数理解析研究所講究録
第 1864 巻 2013 年 50-60

50Catastrophe Risk Derivatives: A New Approach
Mehdi Bekralas Abdessalem
Graduate School of Economics
Osaka University
Masamitsu Ohnishi

Graduate School of Economics & Center for the Study of Finance and Insurance
Osaka University

Introduction

1

We are in a cenmry where the urbanism is in a constant expansion. The modemization and the useof technology in urban and rural areas require more and more investments and expenses. This proliferation is driving the expected value of the catastrophe to attain important sums, which makes it
nowadays very hard for an insurance company to cover these costs. For instance, a big earthquake inTokyo could engender a higher cost than all the capitalization of the Japanese insurance companies.
One of the solutions to this problem is the securitization of the catastrophe risk, since the financial
potential of the stock exchange is huge comparing with these loss amounts.
From their apparition in the CBOT (Chicago Board of Treading) in 1995 the catastrophe derivatives

know an incomparable success. Hence, in the other hand the financial literacy is giving an increasing
interest to the modeling of these derivatives.
A shared point between the catastrophe products is their common underlying that is an index
representing the amount engendered by some catastrophes during a certain period of time. This
index was created to solve a moral hazard problem since the veracity of the catastrophe’s amount is
not observable from the financial statement of an insurance company A good catastrophe derivative
pricing passes inevitably by a good modeling of the loss index. Catastrophe loss indexes displays
the amount of the insurance claims that surpasses a certain amount. That is because the insurance
institutions are interested only by the large amounts (the extraordinary events) as the small ones arecovered by the reserve made for this purpose. For an example, the PCS (Property Claim Services) index
used$5 million as a floor and nowadays this floor moved to $12 million.1
The occurrence of these claims seems to follow ajump process thus some authors modeled theindex as a Poisson process like in Cox, Fairchild, and Pederson (2006)[5]. The inconvenience withthis approach is that it assigns the same size to all claims, while in reality different catastrophes in aninterval of time create gives different claim amounts thus differentjump sizes.
http: $//www$ . iso. $com/Products/Property-Claim-Services/Property$-Cla $im-Services-PCS-info$ -on-losses-from-catastrophes. html
1

51
To correct this, among Jaimungal and Wang (2006) [10] and others, Alexander (2003) [1] used aCompound Poisson process without specifying the distribution of the sizes. Others take the jump
size to be an exponentially distributed random variable, which we will see is not appropriate to theinsurance catastrophe modeling.
Unlike other papers, we do not assume here any probability distribution for the index claims andbut will try to deduce it by using some useful statistical tools. The general methodology of this paper will be as follow: we will start by introducing a simple model and try to derive its induced option
pricing formula. In a second time, we will gradually introduce some more sophisticated models concluding by their induced option pricing.

2 The Loss Index:$A$

general Modeling

Let us denote by
the Loss index process, and by $i=1,2,\ldots$ the random variable representing the -th clam amount.
As usualis the probability space with the statistical (real world) probability measure
Also we assume a constant interest rate that we denote as
Following the existing PCS (Property Claim Services) index, the loss process is characterized by
two distinguished periods:$\{L_{t};t\geq 0\}$

$Y_{i},$$i$$(\Omega,\mathscr{F},\mathbb{P},)$$\mathbb{P}.$$\delta.$.

.

; whereAloss period
represents the first mamrity of the option, that is, for a call as an
example, at the end of this period the index’s level should be more important than the strike sothat the option would have an evenmal payoff.
$[0, t_{1})$$t_{1}$; since we can not assess the real amount of the sinister a period
development period
is left for the option investor so that the index shows the exact amount of claims engendered by
the catastrophes happening between. Note that only the re-estimations of the catastrophe sinister that happen during the Loss period is taken into account. Catastrophes that occur
betweenare not considered.
$A$$[t_{1}, t_{2}]$$[0, t_{1})$$[t_{1}, t_{2}]$For simplicity, we will start by modeling the loss period.
Let the Index process have the following form:$L_{t}=L_{0}\exp\{X_{t}\}, t\geq 0$

,

(1)

where
is a stochastic process (a Compound Poisson Process)satisfying the usual conditionsand
such as:$X_{t}$$X_{t}= \sum_{i=1}^{N_{t}}Y_{i}, t\geq 0$

where$N_{t}$,(2)represents the number ofcatastrophes2 occurring in the interval $[0, t)$ . Hence the sum:

represems the total amount of the claims to be hedged by the institution.

