HDD and CDD Option
Pricing with Market Price of
Weather Risk for Taiwan
Hung-Hsi Huang
Yung-Ming Shiu
Pei-Syun Lin
The Journal of Futures Markets
Vol. 28, No. 8, 790–814 (2008)
HDD and CDD Option Pricing with Market
Price of Weather Risk for Taiwan
 This study extends the long-term temperature
model proposed by Alaton et al. (2002) by taking
into account ARCH/GARCH effects to reflect the
clustering of volatility in temperature.
 The fixed variance model and the ARCH model
are estimated using Taiwan weather data from
1974 through 2003.
 The results show that for HDD/CDD the call
price is higher under ARCH-effects variance
than under fixed variance, while the put price is
lower.
HDD and CDD option pricing with market price
of weather risk for Taiwan
2
HDD and CDD Option Pricing with Market
Price of Weather Risk for Taiwan
 Introduction
 Temperature Modeling and Forecasting Models
 Pricing Weather Derivatives
 Market Price of Risk
 HDD/CDD Option Price
 Differentiating the Option price with respect to the
mean and Standard Deviation of HDD/CDD
 Data
 Empirical Analysis
 Conclusions
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Introduction: 研究背景
 Weather conditions directly affect agricultural
outputs and the demand for energy products,
and indirectly affect retail businesses.
 Among all the weather derivative transactions,
temperature-related deals are the most
prevalent, accounting for more than 80% of all
transactions.
 Although weather derivatives have gradually
gained their importance, there is not yet a
compelling and effective pricing method.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Introduction: 研究背景
 Since the underlying indexes are not traded, it
would be infeasible to employ the arbitrage-free
approach to pricing temperature related
derivatives.
 Moreover, temperature follows a mean reverting
process, instead of a random walk process.
 Further, the weather derivatives market is not
complete. Thus, the Black-Scholes formula is
unsuitable for pricing weather derivatives.
 Also, the actuarial approach to pricing insurance
products is not applicable (Zeng, 2000).
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Introduction: 文獻探討
 To date some approaches have been developed
to price weather derivatives.
 One of the first is Burn Analysis.
 It involves taking long-term historical
temperature values and converting them into
probability distributions for heating degree day
(HDD)/ cooling degree day (CDD).
 The price of the weather option can then be
determined.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Introduction: 文獻探討
 Without taking into account the market price of
risk, Considine (2000) constructs an option
pricing model and illustrates an example for an
HDD put option using a distribution fitted to
historical values of HDD.
 Campbell and Diebold (2005) employ a timeseries approach to modeling and forecasting
daily average temperature, taking into account
the trend, seasonality, and past values of
temperature.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Introduction: 文獻探討
 Based on prior studies, Alaton et al. (2002)
construct a relatively complete model which
takes into account not only temperature trend,
seasonality, and mean reversion but market
price of risk.
 Previous studies such as Cao and Wei (2004)
have shown the market price of risk is important
for weather derivatives.
 In fact, the market price of risk is indispensable
to simulate future temperature under the risk
neutral measure.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Introduction: 研究結果
 Using a sample of 21,900 daily high/low
temperatures over the period of 1974 through
2003 obtained from the Central Weather Bureau,
Taiwan, this study constructs a model which
simultaneously considers the market price of
weather risk, trend, seasonality, mean reversion,
and the clustering of volatility in temperature.
 It is found that the HDD/CDD call price is higher
under ARCH-effects variance than under fixed
variance, while the put price is lower.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Temperature Modeling and Forecasting
Models: Alaton et al. (2002)
 Based on these two characteristics (positive tend and
seaonality), Alaton, Djehiche and Stillberger (2002)
propose the following model to describe the average
temperature at time t:
Tt a  X  Y t  Z sin( wt   )
(1)
 Tt a  X  Y t  1 sin wt   2 cos wt (2)
  2 / 365,   phase angle
 The model for temperature can be described as follows:
Tt   0   1 t   2 sin wt   3 cos wt  t
(3)
t ~ IID (0,  2 )
 The parameters are estimated using the OLS regression.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Temperature Modeling and Forecasting
Models: Alaton et al. (2002)
 Alaton et al. (2002) obtain the following stochastic
differential equation in continuous form:
Time Trend
a
a
dTt  [( dTt / dt )   (Tt  Tt )] dt   t dWt (5) Mean-Reverting
 The above equation can be rewritten in discrete form:
~
T j  T j  (T ja  T ja1 )   T ja1  (1   ) T j 1     j 1 (7)
 j ~ N (0, 1),
j  1, , N  ( N  days in month  )
a
 T j can be estimated based on Equations (2) and (3).
 ̂ can be obtained using the OLS regression and
N

1
~
a
ˆ
ˆ
ˆ  
T


T

j
j 1  (1   ) T j 1
N   2 j 1
2
HDD and CDD option pricing with market price
of weather risk for Taiwan

