跳躍風險與隨機波動度下溫度衍生性商品之評價 - 政大學術集成

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(1)國立政治大學金融學系博士學位論文. 跳躍風險與隨機波動度下溫度衍生性商品之評價政治大. 立. ‧ 國. 學. Pricing Temperature Derivatives under Jump Risks and Stochastic Volatility. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 指導教授：林士貴博士研究生：莊明哲撰. 中華民國一零四年七月.

(2) 跳躍風險與隨機波動度下溫度衍生性商品之評價國立政治大學金融學系博士學位論文指導教授：林士貴. 研究生：莊明哲摘要. 本研究利用美國芝加哥商品交易所針對 18 個城市發行之冷氣指數／暖氣指數衍生性商品與相對應之日均溫進行分析與評價。研究成果與貢獻如下：一、延伸 Alaton,. 治政提出新模型以捕捉更多溫度指數之特徵。二、針對不同模型，分別利用最大概似法、大立期望最大演算法、粒子濾波演算法等進行參數估計。實證結果顯示新模型具有較好之. Djehince, and Stillberg (2002) 模型，引入跳躍風險、隨機波動度、波動跳躍等因子，. ‧ 國. 學. 配適能力。三、利用 Esscher 轉換將真實機率測度轉換至風險中立機率測度，並進一步利用 Feynman-Kac 方程式與傅立葉轉換求出溫度模型之機率分配。四、推導冷氣. ‧. 指數／暖氣指數期貨之半封閉評價公式，而冷氣指數／暖氣指數期貨之選擇權不存在. y. Nat. 封閉評價公式，則利用蒙地卡羅模擬進行評價。五、無論樣本內與樣本外之定價誤. sit. 差，考慮隨機波動度型態之模型對於溫度衍生性商品皆具有較好之評價績效。六、實. n. al. er. io. 證指出溫度市場之市場風險價格為負，顯示投資人承受較高之溫度風險時會要求較高. i Un. v. 之風險溢酬。本研究可給予受溫度風險影響之產業，針對衍生性商品之評價與模型參. Ch. engchi. 數估計上提供較為精準、客觀與較有效率之工具。. 關鍵字：日均溫、冷氣指數／暖氣指數衍生性商品、風險中立評價法、隨機波動度、跳躍風險、粒子濾波演算法、期望最大演算法. i.

(3) Pricing Temperature Derivatives under Jump Risks and Stochastic Volatility Department of Money and Banking, National Chengchi University Ph.D. Thesis Student: Ming-Che Chuang. Advisor: Shih-Kuei Lin. Abstract This study uses the daily average temperature index (DAT) and market price of the. 政治大 in this study: (i) we extend the Alaton, Djehince, and Stillberg (2002)’s framework by 立 CDD/HDD derivatives for 18 cities from the CME group. There are some contributions. introducing the jump risk, the stochastic volatility, and the jump in volatility. (ii) The. ‧ 國. 學. model parameters are estimated by the MLE, the EM algorithm, and the PF algorithm. And, the complex model exists the better goodness-of-fit for the path of the tempera-. ‧. ture index. (iii) We employ the Esscher transform to change the probability measure. Nat. sit. y. and derive the probability density function of each model by the Feynman-Kac formula. io. er. and the Fourier transform. (iv) The semi-closed form of the CDD/HDD futures pricing formula is derived, and we use the Monte-Carlo simulation to value the CDD/HDD fu-. n. al. Ch. i Un. v. tures options due to no closed-form solution. (v) Whatever in-sample and out-of-sample. engchi. pricing performance, the type of the stochastic volatility performs the better fitting for the temperature derivatives. (vi) The market price of risk differs to zero significantly (most are negative), so the investors require the positive weather risk premium for the derivatives. The results in this study can provide the guide of fitting model and pricing derivatives to the weather-linked institutions in the future. Keywords: daily average temperature index, CDD/HDD derivatives, risk-neutral pricing method, stochastic volatility, jump risk, particle filter algorithm, expectation-maximization algorithm. ii.

(4) 誌謝時光飛逝，在國立政治大學求學之四年光陰已接近尾聲，由衷感謝所有授課老師、同學、系辦助理與許多好朋友們在博士班生涯中之教導與照顧，這份情誼皆記錄在腦海中，作為我在博士班生涯美好的回憶。感謝指導老師林士貴博士在論文撰寫過程的指導與建議。雖然老師十分忙碌，但亦很有耐心與不惜辛苦給予論文上的指點與修改，是博士論文得以完成之幕後推手。同時在待人處事上亦獲益良多，修正了許多過去自我感覺良好之不禮貌行為與粗心大意的缺點，這些對於往後無論繼續往學界或業界發展皆有一定的重要性。亦很感謝張智凱博士、王仁和博士、詹芳書博士與孫立憲博士在學位口試時之指教與建議，使我的論文更具有可靠性與完備性。. 政治大陽、春玫；學長姐：育民、安琪、信豪、信瑜、文峰、亭甫、志偉、宗穎；學弟妹：立朝生、偉銘、佩珊。在國立政治大學求學期間給予指教與照顧，在最失意的時候亦給感謝國立政治大學金融系辦助理：淑芳、明潔；同班同學：晉祥、秋練、安興、朝. ‧ 國. 學. 予支持與鼓勵。亦很感謝金融所碩士班眾學弟妹們與統研所：玉華、柏成，在數學與程式上之教導，讓論文實證結果增加其可靠性。. ‧. 最後，要把這份論文獻給我摯愛之家人，感謝您們長久以來的栽培、照顧與支持，. n. al. er. io. sit. y. Nat. 讓我可以無後顧之憂地專注於學業上，在此將這份成就與喜悅與你們一起分享。. Ch. engchi U. iii. 莊明哲謹誌於. v 國立政治大學金融研究所 i n 中華民國一零四年七月.

(5) Contents 1 Introduction. 1. 2 Literature Review. 7. 2.1. Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2. CDD/HDD Derivatives Pricing Model . . . . . . . . . . . . . . . . . . .. 10. 立. 3 The Models. 政治大. 12. ‧ 國. 學. Seasonal Mean-Reversion Model (S-MR) . . . . . . . . . . . . . . . . . .. 12. 3.2. Seasonal Mean-Reversion Model with Seasonal Volatility (S-MR-S) . . .. 13. 3.3. Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Volatil-. ‧. 3.1. y. Seasonal Mean-Reversion Model with Seasonal Stochastic Volatility (S-. io. n. al. er. MR-S-SV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. i Un. v. 16. Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Stochas-. Ch. engchi. tic Volatility (S-MR-JD-S-SV) . . . . . . . . . . . . . . . . . . . . . . . . 3.6. 15. sit. 3.4. Nat. ity (S-MR-JD-S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Stochastic Volatility and Jump Risk (S-MR-JD-S-SVJ) . . . . . . . . . . . . . .. 4 Temperature Derivatives Pricing Formula. 19 21. 4.1. Underlying Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 4.2. Temperature Derivatives Markets . . . . . . . . . . . . . . . . . . . . . .. 23. 4.2.1. Chicago Mercantile Exchange . . . . . . . . . . . . . . . . . . . .. 23. 4.2.2. Over-the-Counter . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 4.3.1. Equivalent Probability Measure . . . . . . . . . . . . . . . . . . .. 26. 4.3.2. Expectation of HDD/CDD Index . . . . . . . . . . . . . . . . . .. 29. 4.3. iv.

(6) 4.3.3. CDD/HDD Futures Pricing Formula . . . . . . . . . . . . . . . .. 35. 4.3.4. CDD/HDD Futures Options Pricing Formula. . . . . . . . . . . .. 35. 4.3.5. CDD/HDD Index Options Pricing Formula . . . . . . . . . . . . .. 37. 5 Estimation Method. 39. 5.1. Maximum Likelihood Estimation (MLE) . . . . . . . . . . . . . . . . . .. 40. 5.2. Expectation-Maximization (EM) Algorithm . . . . . . . . . . . . . . . .. 41. 5.3. Particle Filter (PF) Algorithm . . . . . . . . . . . . . . . . . . . . . . . .. 42. 5.3.1. Monte Carlo Filter . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. 5.3.2. Resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. 5.3.3. Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44. 5.3.4. EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 立. 6 Empirical Analysis. 44 46. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 6.1.1. Temperature Index . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 6.1.2. CDD/HDD Futures and Futures Options . . . . . . . . . . . . . .. 48. ‧. ‧ 國. 學. 6.1. 政治大. Model Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. y. 6.3. 50 51. 6.3.2. Out-of-Sample Pricing Performance . . . . . . . . . . . . . . . . .. 6.3.3. Comparison of Performances cross Regions . . . . . . . . . . . . .. n. er. Market Price of Risk and In-Sample Pricing Performance . . . . .. io. 6.3.1. al. Ch. n U engchi. 49. sit. Estimated Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Nat. 6.2. iv. 53 54. 7 Conclusions. 55. Bibliography. 57. Appendix A Change Measure: Deterministic Volatility Model. 62. Appendix B Change Measure: Stochastic Volatility Model. 64. Appendix C Characteristic Function: Deterministic Volatility Model. 67. Appendix D Characteristic Function: Stochastic Volatility Model. 69. v.

