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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.
∗[email protected]; School of Mathematics & Statistics, University of South Aus-
tralia.†Corresponding author: [email protected]; School of Finance and Economics, Uni-
versity of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia.‡[email protected]; 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 .
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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
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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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