QUANTITATIVE FINANCE RESEARCH CENTRE

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QUANTITATIVE FINANCE RESEARCH CENTRE

Research Paper 223 June 2008

Pricing Financial Derivatives on Weather

Sensitive Assets

Jerzy Filar, Boda Kang and Malgorzata Korolkiewicz

ISSN 1441-8010

Pricing Financial Derivatives on

Weather-Sensitive Assets

Jerzy Filar∗, Boda Kang†& Ma lgorzata Korolkiewicz‡

Abstract

We study pricing of derivatives when the underlying asset is sensi-

tive to weather variables such as temperature, rainfall and others. We

shall use temperature as a generic example of an important weather

variable. In reality, such a variable would only account for a portion

of the variability in the price of an asset. However, for the purpose of

launching this line of investigations we shall assume that the asset price

is a deterministic function of temperature and consider two functional

forms: quadratic and exponential. We use the simplest mean-reverting

process to model the temperature, the AR(1) time series model and its

continuous-time counterpart the Ornstein-Uhlenbeck process. In con-

tinuous time, we use the replicating portfolio approach to obtain partial

differential equations for a European call option price under both func-

tional forms of the relationship between the weather-sensitive asset price

and temperature. For the continuous-time model we also derive a bino-

mial approximation, a finite difference method and a Monte Carlo sim-

ulation to numerically solve our option price PDE. In the discrete time

model, we derive the distribution of the underlying asset and a formula

for the value of a European call option under the physical probability

measure.

∗Jerzy.Filar@unisa.edu.au; School of Mathematics & Statistics, University of South Aus-

tralia.†Corresponding author: boda.kang@uts.edu.au; School of Finance and Economics, Uni-

versity of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia.‡Malgorzata.Korolkiewicz@unisa.edu.au; School of Mathematics & Statistics, University

of South Australia.

1

Key Words: weather-sensitive asset, financial derivatives, diffusion, bino-

mial approximation, numerical methods, time series, actuarial value.

1 Introduction

Weather derivatives are financial instruments with payoffs linked to specific

weather events and are designed to provide protection against the financial

losses that can occur due to unfavorable weather conditions. Weather risk is

an important issue for energy producers and consumers, leisure and hospitality

industry, and in the agricultural industry. Weather derivatives are typically

swaps, futures or options and the underlying variables can be for example

temperature, humidity, rain or snowfall. Since the most common underlying

variable is temperature, only temperature-based derivatives will be considered

here.

The market for weather derivatives has grown rapidly and there is also a

growing body of academic research into pricing of such contracts (e.g. [1, 5,

6, 11, 15, 21, 25]). The academic literature on weather derivatives tends to

generally concentrate on two issues. First, weather derivatives crucially differ

from standard derivatives in that the underlying asset is not tradeable. Pricing

weather derivatives therefore requires new methodology capable of overcoming

this difficulty. Second, for weather derivatives to be effective hedging instru-

ments, good models of weather risks are needed. Rather than studying deriva-

tive instruments with weather as the underlying asset we propose to study

pricing of derivatives where the underlying asset is sensitive to the weather.

Orange juice is an excellent example of a “pure” weather asset. Its produc-

tion requires an extended development time and long-term commitments of

land and labour. As a result, producers are vulnerable to price shocks created

predominantly by adverse weather conditions such as warmer or colder than

normal temperatures. As reported in [22], temperature is the most important

factor influencing price fluctuations for frozen orange juice futures. What is

more, production is influenced primarily by the weather at a single location,

which is an attractive feature for empirical purposes. The methodology devel-

oped in this paper could equally well be applied to address the weather risk

faced by an investor who purchases shares of a company from the hospitality

2

industry, a ski resort for example.

Because of the periodic nature of the climate, we know that the temperature

cannot continue to rise or fall day after day for a long time. We thus require

a model which allows the temperature to deviate from its steady-state value

only for short periods of time. As suggested in [1] and [6], we consider the

simplest model with the mean-reverting property, namely an AR(1) first-order

autoregressive model and its continuous time counterpart, the mean-reverting

Ornstein-Uhlenbeck process. For the present, we choose to work with raw

temperature rather than a temperature index, such as mean summer or winter

temperature, in an attempt to capture as much statistical information about

the behaviour of temperature as possible.

We wish to model prices of derivative contracts on weather-sensitive assets

and hence we need to specify the nature of the relationship between temper-

ature and the underlying asset. In this work we take this relationship to be

deterministic and consider two functional forms, quadratic and exponential.

The exponential form is chosen because of the wide use of the exponential

utility function. An advantage of the quadratic form is that it will allow us to

approximate arbitrary utility functions up to at least second order terms.

We propose to use a binomial tree model together with the replicating

portfolio approach to price options whose underlying asset is weather-sensitive.

