關于天氣衍生品的構建與定價：非洲農業可持續援助

1 Introduction
The United States Department of Commerce has shown in 1997 that the weather impacted 70 per cent of the companies and that the cost of climatic events all included represented more than 200 billions of dollars in 2005. Historical data shows nowadays an increase of extreme weather conditions, in particular with temperature. In addition, temperature volatilities logically tend to affect more countries in which their economy depends strongly on its crops revenues. Unfortunately, agricultural economy is
often synonym of emerging and developing countries, where a bad harvest can engage the virtuous circle of rural livelihoods poverty. Moreover, it appears that natural catastrophes cause more severity for small scale farmers in undeveloped areas. Consequently, a decrease of crop revenue can engender a certain economic and development downturn, raising thus poverty and forming possible humanitarian crisis. In this context, the African continent represents a major agricultural place often stroke by excessive temperatures. Even if financial aids distributed by public organizations constitute the foundation on which nations’ helps come alive, this behavior presents a major default as to give to these nations a long term solution to tackle weather risks. It is on this observation that this article uses weather risk management as a new approach and a possible independent long term plan to help African countries mitigating their weather risk.
1引言
美國商務部已在1997年70％的公司和氣候事件的成本都包括代表2005年的超過200億美元的天氣影響。歷史數據顯示時下極端天氣條件下，特別是隨溫度增加。此外，溫度波動在邏輯上往往會影響到更多的國家，其經濟在很大程度上取決于其作物收入。不幸的是，農業經濟經常新興國家和發展中國家的代名詞，在收成不好可以搞農村生計貧困的良性循環。此外，它出現的自然災害會導致更嚴重的小規模農戶在經濟欠發達地區。因此，減少作物的收入可以使人產生一定的經濟和發展的衰退，從而提高貧困和形成可能的人道主義危機。在這方面，非洲大陸的農業大經常發生中風的溫度過高。即使公共機構的財政援助分布構成的基礎上，國家的幫助來活著，這種行為提出了一個重大的默認值，為這些國家提供長期的解決方案，以應對天氣風險。本文采用這一觀察天氣風險管理作為一種新的方法和可能的獨立的長期計劃，以幫助非洲國家減輕他們的天氣風險。

The emergence of climatic contract arisen during the late 90’s in the United States required researchers to adapt existing valuations models to weather specificities. Indeed, since weather indices cannot be traded on the market, the constitution of a duplicated portfolio is not possible, discrediting thus one ofthe most used model, the Black and Scholes (1973) formula. To get over this difficulty, literature suggests choosing a substitute asset that follows the same characteristics and has a highly correlated price with the unobservable weather contract. German (1999) recommend to build a temperature portfolio replica by picking both temperature and gas derivatives contracts negotiated on financial market at that time. However, its hypotheses became incorrect when Brix, Jewson and Ziehmann (2002) revealed that gas prices are better correlated to the demand than to the temperature. The financial community looked then for other alternatives. One simple method, straight forward to implement and issued from the insurance industry named Actuarial Method or Burn Analysis remains still wildly employed to price climatic contracts. Contrary to earlier developments, the need of creating a hedging portfolio is not necessary. The price of the weather derivative before the maturity date is simply the actualized expected loss that occurred during the period of the contract policy, calculated as the average payoff that the contract would have given upon the same period of time using historical data. At maturity, the price is equal to the historical probability of the underlying to have performed,with an additional risk loading parameter to compensate the seller’s risk taken. Dischel (1999), West (2002), Augros and Moréno (2002), Jewson (2004) and Platen and West (2004) supposed that the price of the risk loading was not relevant for complexity reasons and decided to consider it either null or to give an arbitrary value. Platen and West (2004) explain furthermore that the concurrence between companies incline to decrease this risk parameter.  http://www.mythingswp7.com/shx/
出現在90年代中后期在美國興起的氣候合同要求研究人員對天氣的具體情況，以適應現有的估值模型。事實上，由于天氣指數不能在市場上交易，重復組合的憲法是不可能的，抹黑，從而之一最常用的模型，Black和Scholes（1973）公式。為了克服這個困難，文學建議選擇替代資產，遵循相同的特點，并具有高度相關性的價格與不可觀察天氣合同。德國（1999）建議建立一個溫度組合副本金融采摘溫度和氣體的衍生工具合約談判。當時的市場。然而，它的假設不正確，當糖度，朱森Ziehmann的（2002）透露，天然氣價格是較好的相關性對溫度的需求比。金融界看著其他的替代品。一個簡單的方法，直線前進，以實現從保險業內名為精算方法及發行或刻錄分析仍然仍然廣泛采用到氣候合同訂價。與早期發展，需要創建一個對沖組合是沒有必要的。天氣衍生工具到期日前的價格只是實際使用的預期損失發生期間的合同政策，合同后，將給予一次使用歷史數據同期的平均收益計算。于到期日，價格，補償賣方的風險采取額外的風險加載參數的歷史相關的概率相等。 Dischel（1999年），西（2002），Augros和莫雷諾（2002年），朱森（2004）和壓板和西（2004）假設風險加載的價格是不相關的復雜原因，決定考慮null或給一個任意值。滾筒和西（2004）進一步解釋，同意公司之間傾斜，以減少風險參數。