$\sum_{i=1}^{N_{t}}Y_{i}$symbolizes the filtration generated by the comp und Poisson process
and the-null set ofHere we are not making any assumption regarding the distribution of the claim amounts, but theclassical $i.i.d$. assumption.
$F=\{\mathscr{F}_{t};t\geq 0\}$$0$$\mathscr{F}.$$\mathbb{P}$2

Catastrophes are only the ones generating large sinister (extraordinary amounts).

$X_{t}$

52
Some scientific evidences show that the occurrence of an earthquake opens a hole in the earth.
The opening of the hole may be done by a series of other earthquakes. This part is directly followedby the closing of the hole process via an other set of earthquakes. If the original earthquake has a big
magnimde then the following provoked ones (the replicas) have an important probability to have also

a big magnitude.
This is to say that one big earthquake can generate many others. In the probability of having many
magnimde picks (and thus amounts that aoe measurable by the index) during an interval
where is the time of an earthquake manifestation.
This observation is important in the sense that statistically it allows us to use the strong law of$[t_{i}, t_{j}],$ $i\neq j$$t_{i}$

large numbers. As stated before the correct approach of the index should includes only the high claim
amounts.
For this purpose, the Extreme Value Theory (EVT) seems to provide a very useful set of tools that
could be exploited in our case to set our model.

A Distribution for the Index’s Amounts
In this subsection, we will try to deduct the distribution ofthe amounts that are measured by the index

starting from the sample of all the observable claims by using some statistical approaches.
This distribution differs depending on our definition of the $high$ amount.
According to the EVT, two kinds of definitions could be used.
as the set of the maximum amounts
First, we define the extreme values during a period say$0\leqt_{i}<t_{j}\leq
T.$subsetof eachLet $M_{n};= \max\{Y_{1,\ldots/}Y_{n}\}$ . According to the Fisher, Tippet and Gnedenko Theorem of the Block Max$n=1,2,\ldots$ where $a_{n}>0$ andima, if there exists a sequence of couples)$/$$[0_{J}T]$$[t_{i}, t_{j}],$$(a_{n},b_{n}),$$\lim_{narrow\infty}\mathbb{P}(\frac{M_{n}-b_{n}}{a_{n}}\leq y)=;H(y)$,where $H$ is a non-degenerate cumulative dismbution function, then $H$ has a Generalized Extreme
Value (GEV) distribution which is a distribution that includes one of the following distributions: Weibull,Fr\’echet, or Gumbel that is to say that the normalized maximum has only one of the distributions statedabove.
The GEV distribution has the Following form:$H_{\zeta}(x)=\{\begin{array}{ll}\exp\{-(1+\zeta x)^{-\frac{1}{\zeta}}\} if \zeta\neq 0;\exp\{-\exp(-x)\} if \zeta=0.\end{array}$

Choice of the GEV Distribution
The skewness of the Fr\’echet distribution is given by:$if\alpha>3;$$Skewness_{Fr\’{e} chet}=\{\frac{\Gamma(1-\frac{3}{\alpha})-3\Gamma(1-\frac{2}{\alpha})\Gamma(1-\frac{1}{\alpha})+2\Gamma^{2}(1-\frac{1}{\alpha})}{\infty\sqrt{\{\Gamma(1-\frac{2}{\alpha})-\Gamma^{2}(1-\frac{1}{\alpha})\}^{3}}}$otherwise,

