Utility indifference pricing of insurance catastrophe derivatives
Abstract
We propose a model for an insurance loss index and the claims process of a single insurance company holding a fraction of the total number of contracts that captures both ordinary losses and losses due to catastrophes. In this model we price a catastrophe derivative by the method of utility indifference pricing. The associated stochastic optimization problem is treated by techniques for piecewise deterministic Markov processes. A numerical study illustrates our results.
Keywords
Insurance mathematics Catastrophe derivatives Utility indifference pricing Modeling catastrophe losses Piecewise deterministic Markov processMathematics Subject Classification
G13 G22JEL Classification
91G20 91B70 91B16 93E20 60J751 Introduction
Costly natural catastrophes in the recent past (hurricane Andrew in 1992, hurricane Katrina in 2005, the earthquake and tsunami in Japan 2011 resulting in the nuclear disaster at Fukushima, floods in Thailand 2011) all caused severe stress to the (re)insurance industry. However, these losses are still small relative to losses of the US stock and bond markets. Therefore securitization (i.e. transferring part of the risk to the financial market) is an efficient alternative to reinsuring catastrophe (CAT) losses, cf. [5].
Contracts of this kind are insurancelinked derivatives.^{1} They are usually written on insurance industry catastrophe loss indices, insurerspecific catastrophe losses, or parametric indices based on the physical characteristics of catastrophe events. We focus on the first kind of products; they involve more basis risk, but are less exposed to moral hazard than the others, cf. [4].
Derivatives written on insurance industry catastrophe loss indices were first issued in 1992 by the Chicago Board of Trade; these were futures and later also call and put spread options written on aggregate CATloss indices, cf. [4].
A call spread option is a combination of a call option long and a call option short with a higher strike. Another popular type of catastrophe derivative is the CAT bond. This is a classical bond combined with an option that is triggered by a (predefined) catastrophe event. Note that the buyer of the bond thereby sells the embedded option. The issuer is typically a (re)insurance company that wants to reinsure parts of its risk exposure on the financial market. In return the investor receives a coupon.
CAT derivatives are interesting for investors who seek to diversify their risk, since they are largely uncorrelated with classical financial instruments.
The challenges in pricing CAT derivatives are that the underlying index is not a traded asset, that they are not liquidly traded themselves and, maybe most of all, the modeling of catastrophe events.
In the following we review the existing literature. For a more detailed literature overview we refer to [17].
Geman and Yor [12] study European vanilla call options written on an insurance loss index, which is modeled by a jumpdiffusion. Cox et al. [3] model the aggregate loss of an insurance company by a Poisson process with constant arrival rate of catastrophe events and derive a pricing formula for CATputs. Jaimungal and Wang [14] model the aggregate loss by a compound Poisson process to describe the dynamic losses more accurately. Muermann [18] derives the market price of insurance risk from CAT derivative prices in a compound Poisson model. Leobacher and Ngare [15] use the method of utility indifference pricing to price CAT derivatives written on an insurance loss index modeled by a compound Poisson process.
For catastrophe events, the assumption that the resulting claims occur at jump times of a Poisson process as adopted by most previous studies is not beyond justifiable critique. A generalization was proposed in [9], who model an insurance loss index by a doubly stochastic Poisson process (Cox process), i.e. the arrival rate of claims is a stochastic process itself; they price CAT futures in this model. Lin et al. [16] also model the arrival of CAT events by a doubly stochastic Poisson process. See also [11] for noarbitrage pricing of CAT bonds in this context. Dassios and Jang [6] study the valuation of CAT derivatives by risk neutral valuation, where the underlying is modeled as a Cox process with shot noise intensity.
In this paper, we introduce a novel model for an insurance loss index and for a single insurance portfolio that captures ordinary insurance losses as well as catastrophe losses. We model the ordinary claims in the loss index by a compound Poisson process with constant intensity and we model the arrival of catastrophes by a Poisson process with constant intensity, where a jump triggers another stochastic variable that determines the number of claims in case of a catastrophe.
The claims process of a single insurance company holding a fraction of the total number of contracts is then a dynamic thinning of the process describing the index. Our model has the advantage that the jump height distribution does not need to capture both many small claims and outliers caused by catastrophes, but these outliers are split into many smaller claims.