(7) List of Tables 1. Location of each Weather Station in the U.S. . . . . . . . . . . . . . . . .. 74. 2. Descriptive Statistics of DAT Increments . . . . . . . . . . . . . . . . . .. 75. 3. Mean of DAT in each Month . . . . . . . . . . . . . . . . . . . . . . . . .. 76. 4. Volatility of DAT in each Month . . . . . . . . . . . . . . . . . . . . . . .. 77. 5. The Number of the Zero CDD/HDD Index . . . . . . . . . . . . . . . . .. 78. 6. 政治大 Available Sample of the 立CDD/HDD Derivatives . . . . . . . . . . . . . .. 79 80. 8. Estimated Parameters for Baltimore. . . . . . . . . . . . . . . . . . . . .. 81. 9. Estimated Parameters for Boston . . . . . . . . . . . . . . . . . . . . . .. 82. 10. Estimated Parameters for Chicago . . . . . . . . . . . . . . . . . . . . . .. 83. 11. Estimated Parameters for Cincinnati . . . . . . . . . . . . . . . . . . . .. 12. Estimated Parameters for Dallas . . . . . . . . . . . . . . . . . . . . . . .. 13. Estimated Parameters for Des Moines . . . . . . . . . . . . . . . . . . . .. 86. 14. . . . . . . . . . . .. 87. y. io. 85. er. Nat. 84. sit. ‧. ‧ 國. Estimated Parameters for Atlanta . . . . . . . . . . . . . . . . . . . . . .. 學. 7. al. Estimated Parameters for Houston . . . . . . . . . . . . . . . . . . . . .. 88. 16. Estimated Parameters for Kansas City . . . . . . . . . . . . . . . . . . .. 89. 17. Estimated Parameters for Las Vegas . . . . . . . . . . . . . . . . . . . .. 90. 18. Estimated Parameters for Minneapolis . . . . . . . . . . . . . . . . . . .. 91. 19. Estimated Parameters for New York. . . . . . . . . . . . . . . . . . . . .. 92. 20. Estimated Parameters for Philadelphia . . . . . . . . . . . . . . . . . . .. 93. 21. Estimated Parameters for Portland . . . . . . . . . . . . . . . . . . . . .. 94. 22. Estimated Parameters for Sacramento . . . . . . . . . . . . . . . . . . . .. 95. 23. Estimated Parameters for Salt Lake City . . . . . . . . . . . . . . . . . .. 96. 24. Estimated Parameters for Tucson . . . . . . . . . . . . . . . . . . . . . .. 97. 25. Market Price of Risk and In-Sample Fit for CDD/HDD Futures . . . . .. 98. n. 15. iv Estimated Parameters forC Detroit . . . . . . U . . n. . . hengchi. vi.

(8) 26. Market Price of Risk and In-Sample Fit for CDD/HDD Futures Options. 100. 27. Out-of-Sample Pricing Performance for CDD/HDD Futures . . . . . . . .. 102. 28. Out-of-Sample Pricing Performance for CDD/HDD Futures Options . . .. 103. 29. Comparison of Performances cross Regions . . . . . . . . . . . . . . . . .. 104. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. vii. i Un. v.

(9) List of Figures 1. Location of each City in the U.S. . . . . . . . . . . . . . . . . . . . . . .. 105. 2. Mean of DAT in each Month . . . . . . . . . . . . . . . . . . . . . . . . .. 106. 3. Volatility of DAT in each Month . . . . . . . . . . . . . . . . . . . . . . .. 107. 4. Path of DAT for Atlanta . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. Path of DAT for Baltimore . . . . . . . . . . . . . . . . . . . . . . . . . .. 109. 6. Path of DAT for Boston. 政治大 . . . . . . . . . . . . . . . . . . . . . . . . . . .. 108. 7. Path of DAT for Chicago . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111. 8. Path of DAT for Cincinnati . . . . . . . . . . . . . . . . . . . . . . . . .. 112. 9. Path of DAT for Dallas . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113. 10. Path of DAT for Des Moines . . . . . . . . . . . . . . . . . . . . . . . . .. 114. 11. Path of DAT for Detroit . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. Path of DAT for Houston. 13. Path of DAT for Kansas City . . . . . . . . . . . . . . . . . . . . . . . .. 117. 14. . . . . . . . . . . .. 118. 立. y. 116. er. io. sit. ‧. ‧ 國. 學. Nat. 115. . . . . . . . . . . . . . . . . . . . . . . . . . .. al. 110. Path of DAT for Minneapolis . . . . . . . . . . . . . . . . . . . . . . . .. 119. 16. Path of DAT for New York . . . . . . . . . . . . . . . . . . . . . . . . . .. 120. 17. Path of DAT for Philadelphia . . . . . . . . . . . . . . . . . . . . . . . .. 121. 18. Path of DAT for Portland . . . . . . . . . . . . . . . . . . . . . . . . . .. 122. 19. Path of DAT for Sacramento . . . . . . . . . . . . . . . . . . . . . . . . .. 123. 20. Path of DAT for Salt Lake City . . . . . . . . . . . . . . . . . . . . . . .. 124. 21. Path of DAT for Tucson . . . . . . . . . . . . . . . . . . . . . . . . . . .. 125. n. 15. iv C.h. . . . . . . . . U Path of DAT for Las Vegas . . n. . . engchi. viii.

(10) Chapter 1 Introduction Intergovernmental Panel on Climate Change (IPCC) in 2013 reported that the average of. 政治° 大 warming and the green house effect. 立 It also leads the iceberg to melt with the speed of 226 the global sea surface temperature has increased 0.85 C for one century due to the global. ‧ 國. 學. billion tons per year, and causes the sea surface to become higher year-by-year. In their study period, the sea surface increase 0.19 meters and some countries or regions may be. ‧. flooded in the future. In terms of the temperature, there are more and more occurrences of the extreme temperature along with the climate change. Many countries or regions. Nat. °. sit. y. suffer from the heat wave in the summer such as 50 C in California state in 2013. In. al. er. io. the winter, the cold air mass usually envelops in a wide range. In early 2014, the polar. °. vortex invades Japan and North America, and the lowest temperature is −50 C. Such. n. iv. n U engchi the hedging weather risks increase sharply for the weather-related companies. Thus, the. Ch. extreme weather brings out the huge economic losses and the casualties. The demand of. weather derivatives contract becomes more and more important in the future. Weather derivatives contract is the financial instrument whose underlying asset is the weather indexes like the temperature, the hurricane, the rainfall, and the snowfall. It provides the insurance and reinsurance companies, the hedge funds, the energy companies, the pension funds, the state governments, the retailers, the utility companies, and the individuals to hedge the weather risks. For the temperature derivatives as example, the underlying asset is the sum of the cooling degree day/heating degree day (CDD/HDD) over a specific period. In which, the CDD index is the value of the daily average temperature index (hereafter abbreviated as DAT) being greater than the reference index. The HDD index is the value of the DAT being less than the reference index. The DAT 1.

(11) is computed by the average of the maximum and the minimum temperature in one day,. °. °. and the reference index is usually assumed as 18 C or 65 F. The CDD/HDD indexes are used to measure the demand of the cooling/heading or the refrigerant/warming. For the energy companies, they must pay the more production and maintenance costs for the electric power while the heat wave or the cold air arrives. It is called as the temperature risk. the energy companies have two method for reducing and compensating the related costs and losses. First, the energy companies can shift the temperature risk to the property & casualty insurance companies. The second method is that the energy companies can raise the power price to achieve the purpose. However, the drawback of the former method is that the insurance policy is the pre-contract giving the compensations while the equipments failure, and it cannot reflect and offset the related costs or losses of. 政治大 Therefore, the insurance market and the adjustment of the 立. the energy companies in time. The later method causes the social welfares and economic surpluses to be deprived.. power price bring out the inefficient hedging strategies even if the negative impacts for. ‧ 國. 學. the society. The temperature derivatives are linked to the temperature and reflect the volatility of the temperature directly. It not only gives the energy companies to hedge. ‧. sit. Nat. the social welfares to be destitute while the power price raised.. y. temperature risk in the capital market instead of the insurance market, but also avoids. io. er. For the insurance companies, whatever the roperty & casualty insurance companies and the life insurance companies, give the compensations with more probability while. n. al. Ch. i Un. v. the heat wave, the cold, and the extreme climate arrive. For hedging and shifting the. engchi. temperature risk, they can raise the policy premium and the insure to the reinsurance companies (the insurance market), but the former approach leads to exploit the benefits of the insured institutions and the individuals and the later method causes the more costs for the insurance companies. The temperature derivatives provide the more efficient hedge strategies for the weather-related companies in the capital market. There are some contracts and instruments for evading the weather risks. The catastrophe bond (CatBond) is a debt that the issuers (debtors) have the obligation to pay the periodic coupon to the bond holders (creditors). If the catastrophe occur, the bond issuers’ losses can be compensated by the trigger mechanism which uses the bond principal to offset the losses. At maturity, the bond holders receive the balanced principal. However, there are some drawbacks of the CatBonds: (i) Owing to the higher transaction. 2.

