Content uploaded by Rene A. Carmona
Author content
All content in this area was uploaded by Rene A. Carmona
Content may be subject to copyright.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
International Journal of Theoretical and Applied Finance
Vol. 8, No. 7 (2005) 959–988
c
World Scientific Publishing Company
PRICING PRECIPITATION BASED DERIVATIVES
REN´
E CARMONA
Department of Operations Research and Financial Engineering
Princeton University, Princeton, NJ 08544, USA
and
Bendheim Center for Finance and the Applied and
Computational Mathematics Program
rcarmona@princeton.edu
PAVE L D I KO
Electrabel S.A., Trading & Portfolio Management
Boulevard du R´egent 8, Bruxelles 1000, Belgium
Pavel.Diko@electrabel.com
Received 2 February 2004
Accepted 7 April 2005
We consider the problem of pricing a derivative contract written on precipitation at
a specific location during a given period of time. We propose a jump Markov process
model for the stochastic dynamics of the underlying precipitation. Our model is based on
pulse Poisson process models widely used in hydrology. We develop maximum likelihood
parameter estimation procedures to fit our model to rainfall data. In order to price
derivatives, we assume the existence of a traded asset whose price dynamics are influenced
by the precipitation at the location in question, and we rely on the utility indifference
approach. Two utility functions are considered: exponential and power utility. We derive
explicit solutions for the exponential and bounds for the power utility.
Finally, we apply our model fitting and pricing techniques to a sample rainfall con-
tract in Norway.
Keywords: Weather derivatives; utility indifference pricing; precipitation modeling; jump
Markov processes; maximum likelihood estimation.
1. Introduction
In this paper we concentrate on a specific class of derivatives written on non-traded
assets, precipitation derivatives. They are part of a larger group of weather deriva-
tives used for weather risk mitigation in a wide array of businesses ranging from
agriculture to the entertainment industry.
Pricing of precipitation derivatives poses a great challenge compared to the
other types of weather derivatives, let alone classical equity derivatives. From a
practical view point, precipitation can be quite a localized phenomenon. Theoretical
difficulties lie in the fact that, despite intensive hydrological research, it is not
959
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
960 R. Carmona & P. Diko
straightforward to develop a tractable mathematical model accurately representing
the reality of the precipitation at a specific location.
Moreover, once a model for the precipitation has been established, a second
source of difficulty arises: it is necessary to develop a pricing methodology which can
incorporate the idiosyncracies of the underlying precipitation process. Since precip-
itation cannot be traded directly, the market model is most likely to be incomplete.
Instead of searching for an appropriate risk neutral probability measure, we follow
the utility indifference pricing approach to derive buyer and seller’s prices. This
valuation method can be described in a few sentences as follows. Let us consider an
economic agent whose time-invariant risk preferences are given by a utility function.
Given an initial wealth, she purchases (resp. sells) a derivative written on a non-
traded asset, and she uses her endowment to set up a portfolio in a bank account
and shares of a traded asset, and follow, a self-financing strategy, re-balancing her
portfolio in continuous time. She searches for an optimal strategy which maximizes
her expected utility of wealth at maturity of the derivative. The expected utility
of terminal wealth can also be computed had she not entered the transaction. The
buyer’s (resp. seller’s) price of the derivative is set in such a way as to make the
agent indifferent between the two scenarios — hence the name utility indifference
pricing given to this valuation method.
The topic of utility indifference pricing was first introduced by Hodges and
Neuberger in the early nineties, but it had to wait almost an entire decade to
catch on. It was rediscovered by Davis [8] and used by Henderson [14, 15], and
Musiela and Zariphopoulou [21–23] who considered the case of a derivative written
on a non-traded asset whose price dynamics are given by a geometric Brownian
motion. Most of these contributions are restricted to the case of the exponential
utility function, for in the case of the power utility, one is only able to obtain
bounds on the prices. More generally, Zariphopoulou considers in [28] the problem
of utility optimization when the influence of the non-traded asset on the traded asset
comes through both the correlation between the driving Wiener processes and the
coefficients of the price process SDE. She uses power utility and obtains results for
the scenario without the derivative present. Finally, we mention that Musiela and
Zariphopoulou gave an abstract form of the indifference pricing paradigm on a tree,
and that Carmona and Danilova [3] derived the indifference price for a claim on a
non-traded quantity (temperature) whose dynamics is driven by a diffusion process
when allowed to invest in traded assets whose dynamics are given by diffusions whose
coefficients depend upon the non-traded asset, and when the agent has exponential
utility function. The interested reader is referred to the forthcoming book [10] for
a detailed account of the state of the art in indifference pricing.
The paper is naturally divided into three parts. In the first part, we develop a
model for the precipitation process, and we find maximum likelihood estimators for
its parameters. In the second part, we adapt utility indifference pricing to a class of
security price models including the precipitation model from part one. We consider
exponential and power utility functions. For the exponential utility we obtain an
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 961
explicit formula for the price of the derivative, while for the power utility, we can
only prove lower and an upper bounds. Finally, we conclude the paper with the
detailed analysis of a practical example. Except for the highly publicized example
of the three year deal offered by an amusement park in Paris, and the customized
derivative offered by Aquila to the Sacramento municipal utility to provide power
or cash to purchase power in case of insufficient rainfall for their hydroelectric
generation to operate, not much public information is available on the terms and
conditions of precipitation options and their prices. The example we propose in the
last section of the paper is made up. However, we used our personal experience to
setituptomakeitasrealisticaspossible.
2. Modeling Rainfall Dynamics
2.1. Precipitation data
Data serving as underlying for precipitation contracts come from meteorological
stations around the globe. Although made at discrete time intervals, the observa-
tions come close enough to complete continuous time observation of the rainfall
intensity. A period of rainfall, when its intensity stays constant, can be described
byapair(ξ, β)ofrealnumbers.Hereβ>0 is the length of time for which the rain-
fall intensity stays constant and equal to ξ>0. Let us denote by Mthe sequence
{(ξi,β
i+1)}i=1,2,3,...,n of pairs describing the consecutive periods of constant rain-
fall intensity. The sequence Mcomprises the statistical data for which we want to
formulate a parsimonious yet well-fitting probabilistic model.
Precipitation has been the subject of intensive research for years. Several types
of models have been developed [24], which can be divided into four categories: mete-
orological models which seek to capture the dynamics of the large scale atmospheric
processes controlling precipitation [19], multi-scale models which use multifractal
cascades to describe rainfall [12, 13], statistical models which use purely statisti-
cal techniques to fit the rainfall data to well known distribution types with little
emphasis on underlying physical processes [27] and last stochas tic processes ba sed
models which try to describe the rainfall behavior by a small set of physically mean-
ingful parameters driving a stochastic process. They were introduced by LeCam in
1961 [18] and further developed by Rodriguez-Iturbe et al. [25, 26].
2.2. Precipitation model
A version of the Bartlett-Lewis Poisson-cluster (BLPC) process model of Rodriguez-
Iturbe, Cox and Isham [25] forms the basis of the model which we propose. In this
model, rainfall is assumed to be composed of storms which are in turn composed of
rainfall cel ls. The storms arrive according to a Poisson process; within each storm,
cells arrive according to another Poisson process and the duration of the activity of
the storm is random. Each cell has random duration and random depth. Both storms
and rain-cells can overlap. If two or more cells overlap their depths add up. The
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
962 R. Carmona & P. Diko
best data-fitting version of the model uses exponential distributions for both the
storm and cell duration [6, 25]. Although the BLPC model is physically intuitive,
methods for fitting its parameters are quite limited and predominantly not very
stable. The most widely used approach is based on ‘method-of-moments’ type fitting
as, for example, in Rodriguez-Iturbe et al. [25]. In what follows, we reformulate the
Rodriguez-Iturbe model, and we explain why the new model keeps all the important
features of the old one, while allowing for maximum likelihood estimation (MLE
for short) of parameters. Only then, do we state a rigorous definition of the new
model.
Main obstacles to application of MLE to BLPCs model are the “physically
appealing but mathematically intractable dependencies involved between the rain-
fall intensities at different time points” [4]. Although intractable, the model is very
close to being Markovian. There are two properties which compromise the Markov
property: first, the rectangular shape of the cells implies the memory of the depths
of all active cells at any time and second, the rate of cell extinction at a given
time depends on the number of active cells which is unobservable. The first issue
can be dealt with in the following way: at a cell arrival time, the rainfall process
jumps up by a random amount and at the cell’s extinction time, it jumps down by
a random amount with appropriate distribution. If on average the jumps up are as
large as the jumps down, one can call such a modified model a “randomization”
of the rectangular cell shape. There is one technical point to be addressed — the
possibility of a jump to the negative region. The second issue is also resolvable.
The rate of extinction can be made dependent on the instantaneous intensity of
the rainfall and not the number of active cells. Again this can be thought of as
“smoothing” of the Rodriguez-Iturbe BLPC model. It is noteworthy to realize that
in the case of deterministic cell depths, this approach is identical to the BLPC
model mechanics.