A comprehensive discussion of how binomial trees can be used to price financial

derivatives can be found for example in [7] or [23]. The main advantage of the

binomial model is its flexibility and ease of implementation.

However, we begin with a continuous-time model of weather-sensitive asset

price dynamics given the assumed (deterministic) relationships between the

asset price and temperature. In each case we derive a partial differential equa-

tion (PDE) for the price of a European call option with the weather-sensitive

asset as the underlying. Following the methodology described in [20], we then

construct binomial approximations to asset price dynamics and use the result-

ing binomial trees to calculate the price of the option. Here we rely on the fact

that a binomial tree solution can be seen as equivalent to a numerical solution

to a PDE. We also implement the finite-difference and Monte Carlo simula-

tion methods to solve the derived PDE. Finally, working in discrete time, we

are able to describe the distribution of the weather-sensitive asset price using

3

our assumption of the deterministic relationship between the price and tem-

perature and a time series model for the dynamics of temperature. We then

discuss the pricing of the option under the physical measure induced by the

temperature.

The rest of the paper is organized as follows. In Section 2 we describe

models of temperature dynamics to be used throughout. Then in Section 3

we discuss a continuous-time model of weather-sensitive asset price dynamics

which is used to construct binomial approximations and the resulting binomial

trees to approximately calculate the price of the option. We implement the

finite difference method and Monte Carlo simulation to solve the obtained

partial differential equation in this section as well. The time series model of

the weather-sensitive asset price is presented in Section 5.

2 Models of Temperature Variability

For our model of the relationship between asset prices and weather, we have

chosen temperature as our weather variable of interest. For the dynamics of

temperature, we consider an AR(1) first-order autoregressive model and its

continuous time counterpart, the mean-reverting Ornstein-Uhlenbeck process.

In discrete time we suppose that the temperature at time t, Tt, satisfies the

following recursive equation:

Tt = a0 + a1Tt−1 + ξt, (2.1)

where T0 is known and {ξt} is a sequence of i.i.d. random variables with

ξt ∼ N(0, σ2), t = 1, 2, . . . .

One can rewrite Equation (2.1) as

Tt − Tt−1 = (1 − a1)[a0/(1 − a1) − Tt−1] + ξt = ρ(µ − Tt−1) + ξt,

with ρ = 1 − a1 and µ = a0

ρ, for which the continuous time analogue is known

to be (e.g., see [8], [20]).

dTt = ρ(µ − Tt)dt + σdWt, (2.2)

where {Wt, t ≥ 0} is a standard Brownian motion and ρ > 0 and σ are positive

constants.

4

Note that the drift is ρ(µ − Tt), which means that the temperature Tt is

pulled to its long-run mean value µ while random shocks move it around. Both

processes therefore possess the required mean-reverting property.

3 Continuous-Time Model of Asset Price

In this section, we assume that the price of the weather-sensitive asset {St, t ≥0} is a continuous time stochastic process. In addition, we suppose that the

asset price at time t, St, is a deterministic function of the temperature at

time t, Tt. We consider two functional forms, quadratic and exponential. The

continuous time analogue of the AR(1) model, the mean-reverting Ornstein-

Uhlenbeck process, is used to model the temperature. For each functional form

considered, we derive a partial differential equation for a derivative contract

written on the weather-sensitive asset. For simplicity, we take the derivative

contract to be a European call option which gives its holder the right, but not

the obligation, to buy the underlying weather-sensitive asset at maturity τ at

a pre-specified price K, called the strike price.

3.1 Quadratic Form

For most weather-sensitive assets, extreme temperatures, for instance too low

(freezing) or too high (very hot), have a negative impact on production. Ac-

cording to the law of supply and demand, a significant decrease in supply due

to extreme weather would lead to an increase in demand and hence an increase

in price. A quadratic form therefore appears to be a reasonable first step while

searching for a suitable function to describe the dependence of the asset price

on temperature. Another advantage of the quadratic form is that it will allow

us to approximate arbitrary utility functions up to at least the second order.

3.1.1 Special Case St = T 2t

The following theorem shows the partial differential equation for the price of

a European call option on the weather-sensitive asset when St = T 2t .

5

Theorem 3.1 Suppose that the process of the temperature Tt follows the fol-

lowing stochastic differential equation (SDE),

dTt = ρ(µ − Tt)dt + σdWt,

where {Wt, t ≥ 0} is a standard Brownian Motion, and ρ, µ, σ are positive

constants.

When the price of the weather-sensitive asset satisfies

St = T 2t ,

the price f(t, St) of a European call option with strike price K and maturity τ

on the asset satisfies the following Partial Differential Equation (PDE):

ft + rxfx + 2σ2xfxx − rf = 0,

f(τ, x) = (x − K)+, x ∈ R,(3.1)

where r is a risk-free interest rate, for simplicity assumed to be a constant.