(12) costs and the illiquidity for the bonds, the hedgers cannot hedge efficiently in the bond market. (ii) The CatBonds stipulate that the hedgers can get the compensations only if the losses arriving the upper bound, so the hedgers do not get the substantial compensation while the loss approaching. (iii) The CatBonds cannot hedge the specific natural disasters like the temperature, the rainfall, the intensity of the hurricane, and so on. (iv) The institution suffered from the catastrophe faces the risk of the stock price decreasing. It is difficult to raise funds for the institution. The catastrophe equity put (CatEPut) is also an instrument to hedge weather risks. If the company suffer from the losses of the natural disasters, the option contract gives the buyer with the right to sell the stocks at a strike price while the stock price is less than the strike price and the catastrophe losses exceed a specific threshold. However, the. 政治大 the company to raise funds. According to above, the CatBond and the CatEPut provide 立. CatEPut only gives the company against the risk of the stock pricing falling and facilitate. the new instrument for hedging weather risks, but they exist some drawbacks and are. ‧ 國. 學. inefficient to use.. Owing to the catastrophe insurance, the CatBond, and the CatEPut be not hedge the. ‧. specific natural disasters, there are limited effectiveness of risk to transfer to the capital. sit. y. Nat. market. The Chicago Mercantile Exchange (CME) in the U.S. aims a variety of natural. io. er. disasters to issue the weather related derivatives like the temperature, the rainfall, the intensity of the hurricane, the snowfall. It can give the more flexible instruments for the. n. al. Ch. i Un. v. energy companies and the insurance companies. In a variety of weather-linked derivatives. engchi. in the CME, the temperature derivatives contract is the popular commodities. Namely, the underlying asset of the temperature derivatives is the temperature index like (i) the sum of cooling degree day/heating degree day (CDD/HDD) index and the weekly average temperature (WAT) for the U.S., (ii) the sum of the CDD/HDD index and the cumulative average temperature (CAT) for Canada, (iii) the sum of the HDD index and the CAT index in the winter, and no derivatives in the summer for the Europe, (iv) the sum of the CDD/HDD index for Australia, (v) the CAT index for Osaka, Tokyo, Hiroshima in Japan. In which, the definitions of the CDD index and the HDD index are discussed in the following chapter. There are two derivatives in the U.S.: (i) The CDD/HDD futures contract, and (ii) The CDD/HDD futures options contract. The former is the futures contract whose. 3.

(13) underlying asset is the sum of the CDD index or the HDD index over a period (usually a month). The buyer and seller of the CDD/HDD futures are required to purchase and sell at a delivery date with a predetermined price. On the other hand, the later is the options contracts whose underlying asset is the CDD/HDD futures contract. The buyers of options have the right to exercise the contract at maturity, and purchase/sell the CDD/HDD futures with the strike price. The sellers of options have the obligation to fulfil the requirements of the buyers. The options on futures are European framework. Above two patterns of CDD/HDD derivatives contracts are settled in cash. There are also a variety of temperature derivatives in the over-the-counter (OTC). Especially, the CDD/HDD index options contract is the popular commodity. Comparing to the CDD/HDD futures options contract, the underlying asset of the index option is. 政治大 owing to the DAT be nontradable, the CDD/HDD index options contract are settled in 立 the sum of the CDD/HDD index over a period instead of the futures contract. Also,. cash. However, this study does not discuss the weather commodities in the OTC.. ‧ 國. 學. This study focuses on the model parameters estimating, the pricing method, and pricing performance for the CDD/HDD derivatives in the U.S. including 18 cities like. ‧. Atlanta, Baltimore, Boston, Chicago, Cincinnati, Dallas, Des Moines, Detroit, Houston,. sit. y. Nat. Kansas City, Las Vegas, Minneapolis, New York, Philadelphia, Portland, Sacramento,. io. er. Salt Lake City, and Tucson. The empirical results shows that the DAT for each city in the U.S. increases year-by-year owing to the global warming. Moreover, the volatility of. n. al. Ch. i Un. v. the DAT exist the seasonality and there are many occurrences of the extreme movements. engchi. (called as jump risks). These features of the DAT cannot be ignored while pricing the temperature derivatives. Unfortunately, the DAT is nontradable and the weather market is incomplete. We cannot construct a portfolio to replicate the payoff of the temperature derivatives perfectly. Although there are infinite equivalent probability measures and the price of the temperature derivatives under no-arbitrage framework is not unique, Xu, Odening, and Musshof (2008) supposed that the risk-neutral pricing method is still carried out. Thus, under no-arbitrage framework, we derive the CDD/HDD futures and the CDD/HDD futures options pricing formulas, and estimate the market price of risk by minimizing the sum of squared error between the market price and the model price. The market price of risk is also called as the parameters of the Esscher transform. The empirical. 4.

(14) results illustrates that the market price of risk differs from zero significantly (most are negative) as Alaton, Djehince, and Stillberg (2002), Bellini (2005), Benth, Hardle, and Lopez (2009), Hardle and Lopez Cabrera (2009). That is, the weather market investors require the positive risk premium for the derivatives. Besides, The stochastic volatility and jump diffusion model performs the in-sample and the out-of-sample pricing error in better for the CDD/HDD futures and CDD/HDD futures options. The investors can observe the past market price of risk and predict the future path of the weather derivatives. Then, the traders can use it to determine the trading/hedging strategy for the weather derivatives. Comparing to four regions in the U.S. (Midwest, Northeast, South, and West), (i) for the fitting performance of the historical estimation, the model with jump risks in the. 政治大 and the South regions, and the stochastic volatility model with jump risks of the tem立. temperature index and the stochastic volatility is better in the Midwest, the Northeast,. perature index is better in the West region. (ii) For the in-sample pricing performance. ‧ 國. 學. of the CDD/HDD futures, the model with jump risks in the temperature index and the stochastic volatility exists the better performance in the South and the West regions, and. ‧. the stochastic volatility model with jump risks of the temperature index is better in the. sit. y. Nat. Midwest and the Northeast regions. (iii) For the in-sample pricing performance of the. io. er. CDD/HDD futures options, the jump diffusion model is the better performance in the Midwest region, and others regions do not obey the specific model significantly. (iv) For. n. al. Ch. i Un. v. the out-of-sample pricing performance of the CDD/HDD futures, the stochastic volatility. engchi. model is better in the Midwest, the Northeast, and the West regions, and the jump diffusion model is better in the South region. (v) For the out-of-sample pricing performance of the CDD/HDD futures options, the stochastic volatility model with jump risks of the temperature index exists the better performance in the Midwest, the Northeast, and the West regions, and the model with jump risks in the temperature index and the stochastic volatility exists the better performance in the South region. There are some contributions in this study: (i) we extend the Alaton, Djehince, and Stillberg (2002)’s framework by employing the Merton (1976)’s jump diffusion model, the Heston (1993)’s stochastic volatility model, the Bates (1996)’s stochastic volatility model with jump risks, and Eraker, Johannes, and Polson (2003)’s jump risk in volatility and return model. (ii) The model parameters are estimated by the maximum likelihood. 5.

(15) estimation (MLE) for the jump-free and deterministic (or constant) volatility models, the expectation-maximization (EM) algorithm for the jump and deterministic volatility models, and the particle filter (PF) algorithm for the stochastic volatility models. And, the complex model exists the better goodness-of-fit for the path of the temperature index. (iii) We employ the Esscher transform to change the probability measure and derive the probability density function of each model by the Feynman-Kac formula and the Fourier transform. (iv) The semi-closed form of the CDD/HDD futures pricing formula is derived, and we use the Monte-Carlo simulation to value the CDD/HDD futures options due to no closed-form solution. (v) Whatever in-sample and out-of-sample pricing performance, the type of the stochastic volatility performs the better fitting for the temperature derivatives. (vi) The market price of risk (the parameter of the Esscher transform) differs to. 政治大. zero significantly (most are negative), so the investors require the positive weather risk premium for the derivatives.. 立. This study is organized as follows: Section 2 lists the past important literatures and. ‧ 國. 學. discuss the overview of the development for the temperature model and the CDD/HDD derivative pricing model. Section 3 shows and explain the details of each model. Section. ‧. 4 presents the definitions of the CDD/HDD index and its derivatives pricing formula.. sit. y. Nat. Section 5 discusses the estimation method of the model parameters like the maximum. io. er. likelihood estimation, the expectation-maximization algorithm, and the particle filter algorithm. Section 6 lists the empirical results of the model fitting. Section 7 is the. n. al. Ch. i Un. v. conclusion of this study. The detail proofs are shown in the Appendix.. engchi. 6.

