsurance premiums are described in the literature, some of them assuming a specific dependence structure.
In analogy with the celebrated Black-Scholes formula, we aim in this paper to express the first term of the right-hand side of (1.2) in terms of a building block which represents the distribution function of the terminal loss L T . This feature is hidden in the Black-Scholes model since the terminal value of the stock has an explicit lognormal distribution. More specifically, we aim in computing E L T 1 {L T ∈[K,M ]} by using the building block Note that, on the credit derivative market, the payoff function (1.1) can also be related to Collateralized Debt Obligations (CDOs) where there are several tranches, and so several K and M levels, which are expressed in proportion of the underlying which is the loss of a given asset portfolio.
Stop-Loss contracts are the paradigm of reinsurance contracts, but we aim in dealing with more general payoffs whose valuation involves the computation of the quantity where h : R + → R + is a Borelian map and whereL is of the formL T := N T i=1X i , involving claimsX i which are related to the ones X i of the original loss L T . To be more precise,L T will be the effective loss covered by the reinsurance company whereas L T is the loss quantity that activates the contract. Typical examples will be given in Section 2.1. Once again, this is similar to the valuation of CDOs tranches where the recovery rate is often supposed to be a random variable of beta distribution with mean 40% whereas the realized rate, often revealed only after the formal bankruptcy, does not necessarily match with this value.
In this paper we provide an exact formula for (1.3) in terms of the building block x → E [h(L T + x)] (or of a related quantity for the more general situation (1.3), see (3.4) for a precise statement). This goal will be achieved by using one of the Malliavin calculus available for jump processes. Before turning to the exposition of the model, we emphasize that this methodology goes beyond the analysis of pricing and finds for instance application in the computation of the Expected Shortfall of contingent claims in the realm of risk measures. Indeed, the expected shortfall is a useful risk measure, that takes into account the size of the expected loss above the value at risk. Formally it is defined as As it is well-known, the expected shortfall coincides with Average Value at Risk (AV@R), that is 2 for a precise definition). However, already in the trivial example where the size claims X i are constant equal to 1, this property fails as L T = N T is a Poisson random variable which exhibits a discontinuous distribution function. However, our approach gives an alternative explicit computation of E[L T 1 {L T <β} ] and thus of ES α (−L T ) as We conclude this section with some comments about the modeling of the claims X i andX i . In the classic Cramer-Lundberg model, the claims are independent and identically distributed (i.i.d.) and in addition independent of the counting process N which happens to be an inhomogeneous Poisson process. In this work we consider a doubly stochastic Poisson process N and we allow dependency between the size of the claims, their arrivals and the intensity of N . In particular we do not assume a Markovian setting. The impact of certain dependence structure on the Stop-Loss premium is studied in the reinsurance literature, such as in Albers [1], Denuit et al. [9] or De Lourdes Centeno [8], but those works usually assume dependency between the successive claim sizes and the arrival intervals. Nevertheless, in the ruin theory literature, some contributions already propose explicit dependencies among inter-arrival times and the claim sizes, such as Albrecher and Boxma [2], Boudreault, Cossette, Landriault and Marceau [7] and related works. A general framework of dependencies is proposed by Albrecher, Constantinescu, and Loisel [4] in which the dependence arises via mixing through a so-called frailty parameter. Recently, Albrecher et al. [3] extend duality results that relate survival and ruin probabilities in the insurance risk model to waiting time distributions in the 'corresponding' queueing model. The risk processes have a counterpart in workload models of queueing theory, and a similar mixing dependencies structure is considered in a queueing context. Besides, our framework extends the mixing approach of [4] and [3] by allowing nonexchangeable family of random variables for the claims amounts. In a similar way, in the credit risk modeling we can also suppose that the recovery rate depend on the underlying default intensity such as in Bakshi, Madan and Zhang [5].
We proceed as follows. We first make clear in Section 2 our model for the loss process and present the insurance contracts for which we will propose a pricing formula. The latter will be stated and proved as Theorem 3.5 in Section 3. Particular cases of this result to several types of contracts in insurance are also given in this section. Finally, explicit examples are presented in Section 4.

Model Setup
In this section, we describe the loss process and the associated reinsurance contracts we will study. Throughout this paper, T will denote a positive finite real number which represents the final horizon time.

