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 2 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 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 3 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 4 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 5 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 6 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 7 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.

HDD and CDD option pricing with market price of weather risk for Taiwan 8 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 9 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 10 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.

HDD and CDD option pricing with market price of weather risk for Taiwan 11 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 Time Trend Mean-Reverting

HDD and CDD option pricing with market price of weather risk for Taiwan 12 Temperature Modeling and Forecasting Models: GARCH This study assumes follows GARCH (p, q). Equation (3) can be rewritten as follows:

HDD and CDD option pricing with market price of weather risk for Taiwan 13 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 , 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.

HDD and CDD option pricing with market price of weather risk for Taiwan 14 The Formula for HDD/CDD Option Price (Alaton et al., 2002)

HDD and CDD option pricing with market price of weather risk for Taiwan 15 DATA This study employs a database of daily maximum and minimum temperatures measured in degrees Celsius over the 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 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

HDD and CDD option pricing with market price of weather risk for Taiwan 22 Alaton et al. (2002) estimate the conditional variance

HDD and CDD option pricing with market price of weather risk for Taiwan 23 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 24 Empirical Analysis Based on Equations (22) and (23), and would approximately follow a normal distribution since is assumed to be normally distributed.

HDD and CDD option pricing with market price of weather risk for Taiwan 25 First-Order and Second-Order Moments of H n and C n

HDD and CDD option pricing with market price of weather risk for Taiwan 26

HDD and CDD option pricing with market price of weather risk for Taiwan 27

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 32