关于天气衍生品的构建与定价：非洲农业可持续援助

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
美国商务部已在1997年70％的公司和气候事件的成本都包括代表2005年的超过200亿美元的天气影响。历史数据显示时下极端天气条件下，特别是随温度增加。此外，温度波动在逻辑上往往会影响到更多的国家，其经济在很大程度上取决于其作物收入。不幸的是，农业经济经常新兴国家和发展中国家的代名词，在收成不好可以搞农村生计贫困的良性循环。此外，它出现的自然灾害会导致更严重的小规模农户在经济欠发达地区。因此，减少作物的收入可以使人产生一定的经济和发展的衰退，从而提高贫困和形成可能的人道主义危机。在这方面，非洲大陆的农业大经常发生中风的温度过高。即使公共机构的财政援助分布构成的基础上，国家的帮助来活着，这种行为提出了一个重大的默认值，为这些国家提供长期的解决方案，以应对天气风险。本文采用这一观察天气风险管理作为一种新的方法和可能的独立的长期计划，以帮助非洲国家减轻他们的天气风险。

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.ukthesis.org/shx/
出现在90年代中后期在美国兴起的气候合同要求研究人员对天气的具体情况，以适应现有的估值模型。事实上，由于天气指数不能在市场上交易，重复组合的宪法是不可能的，抹黑，从而之一最常用的模型，Black和Scholes（1973）公式。为了克服这个困难，文学建议选择替代资产，遵循相同的特点，并具有高度相关性的价格与不可观察天气合同。德国（1999）建议建立一个温度组合副本金融采摘温度和气体的衍生工具合约谈判。当时的市场。然而，它的假设不正确，当糖度，朱森Ziehmann的（2002）透露，天然气价格是较好的相关性对温度的需求比。金融界看着其他的替代品。一个简单的方法，直线前进，以实现从保险业内名为精算方法及发行或刻录分析仍然仍然广泛采用到气候合同订价。与早期发展，需要创建一个对冲组合是没有必要的。天气衍生工具到期日前的价格只是实际使用的预期损失发生期间的合同政策，合同后，将给予一次使用历史数据同期的平均收益计算。于到期日，价格，补偿卖方的风险采取额外的风险加载参数的历史相关的概率相等。 Dischel（1999年），西（2002），Augros和莫雷诺（2002年），朱森（2004）和压板和西（2004）假设风险加载的价格是不相关的复杂原因，决定考虑null或给一个任意值。滚筒和西（2004）进一步解释，同意公司之间倾斜，以减少风险参数。

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“消费资本资产定价”卢卡斯（1978年）创建了一个不同的程序。事实上，通过欧拉方程，它们表明，存在一个显着的几个北美城市的温度之间的关系和总的消费水平。相同的结论由理查德·曼弗雷多和桑德斯（2004）在加利福尼亚州弗雷斯诺使用的数据。现在，各地天气衍生工具估值方法是设立一个总体框架，定价他们需要建模的气候变量。

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.