Cao and Wei (2004) created a different procedure based on the “Consumption –Capital Asset Pricing” of Lucas (1978). In fact, via an Euler equation, they show that it exists a significant relation between the temperature of several North American cities and the total level of consumption. The same conclusion were drawn by Richards Manfredo and Sanders (2004) using the data of Fresno in California. Now that a general frame around weather derivative valuation methods is set up, pricing them always requires to modeling the climatic variable.
Cao和Wei（2004）基于“消費資本資產定價”盧卡斯（1978年）創建了一個不同的程序。事實上，通過歐拉方程，它們表明，存在一個顯著的幾個北美城市的溫度之間的關系和總的消費水平。相同的結論由理查德·曼弗雷多和桑德斯（2004）在加利福尼亞州弗雷斯諾使用的數據。現在，各地天氣衍生工具估值方法是設立一個總體框架，定價他們需要建模的氣候變量。

The increasing needs of energy companies to hedge un-forecasted temperature variations have driven researchers’ interest to model principally the temperature movement. The literature shows that the temperature follows a regular movement often represented by a sinusoidal function. It also demonstrates that the variable movement does not diverge far from a mean curve also called the mean reversion process. To take into account the mean reversion, Alaton, Djehiche and Stillberger (2002), Benth and Saltyte-Benth (2005) suggested using a continuous time Ornstein-Uhlenbeck process. We will use this method to value and price the African temperature based contracts we define.

The first objective of this paper is to examine the structure of temperature contract and construct a new weather derivative market for 18 African countries. The historical weather data are gathered frommeteorological centers and serve to calculate fair prices of basic degree day derivatives. The second objective is to assess the hedging effectiveness of the created options against temperature risk bycomparing option premium results with recent agricultural productions and revenues figures. Another prospect is to verify if such derivatives can still be affordable by small farmers. The final seek is to assess the cost of insuring 30 per cent of the three most produced commodity by each country and see if communal derivatives is a viable alternative in the African case.

The remaining of the article is organized as follow. Section 2 provides a general presentation of the weather derivative market, followed by a detailed overview of temperature derivatives. In section 3, we will focus on performing temperature modeling of our 18 African countries using an Ornstein- Uhlenbeck process. Since the estimation of parameters is based on historical data using a martingale process, the data collection description will also be expressed along this part. Section 4 is dedicated to
price the temperature contracts computed using Monte-Carlo simulations. In the same section, the results are presented and discussed around the cost of such temperature derivatives and its hedging efficiency using the value at risk technique among other. Section 5 concludes.

2 Weather derivatives
The first transactions on climatic variables have initially been realized in the United-States in September 1997. The contracts were concluded between two energetic companies, Koch and Enron, on a swap on temperature indices to hedge against warm days in winter. Weather derivatives started at first in the United States for two principle reasons. The first motive came from a deregulation of the energy industry after 1997 that contributed to fuel the expansion of climatic options. To avoid energy prices volatility, the energy industry used weather derivatives as they gave an instant payoff when the demand spread far from their forecast. The second motive concerns the multiple climate disturbances that the country faced in 1997 (For instance, the El Nino1 effect during winter 1997 and the violent precipitations in California). The expansion of the climatic contract had given birth in 1999 to an organized electronic platform launched by the Chicago Mercantile Exchange. The first contracts traded were essentially degree days (DD) temperature contract in 10 cities2. The CME has today enlarged its territory counting 19 cities3 in North America. In 2003, the CME opened a subsidiary in Europe covering European cities4 as well as in Japan with Tokyo and Osaka. Figure 1 shows all principle hedging climatic variables traded in the world with their respective proportions from 2001 until 2005. The obvious observation one can do is the clear domination of temperature contracts named degree day (DD) on this chart, exceeding 60% of the total exchanged amount.

3、Conclusion
We have created in this article, weather derivative contracts based on temperature for 18 African countries. With actual data, we calculated our degree day indexes on the optimum temperature at which crops grow at best. Using an Ornstein-Uhlenbeck process, we modeled the temperature trajectories with a continuous time mean reverted method. The pricing of the created derivatives is made with Monte Carlo simulations and results show two essential findings. First, we observed important correlations between derivatives pay offs and decrease in crops production yields, confirming this paper’s interest. Second, around 80 per cent of the temperature based options cost less than 1000 dollars each, making them affordable for African farmers. Weather derivatives with pay offsdepending on temperature can thus represent a potential sustainable plan in the long run to hedge temperature risk in these African countries, helping to prevent food shortage. Nonetheless, the work undertaken in this study may be extended in several directions, representing some of the challenges the research on weather derivatives still have to take up. First of all, the structure of the temperature options may be redefined, especially with “tailor-made” exotic derivatives, in order to match more precisely the users’ needs. The pricing of such contract could be assessed again using the Monte Carlo simulation as it is already the case for other hybrid contracts. Second, the modeling of the temperature process may be improved by considering the temperature as one variable of a larger climate model including several variables. The development of such models, along with the increasing power of computers, will certainly help better forecast extreme events, and hence better price weather derivatives. Finally, it will certainly be worth spreading these temperature
derivatives to other African countries.