We investigate the valuation and hedging of catastrophe options, whose claim arrival process is modeled by the Cox process or a doubly stochastic Poisson process. Employing the non-arbitrage principle we obtain closed form formula for the pricing of the option. Various hedging parameters are also computed.catastrophe options, Cox process, pricing

Fujita, Takahiko

Ishimura, Naoyuki

Tanaka, Daichi

English

Research Papers in Economics

Hitotsubashi University Repository
Title
An Arbitrage Approach to the Pricing of Catastrophe
Options Involving the Cox Process
Author(s) Fujita, Takahiko; Ishimura, Naoyuki; Tanaka, Daichi
Citation Hitotsubashi Journal of Economics, 49(2): 67-74
Issue Date 2008-12
Type Departmental Bulletin Paper
Text Version publisher
URL http://hdl.handle.net/10086/16521
RightAN ARBITRAGE APPROACH TO THE PRICING OF
CATASTROPHE OPTIONS INVOLVING THE COX PROCESS
＊
TAKAHIKO FUJITA
Graduate School of Commerce and Managements, Hitotsubashi University
Kunitachi, Tokyo 186-8601, Japan
fujita@math.hit-u.ac.jp
NAOYUKI ISHIMURA
Graduate School of Economics, Hitotsubashi University
Kunitachi, Tokyo 186-8601, Japan
ishimura@econ.hit-u.ac.jp
DAICHI TANAKA
Sompo Japan Insurance Inc.
Shinjuku, Tokyo 160-8338, Japan
js-freek_dt93@mug.biglobe.ne.jp
Received February 2008; Accepted May 2008
Abstract
We investigate the valuation and hedging of catastrophe options, whose claim arrival
process is modeled by the Cox process or a doubly stochastic Poisson process. Employing the
non-arbitrage principle we obtain closed form formula for the pricing of the option. Various
hedging parameters are also computed.
Keywords: catastrophe options, Cox process, pricing
JEL Classiﬁcation: G13, C02, G22
Hitotsubashi Journal of Economics 49 (2008), pp.67-74. Ⓒ Hitotsubashi University
＊ We are grateful to Professor Toshio Honda for his valuable suggestions, which causes us to deepen the argument
of our starting attempt of Tanaka (2008). Thanks are also due to the referee for insightful comments, which helps in
improving the manuscript. The project of this research is motivated in part by the program ʻFinancial Engineering
Education for Graduate Course Students in Social Sciences (program leader: Professor Tsunemasa Shiba),ʼ from the
Ministry of Education, Science, Sports and Culture, Japan. The work of the second author (NI) is also partially
supported by the Inamori Foundation through the Inamori Grants for the ﬁscal year 2006-2007.I Introduction
The high level of catastrophic events, such as major typhoons and/or disastrous
earthquakes etc., have had a heavy impact on the reinsurance market. These events are usually
infrequent, hard to predict reliably, and diﬃcult to eﬀectively prevent. In order for insurance
companies to hedge themselves against the resulting risks and also to stabilize insurance
markets, from the 1990s, catastrophe futures and options written on these futures have been
introduced into trading as alternative reinsurance products [e.g., Cummins et al. (2004), Froot
(2001), Jaﬀee and Russell (1997)] . As the ﬁrst successful example of a catastrophe bond
underwritten by a non-ﬁnancial ﬁrm, we recall the contract issued in 1999 by Oriental Land
Company Ltd., which is intended to cover the losses of Tokyo Disneyland caused by a possible
earthquake in the Tokyo area. In recent years, these types of catastrophe (CAT) ﬁnancial
products have established themselves as a special ﬁnancial category.
One reason to consider such products in the markets is that although the insured
catastrophe risks can be very large, its relative size to other risks may be comparable. For
instance, the total loss of the Hanshin-Awaji earthquake having occured in 1994 is estimated to
be 10 trillion yen (Disaster Reduction and Human Renovation Institution (http: //
www.dri.ne.jp)); which is roughly of the same order of a 2 percent change in the value of the
Tokyo Stock Exchange, which is not such an infrequent event.
Typical examples of CAT contracts include the CatEPut, which is the ﬁrst catastrophe
equity put option and a registered trademark of Aon Limited. The payoﬀ function of the
CatEPut at maturity T with exercise value of K has the form
Payoff / 1{XT > L}max{K,ST,0 }/ 
K,ST if ST ? K and XT > L
0i f ST B K or XT C L，
(1)
where S / St denotes the market price and X / Xt the stochastic event. The parameter L is
the trigger level. Here 1{.} stands for the indicator function; namely, 1{XT > L} / 1 if XT > L
and / 0 if XT C L . Thus the option can be exercised, in other words, becomes in-the-
money, only when claims XT exceed L; the owner of the option can hedge against catastrophic
losses exceeding the existing coverage limit during the life of the option. We note that the
CatEPut, in this sense, is classiﬁed among the so-called double trigger options.
The pricing of these structured risk management products should be concerned with
incomplete markets [Vaugirard (2003)]. As to the CatEPut, Cox et al. (2004) made the ﬁrst step
via a non-arbitrage approach when the loss process X follows a Poisson process and obtained a
closed form solution. Later Jaimungal and Wang (2006) extended the results of Cox et al.
(2004) so that X is assumed to follow a compound Poisson process and the risk-free interest
rate is governed by a mean-reverting stochastic process. Furthermore the hedging parameters
Delta, Gamma and Rho are also calculated.
In this note we deal with the pricing of a double trigger CAT option when the claim
arrival process X follows the Cox process or a doubly stochastic Poisson process. Since the Cox