53
where
is Frechet distribution’s shape parameter. So the basicaUy the shape of the distribution
is determined depending on the value of our . Thus the behavior of this distribution is not clear.
Same is for the Weibull distribution.
To remedy this, we follow this reasoning: In one hand, following Proposition 3 of Fraga Alves andNeves (2010) [6], the skewness ofthe GEV distribution is positive for $\zeta>-2.8$ , negative for $zeta<-2.8,$
and approximately equal to 1.14 forIn the other hand the GEV corresponds to the Frechet distribution for, to the Weibull one in, and finally it represents the Gumbel law when
the case where
We deduct that the Fr\’echet distribution is right skewed. Affecting a Fr\’echet distribution to theextreme values of earthquake claims is equivalent to suppose that the maximum amounts are too
close from the ones that appear in a regular simation. In other words this distribution is concentrated
in its small values. While in reality the catastrophes amoum happen to be unexpected, so out of thehistorical data and these amount are larger than we could have in a normal situation.
By the same reasoning, the Weibull distribution assigns a large probability to the values too far
from the end of the original distribution of the claims. This seems to be unrealistic since the earthquake wave vanishes after a certain distance.
Contrariwise the Gumbel law offers a certain uniformity in its distribution thus to begin we will
say that the claim’s extreme values follow a Gumbel distribution.
As a reminder, the Gumbel Cumulative Distribution Function ( $CDF$ ) and Probability Density Function (PDF) are defined consecutively as bellow:$\alpha\in \mathbb{R}^{++}$$\alpha$$\zeta=0.$$\zeta>0$$\zeta<0$$\zeta=0.$$F_{Y}(x)= \exp[-\exp\{-(\frac{x-\mu}{\beta})\}|, -\infty<x<\infty,$$f_{Y}(x)= \frac{1}{\beta}\exp\{-(\frac{x-\mu}{\beta})\}\exp[-\exp\{-(\frac{x-\mu}{\beta}1\}|, -\infty<x<\infty.$

Its mean ad variance are:$E[Y]=\mu+\beta\gamma$

Var$[Y]= \sigma^{2}[Y]=\frac{\pi^{2}\gamma^{2}}{6}$;.

where $\gamma=0.577\ldots$ is the Euler-Mascheroni constant.
Starting from the claims observations we can estimate the parameters of the Gumbel distribution.
The Maximum Likelihood Estimator (MLE) is discussed in Chapter 19 ofForbes, Evans, Hastings, andPeacock (2011) [7]. The system of equations for the shape and scale parameters that is issued bythis method needs to be solved numerically. Many authors suggest instead the use of the Method ofMoments (MM) approach. The MM yields to:$\hat{\beta}=\frac{\sqrt{6}}{\pi}\overline{\sigma}$;$\hat{\mu}=\overline{Y}-\hat{\beta}\gamma,$

and are sample mean and variance, respectively.
Now that we have defined the probabihty dismbution for the claims we can start investigating our
option pricing.

where$\overline{Y}$$\overline{\sigma}^{2}$

54
3 Catastrophe Option pricing
We will start this suction by calculating the Momem generating function of the process
would be of an extreme necessity afterward.

Moment Generating Function of$X_{t}$

since it

$X_{t}$The Moment Generating Function (MGF) of$X_{t}$

is given as follows:$M(z, t)=E[e^{zX_{t}}]$$=E[E[e^{z\Sigma_{i=1}^{N_{t}}Y_{i}1N_{t}]}]$$=E[\{\varphi_{Y}(z)\}^{N_{t}}]$$= \sum_{k=0}^{\infty}\{\varphi_{Y}(z)\}^{k}\frac{(\lambda t)^{k}}{k!}e^{-\lambda t}$$=e^{\lambda t\varphi_{Y}(z)}e^{-\lambda t}$$= \exp[\lambda t\{\Gamma(1-\beta z)e^{\mu z}-1\}], z<\frac{1}{\beta}$