The Loss process
We begin by introducing the loss process L := (L t ) t∈[0,T ] where the size of claims and their arrival times are correlated. Let (N t ) t∈[0,T ] be a Cox process (also called doubly stochastic Poisson process) with random intensity (λ t ) t∈[0,T ] , whose jump times, denoted by (τ i ) i∈N * , model the arrival times of the claims. We suppose that the claim size X i depends on both the cumulated intensity defined by Λ t := t 0 λ s ds and the claim arrival time τ i . Moreover, it will also depend on some random variable ε i where we suppose that (ε i ) i∈N * is a sequence of positive i.i.d. random variables independent of the Cox process N . More precisely, the loss is given by where κ is the discounting factor and f : R 3 + → R + is a bounded deterministic function. We provide several examples as below.
1. In the classic ruin theory, the claim size is often supposed to be independent of the arrival and the intensity process. In this case, we have f (t, ℓ, x) = x.
2. In the second example, we suppose that the dependence of f on the exogenous factor ε is linear and the linear coefficient is a function of the cumulated intensity Λ rescaled by time, i.e., Λt t , which stands for some mean level of the intensity. For instance, let In this example, if ε i follows an exponential distribution with parameter 1, then

Generalized loss process
We can also consider a more general case where the realized claim sizes (X i ) i∈N * are not exactly the ones that are computed to activate the reinsurance contract. More precisely, assume that in addition to the factors (ε i ) i∈N * , there exists a family of i.i.d. positive random variables (ϑ i ) i∈N * which may depend on the random variables ε i 's. Let g : R 4 + → R + be a deterministic bounded function. We can define a modified cumulative loss process aŝ More precisely, although the insurance contract is triggered by the loss process L, the compensation amount can depend on some other exogenous factors (ϑ i ) i∈N * . This would mean for instance that the amounts ϑ i 's are much lower than the ε i 's. A typical example is given by the housing insurance market on the American East Coast. Indeed, this region is seasonally exposed to hurricanes of different magnitudes. Most of the damages impacts the houses of the insured who may as well buy contracts on other belongings such as cars which are much less valuable. After a hurricane episode, the re-insurance Stop-Loss contract will be activated on the basis of the total damages L T on the houses (which are represented by the claims ε i ) whereas the effective damagesL T will also include all other insured belongings (which would be modeled by the ϑ i ). In the special case where the function g does not depend on the fourth variable, the general lossL t reduces to the standard loss defined in (2.1). We give below some examples of the joint distribution (ε i , ϑ i ).
1. The first natural case is that ε i and ϑ i are independent random variables. For example, each of them can follow an exponential distribution (or Erlang distribution) with different positive parameters θ 1 and θ 2 .
2. We can introduce dependence between ε i and ϑ i by using the mixing method in [4]. Let ε i and ϑ i follow Pareto marginal distributions respectively and a dependence structure according to a Clayton copula (according to Example 2.3 in [4], this can be achieved by mixing the two Pareto marginal distributions where the mixing parameter follows a Gamma distribution).
3. Case of explicit dependence : let ε i follow a Pareto distribution and ϑ i follow a Weibull distribution with form or scaling parameter depending of ε i .

Generalized Stop-loss Contrats
We have seen in the introduction the Stop-Loss contract whose payoff is given by Φ(L T ) where Φ has been defined in (1.1) and corresponds to a call spread, that is, the difference of two call functions. Our approach allows us to go beyond the case of the Stop-Loss contract. Consider now a contract where the reinsurance company pays if the a priori loss L T excesses some amount K or belongs to some interval [K, M ]. Then the price of such a contract is :

Expected Shortfall
The expected shortfall is a useful risk measure which takes into account the size of the expected loss above the value at risk. We recall the Expected Shortfall with level α as , which is in particular satisfied if the distribution function of X is continuous (see e.g. [10, Relation (4.38)]). However, the latter property fails already in the case where the size claims X i are constant. Thus one can not rely on the above relation and has to compute directly the conditional expectation ES α (−L T ).
We will provide an alternative expression for the expected shortfall. We denote by β : } So once again the key term to compute turns out to be the expectation E L T 1 {L T <β} .

General payoffs
More generally, we are interested in computing quantities of the form Since in our model, the counting process is given by a Cox process with stochastic intensity, the building block becomes the following mapping by using the conditional expectation Our approach calls for some stochastic analysis material that we present in the next section.

The pricing formulae using the Malliavin calculus
In this section, we establish our main pricing formulae by using the Malliavin calculus. To this end, we first make precise the Poisson space associated to the loss process. Then we provide basic tools for the Malliavin calculus.