We propose to model rainfall intensity within a single storm by a homogeneous
jump Markov process {Yt}whose infinitesimal generator Gis given by
[Gφ](x)=R+
(φ(y)−φ(x))A(x, y)ν(dy),(2.1)
where
A(x, y)=λ1λue−λu(y−x)1(x,∞)(y)¯
λ2(x)λde−λd(x−y)1(0,x)(y)
+¯
λ2(x)e−λdx1{0}(y),
with
¯
λ2(ξ)=1
(0,∞)(ξ)λ(I)
2+λ(II)
2κ(ξ),(2.2)
where νis Lebesgue’s measure on [0,∞) with an added unit point mass at 0, and
κ(ξ) is defined on R+and satisfies κ(x)=xfor 0 ≤x≤K(for some large K), it is
bounded, it is 3-times differentiable, (∂/∂x)κ(x)>0. The model is determined by
the parameter vector (λ1,λ
(I)
2,λ
(II)
2,λ
u,λ
d)=(θ1,θ
2,θ
3,θ
4,θ
5)=θ∈Θ=R5
+.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 963
2.3. Maximum likelihood parameters estimates
The advantage of having a Markovian model for the single storm is the convenience
of being able to write the likelihood function. As stated in Sec. 2.1, we observe
M={(ξi,β
i+1)}i=1,...,n. For the continuous time process Y,βi+1 >0 is the sojourn
time in state ξi>0. Mis an observation of the so called embedded Markov chain
corresponding to the continuous time process Y(see [2]). The log-likelihood function
(with respect to the product of copies of the measure ν) of a particular Mis then
L(M,θ)=lnn−1
i=1
F(ξi,β
i+1,ξ
i+1,θ),
where
F(ξ,β, η,θ)=f(ξ,η, θ)q(ξ, θ)e−q(ξ,θ)β.
Intuitively, the value of Fis the product of two probabilities: q(ξ, θ)e−q(ξ,θ)βwhich
is the density of the sojourn time βof the jump Markov process Yin the current
state ξ,andfwhich is the density of the distribution for the new state ηresulting
from the next jump. For the storm model defined by (2.1)
f(ξ,η, θ)=1
[ξ,∞)(η)λ1
λ1+¯
λ2(ξ)λue−λu(η−ξ)
+1
(0,ξ)(η)¯
λ2(ξ)
λ1+¯
λ2(ξ)λde−λd(ξ−η)
+1
{0}(η)¯
λ2(ξ)
λ1+¯
λ2(ξ)e−λdξ(2.3)
and
q(ξ,θ)=λ1+¯
λ2(ξ).
We derive the values of the estimators by solving the system of first order conditions,
and we check that the obtained solutions are indeed the unique maximum likelihood
estimators of the model parameters. Denote U={i;ξi+1 −ξi>0},D={i:
ξi+1 −ξi<0},D0={i:ξi+1 =0},andΞ
+={i:ξi>0}. The candidate
estimators are then
λd=|D\D0|
i∈D(ξi−ξi+1),λ
1=|U|
n
i=2 βi
,λ
u=|U|
i∈U(ξi+1 −ξi),
and the equations to solve are
i∈D
1
λ(I)
2+λ(II)
2κ(ξi)=
n
i=2
βi,(2.4)
i∈D
κ(ξi)
λ(I)
2+λ(II)
2κ(ξi)=
n−1
i=1
κ(ξi)βi+1.(2.5)
The following proposition states the uniqueness of the estimates for λ(I)
2and λ(II)
2.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
964 R. Carmona & P. Diko
Proposition 2.1. For any realization of M, if the set Dis non-empty and the
system of equations (2.4) and (2.5) has a solution,then it is unique.
Proof. If we use the notation B=n
i=2 βi,andC=n−1
i=1 κ(ξi)βi,weget
Φ(a, b)=
i∈D
1
a+bκ(ξi)−Ba,b>0,(2.6)
Ψ(a, b)=
i∈D
κ(ξi)
a+bκ(ξi)−Ca,b>0.(2.7)
Since Dis non-empty, and since for every iin D,ξi>0, the function Φ is strictly
decreasing both in aand b. Therefore the equation Φ(a, b) = 0 defines an implicit
function a=a(b)forb∈(0,1
Bi∈Dκ(ξi)−1)whichissmoothand
a(b)=−∂Φ/∂b
∂Φ/∂a =−i∈D
κ(ξi)
(a(b)+bκ(ξi))2
i∈D1
(a(b)+bκ(ξi))2
.
Now, let us define g(b)=Ψ(a(b),b). gis only well-defined for b∈
(0,B−1i∈Dκ(ξi)−1). The statement of the proposition is equivalent to saying
that if g(b) = 0 has a solution, then it is unique. It is easy to see that gis smooth.
To show the uniqueness we prove that g<0, and thus gis strictly decreasing
g(b)=
i∈D
−κ(ξi)
(a(b)+bκ(ξi))2(a(b)+κ(ξi))
=
i∈D
−κ(ξi)
(a(b)+bκ(ξi))2i∈D
−κ(ξi)
(a(b)+bκ(ξi))2
i∈D1
(a(b)+bκ(ξi))2
−
i∈D
κ(ξi)2
(a(b)+bκ(ξi))2
=i∈D
κ(ξi)
(a(b)+bκ(ξi))22−i∈D
κ(ξi)2
(a(b)+bκ(ξi))2i∈D1
(a(b)+bκ(ξi))2
i∈D1
(a(b)+bκ(ξi))2
.
The sign of the derivative only depends on the numerator since the denominator is
positive. Note
i∈D
κ(ξi)
(a(b)+bκ(ξj))22
−
i∈D
κ(ξi)2
(a(b)+bκ(ξj))2
i∈D
1
(a(b)+bκ(ξj))2
=
ij
κ(ξi)κ(ξj)−κ(ξi)2
(a(b)+bκ(ξi))2(a(b)+bκ(ξj))2=
i<j
2κ(ξi)κ(ξj)−κ(ξi)2−κ(ξj)2
(a(b)+bκ(ξi))2(a(b)+bκ(ξj))2
=−
i<j
(κ(ξi)−κ(ξj))2
(a(b)+bκ(ξi))2(a(b)+bκ(ξj))2
<0.
To establish that the above estimators are consistent maximum likelihood esti-
mators of the true model parameters, we need to verify certain smoothness prop-
erties of the transition density of the imbedded Markov chain for the process Yin
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 965
order to use the following theorem [2] of Billingsley which we state for the sake of
completeness.
Theorem 2.1 (Billingsley). Suppose that the Markov process {Yt}defined by its
infinitesimal generator (2.1),f,qdefined by (2.3) and the parameter space Θsatisfy
the following conditions:
(1) P(Yt+s=x|Yt=x)is a continuous function of sfor every xand t, the sample
paths of Ytare right-continuous and q(ξ, θ)is a bounded function for any θ∈Θ.
(2) For an y ξ, the set of ηfor which f(ξ, η, θ)>0does not depend on θ.
(3) For an y ξ, β and ηthe F(ξ,β, η, θ)(and thus G(ξ,β, η,θ)=lnF(ξ,η, θ)) has
continuous third order partial derivatives throughout Θ.
(4) For an y θ∈Θthere is a neighborhood N of θsuch that for any three parameters
u, v, w and any state ξ
R+×R+
sup
θ∈N
|Fu(ξ,β, η, θ)|dηdβ < ∞,(2.8)
R+×R+
sup
θ∈N
|Fuv(ξ,β, η,θ)|dηdβ < ∞,(2.9)
Eθsup
θ∈N
|Guvw (z1,ρ
2,z
2,θ
)|<∞,(2.10)
where Fu,F
uv,F
uvw denote the first,second and third order partial derivatives
with respect to parameters θu,θ
v,θ
wand z1is a random variable with the sta-
tionary distribution πθof the Markov chain {ξ1,ξ
2,ξ
3,...}and the conditional
distribution of (ρ2,z
2)given z1is determined by its density F(z1,ρ
2,z
2,θ).
(5) There exists δ>0such that for all u, Eθ[|Gu(z1,ρ
2,z
2,θ)|(2+δ)]<∞and
σuv(θ)defined by σuv (θ)=Eθ[Gu(z1,ρ
2,z
2,θ)Gv(z1,ρ
2,z
2,θ)] form a non-
singular matrix σ(θ)=(σuv (θ)).
(6) For each θ∈Θ,there is exactly one stationary distribution πθ(·)for the imbed-
ded Markov process {ξn},and πθ(ξ,·)πθ(·)for each ξ∈R+.
and that θ0∈Θis the true value of the parameter vector. Then there exists a
solution ˆ
θof ∂L/∂θ =0which gives a consistent estimate of θ0.Ify(t)is a random
vector with components
yu=n−1
2
n
k=1
Gu(ξk,ξ
k+1,β
k+1,θ
0),u=1,...,r,
and l(t)is the random vector with components lu(t)=n1
2(ˆ
θu−θ0
u),u=1,...,r
then as n→∞,y(t)L
−→ N (0,σ(θ0)),l(t)L
−→ N (0,σ(θ0)−1),2[L(ˆ
θ)−L(θ0)] L
→χ2
r.
We check that the conditions of the above theorem are satisfied in the present
situation.
•Condition 1. ¯
λ2is bounded, therefore condition 1 is easily checked.
