Pricing of Catastrophe Insurance Options under Immediate Loss Reestimation
Author(s): Francesca Biagini, Yuliya Bregman and Thilo Meyer-Brandis
Source: Journal of Applied Probability, Vol. 45, No. 3 (Sep., 2008), pp. 831-845
Published by: Applied Probability Trust
Stable URL: http://www.jstor.org/stable/27595989 .
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J.Appl. Prob. 45, 831-845 (2008)
Printed
in England
? Applied ProbabilityTrust2008
INSURANCE
PRICING OF CATASTROPHE
OPTIONS UNDER IMMEDIATE
LOSS REESTIMATION
* **and
FRANCESCA BIAGINI
YULIYA
***
BREGMAN,*
THILO
Universit?t
Ludwig-Maximilians
MEYER-BRANDIS,****
University
M?nchen
of Oslo
Abstract
the initial estimate of each
loss index, where
for a catastrophe
specify a model
loss is reestimated
starting from the
immediately
by a positive martingale
catastrophe
insurance options
random time of loss occurrence. We consider the pricing of catastrophe
written on the loss index and obtain option pricing formulae by applying Fourier transform
works for loss distributions
is that our methodology
techniques. An important advantage
We
with heavy tails, which
discuss
the case when
and provide
Keywords:
formula;
We also
tail behavior for catastrophe modeling.
factors are given by positive affine martingales
of positive affine local martingales.
is the appropriate
the reestimation
a characterization
Catastrophe
insurance
tail; positive
heavy
loss index; Fourier
option;
affine martingale
transform;
option
pricing
2000Mathematics Subject Classification:Primary46N30; 60G07; 91B30
1. Introduction
Over thepast decades the rise in insured losses has exploded from2.5 billion dollars per year
to an average value of the aggregated insurance losses of 30.4 billion dollars per year (2006
prices); see [20] formore details. Table 1 gives a summary of the 10most expensive natural
catastrophes for the last 30 years.
In order
of the vast
options.
to securitize
potential
the catastrophe
of capital markets
Exchange-traded
for example,
reinsurance,
insurance
they offer
risk,
by
insurance
introducing
instruments
low
companies
several
present
transaction
have
exchange-traded
costs,
because
tried to take advantage
insurance
catastrophe
advantages
are
they
with
respect
standardized,
to
and
includeminimal credit risk, because the obligations are guaranteed by the exchange. See [18]
and
[19]
for
the comparison
of
insurance
securities.
In particular,
catastrophe
options
are
standardized contracts based on an index of catastrophe losses, for example, compiled by the
Property Claim Service (PCS), an internationally recognized market authority on property
losses from catastrophes in theUS.
The first index-based catastrophe derivatives were introduced at theChicago Board of Trade
in 1992, but therewas only littleactivity in themarket. A second version of the index, compiled
by the PCS, was introduced in 1995. At the peak, the total capacity created by this version
Received 28 February 2008; revision received 18 July 2008.
*
Theresienstrasse
Universit?t M?nchen,
Postal address: Department of Mathematics,
39,
Ludwig-Maximilians
D-80333 Munich, Germany.
**
Email address: [email protected]
***
Email address: [email protected]
****
Postal address: CMA, University of Oslo, Postbox 1035, Blindem, Norway. Email address: [email protected]
831
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F. BIAGINI ETAL.
832
10 insured catastrophe
1: Top
Table
Insured
losses
(source:
Swiss
Re,
sigma Nr. 2/2007).
loss
(xlO9 dollars)
2005
66.3
Event
Year
Hurricane
Country
US
Katrina:
Gulf ofMexico
floods
dams
1992
23.0
2001
21.4
Bahamas
burst
to oil rigs
damage
Hurricane Andrew:
North Atlantic
US
Bahamas
flooding
Terrorist
attack on World
Trade
US
Center,
Pentagon,
and other buildings
13.7
1994
2004
Northridge
Hurricane
earthquake
Ivan:
US
13.0
2005
to oil rigs
damage
Hurricane Wilma:
US
19.0
US
Caribbean
torrential rain
Mexico
floods
2005
10.4
Hurricane
Jamaica
Haiti
US
Gulf ofMexico
Rita:
floods
damage
to oil rigs
8.6
2004
HurricaneCharley
8.4
1991
1989
TyphoonMireille
Cuba
US
Caribbean
7.4
Hurricane
Japan
US
Hugo
Puerto Rico
of the PCS options amounted to 89 million. Trading in these options slowed down in 1999.
In a separate initiative, theBermuda Commodities Exchange (BCE) was launched in 1997 to
tradeproperty catastrophe-linked option contracts. The BCE suspended itsoperations in 1997.
Trading in the PCS options slowed down in 1999, because of the lack ofmarket liquidity and
of qualified personnel (see, e.g. [18]).