Proof. Following the seminal work of Black, Scholes and Merton [3, 18], we

use the replicating portfolio approach together with the no arbitrage principle

to derive the required PDE. The procedure consists of several steps.

First, we derive the diffusion equation for the price St of the weather-

sensitive asset. We let g(x) = x2, which gives gx = 2x and gxx = 2. Applying

Ito’s lemma to St = g(Tt) allows us to write

dSt = gx(Tt)dTt + gxx(Tt)σ2dt

= (−2ρSt + 2ρµTt + 2σ2)dt + 2σTtdWt

= (−2ρSt + 2ρµ√

St + 2σ2)dt + 2σ√

StdWt

:= µ(St)dt + σ(St)dWt.

(3.2)

Second, we derive the dynamics of the option price. Suppose there is an-

other asset, called Bt, in the market which is riskless and which can be thought

of as a bank account, with price dynamics:

dBt = rBtdt,

where r is a risk-free interest rate.

We form a portfolio consisting of the weather-sensitive asset St and the

riskless asset Bt. Let h0(t) and h1(t) be the number of units of the riskless asset

6

and the weather-sensitive asset held in the portfolio at time t, respectively.

Then the value of the replicating portfolio at time t ≥ 0, Vt, is given by

Vt = h0(t)Bt + h1(t)St.

We want this portfolio to replicate the European call option, which requires

that its terminal value, Vτ , matches the final payoff of the option, i.e. Vτ =

f(τ, Sτ ). The replicating portfolio will need to be (continuously) re-balanced

so that it always “matches” the value of the call. We require that the portfolio

to be self-financing, which means that a change in the value of the portfolio

can only be the result of changes in either the risky or the riskless asset held in

the portfolio. Using Equation (3.2), the self-financing condition can be written

asdVt = h0(t)dBt + h1(t)dSt =

= (rh0(t)Bt + h1(t)µ(St))dt + h1(t)σ(St)dWt.(3.3)

We take the value of the call to be a function of time t and the weather-

sensitive asset price St, namely f(t, St). To prevent arbitrage and following the

Law of One Price, the price of the call must equal the value of the replicating

portfolio, namely Vt = f(t, St). Hence we should have

f(t, St) = h0(t)Bt + h1(t)St. (3.4)

We therefore need to determine h0(t) and h1(t).

Applying Ito’s lemma to the value process Vt = f(t, St) and then substi-

tuting Equation (3.2), we find that

dVt = ft(t, St)dt + fx(t, St)dSt + 12fxx(t, St)σ

2(St)dt =

= [ft(t, St) + fx(t, St)µ(St) + 1/2fxx(t, St)σ2(St)]dt + fx(t, St)σ(St)dWt.

(3.5)

After equating coefficients in Eq. (3.3) and (3.5) and solving for h0(t) and

h1(t), we obtain

h0(t) =ft(t, St) + 1/2fxx(t, St)σ

2(St)

rBt=

ft(t, St) + 2σ2Stfxx(t, St)

rBt,

and

h1(t) = fx(t, St).

7

Substituting the above results into (3.4) (with x = St) we now see that

rf = ft + rStfx + 2σ2Stfxx,

hence by setting x = St we observe that f(t, St) is a solution of the following

PDE,

ft + rxfx + 2σ2xfxx − rf = 0,

with the boundary condition

f(τ, x) = (x − K)+, x ∈ R.

2

For comparison, the famous Black-Scholes PDE is as follows:

ft + rxfx +1

2σ2x2fxx − rf = 0, (3.6)

with the boundary condition

f(τ, x) = (x − K)+, x ∈ R.

Remark 3.1 The diffusion equation for St in (3.2) with the volatility σ(St) =

2σ√

St is related to the so called Constant Elasticity of Variance (CEV) model

(see [2], [9] and Cox and [10]):

dSt = µStdt + σSα/2t dWt, (the elasticity factor 0 ≤ α < 2).

Our model (3.2) is a special case of the above with α = 1.

The analytic solution to the PDE (3.1) is given in [9] as follows: ∀t ≤ τ,

f(t, St) = St

∞∑

n=0

g(n+1, x)G(n+2, kK)−Ke−r(τ−t)

∞∑

n=0

g(n+2, x)G(n+1, kK)

(3.7)

where, g(m, v) is the gamma density function and G(m, v) is the complemen-

tary gamma distribution function and

k =r

2σ2(er(τ−t) − 1), x = kSte

r(τ−t).

8

Remark 3.2 The PDE in Equation (3.1) is clearly different from the Black-

Scholes PDE (3.6) for which there is a closed form solution. In our case, we

provide numerical solutions to (3.1) with the help of binomial tree approxima-

tion, finite-difference methods and Monte Carlo simulation in Section 4. We

note, however, that (3.7) could also be used a basis for approximating numerical

solutions.