The dynamic thinning is reached in a very convenient way (by drawing from a uniform distribution on [0, 1]) that we believe to be applicable in many other situations.
Using this model we present a pricing mechanism for CAT derivatives (like CAT spread options). Since the insurance loss index is not a tradable asset, and since the market for CAT derivatives is not liquid, risk neutral valuation is not applicable. Instead we use the method of utility indifference pricing. For this we need a hedging mechanism, which will be an active management of the risk portfolio. The pricing method requires solving an associated stochastic optimization problem.
Our paper extends [15] by a more realistic modeling approach for the insurance loss index and also for the thinning. In our paper a catastrophe event may partly hit the considered insurance company, whereas in [15] a catastrophe event always only affects one company. Their model for the claims process of a single insurance company is a thinning (a change of the intensity) of the Poisson process driving the number of claims, while ours is a dynamic thinning of the claims for each event and thus has a different distribution of the jumps.
The model presented here is technically harder to handle; we provide the mathematical toolkit in this paper. Using this new model instead of a simpler one is justified by our numerical results, which show that the new model has a significant impact on the price of a CAT derivative as it reflects catastrophes more accurately.
We also introduce a way to compute the utility indifference price of the derivative by Fourier techniques. This method also allows to compute the residual risk and the profitloss distribution and therefore to evaluate coherent risk measures.
The paper is organized as follows. In Sect. 2 we model the insurance loss index and the claims process of a single insurance company. The state process based on which the CAT derivative is priced, is identified as a piecewise deterministic Markov process (PDMP), see [2, 7]. In Sect. 3 we recall the general concept of utility indifference pricing and we solve the associated stochastic optimization problem. In Sect. 4 we show how the utility indifference price and also quantities relevant for risk management can be computed efficiently, and we present a numerical study.
2 The model
Let \((\Omega ,\mathcal{F},\mathbb{P})\) be a probability space carrying all stochastic variables appearing below.
Suppose we have a global claims process C, which keeps track of all property insurance claims in a given country and we consider an insurance company in the same country, so that the index will contain the losses of that particular insurance company among others.
The portfolio income rate consisting of the premium revenues from the risk portfolio is given by a continuous function q of the company’s market share \(\xi \in [0,1]\). The function q is not necessarily linear in \(\xi\), since demand for insurance might depend on the premium the company charges. The wealth process of the insurance company can be controlled by managing the insured portfolio, i.e. by controlling the market share \(\xi\). This allows for optimizing the management strategy for maximizing utility from terminal wealth.
Therefore, we can apply the method of utility indifference pricing for the valuation of CATderivatives.
We assume independence of \(N^1,N^2,(Y_{i,j})_{i,j\ge 1},(\tilde{A}_i)_{i\ge 1},(U_{i,j})_{i,j\ge 1}\).
Assumption 2.1

\({\mathbb {E}}(e^{\eta Y_{1,1}})<\infty\);

\(\limsup _{k\rightarrow \infty }a_{k+1}/a_k<1/{\mathbb {E}}(e^{\eta Y_{1,1}})\).
In contrast to a model where the claims process is a simple compound Poisson process, here assuming the existence of exponential moments of the claim size distribution is not a great restriction, since we model catastrophes as an accumulation of small claims rather than one big claim.
2.1 PDMP characterization

state space \({\mathbb {R}}_0^+ \times {\mathbb {R}}\);

control space [0, 1];

deterministic flow \(d(C_t,X^\xi _t)=(0, q(\xi _t)) dt\) between jumps;

jump intensity \(\lambda\);
 jump kernel Q,where$$\begin{aligned} Q(B(c,x),\xi )= \sum _{k=0}^\infty a_k Q_k(B(c,x),\xi ), \end{aligned}$$and where we use the notation \(B(c,x)=\{(b_1c,b_2x):(b_1,b_2)\in B\}\);$$\begin{aligned} Q_k(B(c,x),\xi )= \sum _{\mathcal{K}\subseteq \{1,\dots ,k\}} \xi ^{\mathcal{K}} \left( 1\xi \right) ^{k\mathcal{K}} \mathbb{P} \left( \left( \sum _{j=1}^k Y_{1,j},\sum _{j\in \mathcal{K}} Y_{1,j} \right) \in B(c,x) \right) , \end{aligned}$$

zero running reward rate;

zero discount rate.