,

(3)

where
$\varphi_{Y}(z)=E[e^{zY}]=\Gamma(1-\beta z)e^{\mu z}, z<\frac{1}{\beta}$

where is the MGF of Gumbel random variable$Y.$

Option Pricing by Esscher Transform
Our index is composed by a series of jumps which means that the market is incomplete. Consequently, there exists not only one measure for which the discounted option price is a martingale.
Accordingly the pricing of our asset is not unique. Here, for the pricing of our catastrophe option, we
propose to use the Esscher transform following Gerber and Shiu (1994) [9] in their seminal paper. For
this purpose, we suppose that there are no transaction costs and no taxes as usual. We also give thepricing of a call option. The put option price afterward can be easily deducted by using the Call/Put
parity.

The Esscher MGF
Recall that the Esscher Moment Generating Function (MGF) is given by:$M(z, t, h);= \frac{M(z+h,t)}{M(h,t)}$where we denote $M(z, t)$ as
Therefore:$X_{t}$,(4)’s MGF3.

here that we are using the same setting as Gerber and Shiu (1994) [9] by modeling the loss as$\overline{3Notice}$$L_{t}=L_{0}\exp\{X_{t}\}.$

55$M(z, t, h)=\exp[\lambda t\{\Gamma(1-\beta(z+h))e^{\mu(z+h)}-e^{\mu h}\Gamma(1-\beta h)\}]$(5)$=\exp[\lambda te^{\mu h}\{\Gamma(1-\beta(z+h))e^{\mu z}-\Gamma(1-\beta h)\}].$The Risk Neutral Esscher Transform
According to the Fundamental Theorem of Asset Pricing (FTAP), a no-arbitrage pricing measuresatisfies:$\mathbb{P}^{*}$$L_{0}=E^{*}[e^{-\delta t}L_{t}] \Rightarrow L_{0}=e^{-\delta t}E^{*}[L_{0}\exp\{X_{t}\}]$$\Rightarrow e^{\delta t}=E^{*}[\exp\{X_{t}I]$(6)$\Rightarrow \delta t=\ln M(1, t, h^{*})$$\Rightarrow \delta=\ln M(1,1, h^{*})$$\Rightarrow \delta=\lambda te^{\mu h^{*}}[\Gamma(1-\beta(1+h^{*}))\{e^{\mu}-\Gamma(1-\beta h^{*})\}]$(7)

This equation has apparently no closed form solution for . Gerber and Shiu (1994) [9] provedin their paper the existence and the uniqueness of the solution of our equation. We thus believe that
depending on the cases some numerical approaches can provide us a solution to this equation.
This
represent the parameter for which the Esscher transform is a martingale. Notice that theuniqueness of does not imply the uniqueness of the martingale measure. This is because the market is highly incomplete because of the jump process, thus, there exist other martingale measures that
can induce us to a fair price. We have to think of the uniqueness of as the existence of a unique Esscher risk neutral transform and that other pricing methods could lead to a different pricing formulas
compared to the one we are trying to derive.
Having our martingale measure we can compute our option price. For a call option on the index
we have we have:$h^{*}$$h^{*}$$h^{*}$$h^{*}$$C_{0}:= E^{*}[e^{-\delta t}(L_{t}-K)^{+}]=e^{-\delta t}L_{0}\int_{\psi}^{\infty}e^{X}f(x, t, h^{*})dx-\int_{\uparrow\psi}^{\infty}e^{-\delta t}Kf(x, t,h^{*})dx,$

where $K$ is the strike price, and V is issued from the argument:$L_{t}-K>0 \Leftrightarrow L_{0}\exp\{X_{t}\}-K>0 =X_{t}>\ln(\frac{L_{0}}{K}1=:\varphi.$

Accordingly, we have$C_{0}=L_{0}\{1-F(\uparrow\mu, t, h^{*}+1)\}-e^{-\delta t}K\{1-F(\gamma/, t, h^{*})\}$.