Remark 3.3 A solution to PDE (3.1) can be obtained using chi-square dis-

tribution as follows (see [16]):

f(t, S) = Se−q(T−t)[1 − χ2(a, b + 2, c)] − Ke−r(T−t)χ2(c, b, a) (3.8)

where

a =4Ke−(r−q)(T−t)

v, b = 2, c =

4S

v, and v =

4σ2

r − q[1 − e−(r−q)(T−t)].

where χ2(z, k, v) is the cumulative probability that a variable with a noncentral

χ2 distribution with non-centrality parameter v and k degrees of freedom is less

than z.

3.1.2 General Case St = aT 2t + bTt + c

More generally, we can set the relationship between St and Tt as follows:

St = aT 2t + bTt + c, a, b, c ∈ R,

where a > 0 and b2 − 4ac < 0.

Letting St = g(Tt), where g(x) = ax2 + bx + c, we have gx = 2ax + b.

Applying Ito’s lemma again allows us to derive price dynamics of the form

dSt = µ(St)dt + σ(St)dWt

with

σ(St) = σ × (2aTt + b) = σ × (2ag−1(St) + b) = σ√

b2 − 4a(c − St). (3.9)

Following the argument outlined in the proof of Theorem 3.1 we can show

that the PDE for the European call value now becomes

ft + rxfx + 12σ2(4ax + b2 − 4ac)fxx − rf = 0,

f(τ, x) = (x − K)+, x ∈ R,(3.10)

with a, b, c ∈ R, a > 0 and b2 − 4ac < 0, r, σ > 0.

9

3.2 Exponential Case St = βe−αTt

Because of the wide applicability of the exponential utility function, we also

consider the case when the price of the risky weather-sensitive asset is an

exponential function of the temperature, namely St = βe−αTt , α ∈ R, β > 0.

The process of the temperature follows the SDE (2.2) as before. The resulting

PDE for the option price is as follows:

ft + rxfx + 12α2σ2x2fxx − rf = 0,

f(τ, x) = h(x) = (x − K)+, x ∈ R, but fixed,(3.11)

where parameters α ∈ R, σ > 0.

Remark 3.4 The difference between PDE (3.11) and the Black-Scholes PDE

(3.6) is the diffusion parameter. The closed form solution of PDE (3.11) is the

solution of Black-Scholes PDE (3.6) with the diffusion parameter ασ instead

of σ.

4 Numerical Solution Methods

In this section we discuss three methods to obtain a solution to PDE (3.10),

namely binomial approximation method, finite-difference method and Monte

Carlo simulation.

4.1 Binomial Approximation

In [20], Nelson and Ramaswamy provided a method to construct binomial

approximations to diffusions commonly employed in financial models. Their

method is “computationally simple” in the sense that the number of nodes

grows at most linearly in the number of time intervals. In this section we

apply their method to Equation (3.2). The binomial model will enable us to

value a European call option on a weather-sensitive asset numerically.

4.1.1 Basic Binomial Approximation Construction

In a binomial model, it is assumed that the asset price can move only either up

or down at n discrete time intervals. Taking the maturity of the option τ as the

10

time horizon of interest, let h = τn

represent the time interval between jumps

in the underlying weather-sensitive asset price. For the corresponding asset

diffusion of the form dSt = µ(t, St)dt + σ(t, St)dWt, binomial approximations

at time t + h are as follows:

S+h (t, S) = S +

√hσ(t, S),

S−

h (t, S) = S −√

hσ(t, S),

qh = 12

+√

hµ(t, S)/2σ(t, S).

where S+h (t, S) and S−

h (t, S) are the up and down jump amount at time t when

the current price is S, and qh is the probability of a jump to S+h (t, S). For a

“computationally simple” binomial tree its branches need to reconnect, which

means in particular that a down move following an up move produces the same

price as an up move following a down move. After two time steps the total

displacement is √h[−σ(t, S) + σ(t + h, S−

h )]

if an up move follows a down move, and it is

√h[σ(t, S) − σ(t + h, S+

h )]

if a down move follows an up move. These displacements are generally not

equal unless the asset volatility σ is a constant.

Recall that in our diffusion (see Equations (3.2) and (3.9)) for the price of

the weather-sensitive asset St, the volatility is not a constant but a function of

St. Hence in our case the two displacements are not equal, so the branches of

the binomial tree do not reconnect and the number of nodes doubles at each

time step. A “computationally simple” binomial tree is constructed next by

an adaptation of the basic procedure.

4.1.2 Computationally Simple Binomial Tree

Analogous to Eq. (25) in [20], we construct a computationally simple binomial

tree for a new process X(St), and then we use the inverse image of X(St) to

construct a computationally simple binomial tree for St.