•Condition 2. It is easily checked because for every θ∈Θ, {η;f(ξ,η, θ)>0}=R+.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
966 R. Carmona & P. Diko
•Condition 3. We can rewrite the function F(ξ,β, η,θ) as function of the vector
parameter θas follows
F(θ)=
¯
λ2(ξ)e−λdξe−(λ1+¯
λ2(ξ))β;η=0
¯
λ2(ξ)λde−λd(ξ−η)e−(λ1+¯
λ2(ξ))β;0<η<ξ
λ1λue−λu(η−ξ)e−(λ1+¯
λ2(ξ))β;ξ≤η
(2.11)
Now it is easy to see that Fpossesses partial derivatives of order three because
¯
λ2does.
•Condition 4. The first two integrability conditions are easy to see from the expres-
sion (2.11). To obtain the required integrability we need Fu(Fuv ) to be bounded
in ηfor η<ξand integrable in for η≥ξ. (We can first integrate w.r. to β.)
Boundedness is apparent because of continuity of the derivatives. The integra-
bility follows from the fact that for η≥ξthe Fu(Fuv ) can be written as:
Fu(η, β)=h(η, β)e−aη−bβ where a, b > 0andh(η, β)isO(ηnβm)forsomeinte-
gers n,m. To check the finiteness of the expectation we prove more, namely that
Guvw is bounded almost surely. Gcan be written as
G(ξ,β, η,θ)=
ln ¯
λ2(ξ)−λdξ−(λ1+¯
λ2(ξ))βη=0,
ln ¯
λ2(ξ)+lnλd−λd(ξ−η)−(λ1+¯
λ2(ξ))β0<η<ξ,
ln λ1+lnλu−λu(η−ξ)−(λ1+¯
λ2(ξ))βξ≤η.
(2.12)
It is clear that only very few of the third order derivatives are non-zero. Namely
Gλ3
1=2
λ3
11{η≥ξ},Gλ3
d=2
λ3
d
1{0<η<ξ},Gλ3
u=2
λ3
u1{η≥ξ}and mixed derivatives
with respect to λ(I)
2and λ(II)
2for 0 <η<ξ. We will denote Gij =G(λ(I)
2)i(λ(II)
2)j.
Then G21 =2κ(ξ)
¯
λ2(ξ)3,G30 =2
¯
λ2(ξ)3,G12 =2κ(ξ)2
¯
λ2(ξ)3,G03 =2κ(ξ)3
¯
λ2(ξ)3.Sincetheformer
are constant and the latter are bounded for all ξ, η, β they satisfy the condition 3.
•Condition 5 is satisfied as long as the following conditions hold true for some
δ>0
Eθ[|ρ2|2+δ]<∞,(2.13)
Eθ[|ρ2z1|2+δ]<∞,(2.14)
as can be seen from the special form of the partial derivatives of G
Gλ1=−βη<ξ
1
λ1−βη≥ξ,G
λd=
−ξη=0
1
λd−(ξ−η)η<ξ
0η≥ξ
,
Gλ(II)
2=κ(ξ)
¯
λ2(ξ)−κ(ξ)βη<ξ
−κ(ξ)βη≥ξ,G
λu=0η<ξ
1
λu−(η−ξ)η≥ξ,
Gλ(I)
2=1
¯
λ2(ξ)−βη<ξ
−β1{ξ>0}η≥ξ.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 967
We first prove (2.13)
Eθ[|ρ2|(2+δ)]=Eθ[|ρ2|(2+δ)|z1=z]πθ(dz)
≤Eθ[|ρ2|(2+δ)|z1=0]πθ(dz)
=Eθ[Λ(2+δ)
1]<∞,
where Λ1is an exponential random variable with parameter λ1. To prove (2.14)
we notice that
Eθ[|z1ρ2|(2+δ)]=Eθ[|zρ2|(2+δ)|z1=z]πθ(dz)
≥Eθ[|zΛM|(2+δ)|z1=z]πθ(dz)
=Eθ[Λ(2+δ)
M]Eθ[z(2+δ)
1],
where ΛMis an exponential random variable with parameter λ1+Mindependent
of z1. Similarly we can show
Eθ[|z1ρ2|(2+δ)]≤Eθ[Λ(2+δ)
1]Eθ[z(2+δ)
1].
Therefore it is enough to prove that Eθ[|z1|(2+δ)]<∞. That however, follows
from Eq. (2.18) of Theorem 2.2 coupled with the drift condition (2.15) below.
Since those imply that the stationary distribution πθpossesses all moments.
We proceed to prove the non-singularity of the matrix σ. First we notice that
σuv =E[GuGv]=−E[Guv].
In this form it is easy to see that the matrix σhas only two non-zero off diagonal
elements, namely
−EGλ(II)
2λ(I)
2=−EGλ(I)
2λ(II)
2=E1{ξ>0}
κ(ξ)
¯
λ2
2(ξ).
Therefore σis non-singular if and only if the matrix
E1{ξ>0}κ(ξ)2
¯
λ2
2(ξ)E1{ξ>0}κ(ξ)
¯
λ2
2(ξ)
E1{ξ>0}κ(ξ)
¯
λ2
2(ξ)E1{ξ>0}1
¯
λ2
2(ξ)
,
is non-singular. This is true since its determinant is positive for all values
of λ(I)
2,λ(II)
2.
•Condition 6. We shall prove this condition in a series of statements. The main
idea will be to show that there exists a “small” subset of the state space towards
which the Markov chain ξhas a geometric drift. Because of Theorem 2.2, this
implies the existence of a unique stationary distribution of ξ. But first we recall
a couple of standard definitions for the sake of definiteness.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
968 R. Carmona & P. Diko
Definition 2.1. Let Pbe the transition kernel of a Markov chain on a measurable
space (X,B(X)). A set C∈B(X) is called small if there exists an m>0, and a non-
trivial measure νmon B(X), such that for all x∈C,B∈B(X)Pm(x, B)≥νm(B).
Definition 2.2. The Markov chain ξis called strongly aperiodic if there exists a
ν1-small set Asuch that ν1(A)>0. Moreover, the chain is said to be φ-irreducible
if for all x∈Xand any A∈B(X) such that φ(A)>0, L(x, A)>0whereL(x, A)
is the probability of reaching Astarting from x.
Proposition 2.2. Any interval [0,x
0]is a small set for the chain ξ.
Proof. We first we show that {0}is a small set. Let m=1andν=Pξ(0,dy)=
f(0,y)(dy+δ0). By definition for any A∈B(R), Pξ({0},A)=ν. For any x∈[0,x
0],
Pξ(x, {0})>0, therefore by Proposition 5.2.4 [20] the interval [0,x
0]isasmall
set too.
Proposition 2.3. There exists x0≥0such that the Markov chain ξhas a geometric
drift towards [0,x
0],i.e.,there exists β>0and b<∞such that for any x>0
Pξ(x, dy)V(y)≤(1 −β)V(x)+b1[0,x0](x),(2.15)
where V(x)=eγx,0<γ<λ
u.
Proof. Let us denote p(x)= λ1
λ1+¯
λ2(x),q(x)= ¯
λ2(x)
λ1+¯
λ2(x).Wehave
PξV=∞
0
Pξ(x, dy)V(y)
=p(x)∞
x
λue−λu(y−x)eγydy +q(x)x
0
λde−λd(x−y)eγydy +e−λdx
=p(x)λu
λu−γeγx +q(x)λd
λd+γ+e−x(λd+γ)1−λd
λd+γ.(2.16)
Let us denote D(γ)= λd
λd+γ,U(γ)= λu
λu−γ. Recall that 0 <γ<λ
u. Rearrang-
ing (2.16) we obtain PξV=eγx{p(x)U(γ)+q(x)[D(γ)+e−x(λd+γ)(1 −D(γ))]}.
Denoting A(x)=D(γ)+e−x(λd+γ)(1 −D(γ)) we get
PξV=V(x)(p(x)U(γ)+q(x)A(x)) = V(x)[p(x)(U(γ)−A(x)) + A(x)].(2.17)
Let us define B(x)= 1−A(x)
Uγ−A(x).B(x) is well defined for all x≥0 because
0<A(x)≤1andU(γ)>1. Taking derivative we get
B(x)=A(x)(1 −U(γ)) = −(λd+γ)e−x(λd+γ)(1 −D(γ))(1 −U(γ)) >0.
Thus B(x) is an increasing function of x. It is easy to see that p(x) is a decreasing
function of xas long as ¯
λ2(x) is increasing. Now
B(γ) = lim
x→∞ B(x)= 1−D(γ)
U(γ)−D(γ),
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 969
and
p(M) = lim
x→∞ p(x)= λ1
λ1+M.
For a ny 0 <γ<λ
uwe can choose Mlarge enough so that there is β1>0such
that p(M)<B(γ)−β1. For any 0 <β<β
1we can then define x0=infH, H =
{x≥0:p(x)<B(x)−β}.His non-empty and bounded from below by 0 so we
have x0≥>0. For x>x
0we have in (2.17) PξV(x)<V(x)(1 −β), therefore the
drift condition (2.15) holds. To show that it also holds for 0 ≤x≤x0, we only need
that [PξV](x) is bounded there, and this is easy to see from (2.17).
Theorem 2.2 (Theorem 15.0.1 [20]). Suppose that the chain ξis ψ-irreducible
and strongly aperiodic. Then the fol lowing conditions are equivalent
(1) The chain is positive recurrent with invariant probability measure π, and there
exists some ν-small set Cwith ψ(C)>0,ρ
C<1and MC<∞,and P∞(C)>0
such that for all x∈C:|Pn(x, C )−P∞(C)|≤MCρn
C.