2
(8)
11
Temperature Modeling and
Forecasting Models: GARCH
 This study assumes  t
follows GARCH (p, q).
Equation (3) can be rewritten as follows:
Tt   0   1t   2 sin wt   3 cos wt  t , t | Ft 1 ~ N (0,  t2 )

p
q
(9)
2
2
         2     ( L) 2   ( L)  2 ,


0
i t i
j
t i
0
t
t
 t
i 1
j 1

HDD and CDD option pricing with market price
of weather risk for Taiwan
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Pricing Weather Derivatives:
HDD/CDD Option Price
 Due to the humid climate in Taiwan, the reference
temperature should be increased to a small extent.
 This study uses 23 ℃, the average temperature of Taipei
during the period 1974-2003, as reference temperature.
 In order to calculate the degree days over a period, we
aggregate the daily degree days for each day during that
period.
H n  t 1 HDDt  t 1 (23 oC  Tt )
(12)
Cn  t 1 CDDt  t 1 (Tt  23 oC )
(13)
n
n
n
n
HDD and CDD option pricing with market price
of weather risk for Taiwan
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The Formula for HDD/CDD Option Price
(Alaton et al., 2002)
ct  e  r (tn t ) E Q [max{ H n  K ,0} | Ft ]
e
 r ( t n t )


K
( x  K ) f H n ( x) dx


 n

 e  r (tn t )  (  n  K ) ( n )  n e 2

2

2
pt  e
 r ( t n t )




E [max{ K  H n ,0} | Ft ]  e
Q
 r ( t n t )
(14)

K
0
( K  x) f H n ( x) dx
2
1  n   


 n2

  

 n    n   2
2   n   
 r ( t n t ) 
e
e e
 ( K   n )  ( n )       


2




n 





(15)
HDD and CDD option pricing with market price
of weather risk for Taiwan
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DATA
 This study employs a database of daily
maximum and minimum temperatures measured
in degrees Celsius over the 1974-2003 period.
 A sample of 21,900 daily highs/lows is obtained
from the Central Weather Bureau, Taiwan.
 Figures 1 depicts the daily maximum and
minimum temperatures at the Taipei Weather
Station, respectively.
 Figure 2 presents the graphs and descriptive
statistics of daily average temperature and
residuals of OLS for equation (3).
HDD and CDD option pricing with market price
of weather risk for Taiwan
15
HDD and CDD option pricing with market price
of weather risk for Taiwan
16
HDD and CDD option pricing with market price
of weather risk for Taiwan
17
HDD and CDD option pricing with market price
of weather risk for Taiwan
18
HDD and CDD option pricing with market price
of weather risk for Taiwan
19
HDD and CDD option pricing with market price
of weather risk for Taiwan
20
HDD and CDD option pricing with market price
of weather risk for Taiwan
21
Alaton et al. (2002) estimate the
conditional variance
E Q [Tt | Fs ]
 (Ts  Tsm )e  (t  s )  Tt m  ( /  )(1  e  )[ s 1   s  2     t ] (18)
Var[Tt | Fs ]  (1 / 2 )(1  e  2 )[ s21   s2 2     t2 ]
(19)
Cov [Tt , Tu | Fs ]  e  ( t  s ) (1 / 2 )(1  e  2 )[ s21   s2 2     t2 ] (20)
0  s  t  u.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Empirical Analysis
 Taking into account ARCH/GARCH effects, the
parameters are estimated in Table III.
 Table IV presents the daily average temperature
statistics for winter and summer months.
 During the winter months, only about 5.03%
daily average temperatures are greater than
23℃, and the ratio of CDD/HDD is only about
1.03%.
 During the summer months, only about 1.38%
daily average temperatures are smaller than
23℃, and the ratio of HDD/CDD is only about
0.24%.
HDD and CDD option pricing with market price
of weather risk for Taiwan
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Empirical Analysis
 Based on Equations (22) and (23), H n and Cn
would approximately follow a normal distribution
since Tt is assumed to be normally distributed.
i
H n  i 1 max( 23  Tti , 0)
n
 i 1 (23  Tti )  23n  i 1 Tti
n
n
(22)
C n  i 1 max( Tti  23, 0)
n
 i 1 (Tti  23)  i 1 Tti  23n
n
n
HDD and CDD option pricing with market price
of weather risk for Taiwan
(23)
24
First-Order and Second-Order Moments
of Hn and Cn


E [ H n | Ft ]  E 23n  i 1 Tti | Ft  23n  i 1 E Q [Tti | Ft ] (24)
Q
Q
n
n
Var [ H n | Ft ]  t 1Var [Tt | Ft ]  2 Cov [Tti , Tt j | Ft ]
n
(25)
i j
E [C n | Ft ]  E
Q
Q


Q
T

23
n
|
F

E
i 1 [Tti | Ft ]  23n
t
i 1 ti
n
n
Var [C n│Ft ]  t 1Var [Tt | Ft ]  2 Cov [Tti , Tt │
Ft ]
j
n
(26)
(27)
i j
HDD and CDD option pricing with market price
of weather risk for Taiwan
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HDD and CDD option pricing with market price
of weather risk for Taiwan
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HDD and CDD option pricing with market price
of weather risk for Taiwan
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HDD and CDD option pricing with market price
of weather risk for Taiwan
28
HDD and CDD option pricing with market price
of weather risk for Taiwan
29
HDD and CDD option pricing with market price
of weather risk for Taiwan
30
HDD and CDD option pricing with market price
of weather risk for Taiwan
31
HDD and CDD option pricing with market price
of weather risk for Taiwan
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