The construction of X is as follows. From Eq. (3.9) and Eq. (25) in [20],

11

we define:

X(s) =

∫ s

(4ac−b2)/4a

dz

σ(z)=

∫ s

(4ac−b2)/4a

dz

σ ×√

b2 − 4a(c − z)=

√b2 − 4ac + 4as

2aσ,

where to guarantee that the volatility of the price of the weather-sensitive as-

set in Eq. (3.9) makes sense, we require the domain of the function σ(z) to

be [(4ac − b2)/4a,∞). For X(St), the volatility is a constant and a computa-

tionally simple binomial tree is readily obtained using the method described

in Section 4.1.1.

It is easy to see that X(s) > 0 when a > 0. Hence for x > 0, the inverse

image of X is given by

S(x) = {s : X(s) = x} =4a2σ2x2 − b2 + 4ac

4a= aσ2x2 +

4ac − b2

4a.

Thus, once a computationally simple binomial tree for X(St) is constructed,

we can use the transform

S(x) =

{

aσ2x2 + 4ac−b2

4a, x > 0

0, otherwise

to construct a computationally simple tree for St. Theorem 2 in [20] guarantees

that as the width h of the interval between jumps goes to 0, the binomial

approximations converge weakly to the continuous process which solves the

SDE of interest, in our case Equation (3.2) for example.

From Eq. (72) in [20], we know that for the purpose of calculating the price

of a European call option on St, we must define the probability of an up-move

in St

ph(x) =

{

[S(x)erh − S−

h (x)]/[S+h (x) − S−

h (x)], if S+h (x) > 0

0, otherwise

where

S±

h (x) = S(x ± k±

h (x)√

h),

and where

k+h (x) =

{

the smallest, odd, positive, integer k such that

S(x + k√

h) − exp{rh}S(x) ≥ 0,

12

and

k−

h (x) =

{

the smallest, odd, positive, integer k such that

S(x − k√

h) − exp{rh}S(x) ≤ 0.

Note that by construction ph is the so-called risk-neutral probability that rules

out arbitrage, as shown in [10]. From Section 2 we know that the European

call option value satisfies the PDE (3.10). In the binomial model, we require

that at every node, the call values satisfy the one-period valuation formula

f(t, St) = exp{−rh}[phf(t + h, S+) + (1 − ph)f(t + h, S−)],

which in effect represents a difference equation corresponding to (3.10).1 To use

this formula one starts at maturity τ of the option and then proceeds backwards

in time along the constructed binomial tree for the underlying weather-sensitive

asset. Theorem 4 in [20] extends this recursive formula to say that the price

at time 0 of a European call option on St with expiration date τ and strike

price K is given by

exp{−rτ}E0,τ [(Shτ − K)+],

where the expected value is calculated along the binomial tree for St using the

risk-neutral probabilities ph and 1 − ph.

4.1.3 Numerical Examples

Following the construction in the previous section, we give some numerical

examples of the construction of a computationally simple binomial tree for the

price of the weather-sensitive asset. In each case, we also calculate the price

of a European call option.

The data and parameter specifications for the examples are as follows. We

are using Adelaide daily mean temperature record from Bureau of Meteorology

from 1/07/1999 to 28/06/2002, which gives us T0 = 10.94C◦. The price per

share today of the underlying weather-sensitive asset is then S0 = $119.63.

For the European call option considered, the time to expiration is one month

(τ = 1/12) and the strike price K = $119.63 so that the option is at-the-money.

The risk-free annual interest rate is taken to be r = 0.05.

1This one-period valuation formula is obtained using the replicating portfolio approach

behind our derivation of the continuous time PDEs in Sections 3.1 and 3.2.

13

Since the option has one month to expiration, we established the volatility

using the data for the month immediately before initiation of the contract,

namely from 29/05/2002 to 28/06/2002 and Equation (2.1). We fit a linear

regression model Tt = a0 + a1Tt−1, and the calculated volatility σ is the sample

standard deviation of the residuals (Tt − Tt). This gives a historical daily

volatility estimate σ = 1.44. The estimates of the intercept and slope are

a0 = 6.29 and a1 = 0.45, respectively. Finally, we construct an n-step binomial

tree with n = 4. Accordingly, we have nh = τ = 1/12 and hence h = 1/48.

Example 4.1 Here we consider a simple quadratic case St = T 2t , or a general

quadratic case with a = 1 and b = c = 0. The binomial tree is shown below.

At each node, the top number is the price of the weather-sensitive asset, while

the number in parentheses is the corresponding price of the call option.