(16) Chapter 2 Literature Review 2.1. 政治大. Temperature Model. 立. Lau and Weng (1995) used Morlet (1983)’s wavelet transform to depict the path of the. ‧ 國. 學. DAT. This method is usually employed in the physics. The purpose of the wavelet transform is that the time series can be decomposed into several wavelet. Differing from. ‧. the Fourier transform which only analyses the average characteristic of a time series and is affected by the extreme value easily, the wavelet transform uses the basis function which. y. Nat. sit. is generalized by the contraction and translation of the prototype function to catch the. al. er. io. features of a time series. It can measure the continuous (stable state) and uncontentious. v. n. change (jump state) path. Lau and Weng (1995) used the monthly North Hemisphere. Ch. i Un. surface temperature (NHST) over period 1945/01 to 1993/07 as sample, by the wavelet. engchi. transform, the path of the DAT can be decomposed into three frequency branches: interannual band (2 to 5 years), inter-decade bands (10 to 60 years), century scale (90 to 180 years). They argued that the results can measure the path of the DAT efficiently. Cao and Wei (1999, 2000, 2004) pointed out that the ”net” DAT (the original DAT is subtracted by average level in current month) exists the feature of the autocorrelation. Thus, they employed the autoregression (AR) model to depict the path and used the sine function to catch the seasonal volatility. Cao and Wei (1999, 2000) adopted the DAT for Atlanta, Chicago, Dallas, Philadelphia over the period 1979 to 1998. The data are obtained from the National Climate Data Center (NCDC) of the National Oceanic Atmospheric Administration (NOAA). The parameters of the AR model are fitted by the maximum likelihood estimation, and the optimal lag period is four period. It shows that 7.

(17) the temperature do exist the autocorrelation, and it cannot be ignored for pricing the temperature derivatives. Alaton, Djehince, and Stillberg (2002) assumed that the long-run mean level of the DAT follows the linear regression model with the time trend and seasonal factor. The time trend measures the global warming caused by the greenhouse effect. The seasonal factor (the DAT is lower in the winter and higher in the summer) is depicted by the sine and the cosine functions. The volatility of the DAT is constant level in one month in their assumption. Moreover, the DAT is along the long-run mean level and it exists the feature of the mean-reversion (MR). According to above characteristics of the DAT, Alaton, Djehince, and Stillberg (2002) supposed the Ornstein-Uhlenbeck (OU) process to depict the path. The OU process is also called as Vasicek model (1977) or Hull-White model. 政治大 years as sample, and estimated the parameters through three steps: (i) For the long-run 立 (1990). They used the DAT of Bromma airport meteorological station in Sweden over 40. mean level, they employed a linear regression model whose dependent variable is DAT. ‧ 國. 學. and the independent variables are the time trend and the seasonal factor. Through the least squares (LS) method, the parameters of the linear regression model are estimated.. ‧. (ii) The monthly volatility is measured by the standard deviation of the ”net” DAT (the. sit. y. Nat. residual of the linear regression model) in the specific month. (iii) The parameter of. io. er. the speed of mean-reversion is computed by the martingale estimation function method developed by Bibby and Sørensen (1995). The empirical results report that the DAT. n. al. Ch. i Un. v. follows the global warming path, the seasonal volatility (higher in the winter and lower. engchi. in the summer), and the mean-revering effect.. Brody, Syroka, and Zervos (2002) also pointed out the autocorrelation of the DAT. The standard Brownian motion or the Gaussian white noise process cannot capture the long memory property. Thus, they introduced the fractional Brownian motion (FBM) into the mean-reverting model, and assume the deterministic processes for the speed of mean-reversion, the long-run mean level of DAT, and the seasonal volatility. In the FBM, the Hurst index (0 ≤ H ≤ 1) measures the correlation between the pre- and post increments of the DAT. If 0 ≤ H < 0.5, the correlation is negative. If 0.5 < H ≤ 1, the correlation is positive. If H = 0.5, the correlation is zero and the FBM reduces to the standard Brownian motion. Differing to Cao and Wei (1999, 2000), Brody, Syroka, and Zervos (2002) used the FBM to depict the risk premium of the autocorrelation directly.. 8.

(18) But, they did not estimate the models parameters, and cannot compare the performance between the FBM and the standard Brownian motion. ˇ Benth and Saltyt˙ e-Bench (2005) advocated that the OU process and the Lévy process are the better model for the DAT, and assumed that the long-run mean level and the volatility follow the truncated Fourier series which is the linear combination of a series of sine and cosine functions and measures the seasonality. They used the DAT of Alta, Bergen, Kristiansand, Oslo, Stavanger, Tromsø, Trondheim in Norway as sample, and pointed out that the DAT in Norway follows the leptokurtic distribution. It implies that the extreme temperature occurs with less probability. Moreover, the empirical results shows that the DAT exists the one-lag positive autocorrelation, the two-lag negative autocorrelation, and the volatility clustering in short term. They suggested that the gen-. 政治大. eralized autoregression conditional heteroskedasticity (GARCH) model can depict these. 立. features.. Campbell and Diebold (2005) also employed the truncated Fourier series to measure. ‧ 國. 學. the long-run mean levels of the DAT and volatility. Apart from the seasonality, the DAT also exists the volatility clustering effect. Thus, they introduced the GARCH model. ‧. with seasonal DAT and volatility to depict the path of the DAT. Campbell and Diebold. sit. y. Nat. (2005) used the DAT of Atlanta, Chicago, Las Vegas, and Philadelphia as sample, the. io. er. empirical results show that the DAT in each city does not follow the normal distribution. Subtracting the time trend, the seasonal factor, and autocorrelated factor, the ”net”. n. al. Ch. i Un. v. DAT obeys the left-skew and platykurtic distribution. Thus, the investors cannot use the. engchi. standard Brownian motion or the Gaussian white noise process to describe the path of the DAT. Moreover, Campbell and Diebold (2005)’s model can predict the path of the DAT exactly in long term than in short term. ˇ Benth and Saltyt˙ e-Bench (2007) also used the truncated Fourier series to measure the long-run mean levels of the DAT and volatility. Obtaining the DAT of Stockholm in Sweden over the period 1961/01/01 to 2006/05/26 as sample, the empirical results are ˇ similar to Benth and Saltyt˙ e-Bench (2005). ˇ Past literatures like Benth and Saltyt˙ e-Bench (2007) usually employed the likelihood ratio test (LRT), the Akaike information criterion (AIC), the Bayesian information criterion (BIC), in-sample and out-of-sample predicting performance to determine the optimal lag period. However, Zapranis and Alexandridis (2008, 2009a, 2009b) used the concepts. 9.

(19) of the wavelet transform and the neural networks which is developed by Lau and Weng (1995) to choose the lag period. The sample includes the DAT of Paris over the period 1971/01/01 to 2000/12/31. The time series of the DAT can be decomposed into 1-year and 7-year wavelets. Huang, Shiu, and Lin (2008) extended Alaton, Djehince, and Stillberg (2002)’s framework. They supposed the GARCH model with the time trend and seasonal factor of the long-run mean level. The sample is obtained from Taiwan Central Weather Bureau over the period 19740/01/01 to 2003/12/31. The GARCH(4,3) performs the better fitting for the DAT. It says that the DAT in Taiwan exists the four lag period unexpected shock effect and three lag period volatility clustering effect. However, under Huang, Shiu, and Lin (2008)’s model, the volatility is negative with positive probability. In the future, the. 政治大 ˇ Benth and Saltyt˙ e-Bench (2011) introduced the continuous-time autoregressive (CAR) 立. parameters will be restricted against the negative volatility.. model which is similar to the Barndorff-Nielsen and Shephard (2006)’s stochastic volatil-. ‧ 國. 學. ity model. Owing to the volatility under the GARCH model be predictable, it does not correspond to reality. And, the volatility exists the characteristics of the autocor-. ‧. relation and seasonality, so the CAR model can depict the DAT more exactly. Using. sit. y. Nat. the DAT of Stockholm of Sweden over the period 1961/01/01 to 2006/05/26 as sample,. io. aurocorrelation and platykurtic distribution.. n. al. 2.2. Ch. engchi. er. ˇ Benth and Saltyt˙ e-Bench (2011) pointed out that the DAT follows the three lag period. i Un. v. CDD/HDD Derivatives Pricing Model. Cao and Wei (1999, 2000, 2004) argued that the DAT is not tradable, so the no-arbitrage and the risk-neutral pricing method cannot be used to price the derivatives. Therefore, they employed Lucas (1978)’s equilibrium pricing method. Given the investors’ utility function, the dynamic process of the DAT, the contingent claim of the temperature derivatives, the price of derivatives is determined by maximizing the utility function. Owing to the CDD/HDD index be non-linear function, it is difficult to derive the dynamic process of the DAT. Therefore, Davis (2001) assumed that CDD/HDD index follows the geometric Brownian motion (GBM), then derived the temperature derivatives pricing formula by Lucas (1978)’s framework. There is a drawback of such method. In. 10.