However, the record losses caused by hurricanes Katrina, Rita, andWilma
a
in creating
catalyst
new
derivative
instruments
to trade
kets. InMarch 2007, theNew York Mercantile Exchange
futures
and
options
again.
These
new
contracts
were
risks
the catastrophe
(NYMEX)
designed
in2005 have been
in the capital
mar
began trading catastrophe
to bring
the transparency
and
liquidity of the capital markets to the insurance sector, providing effectiveways of protecting
against property catastrophe risk and providing the investorswith the opportunity to trade a
new
asset
class
which
has
little or no
correlation
to other
exchange-traded
asset
classes.
The
NYMEX
catastrophe options are settled against theRe-Ex loss index, which is created from
the data supplied by thePCS.
Following the descriptions in [12], [17], and [18], the structureof catastrophe insurance
options is described as follows. The options are written on a loss index that evolves over two
periods: the loss period [0, T\] and the consecutive development period [T\,T2]. During the
contract
occur. However,
loss period,
the index measures
events
that may
specific
catastrophic
at the time of
are
the
losses
estimates
of
the true losses,
occurrence,
catastrophe
reported
only
and these estimates are consecutively reestimated until the end, T2, of the development period.
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Pricing
insurance
of catastrophe
833
options
The loss index thus provides, at any t e [0, T2], an estimation of the accumulation of the final
time (T2) amounts of catastrophe losses thathave occurred during the loss period. Let Nt, t e
denote
[0, T\],
of catastrophes
the number
to time
up
let C//, /=
t, and
1, ...,
denote
Nt,
<
corresponding final amounts of losses at time T2 (which are unknown at time 0
Then the value Lt of the loss index can be expressed as
the
t < T2).
NtATx
Lt=
E[Ui
]P
t e[0,T2],
Ift],
(1.1)
/=i
where the filtration {!Ft, t e
compound
a Poisson
sum with martingales
Poisson
If the number Nt
[0, T2]} represents the information available.
to follow
is assumed
of catastrophes
as
the structure
process,
of
index
the
is thus
a
summands.
In the literature,a fewmodels have been proposed in order tomodel the catastrophe loss
index and to price catastrophe options written on this index. In [12], [13], and [14] theunder
was
index
catastrophe
lying
as
represented
a
Poisson
compound
with
process
nonnegative
jumps. In thismodel, reestimation is not taken into account at all. In [2] and [11] the authors
distinguished between a loss and a reestimation period andmodeled the index as an exponential
over each
is assumed
reestimation
However,
process
period.
Levy
common
is not realistic
because
reestimation
factor. This assumption
at T\ by a
to start exactly
loss
reestimation
happens
individually for each catastrophe and begins almost immediately after the catastrophic event.
In addition, the assumption of an exponential model for accumulated losses during the loss
is rather
period
than earlier
For
unrealistic.
and
catastrophes,
are more
that later catastrophes
implies
starts at a positive
value
(instead
this
example,
that the index
severe
at 0).
of starting
In [1] a more realisticmodel for the loss indexwas proposed and analytical catastrophe option
were
formulae
pricing
but
developed,
was
reestimation
also
done
by
a common
over
factor
the development period only. In [16] a model including immediate reestimation was assumed,
where the reestimationwas modeled through individual reestimation factorsgiven by geometric
Brownian
motion.
no efficient
However,
were
methods
pricing
obtained.
In this paper we specify a realistic model for the loss index that is consistent with (1.1).
a
case,
particular
occurrence
is modeled
As
it comprises
the model
a
Poisson
process,
by
assume
in [16]. We
that catastrophe
proposed
for each
and consider
reestimation
individual
catastrophe where the initial estimate of the zthcatastrophe loss is reestimated immediately by
a
martingale
positive
from
starting
the random
time of loss
occurrence.
then consider
We
the
pricing of catastrophe options written on the index. The main contribution of this paper is to
Fourier
employ
of an
transform
to reduce
manage
of
expectation
random
independent
in order
techniques
to obtain
option
the characteristic
variables.
We
of
function
mention,
pricing
function
of the characteristic
the calculation
the reestimation
in particular,
To
formulae.
this end, we
to the computation
of the index
factor,
in two
evaluated
that our methodology
also
works
for loss distributions with heavy tails, which is the appropriate tail behavior for catastrophe
modeling.
positive
We
then proceed
affine martingales.
to discuss
In this
the case
situation
we
when
provide
the reestimation
factors
a characterization
are
given
of positive
by
affine
(local) martingales.
We believe that the use of exchange-traded insurance derivatives will play a crucial role in
the securitization
of the increasing
catastrophe
risk
in the future.
For
this purpose,
one
essential
task is to develop quantitative tools thathelp to establish liquid trading of these instruments.
We hope that this paper contributes to this aim in that it sets new insights in the pricing of
catastrophe
options.
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F. BIAGINI ETAL.