(2) ThereexistssomesmallsetCwith ψ(C)>0and κ>1such that
supx∈CEx[κτC]<∞.
(3) There exists a small set C, constants b<∞,β>0andafunctionV≥1finite
at some x0∈Xsatisfying PξV(x)≤(1 −β)V(x)+b1C(x),x∈X.
Any of the three conditions imply that the set SV={x:V(x)<∞} satisfies
P(x, SV)=1for al l x∈SV.Hence,there exist constants r>1,R<∞such that
for any x∈SV
n
rn||Pn(x, ·)−π|| ≤ RV (x).(2.18)
Since the Markov chain ξsatisfies the general conditions of Theorem 2.2, and
since condition (2.3) is clearly satisfied, then (2.1) gives the existence of a unique
stationary distribution πθfor each θ∈Θ.
Proposition 2.4. For an y x∈X, πθ(x, ·)πθ(·).
Proof. To prove the statement we only need to show that the measure µ=dy +δ0
on R+is absolutely continuous with respect to πθsince Khas density fwith respect
to µ.
First we show that πθ({0})>0. Clearly, there is a set B∈B(X) such that
πθ(B)>0. Let xB=infB<∞.Then
πθ({0})≥B
πθ(dx)πθ(x, {0})≥B
πθ(dx)e−λdxB≥e−λdxBπθ(B)>0.
If [a, b) is an arbitrary semi-open interval contained in R+then
πθ([a, b)) ≥πθ({0})b
a
λue−λuydy > 0.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
970 R. Carmona & P. Diko
Corollary 2.1. If the first order conditions have a unique solution ˆ
θ, and if the
conditions of Theorem 2.1are satisfied,then ˆ
θis a consistent MLE estimate of the
true parameter θ0.
It is clear from the computations preceding Theorem 2.1 that λ1,λuand λdare
uniquely determined. Proposition 2.1 shows the uniqueness for the other parameters.
One remaining open question is whether the system (2.4) and (2.5) has a solution
at all. It is addressed in the following proposition.
Proposition 2.5. With probability one,for nlarge enough,the system of equations
(2.4) and (2.5) has a solution.
Proof. To prove the statement we show that limn→∞ g(0) >0 almost surely. Since
g(0) = B
|D|
i∈D
κ(ξi)−C=
i∈D
βi+1
1
|D|
i∈D
κ(ξi)−
i∈D
κ(ξi)βi+1
+
i∈U
βi+1
1
|D|
i∈D
κ(ξi)−
i∈U
κ(ξi)βi+1,
then the law of large numbers implies that
lim
n→∞
1
|D|
i∈D
βi+1
1
|D|
i∈D
κ(ξi)−
i∈D
κ(ξi)βi+1
=E(ρ2|z1,z
1−z2>0)E(κ(z1)|z1−z2>0)
−E(ρ2|z1,z
1−z2>0)E(κ(z1)|z1−z2>0) = 0,
and
lim
n→∞
1
|U|
i∈U
βi+1
1
|D|
i∈D
κ(ξi)−
i∈U
κ(ξi)βi+1
=E(ρ2|z1,z
1−z2<0)E(κ(z1)|z1−z2>0)
−E(ρ2|z1,z
1−z2<0)E(κ(z1)|z1−z2<0) >0.
Combining the two we get
lim
n→∞ g(0) = ∞>0.
By this we have established that the MLE estimator which we derived, possesses
all the properties stated in Theorem 2.1.
As mentioned above, in order to get a complete rainfall intensity model the tem-
poral dynamics of storms’ arrivals and durations have to be specified. The inspira-
tion can be taken from the BLPC model which assumes the storms arrive according
to a Poisson process and last for a random (exponentially distributed) time; notice
that they can overlap. In the framework of the proposed model the random number
of storms active at each moment manifests itself in the random cell arrival rate.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 971
This can be captured by replacing the constant cell arrival rate parameter λ1by
a stochastic process ¯
λ1. If the dynamics of the rainfall process conditional on the
realization of stochastic process ¯
λ1is given by (2.1) then MLE can be performed
separately for parameters of the dynamics of ¯
λ1. This gives us freedom to choose a
model for ¯
λ1for which a valid estimation procedure (preferably MLE) is feasible.
A simple model whose MLE is well established is that of switching Markov
process for ¯
λ1. The state space consist of two states {0,λ
1}with transition rates qd
and qp. The ML estimators for λ1,qd,qpin this model have been presented in [1].
Assuming one observes the inter-arrival times U=(U1,U
2,...,U
n) of a Cox process
Ndirected by a switching Markov process M, the likelihood function can be shown
to have the following form
L(U, λ1,q
d,q
p)=
n
i=1
f(Ui,λ
1,q
d,q
p),
f(x, λ1,q
d,q
p)= 1
r2−r1(qd+qp−r1)r1e−r1x−(qd+qp−r2)r2e−r2x
r1=1
2(qd+qp+λ1)− 1
4(qd+qp+λ1)2−l1qd
r2=1
2(qd+qp+λ1)+ 1
4(qd+qp+λ1)2−l1qd.
Once the parameters of ¯
λ1are estimated, the MLE method for estimating the
remaining parameters described above can be used, due to the separation properties
of the likelihood function mentioned above.
2.4. Comparison with Rodriguez-Iturbe’s model
In this section we check numerically that our Markov jump model (MJ) leads to
properties quite analog to those of the widely accepted Poisson pulse (PP) model
of Rodriguez-Iturbe.
We choose three main characteristics of the precipitation to compare the two
models: duration of dry spells, duration of wet spells, and accumulated rainfall
volume during a single wet spell. We perform the comparison at the level of a
single storm because that is where the models differ. We show that the probability
distributions of these quantities are very similar in both MJ and PP models. These
distributions are not explicitly known either for the PP model [25] or for our MJ
model, therefore we rely on Monte Carlo simulations to generate samples from
the two models and compare the resulting empirical distributions of the relevant
characteristics.
In order to set up a reasonable comparison, we have to resolve the issue of
the different parameter sets driving the two models. This calibration can easily be
done because the three parameters of Rodriguez-Iturbe model λ— cells arrival
rate, µx— average cell intensity and η— cell extinction rate can be naturally
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
972 R. Carmona & P. Diko
linked to the jump Markov model in a nearly one-to-one fashion as follows: λ1=λ,
1/λu=1/λd=µx,λ(II)
2=η.
Although the choice of exact parameter values is not important for our compar-
ison, to obtain realistic values for λ,µxand η, we fit PP model to NY Westchester
January data using a form of the method of moments suggested in [25]. Based on
that, we set: λ= 180, µx= 1500, η= 400 and generate 360 months worth of
precipitation data from each model. We set the extra parameter λ(I)
2= 300 so that
the empirical average wet spell length in both models coincide. Samples from both
models are shown in Fig. 1 for the purpose of illustration. There is no obvious dif-
ference in the statistics of the precipitation patterns they produce. As evidenced by
the Q-Q plot in the center pane of Fig. 2, the distributions of the lengths of the dry
spells seem to coincide. The Q-Q plot in the right pane shows that the distributions
of the rainfall volumes also agree in the two models, though to a lesser extend in the
case of the simulations at hand. Finally, the Q-Q plot in the left pane shows that, at
least in the case of the Monte Carlo samples which we generated, the distribution of
the length of a typical wet spell could have heavier tail in the Markov jump model.
0.00 0.05 0.10 0.15 0.20
0 1000 3000 5000 7000
Time
Intensity
Markov Jump model simulation sample
0.00 0.05 0.10 0.15 0.20 0.25
0 2000 4000 6000 8000 10000
Time
Intensity
Poisson pulse model simulation sample
Fig. 1. Monte Carlo samples from the Markov jump model (left) and from the Poisson pulse model
(right).
0.000 0.005 0.010 0.015 0.020 0.025
0.000 0.010 0.020 0.030
MJ
PP
Wet spell duration
0.000 0.005 0.010 0.015 0.020 0.025 0.030
0.000 0.010 0.020 0.030
MJ
PP
Dry spell duration
020 806040
0 20406080
MJ
PP
Rainfall volume per wet spell,
MJ:1049, PP:1128
Fig. 2. Q-Q plots of the wet (left) and dry (right) spell durations from the Markov jump and
Poisson pulse models.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 973
However, the bulk of the data seems to be in a region where the two distributions
agree.
3. Pricing Precipitation Options
3.1. Security price as random evolution
We generalize the standard geometric Brownian motion security price model by
allowing the coefficients µand σto depend on another process Y. Formally we
assume that the dynamics of the traded asset Sare given by a stochastic differential
equation of the form
dSt=St(µ(Yt)dt +σ(Yt)dWt),(3.1)
where the process Y={Yt}is a continuous time Markov process in Rnindependent
of the Wiener process W. Models of this form are common in financial applications:
regime switching models, stochastic volatility models. Also, this is the model used
in [3] to price temperature options. The conditions for the existence and uniqueness
of the solution to (3.1) are discussed in [11]. The relevant results for our purposes
are stated in Theorem 3.1 and its corollary. The pair (St,Y
t) forms a Markov process
which is known in the literature as random evolution.