3 Utility indifference pricing
The method of utility indifference pricing for the valuation of derivatives in incomplete markets has been introduced in [13]. It relies on the fact that even if the derivative cannot be replicated, it may still be the case that much of its variation can be hedged.
In [8] utility indifference pricing is used to price structured catastrophe bonds. However, there is a difference in modeling the hedging possibility. In our setup this is done via managing the insured portfolio. The main idea is that the loss in the portfolio of a single insurance company is necessarily correlated with the insurance loss index. The introduction of the derivative has therefore an influence on the pricing policy of the insurance company.
We will first explain the notion of utility indifference pricing and then apply it to our problem.
Assume the investor has a utility function u and initial wealth x. Define \(J(x,\ell ):=\sup _{X_T}{\mathbb {E}}(u(X_T+ \ell \psi ))\), where the supremum is taken over all possible wealths \(X_T\) that can be generated from x. The random variable \(\psi\) is the payment from a European claim with expiry T, and \(\ell\) is the number of claims that are bought.
The utility indifference bid price \(p^b(\ell )\) is the price at which the investor has the same utility whether she pays nothing and does not receive the claim \(\psi\), or she pays \(p^b(\ell )\) now and receives \(\ell\) units of the claim \(\psi\) at time T. Therefore, \(p^b(\ell )\) is the largest amount of money the investor is willing to pay for buying \(\ell\) units of the claim \(\psi\); it solves \(J(xp^b(\ell ),\ell )=J(x,0)\).
The utility indifference ask price \(p^a(\ell )\) is the smallest amount of money the investor is willing to accept for selling \(\ell\) units of the claim \(\psi\); it solves \(J(x+p^a(\ell ),\ell )=J(x,0)\).
The two prices are related via \(p^b(\ell )=p^a(\ell )\). With this in mind we can define the utility indifference price \(p:=p^b(1)\).
Assumption 3.1

The insurance company has exponential utility \(u(x)=\exp (\eta x)\), \(\eta >0\).

\(X_T\) is of the form \(x+\Gamma ^\xi _T\) for some control \(\xi\) and \(\Gamma ^\xi _T\) does not depend on the initial wealth x.
Note that exponential utility is a natural choice for insurance companies as often such a utility function is used to calculate insurance premia. As an example where exponential utility is used in a stochastic optimal control framework in an insurance context, see [10].
3.1 The stochastic optimization problem
We apply the concept of utility indifference pricing to the model presented in Sect. 2. Our aim is to price a derivative written on the total claims process C with payoff \(\psi (C_T)\), where \(\psi\) is a continuous and bounded function on \({\mathbb {R}}_0^+\).
Example 3.1
Note that the main task in pricing CAT bonds also lies in pricing the embedded spread option, since for exponential utility the price of a CAT bond is the sum of a spread option price and a bond price.
Lemma 3.2
Let W be such that \(V(t,c,x)=u(x)\exp (\eta W(t,c,x))\). Then W is bounded by \(\Vert q\Vert _{\infty } T+\Vert \psi \Vert _\infty\).
Proof
3.2 Verification result
We show that the solution of the HJB equation (9) solves the optimization problem (8). For this we apply results from stochastic control theory for PDMPs; more precisely, a slight variation of the verification theorem [2, Theorem 8.2.8]. For this we recall two definitions from [1].
Definition 3.3
 i.
\(u(x+\psi (c))\le c_u b(c,x)\);
 ii.
\(\int b(\tilde{c},\tilde{x})Q(d\tilde{c}\times d\tilde{x}(c,x),\xi )\le c_Q b(c,x)\) for all \((c,x)\in {\mathbb {R}}_0^+ \times {\mathbb {R}}\), \(\xi \in [0,1]\);
 iii.
\(b(c,x+\int _0^T \int _0^1 q(\xi )r_s(d\xi )ds)\le c_{\text {flow}}b(c,x)\) for all \(r\in \mathcal{R}\).