834
The remaining parts of thepaper are organized as follows. In Section 2 we present themodel
for the loss index. In Section 3 we consider thepricing ofEuropean options in thegeneral model
by using Fourier transformtechniques, and in Section 4 we discuss the special case of positive
affinemartingales as reestimation factors. We conclude with Section 5, where we explicitly
compute prices for spread options, which are the typical instruments in themarket.
2. Modeling
the loss index
Let (Q, 7, P) be a complete probability space. We consider a financialmarket endowed with
a risk-freeasset with deterministic interest rate rt, and the possibility of trading catastrophe
insurance options, written on a loss index. Following the description in the introduction,we
distinguish two time periods:
a loss period [0, T\\\
a
development
T\ <
[T\,T2l,
period
During the contract specific loss period
period,
can
users
option
choose
either
T2 <
oo.
[0, T\], the catastrophic events occur. After the loss
a
or
six month
a
twelve
month
development
period
[T\,T2], where the reestimates of catastrophe losses that occurred during the loss period
continue
to affect
the index.
The
contract
option
matures
at the end
of the chosen
development
period T2.
Motivated by the index structure(1.1) discussed in the introduction,we model the stochastic
=
process L
(Lt)o<t<T2i representing the loss index as follows:
Lt=J2 =
7
1
te[0,T2],
YJALr
where the following notation and hypotheses have been used.
(HI) Ns,
s e [0, T2], is a Poisson process with intensityX > 0 and jump times
ry thatmodels
the number
of catastrophes
occurring
during
the loss period.
(H2) Yj, j = 1, 2, ..., are positive independent and identically distributed (i.i.d.) random
variables with distribution functionFY that represent thefirst loss estimation at the time
occurs.
the y'th catastrophe
(H3)
Ai,
s e
[0, T2],
and
j
=
AJ0
(H4) AJ, Yj, j =
are
1, 2,...,
=
1
i.i.d. martingales
positive
for all
/'=
such
that
1,2,....
1, 2, ..., and N are independent.
In the sequel we will drop the index j and simplywrite Y and A in some formulae, when
only
theprobability distribution of the objects matters.
The
martingales
AJt represent
the unbiased
reestimation
factors.
Here
we
assume
that
reestimation begins immediately after the occurrence of the 7thcatastrophe with initial loss
estimate Yj, individually for each catastrophe.
Here we suppose thatmarket participants observe the evolution of the individual
catastrophe
losses. Note that observing themarket quotes of the catastrophe index L alone is in
general
not sufficientfor theknowledge of the single reestimation factors,A. However, it
might not be
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Pricing
of catastrophe
insurance
to assume
unrealistic
835
options
that market
are able
participants
to obtain
additional
about
information
the
evolution of individual catastrophes. Therefore, we assume thatthemarket informationfiltration
(!Ft)o<t<T2 is the right continuous completion of the filtration generated by the catastrophe
occurrences
N,
factors AK
initial
the corresponding
loss estimates
Yj,
and
the corresponding
reestimation
3. Pricing of insurance derivatives
We now consider the problem of pricing a European option with payoff depending on the
value
of the loss
Lp2
at maturity
index
measure
In the catastrophe
72.
insurance
the underlying
market,
themarket is highly incomplete and the choice of the pricing
index L is not traded. Hence,
is not clear.
we
Here
the common
suppose
that the risk neutral
approach
pricing
measure
is structure
preserving for themodel, except for the fact that thepricingmeasure might introduce a drift into
the reestimation
AJ',
martingales
j
=
we
Here
1,2,....
do not discuss
further
the choice
of the
pricing measure, and, without loss of generality, perform pricing with themodel specification
given under P, where we substitutehypothesis (H3) with
(H3;)
s G
A{,
[0, 72],
and
j
=
I, 2,
AJ0
...,
are
=
1
i.i.d.
positive
for all
=
j
semimartingales
such
that
1,2,....
Consider a European derivative written on the loss index with maturity 72 and payoff
> 0 for a payoff function h :R i-> R+.
Since we have assumed that the interest
h(Lp2)
rate
r
is deterministic,
insurance
derivative
without
loss
in discounted
we
can
of generality,
=
i.e. we can set r
terms,
express
the price
0. Then
option is given by
7Tt=E[h(LT2)
I !Ft],
te[0,T2].
the price
process
process
of
the
of the
(3.1)
In the following we will calculate the expected payoff in (3.1) using Fourier transform tech
niques. To this end, we impose the following conditions.
(Cl) The payoff function h(-) is continuous.
(C2) h(-)
- ke
L2(R)
=
{g :R -> C measurable
| f^
\g(x)\2dx < oo} for some keR.
Remark 3.1. In [1] we could have considered more general options thatdid not necessarily
fulfill condition (C2) by considering dampened payoffs. However, the cost of this approach
is that treating heavy-tailed distributions of claim sizes Y becomes more complicated. The
approach in this paper allows for general claim size modeling, including distributions with
heavy tails. Furthermore, as we will see in Section 5, condition (C2) is, for example, satisfied
by call and put spread catastrophe insurance options, the typical options traded in themarket.