process allows the intensity to be stochastic, it is suitable for the modeling of catastrophic
events. Indeed Dassios and Jang (2003) employ the Cox process with shot noise intensity (see
(2) in §2 below) on this ground, and establish a pricing formulae for a stop-loss reinsurance
HITOTSUBASHI JOURNAL OF ECONOMICS [December 68contract for catastrophic events and catastrophe insurance derivatives. Their technique is based
on utilizing an equivalent martingale probability measure; however, since the shot noise process
possesses many parameters, the calculation becomes rather cumbersome. We additionally
remark that the model of Dassios and Jang (2003) does not contain the market price process.
Here, on the other hand, we adopt an elementary method for both the modeling as well as the
computation and derive a closed form pricing formula by directly clarifying the equivalent
martingale measure. Several generalizations will be indicated together.
Concerning the valuation of other products in this family, we recall, for instance, that
catastrophe insurance futures and call spreads are treated in Cummins and Geman (1995); a
ﬂoating retention insurance is also discussed in Cox et al. (2004), whereas the pricing of a
reinsurance contract with catastrophe bonds is developed in Lee and Yu (2007). We also refer
to the references cited therein.
The paper is organized as follows. In §2 we recall the basic properties of the Cox process
and present our basic model. Based on the non-arbitrage technique, we exhibit a pricing
formula in §3. §4 is devoted to calculating various hedging parameters, namely, Greeks.
Generalizations and discussions of our ﬁndings are examined a little further in §5.
II Preliminary
Firstly, we recall the deﬁnition and properties of our basic process; namely, the so-called
Cox process, or a doubly stochastic Poisson process [e.g., Lando (1998)] . As a general
reference we refer to a comprehensive monograph of Rolski et al. (1998).
It is well observed that the Poisson process has been used as a claim arrival process in
insurance modeling. However, it is also recognized, at the same time, that the homogeneous
Poisson process is inappropriate for modeling the resulting claims for catastrophic events,
because the process has deterministic intensity. Cox processes or doubly stochastic Poisson
processes, on the other hand, provide ﬂexibility in modeling, since the intensity process is
allowed to be stochastic.
Let  : / TλtUt B 0 be an intensity process; namely, a nonnegative, measurable, and
locally integrable stochastic process. A counting process TNPt;QUt B 0 is called a Cox process
or a doubly stochastic Poisson process with intensity  if for each sequence {/i}i = 1,2, …, n of
nonnegative integers, and for 0＜t1 C s1 C t2 C s2 C … C tn-1 C sn-1 C tn C sn, there
holds
Pr 3
n
i =1
 Npsi;"Npti; / /i   = 7
n
i = 1
E 
1
/ir@
si
ti
ludu 
 i
expr"@
si
ti
ludu  .
One typical example, which is favorably used to measure the eﬀect of catastrophic events,
is the shot noise process [e.g., Cox and Isham (1986), Dassios and Jang (2003), Kluppelberg
and Mikosch (1995)]. Suppose 0 ? s1 ? s2 ? … are the points of a Poisson process with the
intensity ρ and {yi}i = 1,2,… are independently and identically distributed nonnegative random
variables with E(yi) ?  , then
lt / l0e
"dt+ 6
i：si C t
yie
"dPt"siQ, (2)
AN ARBITRAGE APPROACH TO THE PRICING OF CATASTROPHE OPTIONS INVOLVING THE COX PROCESS 2008] 69where l0 is the initial value and d denote the rate of exponential decay. In Dassios and Jang
(2003) it is generalized that parameters ρ , d are allowed to be time-dependent. We clearly
imagine that si corresponds to the time at which catastrophe i occurs and yi supplies the jump
size of the catastrophe i.
Now we specify our model, which is represented by the next system.
St / S0 exp ,aN(t;)+rm,
1
2s
2 t+sWt  (3)
Bt / e
rt,
where Wt(t B 0) denotes the standard Brownian motion which is independent of the Cox
process N(t;),a n da, m, s are assigned positive constants. The interest rate r is free of risk
and is kept constant. It is easy to recognize that if a large claim occurs at time t, then the price
changes abruptly from St_ to St % e
"a(N(t;)"N(t_;))St_. If no large claims are made during
the interval, then St follows a typical geometric Brownian motion over the same interval, with
the driftμand the volatilityσ.
Under the process (3) we intend to obtain a pricing of the double trigger catastrophe
options where Xt % N(t;) . We note once again that our formulation deals with an
incomplete market; several choices of equivalent martingale measure will exist for the
valuation. Our current business does not involve deciding what is the suitable one to use; we
merely proceed as simply as possible.
III Pricing Formula
Let C(t) denote the value of the option at time t ( ? T ) whose payoﬀ function at the
maturity T is prescribed by
C(T) / 1{XT > L}max{ST,K,0}. (4)
We assume that Xt % N(t;) with X0 % 0 and the stock price St is described by (3). In this
section we want to estimate an arbitrage-free value of C(t). It is easy to see that the price P(t)
of the CatEPut (1) under the same Xt is proceeded similarly.
We begin by showing the next lemma, which is a straight extension of Lemma 1 in Cox et
al. (2004) and gives a key tool to the valuation.
Lemma 3.1 Let{N(t;)}t B 0 be a Cox process with intensity  / {lt}t B 0. Then
exp{,aN(t;)+log(Mt(/))}t B 0
is a martingale, where /: / 1,e
-a and Mt(}) denotes the moment generating function of
the aggregated process t: / 4
t
0lsds.
Proof.F i r s t l y ,w el e a r nt h a t
E[e
-aN(t;)] / E[E[e
-aN(t;)｜ 