138.507

(18.878)

133.658

(14.153)ր

128.895

(9.515)ր ց 128.895

(9.267)

124.219

(6.016)ր ց 124.219

(7.167)ր

119.629

(3.656)ր ց 119.629

(2.397)ր ց 119.629

(0)

ց 115.125

(1.218)ր ց 115.125

(0)ր

ց 110.708

(0)ր ց 110.708

(0)

ց 106.378

(0)ր

ց 102.134

(0)

t = 0 t = 1 t = 2 t = 3 t = 4

(4.1)

14

Example 4.2 Now we consider the exponential case St = βe−αTt with α =

0.27, β = 1. The binomial tree is as follows:

150.388

(30.759)

142.026

(22.522)ր

134.129

(14.749)ր ց 134.129

(14.501)

126.672

(9.078)ր ց 126.672

(7.167)ր

119.629

(5.371)ր ց 119.629

(3.543)ր ց 119.629

(0)

ց 112.978

(1.751)ր ց 112.978

(0)ր

ց 106.696

(0)ր ց 106.696

(0)

ց 100.764

(0)ր

ց 95.1613

(0)

t = 0 t = 1 t = 2 t = 3 t = 4

(4.2)

Remark 4.1 Note that, in this case, the option price with the provided param-

eters is lower due to less volatility in the weather-sensitive asset price implied

by the exponential model as compared to the quadratic model.

4.2 Finite Difference Methods

In this section, we find the numerical solution to the partial differential equa-

tion (3.1) using finite difference methods as described in [12] or [16].

15

The remaining life of the option τ is divided into N equally spaced intervals

of length ∆t = τ/N , resulting in N + 1 time points 0, ∆t, 2∆t, . . . , τ . Then

a sufficiently high stock price Smax is chosen so that the call option is worth

Smax − K when Smax is reached. Using ∆S = Smax/M we then obtain M + 1

equally spaced stock prices 0, ∆S, 2∆S, . . . , Smax, one of which is the current

stock price.

The time points and stock price points define a grid consisting of a total of

(M + 1)(N + 1) points (i, j), 1 ≤ i ≤ M , 1 ≤ j ≤ N corresponds to time i∆t

and stock price j∆S. The value of the option at the (i, j) point is defined as

fi,j := f(i∆t, j∆S).

For an interior point (i, j) on the grid, ∂f/∂S can be approximated as

∂f

∂S=

fi,j+1 − fi,j

∆S(forward difference approximation), (4.3)

or as∂f

∂S=

fi,j − fi,j−1

∆S(backward difference approximation). (4.4)

Following [16], we use a more symmetric averaged approximation

∂f

∂S=

fi,j+1 − fi,j−1

2∆S. (4.5)

For ∂f/∂t, we use a forward difference approximation

∂f

∂t=

fi+1,j − fi,j

∆t. (4.6)

The backward difference approximation for ∂f/∂S at the (i, j) point together

with the backward difference at the (i, j + 1) point give a finite difference

approximation for ∂2f/∂S2 at the (i, j) point:

∂2f

∂S2=

fi,j+1 − 2fi,j + fi,j−1

∆S2. (4.7)

Substituting equations (4.5), (4.6), and (4.7) into the partial differential

equation (3.1) and noting that S = j∆S gives:

fi+1,j − fi,j

∆t+ rj∆S

fi,j+1 − fi,j−1

2∆S+ 2σ2j∆S

fi,j+1 − 2fi,j + fi,j−1

∆S2= rfi,j

for j = 1, 2, . . . , M − 1 and i = 0, 1, . . . , N − 1. Rearranging terms, we obtain

ajfi,j−1 + bjfi,j + cjfi,j+1 = fi+1,j, (4.8)

16

whereaj = rj∆t

2− 2σ2j∆t

∆S,

bj = 1 + r∆t + 4σ2j∆t∆S

,

cj = −rj∆t2

− 2σ2j∆t∆S

.

We now define the value of the value of the call option along the three edges

of the grid given by S = 0, S = Smax, and t = τ . The value of the call at time

τ is max(Sτ − K, 0), where Sτ is the stock price at time τ . Hence,

fN,j = max(j∆S − K, 0), j = 0, 1, . . . , M. (4.9)

The value of the call option when the stock price is zero is zero. Hence,

fi,0 = 0, i = 0, 1, · · · , N. (4.10)

Finally, we assume that the call option is worth max(S−K, 0) when S = Smax,

so that

fi,M = max(M∆S − K, 0), i = 0, 1, · · · , N. (4.11)

Equation (4.8) is now used to arrive at the value of f at all other points of the

grid. For τ − ∆t, i = N − 1 and j = 1, 2, . . . , M − 1, equation (4.8) gives

ajfN−1,j−1 + bjfN−1,j + cjfN−1,j+1 = fN,j. (4.12)

From equations (4.10) and (4.11) we have

fN−1,0 = 0, (4.13)

fN−1,M = max(M∆S − K, 0). (4.14)

Equations (4.12) can now be solved for the M−1 unknowns fN−1,1, fN−1,2, · · · , fN−1,M−1.

The procedure is repeated until f0,1, f0,2, · · · , f0,M−1 are obtained, one of which

is the required option price.