Now
let
h(u)
=?
1
2n
1??
c~mz(h(z)
/
J_00
-
k) dz
for all u e R
? k and assume that
be theFourier transformof h(-)
(C3) h(') e Ll(R).
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F. BIAGINI ETAL.
836
Note that condition (C2) does not necessarily imply condition (C3). Since conditions (C2)
and (C3) are in force, the following inversion formula holds (cf. [9, Section 8.2]):
h(x)
i:
(3.2)
cluxh(u)du.
Remark 3.2. Note thattheequality in (3.2) is everywhere, not only almost everywhere, because
of condition (Cl). If the probability distribution of Lj2 had a Lebesgue density, an almost
everywhere equality in (3.2) would have been sufficientfor the following computations. How
ever, since the loss index is driven by a compound Poisson process, the distribution of Lj2 has
and we
atoms
an
need
everywhere
to guarantee
equality
(3.3),
below.
By (3.2), and condition (C3), we obtain
7Tt= E[h(LT2)
|rt]
=
\Ft] + k
E[h(LT2)-k
/oo
-oo
exp{iuLT2}h(u)
du
(3.3)
Ft
| !Ft]h(u)du + k,
E[exp{iuLT2}
(3.4)
-L
where in the last equation we could apply Fubini's theorem, because condition (C3) holds.
in order to calculate the price process (7Tt)te[oj2] in (3.4), the essential task is to
compute the conditional characteristic function of Lj2,
Hence,
NT]
ct(u)
:=
=
| !Ft] E
E[exp{iwL;r2}
exp
Ft
l 7= 1
\iuJ^YjAJT2__Tj
u e
(3.5)
for t e [0, T2]. We define the conditional characteristic function of the reestimationmartingale
A-7 as
:= E[exp{uM?}
0 < t < s < T2,
|FtAJ],
ftAI(s, u)
where !FtAJ := o (AJS,0 < s < t) is thefiltrationgenerated by the jth reestimation factor.Then
our main
result
is as follows.
Theorem 3.1. The conditional characteristic function, (3.5), of the loss index
Lj2
1. fort < T\, by
ct(u)
=
U, uY)])}
0(1
exp{-?(ri
E[^oV2
Nt
x
sj, uyj)\Sj=Tj,yj=Yj,
fi= tf-sj(r2
2. for te
u e M;
1
7
[TU T2], by
NT]
ct(u) =
["J
ftA-Sj(T2
=
7
1
-
Sj, uyj)\Sj=tj,yj=Yj,
u E
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isgiven,
Pricing
of catastrophe
insurance
837
options
Here U is a uniformlydistributed random variable on [t,T\] and Y is a random variable with
distributionfunction FY, independent ofU.
Note
that,
estimates,
Yj,
in Theorem
up
the
3.1,
to time
times
t are known
of
catastrophe
occurrence,
Xj,
and
the
initial
loss
data.
3.1. Proof of Theorem 3.1
We distinguish the computations over the two periods.
1. For t g [0, T\], we obtain, by hypothesis (H4) and by the independent increments ofNt,
ct(u)
? E
J2 ' '
1
exp{i?(5>42_r.+
V = l j=N, + \YJAk-r)\
Nl
NTl
} Ft
i i E rexp ?i?
Er ?
explwJ2YjAJT2_Tj\
?
-l;?i
-I
!
L
?;?w
ii
}
F,
YjAJT2_T.\
J
(3.6)
:=bt{u)
:=at{u)
We compute separately the termsat and bt in (3.6). By hypothesis (H4) we have, forat (u), u e
= E
at(u)
Nt
exp in
| Y,YJAT2-Tj
Ft
y=i
=
CXp^iuJ2yjAT2-sj]
Ntt
n=Nt,Sj=Tj,yj=Yj
|Ft]Sj=Tjtyj=Yj
Y\E[cxp{iuyjAJT2_Sj}
=
7
Ft
1
Nt
=
\
F?JSj]Sj=Tj,yj=Yj
Y\E[exp{iuyjAJT2_Sj}
=
7
1
Nt
Wtf-sjWi-Sj.uyj)
=
7
Note
that,
for at(u),
the
1
YjS,
the tys,
and Nt
sj=Tj>yj=Yj
are
known
data,
because
the corresponding
catastrophes happened before time t.
For the second term,bt (u), u G R, we obtain, again by hypothesis (H4) and the independent
increments
bt(u)
=
E
=
E
"
of the Poisson
NTl
?
expliu J2
process
N,
1 Ft
J
1 j=Nt+ \YJAT2-Tj\
exp J2 YJAT2-Tj
1 j=Nt+ \
jiM
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F. BIAGINI ETAL.