Definition 3.4
Theorem 3.5
Let b be a bounding function for our piecewise deterministic Markov decision model with \({\mathbb {E}}(b(C_T,X^\xi _T)\big C_t=c,X_t=x)<\infty\) for all \(\xi ,t,c,x\). Let \(v\in \mathcal{C}^{1,0,1}([0,T] \times {\mathbb {R}}_0^+ \times {\mathbb {R}})\cap {\mathbb {B}}_{b,\gamma }\) be a solution of the HJB Eq. (9) and let \(\alpha ^*\) be a maximizer for (9), leading to the state process \((C,X^{\xi ^*})\).
Then \(v=V\) and \(\xi ^*=\alpha ^*(t,C_{t},X^{\xi ^*}_{t})\) is an optimal feedbacktype Markov policy.
Remark 3.6
In the statement of [2, Theorem 8.2.8] there is another condition required, namely that \(\alpha _b<1\) for a constant \(\alpha _b\) depending on b, Q and the arbitrary \(\gamma\) from Definition 3.4. But it is argued in [1] that for finite horizon problems \(\gamma\) can always be chosen large enough to satisfy \(\alpha _b<1\).
For proving Theorem 3.5, we first need to prove existence of a bounding function.
Lemma 3.7
The function b defined by \(b(c,x):=\exp (\eta x)\) is a bounding function for our piecewise deterministic Markov decision model.
Proof
 i.
\(u(x+\psi (c))=\exp (\eta (x+\psi (c))\) such that \(u(x+\psi (c))=\exp (\eta (x+\psi (c))\le \exp (\eta \Vert \psi \Vert _\infty )b(c,x)\).
 ii.
\(\int b(\tilde{c},\tilde{x})Q(d\tilde{c}\times d\tilde{x}(c,x),\xi ) =\int \exp (\eta \tilde{x})Q(d\tilde{c}\times d\tilde{x}(c,x),\xi ) ={\mathbb {E}}(\exp (\eta xZ_1^{\xi })) \le b(c,x){\mathbb {E}}(\exp (\eta Z_1^{\xi }))\).
 iii.
\(b\left( c,x+\int _0^T\int _0^1 q(\xi )r_s(d\xi )ds\right) =\exp (\eta x+\int _0^T\int _0^1 q(\xi )r_s(d\xi )ds) \le \exp (\eta \Vert q\Vert _{\infty } T)b(c,x)\).
Now we need to show that the backward equation (11) has a solution and hence also (9) has a solution.
Lemma 3.8
For \(\sigma \in \mathcal{C}_b(R_0^+)\) the mapping \(\xi \mapsto {\mathbb {E}}(\sigma (Z_1)\exp (\eta Z_1^{\xi }) )+\lambda /\eta\) is a power series in \(\xi\). Its coefficients are of the form \(h_k(\sigma )={\mathbb {E}}(\delta _k \sigma (Z_1))\) for nonnegative random variables \(\delta _k\) with \(\sum _{k=0}^\infty {\mathbb {E}}(\delta _k)<\infty\) that do not dependent on \(\xi\) and \(\sigma\). The power series converges uniformly on [0, 1].
Proof
Lemma 3.9
The function h is a bounded linear operator.
Proof
We need to prove that for every \(k\in {\mathbb {N}}_0\) the mapping \(h_k\) is a bounded linear operator \(\mathcal{C}_b({\mathbb {R}}_0^+ \times {\mathbb {R}}_0^+)\longrightarrow \mathcal{C}_b({\mathbb {R}}_0^+)\).
Let \(\phi \in \mathcal{C}_b({\mathbb {R}}_0^+ \times {\mathbb {R}}_0^+)\). We show that the mapping \(c\mapsto {\mathbb {E}}(\delta _k \phi (c,Z_1))\) is continuous and bounded on \({\mathbb {R}}_0^+\). Let \(c_n \rightarrow c\) in \({\mathbb {R}}_0^+\). Then \(\phi (c_n,z) \rightarrow \phi (c,z)\) for all \(z\in {\mathbb {R}}_0^+\). The sequence \((\phi (c_n,\cdot )\delta _k)_{n\ge 0}\) is dominated by \(\Vert \phi \Vert _\infty \delta _k\), which is integrable. Hence, \({\mathbb {E}}(\phi (c_n,Z_1)\delta _k)\rightarrow {\mathbb {E}}(\phi (c,Z_1)\delta _k)\) by the dominated convergence theorem. Thus \(h_k(\phi )\) is continuous. Moreover, \(h_k(\phi )\) is bounded, since \(\left {\mathbb {E}}(\phi (c,Z_1)\delta _k)\right \le \Vert \phi \Vert _\infty {\mathbb {E}}(\delta _k)\).