Note that with the help of linear algebra, equations (4.12) can be written

in matrix format for all i = N, . . . , 1 as follows:

b1 c1 0 . . . . . . . . . 0

a2 b2 c2 0 . . . . . . 0

0 a3 b3 c3 0 . . . 0

. . . . . . . . . . . . . . . . . . . . .

0 . . . 0 aM−3 bM−3 cM−3 0

0 . . . . . . 0 aM−2 bM−2 cM−2

0 . . . . . . . . . 0 aM−1 bM−1

fi−1,1

fi−1,2

fi−1,3

...

fi−1,M−3

fi−1,M−2

fi−1,M−1

=

f ∗

i,1

fi,2

fi,3

...

fi,M−3

fi,M−2

f ∗

i,M−1

.

(4.15)

17

wheref ∗

i,1 = fi,1 − a1fi−1,0,

f ∗

i,M−1 = fi,M−1 − cM−1fi−1,M .

Solving the N matrix equations in (4.15) produces the option price.

The method describe above is referred to as the implicit finite difference

method. It has the advantage of being very robust as it always converges to the

solution of the differential equation as ∆S and ∆t approach zero. However,

one of the disadvantages is that M − 1 simultaneous equations have to be

solved in order to calculate the fi,j from the fi+1,j.

The implicit finite difference method can be simplified if the values of

∂f/∂S and ∂2f/∂S2 at point (i, j) on the grid are assumed to be the same as

at point (i + 1, j). Equation (4.5) and (4.7) then become:

∂f

∂S=

fi+1,j+1 − fi+1,j−1

2∆S,

∂2f

∂S2=

fi+1,j+1 − 2fi+1,j + fi+1,j−1

∆S2.

The difference equation is then

fi+1,j − fi,j

∆t+rj∆S

fi+1,j+1 − fi+1,j−1

2∆S+2σ2j∆S

fi+1,j+1 − 2fi+1,j + fi+1,j−1

∆S2= rfi,j

or

fi,j = a∗

jfi+1,j−1 + b∗jfi+1,j + c∗jfi+1,j+1, (4.16)

wherea∗

j = ∆tr∆t+1

(

−rj2

+ 2σ2j∆S

)

,

b∗j = ∆tr∆t+1

(

1∆t

− 4σ2j∆S

)

,

c∗j = ∆tr∆t+1

(

rj2

+ 2σ2j∆S

)

.

The result is the so-called explicit finite difference method.

4.2.1 Numerical Example

Using the same data and parameter specifications as in Section 4.1, we apply

finite-difference methods to calculate the price of a European call option with

one month to expiration and strike price K = 119.63. As before, the annual

risk-free interest rate is assumed to be 0.05.

18

Using the implicit finite-difference approach with M = 200, N = 200, and

Smax = 146.79, we obtain the call price of 3.8457, which is very close to our

binomial approximation price of 3.851. The explicit finite-difference approach

produces the call price of 3.8503. This value is again very close to our binomial

approximation price. For comparison, formula (3.8) gives the option price as

3.874.

4.3 Monte Carlo Simulation

Recall from Section 2 that to model temperature we use the Ornstein-Uhlenbeck

equation

dTt = ρ(µ − Tt)dt + σdWt, (4.17)

where {Wt, t ≥ 0} is a standard Brownian motion and ρ > 0 and σ are positive

constants, and that the price of the weather-sensitive asset satisfies St = T 2t .

Monte Carlo method (see [12, 16]) requires that we first simulate the tem-

perature process. The discretised version of Eq. (4.17) is obtained by dividing

the time interval [0, τ ] into N equal subintervals. Setting h = τN

and for

j = 0, . . . , N − 1 we have:

T(j+1)h = a0 + a1Tjh + ξh, (4.18)

where T0 is known, a0 = µρ, a1 = 1 − ρ and ξh ∼ N(0, (σ√

h)2). A sample

path is simulated according to equation (4.18) M times and this results in

M terminal temperatures T iτ , i = 1, . . . , M. These give rise to prices of the

underlying asset Siτ = (T i

τ )2, i = 1, . . . , M. The price of the European call with

a strike K and time T maturity is then calculated as

C = e−rτ 1

M

M∑

i=1

(Siτ − K)+.

4.3.1 Numerical Example

Using the same data and parameter specifications as in Section 4.1, we apply

the Monte Carlo simulation method to calculate the price of a European call

option with one month to expiration and strike price K = 119.63. As before,

the annual risk-free interest rate is assumed to be 0.05. The Monte Carlo

19

method produces the call price of 3.8178. This is again quite close to the

formula (3.8) price of 3.874.

5 Time Series Model of Asset Price

In this section we propose to work in discrete time and suppose that the

temperature follows an AR(1) process

Tt = a0 + a1Tt−1 + ξt, (5.1)

where T0 is known and {ξt} is a sequence of i.i.d. random variables with

ξt ∼ N(0, σ2), t = 1, 2, . . . .