(8)This expression is familiar in the sense that it resembles the famous Black and Scholes option
pricing formula (Black and Scholes (1973) [2]).
$\lim_{\betaarrow 0+}\Gamma(1-\beta(z+h))\approx 1$, as same as
For the special case where
By then,$\betaarrow 0+,$$M(z, t, h)=\exp[\lambda te^{\mu h}\{e^{\mu z}-1\}]$$\lim_{\betaarrow 0+}\Gamma(1-\beta z)\approx 1.$

,

(9)

56
where thepart would be independent of . In this case, the Esscher transform has a Poisson, or a Poisson process that has an intensity equal to
distribution with an intensity. Then the Esscher $CDF$ is written as:
multiplied by a constant$\lambda te^{\mu h}$$z$$\Lambda=\lambda te^{\mu h}$$\Lambda=\lambda t$$e^{\mu h}$

,

(10)

$F( \uparrow\psi, t,h)=\frac{\Gamma(\lfloor\psi+1\rfloor,\lambda te^{\mu h})}{\lfloor\Psi\rfloor!}$whereis the upper incomplete Gamma function.
For the general case where $\beta>0$ , the Esscher transform PDF is defined as, denoting $f(x, t)$ the
PDF by:$\Gamma(a,b)$$X_{t}$’s(11)$f(x, t,h)= \frac{e^{hx}f(x,t)}{\int_{-\infty}^{\infty}e^{hx}f(y,t)dy}=\frac{e^{hx}f(x,t)}{M(h,t)}$Then the $CDF$ is:$F(x, t, h)= \frac{\int_{-\infty}^{x}e^{uz}f(u,t)du}{M(h,t)}$

.

(12)

Correction of the Claim AmountIn general, in the catastrophe insurance and for the earthquake insurance especially, it is very hard-if
not impossible-to assess the exact amount of the claim caused by a catastrophe on the date of theoccurrence. This amoum is continually revised with time. The first amount is most of the time only
an approximation of the real ultimate amount.
In contradiction with Schradin (1996) [12] who supposes a re-estimation taking place only after thedevelopment period, we follow Jaimungal and Wang (2006) [10], and Biagini, Bregman, and MeyerBrandis $(2008a, 2008b)[3,4]$ . Supposing that the re-estimation befall right after the catastrophe.
Unlike these authors who build their model as a convolution of a semi-martingale process andthe claim size process, we model the re-estimation as a process that do not happen only once per
catastrophe but we allow instead the claim to be re-corrected many times between two jumps.
Starting from this point the loss index is not a pure jump process but rather a jump diffusion
process. We then propose this model:;(13)$Z_{t}=X_{t}+( \mu_{c}-\frac{1}{2}\alpha_{c})t+\alpha_{c}W_{t}, t\geq 0$$L_{t}=L_{0}\exp\{Z_{t}\}, t\geq 0$

,

(14)

where $\{W_{t};t\geq 0\}$ is a,Brownian motion satisfying: $W_{t}\sim N(0, t)$ .
The correction is done through time with a mean and a standard deviation ofus compute the Moment Generating Function of the process for $z=1.$$\mu_{c}$$\alpha_{c}(\geq 0)$. First, let$X_{t}$.

$E[\exp\{X_{t}\}]=\exp\{\lambda t(\varphi_{Y}(1)-1)\} \Rightarrow \exp\{\lambda t\Gamma(1-\beta)e^{\mu}\}=M_{X_{t}}(1)$Proposition 1. llhe process$\{\frac{\exp\{X_{t}\}}{M_{X_{t}}(1)};t\geq 0\}$(15)

is a -martingale.
$\mathbb{P}$Proof. We obviously have to calculate the expectation$\frac{\exp\{X_{t}\}}{M_{X_{t}}(1)}$of our process