t]] / E[ 6
∞
l = 0
e
-alP(N(t;) / l))]
HITOTSUBASHI JOURNAL OF ECONOMICS [December 70/ E  6
 
l = 0
e
-al
l
t
l e
-t  / E[exp{t(e
-a,1)}],
where  

t: % s{λs｜0 C s C t}.
We compute for 0 C s C t C T
E[exp{"aN(t;)log(Mt(/))}｜ 

s]
% E[exp{"a(N(t;)"N(s;))log(Mt(/))"log(Ms(/))}｜ 

s]}
}E[exp{"aN(s;)log(Ms(/))}｜ 

s]
% E[exp{"a(N(t;)"N(s;))log(M(t"s)(/))}｜ 

s]}
}exp{"aN(s;)log(Ms(/))}
% E[exp{"a(N(t;)"N(s;))｜ 

s]M(t"s)(/)}
}exp{"aN(s;)log(Ms(/))}
% M(t"s)("/)M(t"s)(/)exp{"aN(s;)log(Ms(/))}
% exp{"aN(s;)log(Ms(/))},
which completes the proof of Lemma.
For the expression of C(t) we understand that it is the expected discounted value under the
risk-neutral measure Q. Precisely stated, the process {e
− rtSt}t B 0 is a Q-martingale and we have
C(t) % E
Q
t[e
"r(T"t)1{N(T;)＞L}max{ST"K, 0}].
To obtain the measure Q we apply the change of measure. Indicating the original variable by P
like W
P
t: % Wt,w ed e ﬁne
W
Q
t : / W
P
t +
(μ "r)tlog(Mt(/))
s ,
and the Radon-Nikodym derivative process as follows.
dQ
dP % exp "
1
2@
t
0 r
μ "r (s)
s  
2
ds"@
t
0
μ "r (s)
s dW
P
s ,
w h e r ew eh a v ep u tg (t):% E[t exp{/t}]/M(/). Taking Lemma 3.1 into account and
observing that
e
"rtSt % S0 exp{"aN(t;)log(Mt (k))}exp sW
Q
t "
1
2s
2t ,
we conclude that {e
"rtSt} is a Q-martingale. To be explicit, we have for 0 C s C t
E
Q
s[e
"rtSt] % e
"rsSs.
Now we deduce the closed form pricing formula.
AN ARBITRAGE APPROACH TO THE PRICING OF CATASTROPHE OPTIONS INVOLVING THE COX PROCESS 2008] 71Theorem 3.2 The price C (t) of the double trigger option whose payoﬀ function at the
maturity T is (4) is given by
C(t) % 6
∞
l % L1"N(t;)
(Ste
"allog(M(T"t)( ))Φ(dlσ  T"t)"Ke
"r(T"t)Φ(dl))}
}E 
(T"t)
l
l e
"(T"t)］ 
where / : % 1"e
"a and
dl : %
log(St/K)r(T"t)"allog(M(T"t)(/))
s T"t
"
1
2s T,t. (5)
Here Φ(x) denotes the cumulative distribution function for the standardized normal distribution
Φ(2) %
1
 2π@
 