For \(\phi \in \mathcal{C}_b({\mathbb {R}}_0^+ \times {\mathbb {R}}_0^+)\) it holds that \(\Vert h\Vert _\Lambda \le \sum _{k=0}^\infty \Vert h_k(\phi )\Vert _\infty =\sum _{k=0}^\infty \sup _c{\mathbb {E}}(\delta _k\phi (c,Z_1)) \le \Vert \phi \Vert _\infty \sum _{k=0}^\infty {\mathbb {E}}(\delta _k)\). Thus h is bounded by Lemma 3.8. \(\square\)
Lemma 3.10
 1.
\(\tilde{g}\) is defined on \(\ell ^1\) and it is bounded on every normbounded subset of \(\ell ^1\);
 2.
\(\tilde{g}\) is convex;
 3.
\(\tilde{g}\) is Lipschitz on every normbounded subset of \(\ell ^1\).
Proof
For \(x=(x_k)_{k\ge 0}\in \ell ^1\) and \(\xi \in [0,1]\) we have \(\sum _{k=0}^\infty x_k \xi ^k\le \Vert x\Vert _1\). The function \(\xi \mapsto \sum _{k=0}^\infty x_k \xi ^k\), \(\xi \in [0,1]\) is welldefined and continuous as a uniform limit of continuous functions on [0, 1]. Since q is also continuous, the first statement follows.
The proof of the second statement is straightforward.
Following [20] we use the convexity of \(\tilde{g}\) to show that \(\tilde{g}\) is Lipschitz on \(\{x\in \ell ^1:\Vert x\Vert \le R\}\). Let \(\Vert x\Vert ,\Vert y\Vert \le R\) and define \(z:=y+\frac{R}{\Vert yx\Vert }(yx)\). It holds that \(\Vert zy\Vert =R\) and hence \(\Vert z\Vert \le 2R\). By the definition of z we have that \(y=\beta z + (1\beta )x\), where \(\beta =\Vert yx\Vert /(\Vert yx\Vert +R)\). Since \(\tilde{g}\) is convex, \(\tilde{g}(y)\le \beta \tilde{g}(z)+(1\beta )\tilde{g}(x)\) and hence \(\Vert \tilde{g}(y)\tilde{g}(x)\Vert = \Vert \beta (\tilde{g}(z)\tilde{g}(x))\Vert \le 2\beta \sup _{\Vert z\Vert \le 2R}\tilde{g}(z) \le 2\beta c \le (2C/R) \Vert yx\Vert\) for some constant \(c>0\), since \(\tilde{g}\) is bounded on \(\{z\in \ell ^1:\Vert z\Vert \le 2R\}\). \(\square\)
For \(\phi \in \Lambda\) let \(\phi (c):=(\phi _k(c))_{k\ge 0}\) and define the function g by \(g(\phi )(c)=\tilde{g}(\phi (c))\).
Lemma 3.11
The function g is \(\mathcal{C}_b({\mathbb {R}}_0^+)\) valued and locally Lipschitz.
Proof
Let \(\phi =(\phi _k)_{k\ge 0}\in \Lambda\). Let \(c_n\rightarrow c\) in \({\mathbb {R}}_0^+\) and let \(\varepsilon >0\). There exists \(k_0\in {\mathbb {N}}_0\) such that \(\sum _{k\ge k_0} \Vert \phi _k\Vert _\infty <\varepsilon /4\), and for n large enough \(\sum _{k=0}^{k_0} \phi _k(c_n)\phi _k(c)<\varepsilon /2\). Thus, \(\sum _{k=0}^\infty \phi _k(c_n)\phi _k(c) \le \sum _{k=0}^{k_0} \phi _k(c_n)\phi _k(c)+\sum _{k=k_0+1}^{\infty } \phi _k(c_n)\phi _k(c)< \varepsilon\).