838
-
Nti
txpliuj^yj^-sj
= 1
.. .,
Tnt. -Nt,
Nt,Y\,T\,
,
YNt
-Mt
,
^r,
-
7
=
E
-Sj,uyj)
n^o4"7^
=
1
7
-
A^ri
Nt,Y\,x\,...,
,
xnT{ -n(
w,
J
-
=
yj Yj
S j=T;
yy=yy
7Vri
=E
P[ + t?(T2-Tj,uYj)
l
(3.7)
L7=/V,
By Theorem 5.2.1 of [15] we obtain the following result.
Lemma
3.1.
Let
t < T,
0 <
a Poisson
be
Nt
with
process
INT
(rNt+l,...,tNt
has
the same
as
distribution
the order
times
jump
statistics
i.i.d. uniformlydistributed on the interval [t,T].
-
Nt
...,
(?/(i),
Zj,
=
=
j
Then,
1,2,....
for
all
n)
where
U(n)),
j
Uj,
?
1, ...,
n, are
Using Lemma 3.1, and hypothesis (H4) again, we can replace the r7s in (3.7) with the order
statisticsU(j) of i.i.d. uniformly distributed random variables on the interval [t,T\] and obtain
NT]
bt(u)
= E
PI ^(T2-Uu)9uYj)
U
j=Nt + \
Next, we note the following simple, helpful lemma.
Lemma
Un
and
3.2.
Consider
a bounded
statistics
the order
measurable
function
...
E[f(U{i),
Proof.
We
denote
E[/(i/(i),...,
i/(i),
f(x\,
,U{n))]
...,
...
U(n)
,xn)
...
=E[f(U{,
?/(n))]=E
J2
/(^or(l),
of
{1,
random
i.i.d.
symmetric
the set of all permutations
by Ew
ofn
variables
in its arguments.
,Un)l
...,
n}. We
have
- -,
Ua{n)) l{Ua{l)<-<Uain)
oeHn
f(Ul,...,Un)
=
]T
1{Ucr(\)<-<Ucr{n)
E[f(Ui,...,U?)].
By the i.i.d. assumption of the F/s and A-'s, we see that the function
/Bn(s,,...,iB):=E
Y[^(T2-Sj,uYj)
u e
7=1
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U\,
Then
...,
Pricing
insurance
of catastrophe
is symmetric
in s\,...,
839
options
It is then not difficult
sn.
bt(u)= e\e\ Y\^(T2-Sj,uYj)
=
=
=
...,
E[f?(s\,
NTl
to see, using
Lemma
that
3.2,
-Nt,Uri),...9UfNTi-Nt)
n=NTl-Nt,Sj=Uu).
Sn)\n=NTl-N,,Sj=UU)]
. . . ,
U(n))\n=NTl-Nt]
E[/^(?/(i),
E[f2(Ui,...,Un)\n=NTl-Nt]
NT]
= E
*q(T2-Uj,uYj)
Il
\
j=Nt +
NTl
exp\iuJ2 YJAT2-Uj
1
+l
(3.8)
j=Nt
where we substituted the order statisticsU(j) with the i.i.d. uniform variables Uj.
Note that (3.8) coincides with the characteristic function of a compound Poisson process of
the form
NT]
*>
?
j=Nt + \
where
Z-7 =
this case
well
?
j
YjAJT_u_,
known.
Thus,
are
1,2,...,
we
can
NTl
Y AJ
^
IJ/?T2-Uj
1 j=Nt
expjiw
+l
rewrite
=
=
i.i.d.
The
(3.8)
as
exp{-?(Fi
form
-
exp{-?(F!
0(1
0(1
of the characteristic
-
function
is in
E[exp{iuZ[}])}
-
E[i?f?(T2
U, uY)])}.
This completes the proof for the case inwhich t <T\.
2. For
the case
t>
in which
Ct(u)
=
T\, we
obtain
nT{
W
tf-sj
(T2
-
7= 1
as for the term at
in the case
in which
0 <
Sj, uyj)\Sj=Tjiyj = Yj ,
t<
u G R,
T\.
This completes theproof of Theorem 3.1.
Remark
3.3.
In [17]
a
special
case
of our model
was
presented,
where
the reestimation
factor
A was geometric Brownian motion. In this case the conditional characteristic function of the
reestimation factor can be computed by numeric integrationby
i/^(s, u)
?
|
E[exp{iwexp{/?? ^s}} Ft]
?
?
?
?
=
E[exp{iwexp{#r ^t}} txp{Bs Bt ^(s 0}]
?
=
E[exp{iiiu;?exp{5J_?
/
expfiwu^e^Jexpj
-
?(s
-
0}}]L,=exp{B,_,/2)
- (y-(l/2)(s-t))2
2(s
t)
dy
Wt=exp{Bt-t/2}
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F. BIAGINI ETAL.