"∞e
"
y2
2 dy % :@
 
"∞j(y)dy.
It is easy to see that the quantity Ste
-al+log(M(T-t)( ))Φ(dl+s T,t),K}
}e
-r(T-t)Φ(dl) represents the Black-Scholes pricing formula of vanilla call option whose
underlying process is given by {Ste
-al+log(M(T-t)(/))} and the exercise value K; thus
our formula may be understood as an extension of that obtained by Merton (1976, (16)).
Proof. We just calculate the relevant quantity, which is carried out as follows:
We write
ST % e
r(T"t)Stexp{"a(N(T;)"N(t;))log(M(T"t)(/))}}
}exp s(W
Q
T"W
Q
t)"
1
2s
2(T"t) .
Taking account of the Black-Scholes formula and calculating conditionally on the number of
claims N(T；), we infer that
C(t) % E[1{N(T;)＞L}(Ste
"a(N(T;)"N(t;))log(M(T"t)( ))・
}Φ(dN(T;)"N(t;)s T"t)"Ke
"r(T"t)Φ(dN(T;)"N(t;)))]
% 6
∞
l % L1"N(t;)
(Ste
"allog(M(T"t)( ))Φ(dls T"t)"Ke
"r(T"t)Φ(dl))}
}E 
(T"t)
l
l e
"(T"t) .
This ﬁnishes the proof.
IV Hedging Parameters
Here we consider the hedging strategy. In particular we compute various Greeks. The
calculation is rather straightforward. We summarize them as the next proposition, whose proof
will be safely omitted.
HITOTSUBASHI JOURNAL OF ECONOMICS [December 72Proposition 4.1 Let C(t) denote the price of the double trigger option whose payoﬀ function
at the maturity T is given by (4). Subsequently, the Delta, Gamma, Rho, and Vega of C(t) are
computed as the next expressions, respectively.
Δ(t) %
 C(t)
 S % 6
∞
l % L1"N(t;)
e
"allog(M(T"t)( ))Φ(dls T"t)}
}E 
(T"t)
l
l e
"(T"t) 
Γ(t) %
 
2C(t)
 S
2 % 6
∞
l % L1"N(t;)
e
"allog(M(T"t)( ))
Ss T"t
j(dls T"t)}
}E 
(T"t)
l
l e
"(T"t) 
ρ (t) %
 C(t)
 r % 6
∞
l % L1"N(t;)
K(T"t)Φ(dl)}E 
(T"t)
l
l e
"(T"t) 
n(t) %
 C(t)
 s % 6
∞
l % L1"N(t;)
S T− tj(dls T"t)}E 
(T"t)
l
l e
"(T"t) ,
where dl is deﬁned by (5) and j(x) / Φ (x).
V Discussions
We have established a pricing formula of double trigger catastrophe options through the
non-arbitrage principle. Catastrophe ﬁnancial products have been introduced, partly due to the
need to provide alternative hedging tools and thus stabilize insurance markets. For the relevant
claim arrival process, we employ the Cox process or a doubly stochastic Poisson process, which
is customarily used as a model of catastrophic events. The nature of the problem means we are
forced to treat the incomplete market. Invoking the change of measure we evaluate the non-
arbitrage price of options, while hedging parameters are also computed.
Slight extensions of our present results are possible; for example, the risk-free interest rate
r is permitted to be stochastic, say, a mean-reverting process. Confer the related work of
Jaimungal and Wang (2006). We may also specialize the Cox process as, say, the shot noise
process (2), although the parameters would be involved. Observe the elaborate study of Dassios
and Jang (2003). These are accomplished rather directly.
However, several challenging issues remain to be further pursued. For example, there may
be diﬀerent means availbale to handle the incompleteness [e.g., Bjork (2004), Fujita (2002)].
One promising trick in economics theory involves appealing to the optimality of a certain
utility function. It is interesting to ﬁnd the appropriate rule to select the relevant type of utility
functions. In any case we wish to know or ensure how the benchmark price should be. Next,
but no less important, a relatively signiﬁcant research theme involves developing a fast and
accurate algorithm to numerically compute the catastrophe insurance products. Although certain
methods have been introduced, considerable room remains for innovation or improvement. We
AN ARBITRAGE APPROACH TO THE PRICING OF CATASTROPHE OPTIONS INVOLVING THE COX PROCESS 2008] 73will return to these topics in our future investigation.
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An Arbitrage Approach to the Pricing of Catastrophe Options Involving the Cox Process

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An Arbitrage Approach to the Pricing of Catastrophe Options Involving the Cox Process

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