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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.

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