Hedging Corporate Revenues with Weather Derivatives: A Case Study
Master of Science in Banking and Finance - MBF Master’s Thesis
Antoni Ferrer Garcia Franz Sturzenegger
Universit´e de Lausanne Ecole des Hautes Etudes Commerciales HEC - 2001
Abstract This paper searches for the implications in the use of a new generation of financial derivatives known as Weather Derivatives as a form of hedging future corporate revenues. According to the US Department of Commerce about 22 per cent of the US$ 9 trillion GDP in the United States is sensitive to weather. This figure supports the growth in the market that started at the beginning of the 1997s. Likewise, it is estimated that already some 1,800 deals worth roughly US$ 3.5 billion have been transacted in the U.S. An estimated 70 per cent of all businesses face weather risk which extends across geographic and market borders. The current weather derivatives market is still illiquid and several pricing models are being used by financial institutions. On this paper we show the characteristics, pricing models and hedge strategies about such new contracts. Our case study has been done within a multinational corporation that we will be here called XYZ to preserve its confidentiality.
Acknowledgments Before starting, we would like to thank all the people that with their support and understanding have contributed to make this master’s thesis somehow better: Mrs. F. Kafader of Kundendienst-Account, Swiss M´et´eo, Prof. Dusan Isakov (HEC - Gen`eve), Mr. J¨urg Tr¨ub (Swiss Re-insurance), Robert Dischel, Melanie Cao and Jason Wei. Several institutions that have supported us with data, advice or knowledge about the weather derivatives markets: Enron, Aquila Energy, AC Nielsen, Migros, and Koch Energy Trading. Special thanks to company XYZ since it was their idea to write about this topic. It is also theirs most of the data contained on this thesis. With their help and that of Mr. Lagger and Mr. Silen, we started our research. Special thanks, also, to professor Didier Cossin to direct our thesis and to give academical support to such an interesting topic. Last but not least, thanks to our family, girlfriend and friends - Manuel Kast, Dr. Alexander Passow, Beatriz Rueda - that certainly have helped us during the whole MBF Program and during the preparation of this thesis. This thesis is dedicated to them.
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Preface In today’s financial markets, derivative instruments have certainly a granted place on corporate risk management as a way to insure against or hedge business hazards. Derivatives are financial instruments whose values depend on the value of other securities known as the underlying. Those underlyings are often traded assets such as stock, commodities, currencies, bonds but can also be non-traded assets such as stock index. Futures and options are actively traded on major exchanges while forward and swap contracts are evenly traded outside exchange by financial institutions in the over-the-counter market (OTC). Since the study of Black and Scholes, ‘The Pricing of Options and Corporate Liabilities’ and Robert Merton,‘The Theory of Rational Option Pricing’, we have seen an astonishing growth in derivative markets and in the development of more complex instruments that simple plain-vanilla options, such as Asian Options, Lookback Options, Barrier Options, Catastrophic Bonds and others. Nowadays, a new class of derivative securities has been created to offer corporate managers an instrument to hedge their firms against climate conditions’ hazards. They are known as Weather Derivatives and are designed to minimise or avoid the risks due to changes in weather conditions. On the other hand, several questions have been raised for why corporate managers should hedge their business and on what are the consequences of the use of derivatives as a form to offset undesired risks. Sometimes, instead of using derivatives for hedging purposes, managers have traditionally used them to simply speculate1 in financial markets with the intention of profit from market discrepancies. Nonetheless, an uncautious use of derivatives could lead to huge losses that might have an impact not only in the company but be spreaded to the whole financial markets. 1
Whereas hedgers want to avoid an exposure due to price movements, speculators wish to take a position in the markets by betting that the price will go up or down.