The counting process and intensity process
We recall that the loss process involves the Cox process (N t ) t∈[0,T ] with its intensity and jump times, and the family of random variables (ε i ) i∈N * . We begin by introducing a general counting process which will be useful for the construction of (N t ) t∈[0,T ] on a suitable space.
Let Ω 1 be the set of (finite or infinite) strictly increasing sequences in ]0, +∞[ . We define a continuous-time stochastic process C on the set Ω 1 as be the filtration generated by the process C, namely F C t := σ(C s , s ≤ t). It is known that there exists a unique probability measure P 1 on (Ω 1 , F C ∞ ) under which the process C is a Poisson process of intensity 1, that is, for every (s, t) ∈ [0, +∞) 2 , with s < t, the random variable C t −C s is independent of F C s and Poisson distributed with parameter t−s.
T ] be the right-continuous complete filtration generated by the stochastic process λ. Moreover, we set (3.1) Let F ε,ϑ be the σ-algebra generated by (ε i ) i∈N * and (ϑ i ) i∈N * . Note that only (ε i ) i∈N * and (ϑ i ) i∈N * will be involved in the loss process and ε and ϑ are just independent copies which play an auxiliary role. We denote by µ the probability law of the couple (ε i , ϑ i ).
Note that for any t, N t is F C ∞ ⊗ F λ T -measurable random variable. Moreover, for any fixed ω 2 in Ω 2 , N t ( . , ω 2 ) is an inhomogeneous Poisson process on Ω 1 with intensity t → λ t (ω 2 ) with respect to the filtration (F C Λt(ω 2 ) ) t∈[0,T ] which reads as 1 where E denotes the expectation with respect to the measure P. For a process (u t ) t∈[0,T ] such that : we denote by T 0 u s dN s (ω 1 , ω 2 ) the Lebesgue-Stieltjes integral of u(ω 1 , ω 2 ) against the measure N (ω 1 , ω 2 ).
For any i ∈ N, we let τ i be the i-th jump time of the process N , namely with the convention τ 0 = 0.

The Malliavin integration by parts formula
We can now state the Malliavin integration by parts formula on the product space. For any t ∈ [0, T ], and ω 1 ∈ Ω 1 which is of finite length or has a limit greater than t, we define ω 1 ∪ {t} in Ω 1 as the increasing sequence whose underlying set is the union of ω 1 and t. The effect of this operator is to add a jump at time t to the Poisson process N . Finally, for ω := (ω 1 , ω 2 ) ∈ Ω, and t ∈ [0, T ], we set ω ∪ {t} := (ω 1 ∪ {t}, ω 2 ), provided that ω 1 ∪ {t} is well defined. The following lemma is a direct extension of the one presented for example in [13,Corollaire 5] or [14] (see also [15]).

The main result
In this section we present our main result concerning the computation of the quantity 1 By a slight abuse of notation, E · F λ T : where h : R + → R + is a Borelian map with E[h(L T )] < ∞ and where L T andL T are respectively defined in (2.1) and (2.2). We set It might be surprising at first glance to consider the conditional expectation given λ in the building block. In fact, as the intensity λ of N is random, it can be compared to a Black-Scholes model with independent stochastic volatility. In that context the Black-Scholes formula would be written in terms of the conditional law of the terminal value of the stock given the volatility (which would simply be a lognormal distribution with variance given by the volatility). Recall that for the insurance contract presented in Section 2. Before turning to the statement and the proof of the main result, note that Moreover on the set {∆ s N = 0}, one haŝ As Λ is a continuous process,Ẑ satisfies Relation (3.2), provided that E T 0 |Ẑ t |dt < +∞.
We start our analysis with the following lemma.
Proof. We set We first precise the value of L T (ω ∪ {t}) for a fixed element t ∈ (0, T ) and for ω : in Ω such that t ∈ ω 1 and ω 1 ∪ {t} is well defined (the set of such ω has probability 1). By definition, we have that Note that one has Therefore we can write L T (ω ∪ {t}) as the sum of three terms as follows (3.8) By definition, the first term in the sum is just L t (ω). Moreover, by a change of index we can write the third term as Therefore by (3.8) the following equality holds almost surely (3.10) Moreover, from the decomposition formula (3.8) we also observe that ε 1+Nt is independent of In addition, by Assumption 3.1 the conditional law of ε 1+Nt We now compute the characteristic functions of the two random vectors of interest. Let χ be the characteristic function of the random vector Let (u 1 , u 2 , u 3 ) ∈ R 3 . One has Since ε 1+Nt and ϑ 1+Nt are independent of L t + L + T − L + t given F C ∞ ⊗ F λ T , we obtain that where we also use the fact that the probability law of (ε 1+Nt , ϑ 1+Nt ) given F C ∞ ⊗ F λ T coincides with µ (which, we recall, is the probability law of (ε, ϑ)). Moreover, from (3.9) we observe that L t + L + T − L + t has the same law as L T conditioned on F C ∞ ⊗ F λ T . Therefore, we obtain which shows that χ coincides with the characteristic function of the vector The lemma is thus proved.
We now turn to the statement and the proof of the main result of this paper.
Theorem 3.5. Recall that (ε i , ϑ i ) i∈N * and (ε, ϑ) are i.i.d. with common law µ. Under the Assumptions 3.1 and 3.2, it holds that By Relation (3.7) and the fact that the set {∆ t N = 0} is negligeable, we obtain Finally, by Lemma 3.4, the above formula leads to Since ε is independent of F λ T ∨ F ε , one has Therefore as asserted by the theorem.
Remark 3.6. 1. Note that from Equality (3.11), it is clear that our approach only requires the knowledge of the conditional law of L T given λ (via the mapping ϕ λ ) and not the one of the pair (L T ,L T ). This seems to be particularly useful for the numerical approximation of the aforementioned expectation.
2. The theorem above provides us the relation of the pricing formula with respect to the intensity process (λ t ) t≥0 of the counting process.
Relation (3.11) allows us to give a lower (respectively upper) bound on the price if h is assumed to be convex (respectively concave).
Corollary 3.7. Under the assumptions of Theorem 3.5, it holds that : Proof. We prove (i) as statement (ii) follows the same line. As h is assumed to be convex, Jensen's inequality implies that The result is then obtained by plugging this estimate in Relation (3.11).