For the price of the weather-sensitive asset we assume, as before, that it is

a quadratic function of the temperature, namely

St = T 2t , t = 0, 1, . . . . (5.2)

5.1 Distribution of the Weather-Sensitive Asset Price

In this section we specify the distributions of the temperature Tt and the price

of the weather-sensitive asset St. The results are given in the following lemma:

Lemma 5.1 For each t ≥ 1, Tt follows a normal distribution with mean µt

and standard deviation σt, where

µt = at1T0 + c(t), σ2

t = σ2t−1∑

i=0

a2i1 = σ2 1 − a2t

1

1 − a21

,

while St/σ2t has noncentral χ2 distribution with 1 degree of freedom and the

non-centrality parameter

qt =µ2

t

σ2t

. (5.3)

The mean and variance of St are as follows:

E(St) = σ2t (1 + qt) = σ2

t + µ2t , V ar(St) = 2(1 + 2qt)σ

4t = 2σ2

t (σ2t + 2µ2

t ).

20

Proof. Using Equation (5.1) repeatedly we can write that for all t ≥ 1

Tt = at1T0 + c(t) + ε(t),

where

c(t) = a0

t−1∑

i=0

ai1 =

a0(1 − at1)

1 − a1, ε(t) =

t−1∑

i=0

ai1ξt−i.

Since we assume that ξt ∼ N(0, σ2) for all t ≥ 0, we can conclude that Tt ∼N(µt, σ

2t ) with

µt = at1T0 + c(t), σ2

t =

t−1∑

i=0

a2i1 σ2 = σ2 1 − a2t

1

1 − a21

.

Then Tt/σt ∼ N(µt/σt, 1), and according to the definition of non-central χ2

distribution (see p.58 in [14]), T 2t /σ2

t = St/σ2t follows a noncentral χ2 distri-

bution with 1 degree of freedom and the non-centrality parameter qt = µ2t/σ

2t .

The mean and variance of St are the direct consequence of the corresponding

χ2 distribution. 2

Remark 5.1 The density function of the non-central χ2 distribution with 1

degree of freedom and non-centrality parameter qt is, (see p.58 in [14])

φ(x; 1, qt) =

{

exp[−(x+q2

t )/2]

21/2

∑

∞

j=1xj−1/2q2j

t

Γ(j+1/2)22jj!, x > 0,

0, otherwise,(5.4)

where Γ denotes the gamma function

Γ(y) =

∫

∞

0

e−zzy−1dz. (5.5)

5.2 Actuarial Value of a European Call Option

In this section we again appeal to the principle of equivalence from actuarial

theory to value a European call option on the weather-sensitive asset using the

discounted expected cashflow:

exp{−rτ}EP0,τ [(Sτ − K)+]

21

where the expectation is taken under the “physical” measure P . The moti-

vation for the rather simple and intuitively appealing actuarial approach is

our assumption that temperature is the only stochastic determinant of the

weather-sensitive asset price. The explicit description of the distribution of

the weather-sensitive asset price in Section 5.1 allows us to calculate the ex-

pected value of the option payoff using the probability measure induced by the

dynamics of the temperature.

Lemma 5.2 The actuarial price of a European call on the weather-sensitive

asset St with strike price K and time to expiry τ is as follows:

exp{−rτ}EP0,τ [(Sτ − K)+] = exp{−rτ}σ2

τ

∫ +∞

K ′

(x − K ′)φ(x; 1, qτ )dx, (5.6)

where K ′ = K/σ2τ and φ(x; 1, qτ ), the density function of the non-central χ2

distribution with 1 degree of freedom and non-centrality parameter qτ .

Proof. This follows from the definition of the actuarial price and the fact that

Sτ/σ2τ follows the non-central Chi-square distribution with 1 degree of freedom

and non-centrality parameter qτ . 2

Remark 5.2 It remains to be seen to what extent the pricing formula from

Lemma 5.2 holds in practice for prices of derivatives on weather-sensitive as-

sets.

6 Conclusion

We have studied pricing of financial derivatives, for example, European call

option, on weather sensitive assets. We assumed the price of the underlying is

a quadratic function of the temperature modelled by a time series. We derived

the diffusion equation for the price of the underlying asset in continuous time

setting and a partial differential equation for the price of a European call on

the weather-sensitive asset. We constructed a binomial tree to approximate

the underlying diffusion and to enable calculation of the price of a call option

on a weather-sensitive asset. Following [20], we have used a transformation of

the underlying price process St in order to obtain a recombining tree and thus

facilitate the calculation of the option price. We have also implemented finite

22

difference methods and a Monte Carlo simulation to solve our option price PDE

numerically. Resulting option prices are very similar to the price obtained via

the binomial approximation approach. Finally, in discrete time setting, we

have derived the distribution of the underlying asset and the actuarial value

of the corresponding European call option.

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