(20) practice, the CDD/HDD index do not obey a GBM. Although it is easy to derive the pricing formula, the it leads to misprice seriously. Alaton, Djehince, and Stillberg (2002) pointed out that the CDD/HDD index touches zero in their study period with less probability, and it can be ignored. Thus, CDD/HDD index is seen as the linear function. It is helpful to derive the closed-form pricing formula for the CDD/HDD index option contract. Besides, they argued that the DAT is nontradable, but the CDD/HDD derivatives pricing formula can be still valued by adjusting the market price of risk under no-arbitrage and the risk-neutral pricing framework. Brody, Syroka, and Zervos (2002), Bellini (2005) advocated that the investors cannot construct the risk-free portfolio (Black and Scholes, 1973) to replicate the payoff of the CDD/HDD derivatives owing to the DAT be non-tradable. Such market is incomplete and. 政治大 unique. Nonetheless, the market price of risk (or the parameters of the Girsanov’s theorem 立 exists infinite risk-neutral probability measures. That is, the price of the derivatives is not. and the Esscher transform) can be determined by the market price of the CDD/HDD. ‧ 國. 學. derivatives. It reflects the investors’ required risk premium and the future expectation. And, the market price of risk is used to value others temperature derivatives which is the. ‧. present value of the cash flow in future with the risk-adjusted discounted rate.. sit. y. Nat. ˇ Others literature like Benth and Saltyt˙ e-Bench (2005, 2007, 2011), Huang, Shiu, and. io. er. Lin (2008), Zapranis and Alexandridis (2008, 2009a, 2009b) also used the risk-neutral pricing framework to value the temperature derivatives. In which, because the temper-. n. al. Ch. i Un. v. ature model becomes more and more complex, it is very difficult to derive the pricing. engchi. ˇ formula for the temperature derivatives. Benth and Saltyt˙ e-Bench (2011) computed the characteristic function at first by the Feynman-Kac formula. Second, the probability density function (PDF) is calculated by the inverse Fourier transform. Finally, the real number of the expectation of the discounted cash flow is the fair value of the temperature derivatives.. 11.

(21) Chapter 3 The Models 3.1. Seasonal Mean-Reversion 治 Model (S-MR). 立. 政. 大. It is well known that there are some characteristics for the temperature path: (i) The. ‧ 國. 學. periodic cycle of the temperature is lower in winter and higher in summer1 ; (ii) The upward trend of the temperature is caused by the climate change like the global warming,. ‧. the greenhouse effect, and so on; (iii) The volatilities of the temperature are different in each month. Base on above characteristics, Alaton, Djehince, and Stillberg (2002). y. Nat. sit. supposed that the long-run temperature follows a linear function with the time trend. al. er. io. factor and the seasonal factor. The short-run temperature is depicted by a Ornstein-. v. n. Uhlenbeck (OU) process which implies the mean-reversion in short term. On the other. Ch. i Un. hand, because the underlying asset of the temperature derivatives is the accumulated. engchi. value of the HDD/CDD in the specific month, Alaton, Djehince, and Stillberg (2002) assumed that the volatility of the temperature in the specific month is constant. For distinguishing the following models in later sections, in this study, I rename the Alaton, Djehince, and Stillberg (2002)’s model as the seasonal mean-reversion model (S-MR). Consider a probability space, {Ω, F, P}, generated by the Wiener process, Wt ∼ N (0, t), where Ω is the universal set of all outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {Ft }Tt=0 be the filtration including the series of the Wiener process at time t. According to above definition, the DAT, Xt , 1. In this study, I focus on the northern hemisphere only.. 12.

(22) follows the S-MR model, then its dynamic process under the P measure is dXtα α dt + φt dWt , dXt = β (Xt − Xt ) + dt 2πt 2πt α Xt = α0 + α1 t + α2 sin + α3 cos , 365 365    φJan , in January,      φ , in February, F eb φt = .. ..   . .      φ , in December,. (3.1). Dec. where β > 0 is the speed of reversion. Xtα is the long-run mean level where the second term is the trend factor and the third and fourth terms are the seasonal factors. φt =. 政治大 By Itô’s lemma, given the filtration F , s < t and t − s not across a month, the 立. {φJan , φF eb , · · · , φDec } > 0 is the instantaneous volatility in the specific month. s. Xt =. Xtα. + (Xs −. Xsα ) e−β(t−s). Z +. 學. t. φu e−β(t−u) dWu .. (3.2). s. ‧. ‧ 國. solution of the equation (3.1) is. It can be easily observed that the temperature follows the normal distribution with the. Nat. n. al. Ch. φ2s 1 − e−2β(t−s) V ars (Xt ) = . 2β. er. io. Es (Xt ) = Xtα + (Xs − Xsα ) e−β(t−s) ,. sit. y. mean and the variance as below.. i Un. (3.3). v. There are some drawbacks under the S-MR model: (i) The constant volatility in the. engchi. specific month cannot measure the daily change of the volatility, and it misprices the risk during the holding period for the investors; (ii) Under the S-MR model, the temperature follows the normal distribution, and it underestimates the effect of the extreme climate; (iii) If the life of the temperature derivatives cross two or more months, the discontinuoustime volatility brings about the more processing cost.. 3.2. Seasonal Mean-Reversion Model with Seasonal Volatility (S-MR-S). For modifying drawbacks of the constant volatility in the specific month and the discontinuous time volatility across two or more months, I employ a deterministic function 13.

(23) to depict the path of the volatility. Empirical result shows that the volatility is usually lower in the summer and higher in the winter. It implies the volatility with the seasonal cycle. Therefore, similarly to the long-run temperature, the sine function can fit the path of the temperature volatility exactly. Different to Alaton, Djehince, and Stillberg (2002)’s model whose volatility is constant in the specific month, the modified model not only provides the continuous-time volatility but also measure the seasonal cycle of the volatility. It is conducive to price the temperature derivatives accurately. Consider a probability space, {Ω, F, P}, generated by the Wiener process, Wt ∼ N (0, t), where Ω is the universal set of all outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {Ft }Tt=0 be the filtration including the series of the Wiener process at time t. According to above definition, the DAT, Xt ,. 政治大. follows the S-MR-S model, then its dynamic process under the P measure is dXtα α dt + σt dWt , dXt = β (Xt − Xt ) + dt 2πt 2πt α Xt = α0 + α1 t + α2 sin + α3 cos , 365 365 2πt σt = φ0 + φ1 1 + sin + φ2 > 0, 365. 立. (3.4). ‧. ‧ 國. 學. Nat. sit. y. where β > 0 is the speed of reversion. Xtα is the long-run mean level where the second. er. io. term is the trend factor and the third and fourth terms are the seasonal factors. σt is the. al. iv n C the volatility, and φ2 is the location parameter. U the sine function plus one h e n g c Inh iaddition, ensures the positive volatility. n. instantaneous volatility where φ0 > 0 is the based level, φ1 ≥ 0 is the seasonal factor of. By Itô’s lemma, given the filtration Fs , s < t, the solution of the equation (3.4) is Z t α α −β(t−s) Xt = Xt + (Xs − Xs ) e + σu e−β(t−u) dWu . (3.5) s. According to the additivity property of the normal distribution, the third term of the right hand side of the equation (3.5) can be seen as summation of a series normal distribution. Thus, the temperature under the S-MR-S model follows the normal distribution with the mean and the variance as below. Es (Xt ) = Xtα + (Xs − Xsα ) e−β(t−s) , (3.6) 2 Z t 2πu V ars (Xt ) = φ0 + φ1 + φ1 sin + φ2 e−2β(t−u) du = (a) + (b) + (c), 365 s 14.

(24) where. (c) =. φ21. (φ0 + φ1 )2 1 − e−2β(t−s) (a) = , 2β 2 (φ0 + φ1 ) φ1 2π · Cs,t (1, 2) , (b) = 2β · Ss,t (1, 2) − 2π 2 365 4β 2 + 365 1 − e−2β(t−s) φ21 2π − h i 365 · Ss,t (2, 2) + β · Cs,t (2, 2) , 2π 2 4β 4 β 2 + 365. and the functions Ss,t (x, y) and Cs,t (x, y) are 2πs 2πt + φ2 x − sin + φ2 x e−βy(t−s) , Ss,t (x, y) = sin 365 365 2πt 2πs Cs,t (x, y) = cos + φ2 x − cos + φ2 x e−βy(t−s) . 365 365 The variance of the equation (3.5) is computed from the integration by parts. Moreover,. 政治大. if φ1 = 0, the model is the same as the S-MR model which means the volatility without. 立. seasonality in the specific month.. ‧ 國. 學. There are some drawbacks under the S-MR-S model: (i) The oscillations of the volatility follows a fixed cycle and presents the long-run level. It cannot reflect the impacts of the. ‧. short-run variations; (ii) Under the S-MR-S model, the temperature follows the normal distribution, and it underestimates the effects of the extreme climates;. sit. y. Nat. er. Seasonal Mean-Reversion and Jump Diffusion. io. 3.3. n. a. v. i l C Model with Seasonal Volatility U n (S-MR-JD-S). hengchi. Along with some weather circumstance such as the global warming, the greenhouse effect, the climate change, the events of the extreme weather occur with the more and more possibility. The daily temperature raising or falling sharply becomes widespread at everywhere. The sharply variations are called as the jump risks. Therefore, the weatherrelated industries like the energy companies, the agricultural companies, the insurance companies, and the reinsurance companies, suffer the more and more temperature risks, and they can utilize the temperature derivatives to hedge these risks. For this reason, how the jump risks are priced is the important for the derivatives issuers. The comPNt pound Poisson process, j=0 Yj , is usually used to depict the number of occurrences, i.i.d.. Nt ∼ P oi (λt), and the sizes of variations, Yj ∼ N (θ, ν 2 ), j = 0, 1, · · · , ∞, Y0 = 0, for the extreme weather and jump risks. 15.