Theorem 3.1. LetthereexistaconstantKand real functions g(t, y),h(t, y)
such that
|µ(t, x1,y)−µ(t, x2,y)|≤K|x1−x2|
|σ(t, x1,y)−σ(t, x2,y)|≤K2|x1−x2|(3.2)
1
2|σ(t, x, y)|2≤g(t, y)2+K2|x|2
|µ(t, x, y)|≤h(t, y)+K2|x|(3.3)
and for all m=1,2,3,... :E!T
0(g(s, Ys)m+h(s, Ys)m)ds<∞then (3.1) with
the initial data X0=xhas a pathwise unique solution and it is Ft-progressively
measurable.
Corollary 3.1. If Yis real valued and for every m>0
ET
0
|Ys|mds<∞
then conditions (3.3) are equivalent to
|µ(t, x, y)|≤C(|x|+|y|)mand |σ(t, x, y)|≤C(|x|+|y|)m.
3.2. Utility maximization as a stochastic control problem
We now formalize the problem described in the introduction. We are given a filtered
probability space (Ω,F,{Ft}0≤t≤T,P). Ftis the sigma field σ((Ss,Y
s):0≤s≤t)
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
974 R. Carmona & P. Diko
augmented by the null-sets of F=FT. Trading is allowed in a risky asset whose
price process is a random evolution (St,Y
t) defined in Sec. 3.1 and a bank account
for which we assume that the interest rate is zero, and we assume that the agent
possesses initial wealth x. We look for a self financing trading strategy φ—the
amount of money invested in the risky asset — that maximizes the utility of the
terminal wealth. For any strategy, the dynamics of the wealth process are given by
dXt=φt(µ(Yt)dt +σ(Yt)dWt) (3.4)
and the objective is to maximize
E[U(XT)] (3.5)
Clearly, for any admissible strategy, the pair (X,Y ) also forms a random evolution.
The problem of maximizing (3.5) for the process (3.4) above falls in the framework
of controlled Markov processes. As such, it can be solved by dynamic programming,
and the conditions for the existence of the optimal solution are given in verification
theorems [11].
3.3. HJB equation for random evolutions
Candidates for the solution of problem (3.5) among Markovian controls can be found
by solving the corresponding HJB equation. The controlled backward evolution
operator Auof the random evolution (X, Y )reads
AuV(t, x, y)= ∂V
∂t +uµ(y)∂V
∂x +1
2u2σ2(y)∂2V
∂x2+GV (t, x, y),(3.6)
where Gis the infinitesimal generator of the process Y. Hence the HJB equation
for the value function Vof the optimization problem (3.5) is
0=sup
u
AuV=∂V
∂t +GV +sup
u"uµ(y)∂V
∂x +1
2u2σ2(y)∂2V
∂x2#.(3.7)
We can perform the maximization with respect to uexplicitly to get the optimal
control u∗and an integro-differential HJB equation
u∗=−µ(y)
σ2(y)
∂V/∂x
∂2V/∂x2,(3.8)
0=∂V
∂t +GV −1
2
µ2(y)
σ2(y)
(∂V/∂x)2
∂2V/∂x2.(3.9)
The terminal condition for the backward HJB equation (3.9) is provided by the
utility function U,namely:V(T,x,y)=U(x). If we assume the process Yis the
jump Markov process storm model developed in Sec. (2.2) we obtain in particular
0= ∂V
∂t +λ(y)R
[V(t, x, z)−V(t, x, y)]Π(y, dz)−1
2
µ2(y)
σ2(y)
(∂V/∂x)2
∂2V/∂x2,(3.10)
where λis the jump rate function and Π is the jump transition kernel for the
process Y.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 975
In order for a solution Vof the HJB equation to be the value function of the
stochastic control problem it needs to satisfy the conditions of the verification theo-
rem [11]. In particular it needs to belong to Dthe common domain of the backward
evolution operators Au;u∈U. Before we proceed to obtain explicit solutions to
(3.9) we characterize the set Dfor the Eq. (3.10).
Proposition 3.1. The domain Dof the HJB equation (3.10) contains all continu-
ous functions Ψ(t, x, y)defined on [0,T]×R2,such that:
(1) ∂Ψ/∂t, ∂Ψ/∂x, ∂2Ψ/∂x2are continuous.
(2) Ψ,∂Ψ/∂t, ∂Ψ/∂x, ∂2Ψ/∂x2have polynomial growth,i.e.,for some Kand m
|Ψ(t, x, y)|+|∂Ψ/∂t(t, x, y )|+|∂Ψ/∂x(t, x, y)|+|∂2Ψ/∂x2(t, x, y)|
≤K(1 + |x|+|y|)m.
Proof. We will show that for all u∈Uany Ψ satisfying the conditions of the
proposition belongs to the domain D(Au) of the operator Au(along the lines of [11],
Appendix B). Let us fix an arbitrary u∈U. For all Ψ satisfying the conditions 1
and 2 the operator Auis well defined. By condition 1 all Ψ, ∂Ψ/∂t and AΨarecon-
tinuous functions. Condition 2 guarantees that for any s>t:Etx|Ψ(s, Xs,Y
s)|<∞
and !s
t|AΨ(r, Xr,Y
r)|<∞. It remains to show that such functions satisfy
the Dynkin formula Etxy[Ψ(s, Xs,Y
s)] −Ψ(t, x, y)=Etxy !s
tAuΨ(r, Xr,Y
r)dr;
t<s≤T
We can decompose the operator Auin two parts AuΨ=Au
0Ψ+HΨwhere
Au
0Ψ(t, x, y)= ∂Ψ
∂t µ(y)∂
∂xf +1
2σ2(y)∂2
∂x2f,
HΨ(t, x, y)=R
[Ψ(t, x, z)−Ψ(t, x, y)]Π(y, dz).
Let t1<t
2<··· <t
Nbe the jump times of the process Yon [t, s]andletusset
t0=tand tN+1 =s.Then
Ψ(s, Xs,Y
s)−Ψ(t, x, y)=
N
i=0
[Ψ(ti+1,X
ti+1 ,Y
ti)−Ψ(ti,X
ti,Y
ti)]
+
N
i=1
[Ψ(ti,X
ti,Y
ti)−Ψ(ti,X
ti,Y
ti−1)].
Since on Yis constant on [ti,t
i+1)fori=0,1,...,N by the results known for Ito
diffusions Ψ(ti+1,X
ti+1 ,Y
ti)−Ψ(ti,X
ti,Y
ti)=!ti+1
tiAu
0Ψ(r, Xr,Y
r)dr. Hence
Etxy N
i=0
[Ψ(ti+1,X
ti+1 ,Y
ti)−Ψ(ti,X
ti,Y
ti)]$=s
t
Au
0Ψ(r, Xr,Y
r)dr.
To finish the proof we notice, that we can write
Eti,y[f(ti+1)]=λ(y)Etiti+1
ti
f(t)dt,
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
976 R. Carmona & P. Diko
for a continuous function f(t). Then
Eti−1,y[Ψ(ti,X
ti,Y
ti)−Ψ(ti,X
ti,y)]
=Eti−1,y[Eti[Ψ(ti,X
ti,Y
ti)−Ψ(ti,X
ti,y)]]
=Eti−1,y[HΨ(ti,X
ti,y)/λ(y)]
=Etiti
ti−1
HΨ(t, Xt,Y
t).
Implicitly we conditioned by the Wiener process Wwhich is however independent
of Ytherefore we omitted it in the notation. Now using the strong Markov property
of Ywe get
Etxy N
i=1
[Ψ(ti,X
ti,Y
ti)−Ψ(ti,X
ti,Y
ti−1)]$=Etxy s
t
HΨ(r, Xr,Y
r)dr).
3.4. Optimal hedging strategies
In this subsection we solve the HJB equation (3.9) in the case of the exponential
utility function U(x)=−e−αx with α>0. We can get a explicit solution in this case
by linearizing the HJB equation by a Hopf–Cole transformation. We hypothesize
the form of the solution as V(t, x, y)=−e−αx F(t, y). Substituting in (3.9) we get
F(t, y)1
2
µ2(y)
σ2(y)=∂F
∂t (t, y)+GF, (3.11)
with the terminal condition F(T, y) = 1. The solution of this equation is given by
the so-called Feynman–Kac formula. Indeed, its right hand side is the backward
evolution operator of Y(the justification will be given below). Such a formula
already appeared in [3] where the zero order term found in the exponential was
called the traded risk premium. The solution is
F(t, y)=Ety exp −1
2T
t
µ2(Ys)
σ2(Ys)ds.
Substituting out we see that the candidate for the expression of the value function
solving the HJB equation (3.9) with exponential utility is
V(t, x, y)=−e−αxEty exp −1
2T
t
µ2(Ys)
σ2(Ys)ds,
the corresponding optimal trading strategy being given by the time varying Sharpe
ratio
φt=µ(Yt−)
ασ2(Yt−),
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 977
where Yt−= limh0Yt−his the left hand limit assuring the predictability of the
trading strategy. To check that the candidate solution is indeed the optimal solution
of the optimization problem we need to check if it satisfies the verification theorem
[11], and hence is a bona-fide optimal solution of the original problem. We will
proceed to do this for the case when Yis the jump Markov storm model.
Proposition 3.2. If σ(y)>>0for all ythen the trading strategy φis the optimal
trading strategy in the utility maximization problem with Ybeing the jump Markov
process storm model.