840
For
further
of
function
the characteristic
about
results
random
lognormal
we
variables,
refer
the reader to [10].
the next
In
turn our
we
section
to a class
attention
of
reestimation
where
processes
the
conditional characteristic function is numerically tractable and in some cases analytically
For
affine processes.
obtainable:
on affine processes
information
further
their applications
and
tomathematical finance, we refer the reader to [5], [6], and [7].
4. Reestimation with positive affine processes
suppose that the reestimation factors are given by positive affine processes.
We
constitute
processes
a reach
class
of processes
to model
suitable
a wide
Affine
of phenomena.
range
At the same time, the advantage is that the conditional characteristic function can be obtained
explicitly up to the solution of twoRiccati equations.
Definition 4.1. A Markov process A = (At, Px) on [0, oo] is called an affineprocess if there
exist C-valued functions (?)(t,u) and \?/(t,u),defined on R+ x R, such that
=
| !Ft] exp{0(r2
E[exp{iwAr2}
assume
We
t, u) + ir(T2
-
for t > 0.
t, u)At]
(4.1)
that
(Al) A is conservative, i.e. for every t > 0 and x > 0
=
PX[A, <oo]
l;
(A2) A is stochastically continuous for every Px.
1.1 of [8], assumption (A2) is equivalent to the assumption that (p(t, u) and
By Proposition
\?r(t, u)
are continuous
in t for each
u.
In the framework of our model, the computation of the conditional characteristic function
reduces to the computation of 0 and t/>.In some cases these are explicitly known, otherwise they
can be obtained
In the particular
numerically.
affine martingales
under
the pricing
case when
we
measure,
the reestimation
are able
to prove
factors
the following
remain
positive
characterization,
which provides a useful simplification of the conditional characteristic function.
Theorem 4.1. Let A be an affineprocess satisfying assumptions (Al) and (A2). The affine
process A is a positive local martingale ifand only ifA admits thefollowing semimartingale
characteristics
(B, C,
Bt = ?
v):
I Asds,
Ct
Jo
= a
and
/ As ds,
Jo
where
v(dt, dy)
=
Atp(dy)
dt,
/oo
yp{?y),
a
>
0, and
?jl is a Levy
measure
on
(0, oo).
Proof Since At satisfies assumptions (Al) and (A2), by [6, Theorem 2.12] and [8, The
orem 1.1], At is a positive affine semimartingale if and only if At admits the following
characteristics (B,C,v)\
Bt =
/
Jo
(b + ?As)ds,
Ct=a
Jo
Asds,
and
v(dt, dy) =
(m(dy) + Atp(dy))
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dt,
Pricing
insurance
of catastrophe
for every
841
options
Px, where
b = b+
/
7(0, oo)
(1A y)m(dy),
a, b > 0, ? G R, and m and /xare Levy measures on (0, oo), such that
/
7(0,oo)
< oo.
(vA l)ra(d_y)
See also [8]. By assumption (A2) and Theorem 7.16 of [3], the operator L,
(f(x+ y) f(x))m(dy)
Lf(x)= \uxf"(x)+ (b+ ?x)f'(x)+z f
7(0, oo)
+ * /" (/(*+ y)- f(x) - f(x)(\ A y))p(dy) (4.2)
7(0,oo)
on C2(R+),
is a version of therestrictionof theextended infinitesimalgenerator ofA toC2(R+).
(An operator L with domain ?>i is said to be an extended generator forA if?>l consists of
those Borel functions / forwhich thereexists a Borel function Lf such that theprocess
L{
= f(At)-f(A0)-
[ Lf(Xs)ds
Jo
is a local martingale.) Then A is a local martingale ifand only if
Lf(x)
Substituting f(x)
= 0
forf(x) =x.
= x into (4.2), we obtain
Lx = b+ ?x +
ym(dy)+x
/
?/(0,oo)
=
I
7(0,oo)
(y (1A y))p(dy)
+ b+ / ym(dy).
(y- l)v>(dy))x
V / 7i
7(o,oo)
(?+
Hence, A is a local martingale ifand only if
\?+
(y-l)ii(dy))x
+ b+
/ V7l
7(0,oo)
ym(dy)
= 0
foranyxGR+.
(4.3)
Since b > 0 and m is a nonnegative measure, condition (4.3) means that
/oo
yfi(dy).
(4.4)
Let A be an affine process satisfying assumptions (Al) and (A2). By Theorem 4.3 of [7],
the conditional characteristic function of A satisfies (4.1), where 4>(t,u) and \//(t,u)solve the
equations
(?)(t,u)= i F(x//(s,u))ds,(4.5)
7o
dtx?/(t,u)
=
R(f(t,
u)),
V(0, u)
=
iu,
(4.6)
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F. BIAGINI ETAL.