the filtration generated by the compound Poisson process up to time$s.$

$X_{t}$

conditionally on$\mathscr{F}_{s}$57
For $0\leq s<t$, we have:$E[\frac{\exp\{X_{t}\}}{M_{X_{t}}(1)}|\mathscr{F}_{s}]=E[\exp\{X_{t}-\ln M_{X_{t}}(1)\}|\mathscr{F}_{s}]$(16)$= E[\exp\{\sum_{i=1}^{N_{t}}Y_{i}-\ln M_{X_{t}}(1)\}|\mathscr{F}_{s}|$$E[\exp\{\sum_{i=1}^{N_{t}}Y_{i}-\ln M_{X_{t}}(1)\}|\mathscr{F}_{s}]$$= E[\exp\{\sum_{i=1}^{N_{t}}Y_{i}-\ln M_{X_{t}}(1)-\sum_{i=1}^{N_{s}}Y_{i}+\ln M_{X_{s}}(1)\}|\mathscr{F}_{s}|\cdot E[\exp\{\sum_{i=1}^{N_{s}}Y_{i}-\ln M_{X_{s}}(1)\}|\mathscr{F}_{s}]$$= E|\exp\{\sum_{i=N_{s}+1}^{N_{f}}Y_{i}-\ln M_{X_{t}}(1)+\ln M_{X_{S}}(1)\}|\mathscr{F}_{s}]\cdot\exp\{\sum_{i=1}^{N_{s}}Y_{i}-\ln M_{X_{S}}(1)\}$

,

(17)

(18)

where the deviation from Eq. (17) to Eq. (18) is because the last part of the expression Eq. (18) is-measurable.
The other part is:$\mathscr{F}_{s}$$E[\exp\{\sum_{t=N_{s}+1}^{N_{f}}Y_{i}-\ln M_{X_{t}}(1)+\ln M_{X_{s}}(1)\}|\mathscr{F}_{s}]=E[\exp\{\sum_{i=N_{s}+1}^{N_{t}}Y_{i}-\lambda(s-t)(\varphi_{Y}(1)-1)\}|\mathscr{F}_{s}|$(19)$= E[\exp\{\sum_{i=N_{s}+1}^{N_{t}}Y_{i}\}|\mathscr{F}_{s}|\exp\{-\lambda(s-t)(\varphi_{Y}(1)-1)\}$$\sum_{i=N_{s}+1}^{N_{t}}Y_{i}\}$is the sum of the jumps with size

$Y_{i}$

that occurs in the time interval $(s, t]$ , and because

the jumps are independent between the intervals and the only thing that matters is the length of theinterval we have that: $N_{t}-N_{s}\infty N_{t-s}$ . This leads to he fact that:$\sum_{t=N_{s}+1}^{N_{t}}Y_{i}=d\sum_{i=1}^{N_{t-s}}Y_{i}$, So:$E|\exp\{\sum_{i=N_{s}+1}^{N_{t}}Y_{i}\}|\mathscr{F}_{s}]\cdot\exp\{-\lambda(s-t)\{\varphi_{Y}(1)-1)\}$$= E[\exp\{\sum_{i=1}^{N_{t-s}}Y_{i}\}]\cdot\exp\{-\lambda(s-t)(\varphi_{Y}(1)-1)\}$$=M_{X_{t-s}}(1)\exp\{-\lambda(s-t)(\varphi_{Y}(1)-1)\}$$=\exp\{-\lambda(t-s)(\varphi_{Y}(1)-1)\}\cdot\exp\{-\lambda(s-t)(\varphi_{Y}(1)-1)\}=1.$

Finally,.