Proof. We first show that the strategy φhas all moments and therefore is
admissible:
ET
0
|φt|mdt=ET
0%%%
µ(Yt)
σ2(Yt)α%%%dt
≤KET
0
|Yt|mdt
=(KT)E1
TT
0
|Yt|mdt
<(KT)E[|Y∞|m],
where the last inequality follows from the ergodicity and existence of all moments of
the process Y(see Theorem 2.2). It remains to show that V∈D. We will show that
Vsatisfies the conditions of Proposition 3.1. It is easy to see that all the conditions
are satisfied if and only if
Ety exp −1
2T
t
µ2(Ys)
σ2(Ys)ds<C|y|m,
which is obvious since the left hand side is bounded.
We can solve the HJB equation (3.9) along the same lines in the case of the power
utility function U(x)=xα/α with α<1. Again we search for a value function in the
form V(t, x, y)=(xα/α)F(t, y), and substituting out we get the linearized equation
F(t, y)α
2(1 −α)
µ2(y)
σ2(y)=∂F
∂t (t, y)+GV, (3.12)
with terminal condition F(T,y) = 1. Again, invoking the Feynman–Kac formula we
get the solution
F(t, y)=Ety exp α
2(1 −α)T
t
µ2(Ys)
σ2(Ys)ds,
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
978 R. Carmona & P. Diko
and after substitution, we get the candidate for the solution of (3.9) with power
utility to be
V(t, x, y)= xα
αEty exp α
2(1 −α)T
t
µ2(Ys)
σ2(Ys)ds,
with the corresponding candidate
φt=µ(Yt−)
σ2(Yt−)(1 −α)Xt,
for trading strategy. Verification can be done along the same lines as with exponen-
tial utility.
3.5. Utility maximization with a derivative
We are ready to move onto maximizing the expected utility with a derivative written
on the underlying process Y. In precipitation market practically all derivatives have
Asian structure since the instantaneous precipitation intensity has little practical
importance. Two main types of precipitation derivatives can be encountered. The
payoff can depend upon the amount of rain during a given time period f1(Y)=
!t
tYtdt −K+, or upon the amount of time it rains (does not rain) during a given
time period f2(Y)=!t
t1{Yt>}dt −K+,whereis the minimal precipitation
intensity that constitutes a “rainy day”. For simplicity we first assume K=0,
later we will show how we can extend our results to the case K>0. Under this
simplifying assumption both payoffs can be expressed under a common framework
ξ=!t
th(Ys)ds 0≤t≤t ≤T. For type 1 derivative h(y)=y,fortype2in
turn h(y)=1
(,∞)(y). The results below also hold for any function h≥0with
polynomial growth.
When maximizing the utility two points of view have to be considered: the
buyer’s and the seller’s. The buyer’s ob jective function is: E[U(XT+ξ)]. The seller’s
in turn is: E[U(XT−ξ)]. We will solve only the buyer’s problem. The sellers prob-
lem can be solved along the same lines. We transform the optimal portfolio problem
with the derivative present to an equivalent problem of optimal investment with ran-
dom endowment — for which a rather extensive body of literature exists [7, 9, 17].
Proposition 3.3. The solution (φ∗,V∗)of buyer ’s problem is the same as the
solution (ˆ
φ∗,ˆ
V∗)of the following random endowment optimal investment problem
max
φEU(ˆ
XT),
dˆ
Xt=g(t, Yt)dt +ˆ
φt(µ(Yt)dt +σ(Yt)dWt),(3.13)
where g(t, y)=1
(t,t)(t)h(y).
Proof. Let φbe an admissible strategy for the buyer’s problem. Let us denote
V(φ), X(φ), ˆ
V(φ), ˆ
X(φ) the value functions and wealth processes in the original
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 979
problem and (3.13) respectively corresponding to the strategy φ. It is clear that φ
is an admissible strategy for problem (3.13) and V(φ)≤ˆ
V(φ)≤ˆ
V∗and therefore
V∗≤ˆ
V∗. Now let us consider the optimal strategy ˆ
φ∗for the problem (3.13) and
define a new strategy ψ=(ψ0,ψ
1); it is necessary to look at both components
of the strategy, the amount of money invested in the risky asset and the riskless
bond. Let ψ1=(
ˆ
φ∗)1and ψ0
t=(
ˆ
φ∗)0
t−!t
0g(s, Ys)ds.Then ˆ
X(ψ)satisfiesdˆ
Xt=
ψ1
t(µ(Yt)dt +σ(Yt)dWt) and therefore XT(ψ)= ˆ
XT(φ∗)−!T
0g(t, Yt)dt and hence
V∗≥V(ψ)=EUXT(ψ)+T
0
g(t, Yt)dt=E[U(XT(φ∗))] = ˆ
V∗.
Proposition 3.4. The solution (φ∗,V∗)of seller’s problem is the same as the
solution (ˆ
φ∗,ˆ
V∗)of the following random endowment optimal investment problem
max
φE[U(ˆ
XT)],
dˆ
Xt=−g(t, Yt)dt +ˆ
φt(µ(Yt)dt +σ(Yt)dWt),(3.14)
where g(t, Yt)=1
(t,t)(t)h(Yt).
The proof of this proposition goes along the same lines as Proposition 3.3.
Now we can write the HJB equation for problem (3.13)
0= ∂V
∂t +∂V
∂x g(t, y)+GV +sup
u"uµ(y)∂V
∂x +1
2u2σ2(y)∂2V
∂x2#.(3.15)
We can perform the maximization with respect to uexplicitly to get the candidate
optimal decision u∗and explicit HJB equation
u∗=−µ(y)
σ2(y)
∂V/∂x
∂2V/∂x2,(3.16)
0=∂V
∂t +∂V
∂x g(t, y)+GV −1
2
µ2(y)
σ2(y)
(∂V/∂x)2
∂2V/∂x2,(3.17)
with the terminal condition V(T, x,y)=U(x). Specifically for the jump Markov
process storm model (3.17) becomes
0=∂V
∂t +∂V
∂x g(t, y)+λ(y)R
[V(t, x, z)−V(t, x, y)]Π(y, dz)−1
2σ2(y)(∂V/∂x)2
∂2V/∂x2.
(3.18)
The domain Dfor this equation is the same as for Eq. (3.9).
The exponential utility allows us to separate variables in the equation.
As before we search for an expression of the value function in the form
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
980 R. Carmona & P. Diko
V(t, x, y)=−e−αxF(t, y). The HJB equation then reduces to
F(t, y)αg(t, y)+ 1
2
µ2(y)
σ2(y)=∂F
∂t (t, y)+GF, (3.19)
with terminal condition F(T,y) = 1, and using the Feynman–Kac representation
formula we obtain
F(t, y)=Eye−RT
tαg(s,Ys)+ 1
2
µ2(Ys)
σ2(Ys)ds.
Thus the candidate solution is
V(t, x, y)=−e−αxEye−RT
tαg(s,Ys)+ 1
2
µ2(Ys)
σ2(Ys)ds,
with the corresponding trading strategy φt=µ(y)/(σ2(y)α). We have a verification
result.
Proposition 3.5. If σ(y)>>0for all ythe trading strategy φis the optimal
trading strategy in the utility maximization problem (3.13) with Ybeing the jump
Markov process storm model.
The proof of this verification result is analogous to the proof of Proposition 3.2.
The situation is more complicated in the case of power utility. It does not possess
the convenient separability property of the exponential utility therefore adding the
derivatives causes the separation of variables argument to fail. A lot of attention has
been given to this problem. For example, existence of the solution has been proved
in a very general setting in [7]. Henderson [14] derived lower and upper bounds for
the value function in the case where the derivative payoff is small compared to the
wealth of the agent. Her approach is based on convex duality (to obtain the upper
bound) and expansion of the utility around the no-derivative solution (to get the
rate of convergence of the lower and an upper bound). We will apply the approach
of Henderson, to obtain bounds on the value function in our case.
•Lower Bound. A trivial lower bound is given by the value function of the problem
without the derivative since the derivative payoff is always non-negative. In the
present situation, we can improve this bound by using Proposition 3.3. We use the
optimal strategy from the problem without the derivative (3.5) in the transformed
random endowment problem (3.13). An elementary calculation shows that the
value function for this strategy is
VL(0,x,y)= 1
αEy"exp T
0
µ2(Yt)
σ2(Yt)
1
1−α1−1
2(1 −α)dt
×x+T
0
g(t, Yt)exp−t
0
µ2(Ys)
σ2(Ys)
1
1−α1−1
2(1 −α)ds
−1
1−αt
0
µ(Ys)
σ(Ys)dWs#α
,
which by Proposition 3.3 is greater than the no-derivative value function.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 981
•Upper Bound. We get an upper bound by employing the convex duality theory.
Started by Karatzas et al. [16] in the Ito processes setting, it was extended by
Cvitanic [7] to the more general semimartingale setting. Our upper bound is a
direct consequence of the main result of [7] when we choose a particular equivalent
martingale measure Q0. A natural candidate is the one suggested by Girsanov’s
theorem. It is defined by its density with respect to P
dQ0
dP=exp
−T
0
µ(Ys)
σ(Ys)dWs−1
2T
0
µ2(Ys)
σ2(Ys)ds.