842
where, for z e {C | Rez
< 0},
R(z) = \az2 + ?z+ 2 f
(zzy-1-z?A
.7(0,00)
(4.7)
l))/x(dy),
F(z) = bz+ [
(ezy
l)m(dy),(4.8)
.7(0, oo)
and a, ?, b,m, and /xare parameters of infinitesimal generator (4.2) of A.
martingale thenby (4.4) we can simplify (4.7) and (4.8) as follows:
If A is a local
+ f
-zy- Dp(dy), (4.9)
R(z)= \az2
(tzy
2
J(0,oo)
F(z)=0.
(4.10)
From (4.5) and (4.10), we immediately obtain, for positive affine local martingales,
(j)(t, u)
= 0.
In order to determine i/?we have to, in general, solve (4.5) numerically. For some special
cases,
however,
we
can
compute
We
y?ranalytically.
two examples.
give
Example 4.1. If A has no jump part thenA is called a Feller diffusion (see, e.g. [6]). In this
case thepositive affinemartingale dynamics are given by
dA,
=
Ja~?tdWt,
where Wt is a standard Brownian motion. Consequently, we have /x= 0 in (4.9), and we can
rewrite (4.6) as
f't
=
\af}.
(4.11)
Solving the differential equation (4.11), we obtain
\J/(t,u)= 0
where C(u)
the expression
or
\/f(t,u)=-,
mgM,
at/2 + C(u)
can be found from the boundary condition \j/(0,u) =
for
we
\?/,
\u. Substituting C(u)
into
obtain
iJf(t,u)==Q
or
\jf(t,u)
=-,
at/2 +
\/u
u e R.
Note that ifwe have no jump part then A has positive probability of being absorbed at 0.
However, itmay stillbe of interestto also consider the case of positive probability of absorption
at 0, ifwe wish to include thepossibility of fraud or falsified
reportingof claims into themodel.
In this case, reestimationmight discover the fraud and the previous fake evaluation will be set
toO.
Example 4.2. In order to give an example of a positive affinemartingale including jumps
where we can solve for j//explicitly,we specify the jump density /x(dy) in the semimartingale
characteristics inTheorem 4.1 as
3
4V^
dy
"
y5/2
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of catastrophe
Pricing
insurance
843
options
Then some calculations give R(z)
in (4.7):
1 ?
3
R(z) = -az2 + ?=
2
=
4V7T
{az2
+
dy
1)-^
f
(^ -zy\
y5/I
7(0,00)
(~z)3/2
for z g {C | Rez < 0}.
Consider r?(t,u) := ?i/r(t, u). By (4.6) we thenhave
-v?t
The solutions to (4.12) are n(t, u) =0
r?(i, u)
=
+
\oLr)2 r?3/2. (4.12)
and
= -,(1
or
+ W(-C(w)e-r/?;))-2,
(4.13)
is defined to be
where W(-) is the Lambert W function. (The Lambert W function, W(z),
= z, z G C. See also [4] formore details on theLambert
the function satisfyingW(z)QW^
=
= iu
function.) The boundary condition n(0, u)
?\?/(0,u)
yields
=
+
C(M)
-(-1^)exp{-I
=
into (4.13), we obtain, for \?r(t,
u)
?rj(t, u),
Substituting C(u)
^(t,u)=0
+
^)
or
\lf(t,u)=-2(1
(l
+
+
+
+
W/fl~1
w((-\
-7-)
expi--
- 1+
-7
5. Pricing of call and put spreads
We conclude thispaper by showing thatwe can apply thedeveloped pricingmethod to spread
options, which are the typical catastrophe options traded in themarket. A call spread option
with strikeprices 0 < K\ < K2 is a European derivative with thepayoff function atmaturity
given by
0
h(x)
x - K\
if0<x<^i,
ifK\ < x < K2,
K2-Ki
ifx>K2.
is satisfied for k := K2 ? K\. In particular,
The integrabilitycondition h(-) ? k g L2(R+)
h(-)-ke
L,(M+).
To satisfy conditions (Cl) and (C3), we continuously extend h fromR+ toR by
h(x)
ifx<0,
:? \h(-x)
{
ifx>0.
\h(x)
Note that theprice processes of the two corresponding options with payoffs h(Lp2) and h(Lp2)
remain
the same,
because
Let
h(u)
Lp2
:=?
>
0.
/
27T 7-oo
e-mz(h(z)
-
k) dz
for all u e R
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F. BIAGINI ETAL.
844
? k. Then
be theFourier transformof h
?w-?(?'
1 1
=-~(exp{
1 1
(Re(cxp{iu K2})
1 11
?
=-x(cosuK2
n uz
-
-
K2)dx
K2)dx
?
+ cxp{iuK2]
?iuK2}
2tt uz
71 UZ
c(Ki
+
K2)dx
J-K\
c(x
=-^
rKx
-
*(-x
?
exp{?iuK\}
?
cxp{iuK\})
Rc(cxp{iuK\}))
e L
cosuK\)
(K),
and, by applying the inversion formula (3.2) to h(x) forx > 0, we find that (3.2) also holds for
h, since h(x) = h(x) forx > 0.
