HDD and CDD Option Pricing with Market Price of Weather Risk for Taiwan Hung-Hsi Huang Yung-Ming...

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

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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.


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HDD and CDD Option Pricing with Market Price of Weather Risk for Taiwan

IntroductionTemperature Modeling and Forecasting ModelsPricing Weather Derivatives

Market Price of RiskHDD/CDD Option PriceDifferentiating the Option price with respect to the

mean and Standard Deviation of HDD/CDD DataEmpirical AnalysisConclusions


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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.


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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).


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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.


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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 time-series approach to modeling and forecasting daily average temperature, taking into account the trend, seasonality, and past values of temperature.


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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.


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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.


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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:

The model for temperature can be described as follows:

The parameters are estimated using the OLS regression.

angle phase,365/2

)2(cossin

)1()sin(

21

wtwttYXT

twZtYXTa

t

at

),0(~

)3(cossin2

3210

IID

wtwttT

t

tt


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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:

The above equation can be rewritten in discrete form:

can be estimated based on Equations (2) and (3).

can be obtained using the OLS regression and

)5()]()/[( ttta

ta

tt dWdtTTdtdTdT

ajT

)8()ˆ1(ˆ~

2

1ˆ

1

2

112

N

jj

ajj TTT

N

)month in days ( ,,1),1,0(~

(7) )1()(~

1111

NNjN

TTTTTT

j

jjaj

aj

ajjj

Time TrendMean-Reverting


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Temperature Modeling and Forecasting Models: GARCH

This study assumes follows GARCH (p, q). Equation (3) can be rewritten as follows:

)9(,)()(

),0(~|,cossin

220

1

2

1

20

2

213210

tt

q

jitj

p

iitit

ttttt

LL

NFwtwttT

t


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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.

n

t ton

t tn TCHDDH11

)12()23(

n

t

ot

n

t tn CTCDDC11

)13()23(


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The Formula for HDD/CDD Option Price(Alaton et al., 2002)

)15(

2)()(

)()(]|}0,[max{

(14)2

)()(

)()(

]|}0,[max{

22

2

2

1

2)(

0

)()(

2)(

)(

)(

n

nn

n

n

nn

n

n

n

n

n

eeKe

dxxfxKeFHKEep

eKe

dxxfKxe

FKHEec

n

n

nnn

ttr

K

Httr

tnQttr

t

nnn

ttr

K Httr

tnQttr

t


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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).


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18


19


20


21


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Alaton et al. (2002) estimate the conditional variance

.0

)20(])[1)(2/1(]|,[

)19(])[1)(2/1(]|[

)18(])[1)(/()(

]|[

222

21

2)(

222

21

2

21)(

uts

eeFTTCov

eFTVar

eTeTT

FTE

tssst

sut

tssst

tssm

tstm

ss

stQ


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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%.


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Empirical Analysis

Based on Equations (22) and (23), and would approximately follow a normal distribution since is assumed to be normally distributed.

nH nC

itT

)23(23)23(

)0,23max(

)22(23)23(

)0,23max(

11

1

11

1

nTT

TC

TnT

TH

n

i t

n

i t

n

i tn

n

i t

n

i t

n

i tn

ii

i

ii

i


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First-Order and Second-Order Moments of Hn and Cn

)27(]│,[2]|[]│[

)26(23]|[|23]|[

)25(]|,[2]|[]|[

)24(]|[23|23]|[

1

11

1

11

jittt

n

t tttn

n

i ttQ

t

n

i tQ

tnQ

jittt

n

t tttn

n

i ttQ

t

n

i tQ

tnQ

FTTCovFTVarFCVar

nFTEFnTEFCE

FTTCovFTVarFHVar

FTEnFTnEFHE

ji

ii

ji

ii


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27


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29


30


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HDD and CDD Option Pricing with Market Price of Weather Risk for Taiwan Hung-Hsi Huang Yung-Ming...

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Transcript of HDD and CDD Option Pricing with Market Price of Weather Risk for Taiwan Hung-Hsi Huang Yung-Ming...