(25) Consider a probability space, {Ω, F, P}, generated by the Wiener process, Wt ∼ P t N (0, t) and the compound Poisson process, N j=0 Yj , where Ω is the universal set of all outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. The Wiener process {Wt }Tt=0 , the jump frequency {Nt }Tt=0 , and the jump T amplitude {Yj }∞ j=0 are independent. Let {Ft }t=0 be the filtration including the series of. the Wiener process and the compound Poisson process at time t. According to above definition, the DAT, Xt , follows the S-MR-JD-S model, then its dynamic process under the P measure is dXt = β. Xtα. (Xtα. Nt X dXtα − Xt ) + dt + σt dWt + d Yj − λθdt, dt j=0. (3.7). 2πt 2πt + α3 cos , = α0 + α1 t + α2 sin 365 365 2πt + φ2 > 0, σt = φ0 + φ1 1 + sin 365 . 立. 政治大. ‧ 國. 學. where β > 0 is the speed of reversion. Xtα is the long-run mean level where the second term is the trend factor and the third and fourth terms are the seasonal factors. σt is the. ‧. instantaneous volatility where φ0 > 0 is the based level, φ1 ≥ 0 is the seasonal factor of the volatility, and φ2 is the location parameter. In addition, the sine function plus one. y. Nat. sit. ensures the positive volatility.. al. er. io. Although the S-MR-JD-S model can measure the extreme temperature with more. v. n. accurately, the volatility is decomposed into a seasonal volatility and jump volatility, and. Ch. i Un. it only presents the long-run level. For modifying this drawback, we will introduce Heston. engchi. (1993)’s stochastic volatility model in the next section.. 3.4. Seasonal Mean-Reversion Model with Seasonal Stochastic Volatility (S-MR-S-SV). The S-MR model, the S-MR-S model, and the S-MR-JD-S model employed the deterministic function to measure the path of the temperature volatility. These model can only depict the long-run mean level of the volatility, and cannot reflect the climate change immediately in the short run. Past literatures employed the GARCH family models to predict the volatility. If the current temperature increment enlarges/shrinks, the volatility goes upward/downward in the future. That is, under GARCH family models, the 16.

(26) volatility is determined by the historical and the realized temperature. However, the volatility is interfered by not only historical and the realized temperature but also other factors like the geographical environment, global climate change, sea surface temperature, and so on. The GARCH family models are likely to bring about the misestimated volatility. For this reason, the volatility of the temperature is unpredictable. The stochastic volatility model plays the fairish framework to depict the unpredictable volatility. On the other hand, according to Nelson and Foster (1994), the variance equation of the GARCH model converges to the square-root process while the time-step closes to zero. Like in Feller (1951), Cox, Ingersoll, Ross (1985), Heston (1993), they assumed that the variance obeys the square-root process against the negative volatility. Thus, I employ the Heston (1993)’s framework and lead the long run seasonality into the stochastic volatility model.. 政治大 ∼ N (0, t), Corr (W , W ) = ρ, where Ω is the universal set of all 立. Consider a probability space, {Ω, F, P}, generated by two Wiener processes, WX,t ∼ N (0, t) and WV,t. X,t. V,t. outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability. ‧ 國. 學. measure. Let {Ft }Tt=0 be the filtration including the series of the Wiener process at time t. According to above definition, the DAT, Xt , follows the S-MR-S-SV model, then its. ‧. n. al. er. io. sit. y. Nat. dynamic process under the P measure is p dXtα α dXt = β (Xt − Xt ) + (3.8) dt + Vt dWX,t , dt " # dV φ p dVt = κ Vtφ − Vt + t dt + σ Vt dWV,t , dt 2πt 2πt α Xt = α0 + α1 t + α2 sin + α3 cos , 365 365 q 2πt φ Vt = φ0 + φ1 1 + sin + φ2 > 0, 365 where, for the part of the temperature process, β > 0 is the speed of reversion. Xtα is. Ch. engchi. i Un. v. the long-run mean level where the second term is the trend factor and the third and fourth terms are the seasonal factors. For the part of the variance process, Vt is the instantaneous volatility. κ > 0 is the speed of reversion. Vtφ is the long-run mean level where φ0 > 0 is the based level, φ1 ≥ 0 is the seasonal factor of the volatility, and φ2 is the location parameter. Also, the sine function plus one ensures the positive volatility. √ σ Vt is the volatility of the variance. There are some drawbacks under the S-MR-S-SV model: (i) It cannot measure the extreme movements of the DAT. (ii) The shocks of the DAT coming from the natural 17.

(27) disasters not only cause the jump movement in the DAT, but also bring out the volatility increasing sharply and rapidly. Under the S-MR-S-SV model, the volatility can only moves gradually, and it cannot measure the rapid shock in the volatility.. 3.5. Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Stochastic Volatility (S-MR-JD-S-SV). Owing to the global warming, the greenhouse effect, the climate change, the jump movements of the DAT occur with more probability. The daily temperature increasing or. 政治大 more weather risk. For this reason, the jump diffusion must be considered while pricing 立 decreasing sharply becomes widespread at everywhere. The investors face the more and. the derivatives.. ‧ 國. 學. Consider a probability space, {Ω, F, P}, generated by two Wiener processes, WX,t ∼. ‧. N (0, t), WV,t ∼ N (0, t), Corr (WX,t , WV,t ) = ρ, and the compound Poisson process, PNt j=0 Yj , where Ω is the universal set of all outcomes, F is the σ-field of the subset be-. y. Nat. long to Ω, and P is the physical probability measure. The Wiener processes {WX,t }Tt=0 ,. io. sit. {WV,t }Tt=0 , the jump frequency {Nt }Tt=0 , and the jump amplitude {Yj }∞ j=0 are indepen-. n. al. er. dent. Let {Ft }Tt=0 be the filtration including the series of the Wiener processes and the. Ch. i Un. v. compound Poisson process at time t. According to above definition, the DAT, Xt , follows. engchi. the S-MR-JD-S-SV model, then its dynamic process under the P measure is Nt X p dXtα α dXt = β (Xt − Xt ) + Yj − λθdt, dt + Vt dWX,t + d dt j=0 ". (3.9). # φ p dV dVt = κ Vtφ − Vt + t dt + σ Vt dWV,t , dt 2πt 2πt α Xt = α0 + α1 t + α2 sin + α3 cos , 365 365 q 2πt φ Vt = φ0 + φ1 1 + sin + φ2 > 0, 365 . . where, for the part of the temperature process, β > 0 is the speed of reversion. Xtα is the long-run mean level where the second term is the trend factor and the third and fourth terms are the seasonal factors. For the part of the variance process, Vt is the 18.

(28) instantaneous volatility. κ > 0 is the speed of reversion. Vtφ is the long-run mean level where φ0 > 0 is the based level, φ1 ≥ 0 is the seasonal factor of the volatility, and φ2 is the location parameter. Also, the sine function plus one ensures the positive volatility. √ σ Vt is the volatility of the variance. The shocks of the DAT coming from the natural disasters not only cause the jump movement in the DAT, but also bring out the volatility increasing sharply and rapidly. Under the S-MR-JD-S-SV model, the volatility can only moves gradually, and it cannot measure the rapid shock in the volatility. Thus, the next model will modify this drawback.. 3.6. Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Volatility 治政 Stochastic. 大. 立 (S-MR-JD-S-SVJ) and Jump Risk. ‧ 國. 學. The shocks of the DAT coming from the natural disasters not only cause the jump movement in the DAT, but also bring out the volatility increasing sharply and rapidly. Eraker. ‧. Johannes, and Polson (2003) argued the diffusive volatility can only move gradually and. sit. y. Nat. connot measure the shock in short term. Thus, jump in volatility revises this drawback,. io. er. and captures the rapid movements of jumps.. Consider a probability space, {Ω, F, P}, generated by two Wiener processes, WX,t ∼. n. al. Ch. i Un. v. N (0, t), WV,t ∼ N (0, t), Corr (WX,t , WV,t ) = ρ, where Ω is the universal set of all. engchi. outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {Ft }Tt=0 be the filtration including the series of the Wiener process at time t. According to above definition, the DAT, Xt , follows the S-MR-S-SV model, then its dynamic process under the P measure is NX,t X p dXtα α dXt = β (Xt − Xt ) + dt + Vt dWX,t + d YX,j − λX θX dt, dt j=0 # NV,t X p dVtφ dVt = κ − Vt + dt + σ Vt dWV,t + d YV,j , dt j=0 2πt 2πt α Xt = α0 + α1 t + α2 sin + α3 cos , 365 365 q 2πt φ Vt = φ0 + φ1 1 + sin + φ2 > 0, 365 ". . Vtφ. . 19. (3.10).