$E[\frac{e^{X_{t}}}{M_{X_{t}}(1)}|\mathscr{F}_{s}]=\exp\{\sum_{i=1}^{N_{s}}Y_{i}-\ln M_{X_{s}}(1)\}\cdot 1=\frac{e^{X_{s}}}{M_{X_{s}}(1)}, 0\leq s<t$(20)$\square$Under the risk neutral probability measure
lowing form:$\mathbb{Q}$the Brownian motion noted$W_{t}^{\mathbb{Q}}:=W_{t}+ \frac{(\mu_{c}-\delta)t-\ln M_{X_{t}}}{\alpha_{c}}, t\geq 0.$$\{W_{t}^{\mathbb{Q}};t\geq 0\}$has the fol-

58
It follows that the Radon-Nikodym derivative process is:$\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\ovalbox{\tt\small REJECT}_{t}}=\exp\{-\frac{1}{2}\int_{0}^{t}(\frac{\mu_{c}-\delta+\omega}{\alpha_{c}})^{2}ds-\int_{0}^{t}(\frac{\mu_{c}-\delta+\omega}{\alpha_{c}})dW_{s}\}, t\geq 0$

,

(21)

where
$\omega;=\lambda\{\Gamma(1-\beta)e^{\mu}-1\}.$

Theorem3.1. The price $ofa$ catastrophe call on the loss index at $t_{i}met=0$ is given by:$\sum_{k=0}^{\infty}E^{\mathbb{Q}}|\kappa(t,L_{0}\exp\{\ln M_{X_{t}}(1)\}\exp\{\sum_{i=0}^{k}Y_{i}\})|e^{-\lambda}\frac{(\lambda t)^{k}}{k!}$

,

(22)

where
$k(t,x)=xN(d_{+}(t,x))-Ke^{-\delta t}N(d_{-}(t,x))$$d_{\pm}(x, t)= \frac{1}{\alpha_{c}\sqrt{t}}[\ln(\frac{x}{K})+(\delta\pm\frac{\alpha^{2}}{2})t|$

Proof. Recall that, from Eqs. (13), (14), we have:$L_{t}=L_{0} \exp\{(\mu_{c}-\frac{1}{2}\alpha_{c}^{2})t+\alpha_{c}W_{t}\}\exp\{X_{t}\},$

and$C_{t}= E|E^{Q}\Vert L_{0}\exp\{(\mu_{c}-\frac{1}{2}\alpha_{c}^{2})t\}\exp\{\sum_{j=1}^{N_{t}}Y_{\dot{j}}\}-K)^{+}|\sigma(\exp\{\sum_{j=1}^{N_{t}}Y_{j}\}\Vert]$

In one hand, since the jump size and the diffusion process are supposed to be independent of eachother, we could separate the compound Poisson process exponent from the geometric Brownian motion to get two expectations. In the other hand, inside the second expectation (the risk neutral one)4is$\sigma(\exp\{\sum_{j=1}^{N_{t}}Y_{j}\})$-measurable. Hence the quantity in the first expectation is the Black and Scholes

formula for a call price (Black and Scholes (1973) [2]). Calculating the first expectation as usual for a
Poisson process, we got our final pricing formula.
$\square$This pricing formula needs to be calculated via simulation.

4 An Esscher Transform Pricing for the Claims with Correction
We proceed as same as with the firs case without continuous corrections.
$M_{Z}(z, t)=E[e^{zZ_{t}}]$$= E|e^{zX}te^{z\alpha_{c}W}texp\{z(\mu_{c}-\frac{1}{2}\alpha_{c}^{2})t\}|$$=M_{X}(z, t)M_{W}(z \alpha_{C/}t)\exp\{z(\mu_{c}-\frac{1}{2}\alpha_{c}^{2})t\}$$= \exp\{\lambda t\{\Gamma(1-\beta z)e^{\mu z}-1\}+\mu_{c}tz+\frac{1}{2}\sigma^{2}z^{2}\}$as-measure$\overline{4}$We used the
Merton (1976) [11]$\mathbb{P}$expectation here

the intensity does not change in the risk neutral world according to

59
Thus:
$M(z, t, h)= \exp\{\lambda te^{\mu h}\{\Gamma(1-\beta(z+h))e^{\mu z}-\Gamma(1-\beta h)\}+(\alpha_{c}h-\mu_{c})tz+\frac{1}{2}\alpha_{c}^{2}t^{2}z^{2}\},$

we then find that:$M(1,1,h^{*})= \delta=\exp\{\lambda e^{\mu h^{*}}\{\Gamma(1-\beta(1+h^{*}))e^{\mu}-r(1-\beta h^{*})\}+(\alpha_{c}h^{*}-\mu_{c})+\frac{1}{2}\alpha_{c}^{2}\}$extracting$h^{*}$

and replacing it in the formula will give us our option price.