The claim that g is locally Lipschitz follows from Lemma 3.10: let \(R>0\) and let \(\phi ^1,\phi ^2\in \Lambda\) with \(\Vert \phi ^1\Vert _\Lambda \le R\) and \(\Vert \phi ^2\Vert _\Lambda \le R\). \(\tilde{g}\) is Lipschitz on the ball with radius R in \(\ell ^1\). Denote the corresponding Lipschitz constant by \(L_R\). Then \(\phi ^1(c),\phi ^2(c)\) lie in the ball with radius R in \(\ell ^1\). Hence, \(\Vert g(\phi ^1)g(\phi ^2)\Vert _\infty =\sup _c\tilde{g}(\phi ^1(c))\tilde{g}(\phi ^2(c))\le L_R \Vert \phi ^1(c)\phi ^2(c)\Vert _1 \le L_R\Vert \phi ^1\phi ^1\Vert _\Lambda\). \(\square\)
Finally, define \(f:\mathcal{C}_b({\mathbb {R}}_0^+) \longrightarrow \mathcal{C}_b({\mathbb {R}}_0^+ \times {\mathbb {R}}^+)\), \(f(w)(c,z):=\exp (\eta (w(c+z)w(c))\) and note that f is locally Lipschitz.
Lemma 3.12
Let \(\mathcal{H}\) be defined as in (12). If \(\varphi \in \mathcal{C}_b({\mathbb {R}}_0^+)\) , then \(\mathcal{H}\varphi \in \mathcal{C}_b({\mathbb {R}}_0^+)\) and \(\mathcal{H}\) is locally Lipschitz.
Proof
We have \(\mathcal{H}=g \circ h \circ f\). The first claim follows from Lemma 3.11. Further, \(\mathcal{H}\) is locally Lipschitz as a concatenation of locally Lipschitz functions. The latter follows from Lemmas 3.9 and 3.11. \(\square\)
Now we prove that (11) has a unique maximal local solution.
Lemma 3.13
Let \(\psi \in \mathcal{C}_b({\mathbb {R}})\). Then the backward equation (11) has a unique maximal local solution.
Proof
The function \(w:[T\varepsilon , T]\times {\mathbb {R}}_0^+\longrightarrow {\mathbb {R}}\) defined by \(w(t,c)=\varphi (t)(c)\) is the unique maximal local solution of (11). \(\square\)
Proof (Proof of Theorem 3.5)
Since for every \(\varphi \in \mathcal{C}_b({\mathbb {R}}_0^+)\) and \(c\in {\mathbb {R}}_0^+\) the function \(\xi \mapsto q(\xi ) +(\tilde{\mathcal{A}}^\xi w)(c)\) is continuous on [0, 1] by Lemma 3.8, there exists a maximizer for (9). By Lemmas 3.7 and 3.13 the assumptions of Theorem 3.5 are satisfied on \([T\varepsilon ,T]\times {\mathbb {R}}_0^+ \times {\mathbb {R}}\longrightarrow {\mathbb {R}}\), where \(\varepsilon\) is as in the proof of Lemma 3.13. Along the lines of the proof of [2, Theorem 8.2.8] it can be shown that \(v:[T\varepsilon ,T]\times {\mathbb {R}}_0^+ \times {\mathbb {R}}\longrightarrow {\mathbb {R}}\) with \(v(t,c,x)=u(x)\exp (\eta w(t,c))\) solves the optimization problem (8) for \(t\in [T\varepsilon ,T]\), i.e. \(v(t,\cdot ,\cdot )=V(t,\cdot ,\cdot )\) for \(t\in [T\varepsilon ,T]\).
By Lemma 3.2 it holds that \(w(t,c)=1/\eta \log (v(t,c,x)/u(x)) =1/\eta \log (V(t,c,x)/u(x))=W(t,c,x)\le \Vert q\Vert _{\infty } T+\Vert \psi \Vert _\infty\) for \(t\in [T\varepsilon ,T]\). Therefore, \(\Vert w(T\varepsilon ,.)\Vert _\infty <2( \Vert q\Vert _{\infty } T+\Vert \psi \Vert _\infty )\) and hence \(\varepsilon =T\). Thus w solves (11) on the whole of \([0,T]\times {\mathbb {R}}_0^+\) and therefore v solves the optimization problem (8) in the whole of \([0,T] \times {\mathbb {R}}_0^+ \times {\mathbb {R}}\). \(\square\)
3.3 Utility indifference price
Lemma 3.14
Proof
4 Computations
In this section, we present a convenient numerical method for computing the expected value in (17) for the case where the distribution of \(Y_{1,1}\) has a smooth density. Denote by \(f^{*k}\) the kfold convolution of a function f with itself, i.e. \(f^{*2}=f*f\) and \(f^{*(k+1)}=f*f^{*k}\). Let \(\hat{F}\) denote the Fourier transform. It holds that \(\hat{F}(f^{*k})=\hat{F}(f)^k\). The following lemma gives an efficient method for computing \({\mathbb {E}}(\exp (\eta w(t,c+Z_1))\exp (\eta Z_1^{\xi }))\).