(29) where, for the part of the temperature process, β > 0 is the speed of reversion. Xtα is the long-run mean level where the second term is the trend factor and the third and fourth terms are the seasonal factors. For the part of the variance process, Vt is the instantaneous volatility. κ > 0 is the speed of reversion. Vtφ is the long-run mean level where φ0 > 0 is the based level, φ1 ≥ 0 is the seasonal factor of the volatility, and φ2 is the location parameter. Also, the sine function plus one ensures the positive volatility. √ σ Vt is the volatility of the variance.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 20. i Un. v.

(30) Chapter 4 Temperature Derivatives Pricing Formula 立. 政治大. In this chapter, we narrate the underlying asset of the temperature derivatives contract at. ‧ 國. 學. first. As its name implies, the payoff of the temperature derivatives is linked to the DAT. However, the heating degree day (HDD) and cooling degree day (CDD) is used generally. ‧. instead of the DAT. in the section 4.1, the definitions of the HDD, the CDD, and the underlying asset of the temperature derivatives contract will be shown in detail. Second, the. y. Nat. sit. temperature derivatives are usually traded at the Chicago Mercantile Exchange (CME). al. er. io. and the over-the-counter (OTC). The payoff patterns of the temperature derivatives at. v. n. maturity are different in each market. We aim these different payoffs to discuss in the. Ch. i Un. section 4.2. Finally, as in Bates (1996), Heston (1993), Duffie, Pan, and Singleton (2000),. engchi. Wong and Lo (2009), Pillay and O’Hara (2011), we derive the characteristic function of each model by the partial integro-differential equation (PIDE) which is also called as Kolmogorov backward equation (KBE) or Fokker-Planck equation (FPE). Then, the probability density function (PDF) can be computed by the inverse Fourier transform. Given the payoff at maturity, the CDD/HDD futures can be priced. The pricing formula is reported in the section 4.3. Unfortunately, there is no closed-form solution for the CDD/HDD futures options and the Monte-Carlo simulation pricing method is used to value the premium of the options.. 21.

(31) 4.1. Underlying Asset. The HDD is computed by the difference between the daily average temperature and the reference index only if the daily average temperature is less than the reference index. The larger HDD implies the more demand of the heater in the winter and the more temperature risk for the energy companies, the insurance companies, the agricultural companies, and so on. On the other hand, the CDD is calculated by the difference between the daily temperature and the reference index only if the daily temperature is greater than the reference index. The higher CDD means the more demand of the refrigerant in the summer and the more temperature risk for the climate-related companies. In which, the daily average temperature is the average of the minimum level and the maximum. ° ° 治政 study, we use the degree of Centigrade (°C) as the unit.大 According to above description, 立HDD at time T can be expressed by the formula of the CDD and the level in one day, and the reference index is usually stipulated by 18 C (or 65 F). In this. i. HDDTi = max (18. Ti , 0) .. (4.1). ‧. ‧ 國. −X. 學. . CDDTi = max (XTi − 18 , 0) ,. (4.2). io. sit. Nat. . CDDTi − HDDTi = XTi − 18 .. y. And, the relationship between the CDD and the HDD indexes is. n. al. er. The temperature derivatives is the contract whose underlying asset is the sum of the. i Un. v. HDD or the CDD index over a period. Let the notations H (T1 , Tn ) and C (T1 , Tn ) be. Ch. engchi. the sum of the HDD or the CDD index over the period [T1 , Tn ], respectively. Consider a temperature derivatives contract with maturity Tn and the releasing period [T1 , Tn ]. The payoff of the CDD and HDD derivatives at maturity are C (T1 , Tn ) = δ. H (T1 , Tn ) = δ. °. Tn X. CDDk = δ. Tn X. k=T1. k=T1. Tn X. Tn X. HDDk = δ. k=T1. max (Xk − 18, 0) ,. (4.3). max (18 − Xk , 0) ,. k=T1. . where δ is the dollars per 1 C ($/ ) or the tick size. According to the temperature. . derivatives issued by the CME, the tick size is δ = $20/ . The equation (4.3) is the non-linear function, and it is difficult to deduce the distribution and the dynamic process of the cumulative value. However, Alaton, Djehince, and Stillberg (2002) pointed out the 22.

(32) zero level of the CDD and HDD in the specific with much lower probability. They assumed that the cumulative value, H (T1 , Tn ) and C (T1 , Tn ), is the linear function. Therefore, The equation (4.3) can be rewritten as Tn X. H (T1 , Tn ) = δ 18n −. ! Xk. ,. Tn X. C (T1 , Tn ) = δ. k=T1. ! Xk − 18n .. (4.4). k=T1. This assumption is the more straightforward to depict the distribution of the cumulative. °. value. However, there are many days for the DAT greater than 18 C in the winter and. °. less than 18 C in the summer. The zero or negative level for the CDD/HDD index cannot be ignored. Therefore, this study modifies Alaton et al. (2002)’s assumption and derive the CDD/HDD derivatives pricing method.. 立. Chicago Mercantile Exchange. 學. 4.2.1. 治. 政 Temperature Derivatives Markets 大. ‧ 國. 4.2. There are two patterns of the temperature derivatives: (i) HDD/CDD futures contract. ‧. and (ii) HDD/CDD futures options contract. The former is the futures contract whose. y. Nat. underlying asset is the sum of the HDD index or the CDD index over a period (usually. io. sit. a month). The buyer and seller of the HDD/CDD futures are required to purchase and. n. al. er. sell at a delivery date with a predetermined price. On the other hand, the later is the. i Un. v. options contracts whose underlying asset is the HDD/CDD futures contract. The buyers. Ch. engchi. of options have the right to exercise the contract at maturity, and purchase/sell the HDD/CDD futures with the strike price. The sellers of options have the obligation to fulfil the requirements of the buyers. The options on futures are European framework. Above two patterns of HDD/CDD derivatives contracts are settled in cash. According the first fundamental theorem of asset pricing, the market exists no arbitrage opportunities, if and only if the equivalent probability measure can be found such that the discounted price of the underlying asset is a martingale. The equivalent probability measure is called the risk-neutral measure and is denoted Q (Q ∼ P). Owing to the DAT be non-tradable, the weather market is incomplete. Although there are infinite equivalent probability measures and the price of the temperature derivatives under no arbitrage is not unique, Xu et al. (2008) supposed that the risk-neutral pricing method is still carried out. 23.

(33) Consider a HDD/CDD futures contract with maturity Tn and the underlying asset H (T1 , Tn ) or C (T1 , Tn ) which is the sum of the HDD or CDD over a period [T1 , Tn ]. Under no arbitrage opportunities, the risk-neutral value at time t of a position which receives the payments FH (t, T1 , Tn ) − FH (Tn , T1 , Tn ) at maturity Tn is zero. That is, e−r(Tn −t) EtQ [FC (Tn , T1 , Tn ) − FC (t, T1 , Tn )] = 0,. (4.5). e−r(Tn −t) EtQ [FH (Tn , T1 , Tn ) − FH (t, T1 , Tn )] = 0, with FC (Tn , T1 , Tn ) = C (T1 , Tn ) = δ. Tn X. CDDk ,. k=T1 Tn X. HDD . 治政大 In which, r is the risk-free interest rate and E (·) is conditional expectation under the 立 FH (Tn , T1 , Tn ) = H (T1 , Tn ) = δ. k. k=T1. Q t. ‧ 國. 學. Q measure given the filtration Ft . In the other words, given the filtration Ft , the value of the HDD/CDD futures at time t is. FC (t, T1 , Tn ) = EtQ [C (T1 , Tn )] ,. ‧. FH (t, T1 , Tn ) = EtQ [H (T1 , Tn )] ,. (4.6). Nat. sit. y. where EtQ (·) is conditional expectation under the Q measure given the filtration Ft .. al. er. io. As the previous mentions, the HDD/CDD futures options contract is the option with. n. the HDD/CDD futures contract as the underlying asset. Consider a HDD/CDD futures. Ch. i Un. v. options contract with maturity T , the strike price K, and the Tn -maturity futures. Given. engchi. the filtration Ft , the value of the HDD/CDD futures options at time t is . COi,t = EtQ e−r(T −t) [Fi (T, T1 , Tn ) − K] 1{Fi (T,T1 ,Tn )>K} ,. (4.7). . P Oi,t = EtQ e−r(T −t) [K − Fi (T, T1 , Tn )] 1{Fi (T,T1 ,Tn )<K} , where COi,t and P Oi,t is the call/put futures options with i underlying asset. i ∈ {H, C} presents the CDD and HDD futures options. r is the risk-free interest rate. It is worth noting that the maturity of the HDD/CDD futures contract (underlying asset) must be greater than the HDD/CDD futures options, T < Tn . Otherwise, if T > Tn , the underlying asset is terminated before the options exercised, and the option is also vanished simultaneously.. 24.