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The objective of this research is twofold: i) first, present the explanations for corporate hedging; the theory behind it and attempt to shed some light on the use of Weather Derivatives as a form of hedging volumetric risks for corporate institutions and ii) second, a case study where weather derivatives are used to hedge potential risks due to climatological effects on a company’s business. To achieve these objectives, we conduct our case study with the help of a multinational company that has provided us with data on sales for juice and milk in Switzerland, Germany, France and U.K. To keep confidentiality, we will call this company “XYZ”. For this reason, this paper is organised in three chapters. Chapter 1 introduces the theory behind corporate hedging as well as the weather derivatives. Section 1 introduces the concepts of why corporations should hedge risks. Section 2 shows the characteristics of the weather derivatives markets. Section 3 introduces the impact of weather conditions in real businesses. Section 4 shows the methods of forecasting weather. Section 5 introduces a literature survey of weather derivatives models. Section 6 ends this chapter with mathematical applications and modelling of weather derivatives as well as some extention of the model. Chapter 2 presents the case study using data from company XYZ. Section 1 introduces the company’s business. Section 2 explains the correlations between sales and weather in Switzerland and abroad. Section 3 presents the market exposure of XYZ’s sales under climatological changes. Section 4 shows the instruments to hedge weather risks. Section 5 contains the hedge strategy using weather derivatives. Chapter 3 concludes this study and propose further ideas about the topic. References and appendices are presented at the end.
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Contents 1 Weather Derivatives: A literature Review 1.1 Corporate Hedging . . . . . . . . . . . . . . . . . . . . . . . 1.2 Weather derivatives: Market and characteristics . . . . . . . . 1.3 Impact of weather conditions on the economy . . . . . . . . . 1.3.1 Extreme weather conditions and natural catastrophes . 1.3.2 Abnormal weather . . . . . . . . . . . . . . . . . . . 1.4 Forecasting the weather . . . . . . . . . . . . . . . . . . . . . 1.5 An analysis of weather models: A literature survey . . . . . . 1.5.1 Understanding the weather evolution models . . . . . 1.6 Mathematical applications / Modelling . . . . . . . . . . . . . 1.6.1 Modelling weather derivatives . . . . . . . . . . . . . 1.6.2 The “burn analysis” model . . . . . . . . . . . . . . . 1.6.3 Auto-correlated regression model. The case for Geneva 1.6.4 Extension to the model: Some US cases . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
2 The Case Study 2.1 The company business . . . . . . . . . . . . . . . . . . . . . . . 2.2 Understanding the correlation between sales and weather . . . . . 2.2.1 The correlation of XYZ’s sales with the temperature . . . 2.2.2 Lagging data to make it more consistent . . . . . . . . . . 2.2.3 Country specificity . . . . . . . . . . . . . . . . . . . . . 2.3 Market exposure of XYZ’s sales under changing environmental conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The hedge procedure . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Natural hedging . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Weather Derivatives Instruments . . . . . . . . . . . . . . 2.5 Designing a hedge strategy using weather derivatives . . . . . . .