In particular,
since
cs
n^
>
Lj2
0 almost
surely
:/:
spread,
we
can write
\Ft] + k
dw
expfiwL^j/^w)
E[txp{iuLr2}
Ft
| !Ft]h(u)du + &
du -\-k
ct(u)h(u)
JT J-oo
of the call
|Ft]+k
=E[h(LT2)-k
=
E[h(LT2)-k
= E
for the price
(cos uK2
M2
? cos
?
K\
uK\) du + K2
(5.1)
=
where ct(u) is defined in (3.4). Note that the integral in (5.1) is real, since Im(cr(?u))
?
definition
of
ct.
ln\(ct (u)) by
Analogously, for the put spread catastrophe option with payoff at thematurity given by
8(x)
if0 <*<#!,
ifK\ < x < K2,
ifx > K2,
K2-K\
- x
K2
0
we
obtain
nt
PS =?
1 f00 ct(u)
I
?5?
IT J_00
UZ
?
(cosuK\
cos
uK2)
du.
Note that the call-put parity is satisfied:
7Tt
=
K2
?
K\
?
7Tt
.
Acknowledgements
We wish to thank theNew YorkMercantile Exchange for the informationprovided concerning
catastrophe
insurance
options
and Damir
Filipovic
for interesting
comments
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and
remarks.
Pricing
of catastrophe
insurance
options
845
References
T. (2008). Pricing of catastrophe insurance options written on
[1] BiAGiNi, F., Bregman, Y. and Meyer-Brandis,
a loss index. To appear in Insurance Math. Econom.
C. V. (1999). A new model for pricing catastrophe insurance derivatives. CAF Working Paper
[2] Christensen,
Series No. 28.
E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov processes. Prob.
[3] ?inlar,
Theory Relat. Fields 54, 161-219.
R. M. et al. (1996). On theLambert W function. Adv. Comput. Math. 5, 329-359.
[4] Corless,
K. (2000). Transform analysis and asset pricing for affine jump diffusions.
[5] Duffie, D., Pan, J. and Singleton,
Econometrica 68, 1343-1376.
W. (2003). Affine processes and applications in finance. Ann.
[6] Duffie, D., Filipovic, D. and Schachermayer,
Appl. Prob. 13,984-1053.
[7] Filipovic, D. (2001). A general characterization of one factor affine term structuremodels. Finance Stoch. 5,
389^12.
S. (1971). Branching processes with immigration and related limit theorems.
K. and Watanabe,
[8] Kavazu,
Theory Prob. Appl. 16, 36-54.
K. (1993). Analysis 2. Springer, Berlin.
[9] K?nigsberger,
[10] Leipnik, R. B. (1991). On lognormal random variables. I. The characteristic function. J.Austral. Math. Soc. Ser.
B 32, 327-347.
M. (1996). Pricing PCS-options with the use of Esscher-transforms. AFIR Colloquium N?rnberg,
[11] M?ller,
October 1-3 1996. Available at http://www.actuaries.org/AFIR/colloquia/Nuernberg/papers.cfm.
[ 12] M?RMANN, A. (2001 ). Pricing catastrophe insurance derivatives. Financial Markets Group Discussion
Available at http://ssrn.com/abstract=31052.
[13] M?RMANN,
A.
(2003). Actuarially
consistent valuation of catastrophe derivatives. Working
Paper 400.
paper. Available
http ://knowledge.wharton. upenn.edu/muermann/.
[14] M?RMANN, A. (2006). Market price of insurance risk implied by catastrophe derivatives. Working
Available at http://irm.wharton.upenn.edu/muermann/.
at
paper.
J. L. (1999). Stochastic Processes for Insurance and
[15] Rolski, T., Schmidli, H., Schmidt, V. and Teugles,
Finance. JohnWiley, Chichester.
[ 16] Schmidli, H. (2001 ).Modeling PCS options via individual indices.Working paper 187, Laboratory ofActuarial
Mathematics, University of Copenhagen.
new instrument for managing catastrophe
insurance options?a
H. R. (1996). PCS Catastrophe
[17] Schradin,
risk. Contribution
to the 6th AFIR
International Colloquium
N?rnberg,
October
1-3
1996. Available
http://www.actuaries.org/AFIR/colloquia/Nuernberg/papers.cfm.
[18] Sigma Nr. 3 (2001). Capital Market Innovation in the Insurance Industry. Swiss Re, Z?rich.
[19] Sigma Nr. 7 (2006). Securitization?New
Opportunities for Insurers and Investors. Swiss Re, Z?rich.
in 2006. Swiss Re, Z?rich.
[20] Sigma Nr. 2 (2007). Natural Catastrophes and M an-Made Disasters
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at