(34) 4.2.2. Over-the-Counter. Apart from the CME, most temperature derivatives are traded on the OTC market. The popular pattern is the European call/put option with the sum of the HDD index or the CDD index over a period (usually a month) as the underlying asset. Differentiating from the HDD/CDD futures option contract on the CME, the underlying asset of the options on the OTC market is the temperature-linked index instead of the futures contract. Additionally, because the underlying asset of the HDD/CDD options is non-tradable, the options are only settled in cash. Also, Xu et al. (2008) supposed that the riskneutral pricing method is still carried out under the incomplete market and the case of the non-tradable underlying asset. Consider a HDD/CDD options contract with maturity Tn and the strike price K.. 政治大. Given the filtration Ft , the value of the HDD/CDD options at time t is. 立. . COH,t = EtQ e−r(Tn −t) [H (T1 , Tn ) − K] 1{H(T1 ,Tn )>K} ,. (4.8). ‧ 國. 學. . COC,t = EtQ e−r(Tn −t) [C (T1 , Tn ) − K] 1{C(T1 ,Tn )>K} ,. ‧. . P OH,t = EtQ e−r(Tn −t) [K − H (T1 , Tn )] 1{H(T1 ,Tn )<K} ,. sit. y. Nat. . P OC,t = EtQ e−r(Tn −t) [K − C (T1 , Tn )] 1{C(T1 ,Tn )<K} ,. al. n. the CDD and HDD options. r is the risk-free interest rate.. 4.3. Ch. engchi. er. io. where COi,t and P Oi,t is the call/put options with i underlying asset. i ∈ {H, C} presents. i Un. v. Pricing Formula. In this section, we discuss the pricing formula of the temperature derivatives on the CME and the OTC market. Alaton, Djehince, and Stillberg (2002) pointed out the zero level of the CDD and HDD in the specific with much lower probability. They ignored the feature of the positive HDD/CDD index and assumed that the cumulative value, H (T1 , Tn ) and C (T1 , Tn ), is the linear function for deriving pricing formula more easily. However, there are more zero indexes over the sample period at each city which are shown in the following section. The fair value of the HDD/CDD derivatives under the Alaton, Djehince, and Stillberg (2002)’s framework will be misestimated seriously. The section proceeds as follows: First, owing to the incomplete market for the temperature derivatives, we use 25.

(35) the Girsanov’s theorem and the Esscher transform to change and to find the equivalent probability measure. Second, for the HDD/CDD futures contract, the pricing formula can be derived by the Black-Scholes-Merton’ approach under S-MR, S-MR-S, S-MR-JD-S models and by the characteristic function and the Fourier transform under the stochastic volatility models like in Heston (1993), Bates(1996), Duffie, Pan, and Singleton (2000), Wong and Lo (2009), Pillay and O’Hara (2011). Finally, the HDD/CDD futures options contract and the index options do not exist the close-form solution, and we employ the Monte-Carlo simulation pricing method to value the temperature derivatives.. 4.3.1. Equivalent Probability Measure. The Girsanov’s theorem and the Esscher transform are the popular ways to find the. 政治大. equivalent probability measure for the financial derivatives pricing. According the first. 立. fundamental theorem of asset pricing, the market exists no arbitrage opportunities, if and. ‧ 國. 學. only if the equivalent probability measure can be found such that the the relative price between the underlying asset and the money market account is martingale. On the other. ‧. hand, the second fundamental theorem of asset pricing tells that the market is complete if and only if the equivalent probability measure is unique.. y. Nat. sit. However, the underlying asset of the temperature derivatives is nontradable, then the. er. io. market is incomplete. For financial view, the market participantors cannot construct the. al. iv n C volatility risk and the jump risk. It means the perfect hedging strategy is difficult h e nthat hi U c g to implement. For the mathematical view, the distributions of the stochastic process are n. portfolio to duplicate the cash flow of the temperature derivatives exactly even if the. altered if changing the measure, and there are infinite equivalent probability measure. In this subsection, we report the change of measure for the deterministic volatility model and the stochastic volatility model, respectively. Deterministic Volatility Model The S-MR-JD-S model like the equation (3.7) is the example of the deterministic volatility model. Let ηt be the Radon–Nikod´ ym derivative with Esscher transform parameters h1 σt for the Brownian motion and h2 for the jump term. P Nt. Rt. e 0 h1 σu dWu +h2 j=0 Yj dQt . ηt = = Rt P Nt h1 σu dWu +h2 j=0 Yj dPt 0 E e 26. (4.9).

(36) Because the Brownian motion and the jump term are independent, we can change the probability measure separately. That is, the equation (4.9) can be decomposed into two Radon-Nikod´ ym derivatives and be denoted η1,t and η2,t . Z t Z 1 t 2 2 h1 σu dWu , h σ du + η1,t = exp − 2 0 1 u 0 " # Nt X h2 ν2 2 η2,t = exp −λ eh2 θ+ 2 − 1 t + h2 Yj .. (4.10). j=0. According to the Esscher transform, under the Q measure, the distributions of the Brownian motion and the jump term are as follows. dWtQ = dWt − h1 σt dt ∼ N (0, dt) , h2 ν 2 i.i.d. h2 θ+ 22 t , Yj ∼ N θ + h2 ν 2 , ν 2 . Nt ∼ P oi λe. 立. (4.11). 政治大. Therefore, the dynamic processes of the DAT for the S-MR-JD-S model under the Q. (Xtα. Nt X dXtα 2 − Xt ) + Yj − λθdt, + h1 σt dt + σt dWtQ + d dt j=0. ‧. dXt = β. ‧ 國. . 學. measure are. (4.12). sit. y. Nat. where h1 is the temperature risk premium and h2 is the jump risk premium. The detail. io. er. proof are shown in the Appendix A. The change of measure for others deterministic models such as the S-MR model and the S-MR-S model is similar to the S-MR-JD-S. al. n. model.. Ch. engchi. i Un. v. Stochastic Volatility Model The S-MR-JD-S-SVJ model like the equation (3.10) is the example of the stochastic volatility model. Recall the equation (3.10) and the dependent Brownian motions, WX,t and WV,t , can be decomposed into two independent Brownian motions, B1,t and B2,t , by the Cholesky decomposition. That is, the dependent Brownian motions are rewritten as dWX,t = dB1,t ,. dWV,t = ρdB1,t +. p 1 − ρ2 dB2,t .. (4.13). Thus, under P measure, the equation (3.10) can be rewritten as following. dXt = β. (Xtα. NX,t X p dXtα − Xt ) + dt + Vt dB1,t + d YX,j − λX θX dt, dt j=0 27. (4.14).

(37) ". . dVt = κ Vtφ − Vt. . # NV,t X p p dVtφ 2 dt + σ Vt ρdB1,t + 1 − ρ dB2,t + d + YV,j . dt j=0. √ Let ηt be the Radon–Nikod´ ym derivative with Esscher transform parameters h1 Vt , √ h2 Vt for the new Brownian motions and h3 , h4 for two jump terms. dQt = ηt = dPt. Rt. e . 0. R √ √ PNX,t PNV,t h1 Vu dB1,u +h3 j=0 YX,j + 0t h2 Vu dB2,u +h4 j=0 YV,j. Rt. E e. 0. R √ √ PNX,t PNV,t h1 Vu dB1,u +h3 j=0 YX,j + 0t h2 Vu dB2,u +h4 j=0 YV,j. .. (4.15). Because the new Brownian motions and the jump terms of the DAT and the variance are independent, we can alter the probability measure separately. That is, the equation (4.15) can be decomposed into four Radon-Nikod´ ym derivatives and be denoted η1,t , η2,t , η3,t , η4,t .. 政治大. Z Z t p 1 t 2 η1,t = exp − h Vu du + h1 Vu dB1,u , 2 0 1 0 Z Z t p 1 t 2 η2,t = exp − h Vu du + h2 Vu dB2,u , 2 0 2 0   NX,t 2 X h2 ν 3 X YX,j  , η3,t = exp −λX eh3 θX + 2 − 1 t + h3. 立. (4.16). θV θV − h4. y. η4,t = exp −λV.  NV,t X − 1 t + h4 YV,j  .. io. sit. Nat. . ‧. ‧ 國. 學. j=0. . j=0. n. al. er. According to the Esscher transform, under the Q measure, the distributions of two inde-. i Un. v. pendent Brownian motions and two jump terms are as follows.. Ch. e npg c h i. Q dB1,t = dB1,t − h1. NX,t. Vt dt ∼ N (0, dt) ,. (4.17). p Q dB2,t = dB2,t − h2 Vt dt ∼ N (0, dt) , h2 ν 2 i.i.d. h3 θX + 32 X 2 2 ∼ P oi λX e t , YX,j ∼ N θX + h3 νX , νX , NV,t ∼ P oi. λV θV t θV − h4. ,. i.i.d.. YV,j ∼ exp (θV − h4 ) .. Also, the independent Brownian motions can revert to the dependent Brownian motions under the Q measure like the equation (4.13). Therefore, the dynamic processes of the DAT and the variance for the S-MR-JD-S-SVJ model under the Q measure are dXt = β. (Xtα. NX,t X p dXtα Q + h1 Vt dt + Vt dWX,t + d YX,j − λX θX dt, − Xt ) + dt j=0 28. (4.18).