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6 6 8 11 12 13 14 16 17 22 22 23 25 28 30 30 30 31 35 41 47 51 51 56 61
3 Conclusion and further ideas 3.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Human Physiology
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B Exhibits
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Chapter 1 Weather Derivatives: A literature Review 1.1
Corporate Hedging
Several financial economists have provided us with many theories as to why managers would hedge corporate revenues. Modigliani and Miller, according to their seminal paper, pointed out the lack of need for corporate hedging. After this statement many academics have taken up the challenge of explaining why we see this phenomenon. In the area of corporate hedging Clifford Smith, David Mayers and Ren´e Stulz have certainly contributed by giving us some of the reasons. In 1982 Mayers and Smith published the earliest work on hedging corporation in their Journal of Business article, “On the Corporate Demand for Insurance”. In this article the authors suggest seven possible explanations for why corporations would insure their assets, even if their shareholders are well diversified. However the focus was on property and liabilities rather than in derivatives. Concerning the use of derivatives as a hedge strategy the main reasons that make these instruments attractive to hedge are: 1. Non-diversifiable stakeholders (employees, customers and suppliers) will demand expensive terms in contracts with a risky firm since they would be over-exposed to the fluctuation of cash flows without being offset by other external non-correlated revenues. 2. The probability of costly bankruptcy can be reduced. Volatility-reduction can be achieve by hedging and consequently increasing the recovery rate on defaulted debt which leads to decrease bankruptcy costs. 6
3. Tax reasons: Progressive corporate tax rates induce firms to smooth their profits. Under this tax regime, companies would pay more taxes if their revenues are for example 30 and 70 than 50 and 50 in some years. Limited or delayed deductibility of large losses, due to time-limits on loss carry-backs and carry-forward and due to government’s abstention from participating in the firm’s losses. Furthermore other articles have also provided explanations for corporate hedging: Leverage motivation - Over investment problem: Asset substitution(Jensen & Meckling 1976) where debt creates an incentive to take risky projects since the debtholders will bear all the downside risk of a project. Equity can be seen as a call option on the firms assets since shareholders are entitle to have limited responsibility on the liabilities. Stockholders might exercise their option to default when firm gets in trouble. Requiring hedging in those bonds can reduce the firms’ incentive to increase risk and consequently reduce bondholders’ discount of those securities. - Under investment problem: Debt overhang (Myers 1977) where investing in positive NPV projects imply transferring value from equityholders to debtholders because the latter have to share the profits but not the costs. This creates an incentive to forgo positive NPV projects. A bond covenant requiring hedge may act in favour of undertaking positive NPV projects and consequently reducing the cost of debt. Assymmetric information issues - Costly external financing (Froot, Scharfstein and Stein 1993). External financing is costly because potential investors are less informed about the project. Hedging adds value to the firm to the extent that it helps ensure that a corporation has sufficient internal funds available to take advantage of attractive investments opportunities in the future. In order to maintain a cheap access to capital, corporations may need a risk management program.
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Although the use of derivatives in the corporate sector and its consequences have been widely studied1 , numerous articles have criticised its use. As an example, the use of derivatives have caused spectacular losses for financial institutions (Barings Bank), non-financial companies (Procter & Gamble, Landis & Gyr, etc) and government institutions (Orange County), which put the whole financial markets in alert due to the value of the losses. Notice that those cases occurred by simple speculation or a misunderstanding in the features of derivatives without taking in consideration its implications in case of not been possible to market margin calls.
1.2
Weather derivatives: Market and characteristics
According to the Department of Commerce about 22 per cent of the 9 trillion USD gross domestic product in the United States is sensitive to weather. This figure supports the growth in the market for weather derivatives that started at the beginning of the 1997s and it is estimated that some of 1,800 deals worth roughly $3.5 billion have been done in the USA. An estimated 70 per cent of all businesses face weather risk which extends across geographic and markets borders. Before 1997 utility companies managed their earnings stabilisation primarily through price hedging derivatives while volumetric risks were largely left unhedged. However, the recent deregulation in the energy sector increased competition leading to hedge volumetric risks caused by unexpected weather conditions with derivatives. Whilst the weather is still an uncontrollable variable, a new class of financial instrument - weather derivatives - enable companies to have a more active approach to manage weather risk. Weather derivatives are nowadays not only traded in over-the-counter (OTC) market. Standardised contracts are being traded on the Chicago Mercantile Exchange (CME) which provides contracts based on the temperature for many US cities. I-Wex.com, a LIFFE-backed (London International Financial Futures and Options Exchange) and Internet companies Wire and Intelligent Financial System have joined forces to create an internet-based weather derivatives exchange2 . 1