Applications and examples
In this section, we provide some application examples of our main result, in particular for the (generalized) stop-loss contract. Such explicit computations will also be useful for the CDO tranches and expected shortfall risk measure.

Computation of the building block
We first focus on the building block ϕ h λ (defined in (3.4)) when h : which corresponds to the payoff of a stop-loss contract or a CDO tranche. Let F ε := σ(ε i , i ∈ N * ). For any i ∈ N * , we set X i := f (τ i , Λ τ i , ε i ) and x in R + , we have where S k := {0 < t 1 < · · · < t k ≤ T }, X (1:k) := (X 1 , . . . , X k ) and It just remains to compute the joint distribution of the claims X (1:k) in different situations.
In particular, we provide below an explicit example.
Model on (ε i , ϑ i ) : We assume that (ε i , ϑ i ) i∈N * are i.i.d. random vectors, with marginal distributions following Pareto distributions P(α ε , β ε ) and P(α ϑ , β ϑ ) (for a set of parameters α ε , β ε , α ϑ , β ϑ > 0) respectively. The dependence structure is modeled through a Clayton copula with parameter θ > 0. We recall that the Clayton copula is C(u, v) and the density c of the Clayton copula is given by The joint distribution of (ε 1 , ϑ 1 ) is then given by Joint law of (λ t , Λ t ) : The final step in the computation of Relation (3.11) is to make precise the joint law of (λ t , Λ t ). More precisely, we need to compute Assume the intensity process (λ t ) t∈[0,T ] is given by where W is a Brownian motion, and β a constant (non null). Then the cumulative intensity is By Borodin and Salminen [6] (page 169), the joint law of (Λ t , W t ) is given by where the function i y (z) = ze π 2 4y π √ πy ∞ 0 exp −zch(x) − x 2 4y sh(x) sin( πx 2y )dx.

A Black-Scholes type formula for generalized Stop-Loss contracts in the Cramer-Lundberg
As an illustration, we conclude our analysis by specifying our result in the classic Cramer-Lundberg model. More precisely, we assume that the Cox process is an homogeneous Poisson process with constant intensity λ 0 > 0 and set h : In that case we omit the dependency on Λ for the mappings f and g (as Λ t = tλ 0 ). (recall that µ := L (ε,ϑ) ).
If we assume furthermore that f (t, x) = g(t, x, y) = x and κ = 0, then the loss process L T corresponds to the cumulated loss of the classic Cramer-Lundberg model. In this context, a huge literature deals with the computation of the ruin probability and related quantities, such as the discounted penalty function at ruin (Gerber-Shiu function). Others papers are concerned with the pricing of Stop-Loss contract. The pricing relies on the computation of a term of the form M K ydF (y) with F being the cumulative distribution function of the loss process L T , and the discussion in the literature mainly focuses on the derivation of the compound distribution function F (usually calculated recursively, using Panjer recursion formula and numerical methods/approximations) cf. [12] and [11]. Our Malliavin approach provides another formula which reads as Note that our result indeed coincides with the one obtained in [11]. Indeed, if one translates in a general setting Formula [11, (6)] (as µ is constrained to have a finite support in N in [11]) the distribution F satisfies which is exactly (4.3).