Lemma 4.1
Assume that the distribution of \(Y_{1,1}\) has a piecewise continuous density \(\mu\). Denote by \(\tilde{\mu }(z)=\exp (\eta z) \mu (z)\). Let \(\sigma\) be measurable and bounded.
Proof
Thus \(\sum _{k=1}^\infty a_k\left( \xi \tilde{\mu }+(1\xi )\mu \right) ^{*k}=\hat{F}^{1}\left( \hat{F}\left( \sum _{k=1}^\infty a_k\left( \xi \tilde{\mu }+(1\xi )\mu \right) ^{*k}\right) \right)\) a.e. on \({\mathbb {R}}_0^+\). \(\square\)
4.1 Numerical experiments
Our aim is to price a CAT (spread) option, i.e. \(\psi (C_T)=\max (0,\min (C_TK,LK))\).
We want to study two effects: the effect that holding a derivative has on the risk loading (which depends on the optimal market share \(\xi ^*\)) and its change over time, and the effect of our model [the clustered claims (CC) model] on the utility indifference price of the derivative and on the risk loading in comparison to the model where the claims process is a simple compound Poisson process as, e.g., in [15] [the single claim (SC) model].
For the SCmodel we choose \(\lambda _1=100, \lambda _2=0\).
In order to be able to compare the two models we adapt \(\lambda _1,\lambda _2\) such that the expected annual claim size per contract a stays constant, yielding \(\lambda _1=69, \lambda _2=1\).
Figure 2 shows the risk loading corresponding to the optimal market share \(\xi ^*\) in dependence of the value c of the claims process. For small c the risk loading is the same as in the case of no derivative held, since the probability that the derivative has a positive payoff is small. For \(c>L\) a further increase of c does not change the payoff and hence the situation is the same as for holding no derivative.
As time increases the probability that the payoff of the derivative grows in c and hence compensates losses during the remaining time decreases, but also the expected number of claims before T decreases. An interesting observation is that for small c the first effect dominates and hence the risk loading increases, whereas for large c the latter effect dominates and hence the risk loading decreases.
In the CCmodel the risk loading is in general higher than in the SCmodel, since we imposed risk aversion. The risk loading decreases significantly when a derivative is bought.
We observe the same effects when comparing our CCmodel to the SCmodel for the example of [15], where a bounded claim size distribution was used.
4.2 Concluding remarks
The introduction of a derivative serves as an effective alternative to classical reinsurance and leads to significantly smaller insurance premia. The CCmodel introduced in this paper has a significant impact on the price of a CAT derivative as well as on the optimal average risk loading. It reflects catastrophes more accurately.
4.2.1 Risk management
In this section we compute the profitloss distribution and the residual risk of an insurance company holding a CAT derivative. The former is useful for the derivation of coherent risk measures, the latter quantifies the efficiency of the hedge.
4.2.2 Profitloss distribution
The profitloss distribution is the distribution of the (optimally controlled) wealth \(\rho :=X_T^{\xi ^*}+\psi (C_T)p\) in case the company holds a CAT derivative.
4.2.3 Residual risk
The numerical experiments in Sect. 4.1 showed that the optimal market share \(\xi ^*\) of an insurance company that holds a CAT derivative is higher than for a company that does not (\(\xi ^0\)). The change in the strategy \(\xi ^*\xi ^0\) when an insurance company buys a CAT derivative, is also the strategy used for hedging the derivative itself.
Footnotes
 1.
Details on currently listed insurancelinked derivatives can be found at http://www.artemis.bm/deal_directory.