Clearly the measure Q0makes the traded asset price Sand therefore the wealth
process X, a martingale. Moreover, it is equivalent to P. Hence, it is in ∆. We
can get the value of the dual objective function for this measure
E˜
UydQ
dP=1−α
αyα
α−1Awhere
A=Eexp 1
2
α
(1 −α)2T
0
µ2(Ys)
σ2(Ys)ds.
Therefore: D(y)=1−α
αyα
α−1A+y(x+E0[ξ]), and the upper bound is obtained
by minimizing D(y)overy.Weget
1−α
α(x+E0[ξ])αA1−α=V0(t, x, y)1+ E0[ξ]
x1−α
,(3.20)
where V0is the value function of the problem (3.5) with no derivative present.
It is easy to see that these bounds converge as the wealth level becomes large
compared to the derivative payoff.
3.6. The indifference prices
As we mentioned in the introduction the solution to the optimization problems
without a derivative claim and with it allow us to set a price on the derivative claim
by the principle of indifference. Let us denote V0the value function of (3.5), Vξ
the value function of (3.13) and pthe price of the derivative claim with payoff ξ.
The indifference principle states V0(0,x+p, y)=Vξ(0,x,y). Solving this implicit
equation for pwill allow us to price the derivative. We will be able to give quite an
explicit formula in the exponential utility case. With the power utility we will only
be able to give bounds on the price, as expected.
Using the results of Sec. 3.4 and plugging the value functions in the indifference
equation we obtain for the exponential utility
p=1
αln
Eye−RT
0
1
2
µ2(Ys)
σ2(Ys)ds
Eye−RT
0αg(s,Ys)+ 1
2
µ2(Ys)
σ2(Ys)ds.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
982 R. Carmona & P. Diko
Notice that this value of pis independent of the initial wealth. This is very specific to
the particular choice of the exponential utility function. Also pis in practically closed
form since it can be easily evaluated by Monte Carlo simulation of the precipitation
process Y, whose sample paths can be simulated without error.
In the case of power utility, we get bounds based on the bounds on the value
function
αVL(0,x,y)
EtyeRT
t
α
2(1−α)
µ2(Ys)
σ2(Ys)ds
1/α
−x≤p≤x
1−
1+
Ey!T
0g(t, Yt)dtdQ0
dP
x
α−1
α
.
3.7. Derivatives with non-zero strikes
To simplify the notation, in the previous sections we only considered derivatives
with the payoff ξof the form: ξ=!t
th(Ys)ds 0≤t≤t ≤T. We will show that
our results directly extend to the payoffs of the form ξ=!t
th(Ys)ds −K+0≤
t≤t ≤Twhere the derivative starts paying off only after a certain strike amount
K>0 is reached. To deal with this case let us define a process Rt=!t
th(Ys)ds for
t≥tand 0 otherwise and a payoff rate function ˆ
h(Yt,R
t)=1
{Rt>K}h(Yt). It is
easy to see that the payoff ξcanbewrittenintermsofˆ
has ξ=!t
tˆ
h(Ys,R
s)ds 0≤
t≤t ≤T.
We were able to eliminate the strike Kat the expense of increasing the dimension
of the underlying by one. Let us denote by ˆ
Y=(Y,R) this new underlying process.
The optimization problem we are facing now is as follows
max
φE[U(XT)],(3.21)
subject to: dXt=ˆg(t, ˆ
Yt)dt +φt(µ(Yt)dt +σ(Yt)dWt)whereˆg(t, y, r)=
1(t,t)(t)ˆ
h(y, r)). Clearly ˆ
Yis a random evolution with infinitesimal generator
ˆ
Gf(y, r)=G+∂f
∂r g(t, y). Because ˆ
Yis a random evolution and hence a Markov
process, we can apply the theory developed in previous sections to the solution of
problem (3.21) and obtain equivalent results.
For the exponential utility the value function is
V(t, x, y, r)=−e−αxEy,r e−RT
tαˆg(s, ˆ
Ys)+ 1
2
µ2(Ys)
σ2(Ys)ds,
with the corresponding trading strategy: φt=µ(y)
σ2(y)αand the indifference price
p=1
αln
Eye−RT
0
1
2
µ2(Ys)
σ2(Ys)ds
Ey,re−RT
0αˆg(s, ˆ
Ys)+ 1
2
µ2(Ys)
σ2(Ys)ds.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 983
For the power utility we again get bounds based on the bounds on the value function
αVL(0,x,y)
Ety eRT
t
α
2(1−α)
µ2(Ys)
σ2(Ys)ds
1/α
−x≤p≤x
1−
1+
Ey!T
0ˆg(t, ˆ
Yt)dtdQ0
dP
x
α−1
α
.
4. A Practical Application
In this section we apply the techniques developed in this paper to price rainfall
sample contracts which we chose for the purposes of this study. We chose a location
in Norway and electric power prices because Norway produces most of its electricity
from hydroelectric powerplants (hence power can serve as traded asset in our model)
and because of the success of deregulation and the availability of wholesale electricity
prices in the NordPool.
Consider a sample contract: a call option on the amount of rainfall during June
and July 2004, as recorder by Bergen weather station in Norway with a strike of
K mm and tick price 1 NOK, i.e., the seller of the contract pays to the buyer one
NOK for each millimeter of cumulative rainfall above K mm during June and July
2004 in Bergen, Norway.
We need to fit the jump Markov model to rainfall in Bergen, find a traded asset
influenced by the amount of rainfall in Bergen in June and July, fit a model of the
type considered in the paper, determine the exact nature of the functions µand σ
in (3.1), and calibrate the model to empirical data. The Norwegian Meteorological
Institute kindly provided us with high-frequency rainfall intensity data records from
the “pluviometer” equipped weather station in Bergen for calendar year 2002. Plu-
viometers record times (with one minute precision) of tips of a bucket with known
(small) volume, and therefore provides practically continuous time observations of
the rainfall intensity process. A time series plot of these data is given in Fig. 3. We
examined the influence of the amount of rainfall on prices of several instruments
liquidly traded on the NordPool power exchange, looking for a strong relationship
between Fourth Quarter 2002 forward power contract and rainfall in Bergen dur-
ing the course of the whole year. NordPool is a name of a common electric power
exchange of four Nordic countries: Norway, Sweden, Finland and Denmark, estab-
lished after the liberalization of their power markets. Nowadays it is one of the
most liquid European power exchanges offering spot and forward electricity prod-
ucts. Finally we calibrated the functional form of µand σto the data, and we used
Monte Carlo computations to arrive at utility indifference prices for the contracts.
The details of the exercise follow.
4.1. Bergen rainfal l model
Rainfall data from May to July 2002 were selected to fit our rainfall model. The
period was chosen to closely follow the underlying period of the weather derivative
since the rainfall characteristics vary over the course of the year [24]. The data are
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
984 R. Carmona & P. Diko
0 500 1000 2000
Time
Intensity mm/day
May Jun Jul Aug
Fig. 3. Bergen, Norway May–July 2002 rainfall intensity.
Table 1. Parameter estimates for Bergen data.
Parameter Value C.I. (95%)
λ178.6day
−1see text
qd0.85 day−1see text
qp5.03 day−1see text
λ(I)
20.00 day−1(−0.081, 0.081)
λ(II)
22.82 mm−1(2.702, 2.935)
λu0.012 day mm−1(0.0116, 0.0131)
λd0.011 day mm−1(0.0102, 0.0118)
plotted in Fig. 3. The maximum likelihood parameters estimates for the multiple
storm model fitted to these data are listed in Table 1. We computed approximate
confidence intervals using result 6 of Theorem 2.1. We could not obtain confidence
intervals for parameters of the process ¯
λ1only for the mean of its limiting distri-
bution. The maximum likelihood mean estimate is 11.36 with approximate 95%
c.i. (10.68,12.05). The confidence intervals are quite tight given the short period
of data used for fitting the model. This is because of the very high time resolution
of the data (the sample length was 2150). Due to the measurement methodology of
pluviometers, the rainfall intensity is always greater than 0. The cutoff level under
which the rainfall intensity is considered to be 0 was set to 1 mm/day.
4.2. The influence of Bergen’s rainfall on NordPool power prices
We examined several products traded on NordPool power exchange to see whether
their prices were influenced by rainfall in Bergen. Of the examined products forward
Fourth Quarter Baseload 2002 (Q402) seemed to be the electricity product most
influenced by rainfall during the summer period. Fourth Quarter 2002 Baseload
forward contract is a contract for delivery of constant power capacity on Norwegian
power grid during 24 hours, every day of the period starting 1/10/2002 ending
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 985
Fig. 4. Q402 NordPool forward price and Bergen, Norway January–July 2002 rainfall intensity
(left) and Bergen log-rainfall-intensity versus Q402 log-returns.