5 Pricing in the Development Period
During the development period we assist only to the correction of the claims. The index on which
our option is based, by convention, do not take into consideration the new jumps even if a catastrophe may happen during this period. Thus, it tums out that the loss index become a pure geometric
Brownian motion during the development interval. We can then, rewrite our loss index ass follow:.

For the simple case (without corrections in the loss period):$L_{t}=L_{0}e^{X_{t}^{*}}, t\geq 0$

,

(23)

.

$X_{t}^{*}=1_{\{t\leq T_{1}\}} \sum_{i=1}^{N_{t}}Y_{i}+0_{\{t>T_{1}\}}(\mu_{c}-\frac{1}{2}\alpha_{c}^{2})t+\alpha_{c}W_{t}, t\geq 0$(24)

For the case with corrections:$Z_{t}^{*}= T_{\{t\leq T_{1}\}}\sum_{i=1}^{N_{t}}Y_{i}+(\mu_{c}-\frac{1}{2}\alpha_{c}^{2})t+\alpha_{c}W_{t}, t\geq 0$

.

(25)

Observe that, in this case, the Brownian motion is present with a probability equal to 1 a.s.
The pricing of the option during this period leads to the same results for the two models, since,
during this interval, these two models are equal and are represented by a geometric Brownian motion
only.

Pricing Formula for the Development Period
Using the Esscher Transform Method,$\mathcal{W}e$have the following result:$M(z, t,h)= \exp\{(\mu_{c}+h\alpha^{2}z+\frac{1}{2}\alpha_{c}^{2}z^{2})t\}, t\geq 0$

The transformed process is then normally distributed and our new$h^{*}$.

(26)and call option price are:$\delta-\alpha_{c}^{2}-\mu_{c}\underline{1}$$h^{*}= \frac{2}{\alpha_{c}^{2}}$;$C_{0}=L_{t_{1}} \varphi(\frac{-\varphi+(\delta+\frac{1}{2}\alpha_{c}^{2})\tau}{\alpha_{c}\sqrt{\tau}})-e^{-\delta\tau}K\varphi(\frac{-\uparrow l’+(\delta-\frac{1}{2}\alpha^{2})\tau}{\alpha\sqrt{\tau}}1\prime$

(27)

(28)

60
where $\tau:=t-T_{1}$ for $t>T_{1}.$This pricing formula is nothing else then the Black and Scholes formula for a call option (Blackand Scholes (1973) [2] . Obviously, using the martingale approach will lead to the sane result5.
$)$

References[1] Alexander, M., Actuarially Consistent Valuationof Catastrophe Derivati ves, Working Paper, 03-18,

The Wharton Financial Institutions Center, University of Pennsylvania, July 2003.
[2] Black, F and Scholes, M., The Pricing ofOption and Corporate Liabilities, Journalomy, Volume 81, pp. 637-659, 1973.

ofPolitical Econ-[3]

Biagini, E, Bregman, Y., and Meyer-Brandis, T., Pricing of Catastrophe Insurance Options Written
on a Loss Index with Reestimation, Insurance; Mathematics and Economics, Volume 43, Issue 2,
pp. 214-222, 2008a.

[4]

Biagini, E, Bregman, Y., and Meyer-Brandis, T., Pricing of Catastrophe Insurance Options under
Immediate Loss Reestimation, Journal ofApplied Probability, Volume 45, Number 3, pp. 831-845,
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5 This is because in the development period we have only one source of randomness issued from the Brownian motion
and one asset. Hence according to the meta theorem the market is complete and thus the martingale measure is unique