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). A. Eichler is supported by the Austrian Science Fund (FWF): Project P21196. G. Leobacher is supported by the Austrian Science Fund (FWF): Project F5508N26, which is part of the Special Research Program “QuasiMonte Carlo Methods: Theory and Applications” and by the Austrian Science Fund (FWF): Project P21196. This paper was written while G. Leobacher was member of the Department of Financial Methematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria. M. Szölgyenyi is supported by the Vienna Science and Technology Fund (WWTF): Project MA14031.
References
 1.Bäuerle N, Rieder U (2010) Optimal control of piecewise deterministic markov processes with finite time horizon. In: Piunovskiy A (ed) Modern trends in controlled stochastic processes: theory and applications. Luniver Press, Frome, pp 123–143Google Scholar
 2.Bäuerle N, Rieder U (2011) Markov decision processes with applications in finance. Universitext. Springer, BerlinCrossRefMATHGoogle Scholar
 3.Cox SH, Fairchild JR, Pedersen HW (2004) Valuation of structured risk management products. Insur Math Econ 34(2):259–272CrossRefMATHGoogle Scholar
 4.JD Cummins (2006) Should the government provide insurance for catastrophes? Fed Reserve Bank St. Louis Rev 88(4):337–380Google Scholar
 5.Cummins JD, Lalonde D, Phillips RD (2004) The basis risk of catastrophicloss index securities. J Financ Econ 71(1):77–111CrossRefGoogle Scholar
 6.Dassios A, Jang JW (2003) Pricing of catastrophe reinsurance and derivatives using the cox process with shot noise intensity. Financ Stoch 7:73–95MathSciNetCrossRefMATHGoogle Scholar
 7.Davis MHA (1993) Markov models and optimization. Monographs on statistics and applied probability. Chapman & Hall, LondonCrossRefGoogle Scholar
 8.Egami M, Young VR (2008) Indifference prices of structured catastrophe (CAT) bonds. Insur Math Econ 42(2):771–778CrossRefMATHGoogle Scholar
 9.Embrechts P, Meister S (1997) Pricing insurance derivatives: the case of CAT futures. In: Securization of insurance risk: the 1995 Bowles symposium, pp 16–26Google Scholar
 10.Fernández B, HernándezHernández D, Meda A, Saavedra P (2008) An optimal investment strategy with maximal risk aversion and its ruin probability. Math Methods Oper Res 68:159–179MathSciNetCrossRefMATHGoogle Scholar
 11.Fuita T, Ishimura N, Tanake D (2008) An arbitrage approach to the pricing of catastrophe options involving the cox process. Hitotsubashi J Econ 49:67–74Google Scholar
 12.Geman H, Yor M (1997) Stochastic time changes in catastrophe option pricing. Insur Math Econ 21(3):185–193MathSciNetCrossRefMATHGoogle Scholar
 13.Hodges S, Neuberger A (1989) Optimal replication of contingent claim under transaction costs. Rev Futures Mark 8:222–239Google Scholar
 14.Jaimungal S, Wang T (2006) Catastrophe options with stochastic interest rates and compound poisson losses. Insur Math Econ 38(3):469–483MathSciNetCrossRefMATHGoogle Scholar
 15.Leobacher G, Ngare P (2016) Utility indifference pricing of derivatives written on industrial loss indexes. J Comput Appl Math 300:68–82MathSciNetCrossRefMATHGoogle Scholar
 16.Lin S, Chang C, Powers MR (2009) The valuation of contingent capital with catastrophe risks. Insur Math Econ 45(1):65–73MathSciNetCrossRefMATHGoogle Scholar
 17.Muermann A (2004) Catastrophe derivatives. In: Teugels JL, Sundt B (eds) Encyclopedia of actuarial science, vol 1. Wiley, Chichester, pp 231–236Google Scholar
 18.Muermann A (2008) Market price of risk implied by catastrophe derivatives. N Am Actuar J 12(3):221–227MathSciNetCrossRefGoogle Scholar
 19.Olofsson P, Andersson M (2012) Probability, statistics, and stochastic processes, 2nd edn. Wiley, HobokenCrossRefMATHGoogle Scholar
 20.Roberts AW, Varberg DE (1974) Another proof that convex functions are locally Lipschitz. Am Math Mon 81:1014–1016MathSciNetCrossRefMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.