31/12/2002. The price of this contract is reported with daily granularity. The plot
of precipitation intensity together with daily prices of Q402 is given in Fig. 4. It is
clear from these plots that there is an apparent upwards trend in the prices during
dry periods and downward trend during wet periods. A commonly used model for
electricity forward prices in the energy derivatives literature (see for example [5])
is the geometric Brownian motion described by the stochastic differential equation
(3.1). To complete the model we need to express the relationship between power
price and rainfall via the functional forms of the drift and volatility coefficients µ
and σof the stochastic differential equation (3.1) driving the dynamics of the Q402
forward contract. Let us denote by Fdthe price of Q402 contract on a day d,andlet
us denote by Rdthe aggregate amount of rainfall in Bergen on that day. We estimate
the drift coefficient of ¯µdon day dby the average of Q402 log-returns on over an
11 day window centered on day d. Accordingly, we estimate the rainfall intensity
¯
Rdon the same day by the average rainfall over the same period. Special treatment
is necessary for weekends. They are considered as one day and are included in the
rainfall intensity average and not in price log-returns average as there are no prices
reported on weekends. The relationship between ¯
Rdand ¯µdis non-linear. Taking
logarithm of ¯
Rdmakes the relationship linear as we can see from Fig. 4. From this
analysis, we obtain a functional dependence of Q402 drift coefficient on the rainfall
intensity: µ(y)=a∗ln(+y)+bwhere represents the cutoff level for no rainfall
a,bare obtained by ordinary least squares regression. The values of the three
parameters together with the estimate of the volatility are given in Table 2. (Where
relevant, the confidence intervals are given.) Rather wide confidence intervals are
a direct consequence of scarcity of data. The power markets are rather young and
substantial historical price track record has not been generated yet.
4.3. Utility indifference price for Bergen rainfall call
We use the price formula derived in Sec. 3.7 and we rely on Monte Carlo simulation
of the rainfall process to compute the expectations and put a price on the sample
contract. Since we derived exact pricing formulae in this case, we assume that buyer
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
986 R. Carmona & P. Diko
Table 2. Parameter estimates for Q402 drift
and volatility.
Parameter Value C.I. (95%)
a−0.55 (−0.67,−0.43)
b0.40 (0.20,0.60)
0.10 N/A
σ0.20 (0.18,0.22)
Table 3. Bergen rainfall pricing results.
Strike Risk Utility Indifference Price Seller’s Risk
Aversion Hedging Neutral
With Power Hedge Without Power Hedge Discount Price
Seller Buyer Seller Buyer
200 0.1 232.68 33.54 235.20 36.51 99% 145.97
0.01 158.63 117.26 167.00 124.85 95% 145.21
0.001 133.62 129.96 140.99 137.15 95% 139.05
250 0.1 167.90 27.94 169.87 29.95 99% 98.98
0.01 111.48 68.15 122.32 75.69 91% 96.07
0.001 84.16 80.90 91.19 87.82 92% 89.49
300 0.1 174.13 11.31 177.71 12.72 98% 57.29
0.01 60.81 34.12 70.52 39.01 86% 50.58
0.001 43.82 41.43 49.52 46.84 88% 48.16
350 0.1 162.90 4.37 167.38 5.18 97% 26.21
0.01 24.82 14.68 28.79 17.15 86% 21.92
0.001 27.82 25.90 34.17 31.86 81% 32.99
400 0.1 116.06 1.89 120.36 2.31 96% 9.09
0.01 6.37 3.60 8.20 4.61 78% 5.99
0.001 6.44 6.15 8.04 7.68 80% 7.86
450 0.1 14.53 0.36 17.01 0.48 85% 1.65
0.01 2.90 1.25 4.30 1.81 68% 2.65
0.001 3.27 2.97 4.98 4.50 66% 4.73
500 0.1 0.38 0.07 0.54 0.10 71% 0.20
0.01 0.38 0.30 0.55 0.43 70% 0.48
0.001 0.21 0.21 0.30 0.29 72% 0.30
and seller have exponential utility functions. Also, our bound on the seller’s price
would be indentically infinity which is another reason to work with exponential
utility. Table 3 lists the buyer/seller prices for various strike prices and risk aversion
levels. It compares them to the utility indifference prices without trading the power
contract, and shows the relative decrease in the price due to hedging with the
electricity contract. Also, we give a risk-neutral price under a measure which makes
price process Sa martingale and preserves the historical statistics of the rainfall
process Y. As expected the buyer’s price without hedging is lower than the risk-
neutral price while the seller’s is higher. The difference increases with increasing
risk aversion (i.e., concavity of utility function). Both sellers’s and buyer’s price
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
Pricing Precipitation Based Derivatives 987
with power hedging is lower than without it. The gap between buyer’s and seller’s
price decreases but remains positive. It is interesting to notice, however that for
low risk aversion levels, the seller price with hedging is lower than the buyer price
without hedging. Hence, if in reality a seller with knowledge and access to power
markets meets a buyer without the access or knowledge a mutually acceptable
weather-derivative deal can be done if the seller hedges herself in the power market.
A word of caution is in order at this point. Applying the method in practice
must be done with caution. The rainfall model parameters but mainly the rainfall
influence on traded asset price evolution cannot be estimated without error. As could
be seen from the approximate confidence intervals computation the uncertainty can
be as much as 20%, and the resulting uncertainty in price can be of the same order
of magnitude as the bid/ask spread.
Acknowledgment
The authors would like to thank an anonymous referee for constructive suggestions
which led to a significant improvement of the paper.
References
[1] C. W. Anderson, N. Mole and S. Nadarajah, A switching poisson process model for
high concentrations in short-range atmospheric dispersion, Atmospheric Environment
31(6) (1997) 813–824.
[2] P. Billingsley, Statistical Inference for Markov Processes (The University of Chicago
Press, 1961).
[3] R. Carmona and A. Danilova, Optimal utility hedging for markets with basis risk:
The case of temperature options, Technical Report, Princeton University, Princeton,
NJ (2003).
[4] R. E. Chandler, A spectral method for estimation parameters in rainfall models,
Bernoulli 3(3) (1997) 301–322.
[5] L. Clewlow and C. Strickland, Valuing energy options in a one factor model fitted to
forward prices, working paper (1999).
[6] P. S. P. Cowpertwait, A generalized point process model for rainfall, P roceedi ngs : of
the Royal Society in London, Series A 447 (1994) 23–37.
[7] J. Cvitanic, W. Schachermayer and H. Wang, Utility maximization in incomplete
markets with random endowment, Finance and Stochastics 5(2) (2001) 259–272.
[8] M. H. A. Davis, Optimal hedging with basis risk, Preprint.
[9] D. Duffie, W. H. Fleming, H. M. Soner and T. Zariphopoulou, Hedging in incomplete
markets with HARA utility, Journal of Economic Dynamics & Control 21(4–5)
(1997) 753–782.
[10] R. Carmona (ed), Indifference Pricing (Princeton University Press, 2005).
[11] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions.
Applications of Mathematics, Vol. 25 (Springer-Verlag, 1993).
[12] V. K. Gupta and E. C. Waymire, Multiscaling properties of spatial rainfall and river
flow distributions, Journal of Geophysical Research 95(D3) (1990) 1999–2010.
[13] V. K. Gupta and E. C. Waymire, A statistical analysis of mesoscale rainfall as a
random cascade, Journal of Applied Meteorology 32 (1994) 251–267.
October 19, 2005 11:1 WSPC-104-IJTAF SPI-J071 00331
988 R. Carmona & P. Diko
[14] V. Henderson, Valuation of claims on nontraded assets using utility maximization,
Mathematical Finance 12(4) (2002) 351–373.
[15] V. Henderson and D. G. Hobson, Real options with constant relative risk aversion,
Journal of Economic Dynamics & Control 27 (2002) 329–355.
[16] I. Karatzas, J. P. Lehoczky, S. E. Shreve and G. Xu, Martingale and duality methods
for utility maximization in an incomplete market, SIAM Journal of Control and
Optimization 29(3) (1991) 702–730.
[17] N. El Karoui and M. Jeanblanc-Picque, Optimization of consumption with labor
income, Finance and Stochastics 2(4) (1998) 409–440.
[18] L. LeCam, A stochastic description of precipitation, Proceedi ngs of the Fou rth B erke-
ley Symposium on Mathematical Statistics and Probability, Vol. 3 (1961), pp. 165–186.
[19] J. Mason, Numerical weather prediction, Proceedi ngs o f Roya l Soci ety i n London ,
Series A 407 (1986) 51–60.
[20] S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability (Springer-
Verlag, 1993).
[21] M. Musiela and T. Zariphopoulou, Indifference prices and related measures, technical
report (2001).
[22] M. Musiela and T. Zariphopoulou, Pricing and risk management of derivatives written
on non-traded assets, preprint (2001).
[23] M. Musiela and T. Zariphopoulou, An example of indifference prices under exponen-
tial preferences, technical report (2003).
[24] C. Onof, R. E. Chandler, A. Kakou, P. Northrop, H. S. Wheater and V. Isham, Rain-
fall modelling using poisson-cluster processes: A review of developments, Stochastic
Environmental Research and Risk Management 14 (2000) 384–411.
[25] I. Rodriguez-Iturbe, D. R. Cox and V. Isham, Some models for rainfall based on
stochastic point processes, P roceedin gs of th e Ro ya l S ociety in Londo n, Serie s A
410(1839) (1987) 269–288.
[26] I. Rodriguez-Iturbe, D. R. Cox and V. Isham, A point process model for rainfall:
Further developments, Proceed ing s o f the Ro yal S ocie ty in Lon don, Ser ies A 417
(1988) 283–298.
[27] D. S. Wilks and R. L. Wilby, The weather generation game: A review of stochastic
weather models, Progress in Physical Geography 23(3) (1999) 329–357.
[28] T. Zariphopoulou, A solution approach to valuation with unhedgeable risks, Finance
and Stochastics 5(2002) 61–82.