See, for example,recent surveys by Wharton School and Chase Manhattan Bank (1995) and by Ernst and Young (1995) 2 Risk Magazine, March 2000
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A weather derivative or weather option is a financial instrument that has a payoff derived from variables such as temperature, snowfall, humidity and rainfall. However, the industry has set up temperature as the common underlying for those contracts. Unlike insurance and catastrophe linked-instruments, which cover high-risk and low probability events, weather derivatives shield revenues against low-risk and high probability events (such as mild winters). Temperature contracts are more specifically traded in what is called Heating Degree-Days (HDD) or Cooling Degree-Days (CDD) defined on daily average temperatures. The number of heating degree-days is the difference between 65 degrees Fahrenheit and the daily average temperature whilst the number of cooling degree-days is the difference between the daily average temperature and 65 degrees Fahrenheit. HDD and CDD can never be negative. Daily average temperatures are the arithmetic average of the minimum and maximum records in a midnight to midnight basis. A more elegant description of HDD and CDD is done below:
HDD = max 65 degree Fahrenheit - daily average temperature, 0 CDD = max daily average temperature - 65 degree Fahrenheit, 0 Typical contracts are written on cumulative HDD/CDD structured by options, futures, swaps and collars for a given period. HDD contracts last during the November to March period whilst CDD for May to September. One could define four basic elements in options or futures/swap contracts: i) the underlying variable HDD/CDD; ii) the accumulation period: a season or a month; iii) the specific weather station that record the daily temperature and iv) the tick size assigned to each HDD/CDD. The world’s first exchange-traded, temperature-related weather derivatives, which started trading on September 22 , 1999, on the Chicago Mercantile Exchange, remains the only major exchange where weather products are traded. The CME introduced the electronic trading of weather derivatives on its Globex system with the intention of enlarging the size of the market and remove credit risk by trading on weather contracts. CME contracts have attracted new participants and increased liquidity in the weather derivatives market for a number of reasons:first, it allows small transaction sizes which leads to increment the number of investors; second, it provides price discovery since weather options and futures are quote in real time and can be accessed by everyone; third, it ensures low trading costs on the Globex system by using an electronic system that needs less personal to operate; and fourth, it eliminates credit risk for participants which is bypassed to 9
the clearing house system. Table 1.1 provides an example of contracts traded on the Chicago Mercantile Exchange for HDD option for Atlanta and CDD futures contract for Chicago. Chicago CDD (Future contract) Contract Size Measuring Station Contract Month
100 x the CME degree day index O’Hara Airport (ORD) 12 consecutive calendar months
Minimum Tick Size Regular Strike Price
1.00 (HDD/CDD) index point =$100 Not Applicable
Exercise Final Settlement Price Initial Strike Range
Not Applicable The Exchange will settle the contract to the CME degree day index of the contract month by EarthSat Not Applicable
Position Limits Trading Hours
10,000 futures contracts 3.45 p.m. To 3.15 p.m. (next day)
Atlanta HDD (Option contract) One futures contract Hartsfield Airport (ATL) 5 months (HDD): Nov–Mar 5 months (CDD): May–Sep 1.00 (HDD/CDD) index point =$100 HDD: 50 index points CDD:25 index points European Style Not Applicable
150 HDD / 75 CDD index points up and down from at-the-money (Futures equivalents) Same as futures
Table 1.1
Although CME weather derivatives contracts are traded for 11 cities in the US, most of the deals are executed by OTC participants under ISDA (International Swap and Derivatives Association) Master Agreements standards, that provide tailor-made products to suit clients’ needs. ISDA’s standardised documentation allows any firm to enter into a contract with another firm readily, if both have derivatives market experience. The first transaction in the weather derivatives market took place in 1997, when Aquila Energy included an option in a weather contract. After that Aquila Trading and Risk Management and Koch Supply and Trading have joined, transacting some 140 and 80 contracts, respectively. Enron Corp. has been active in the market with some 70 deals since it announced a weather contract to Northeast utility on the same year. Another important player has been Willis Risk Management, the London based risk management group, that until now has structured 92 weather option strategies. Others have also started to offer customised weather hedge contracts such as Worldwide Weather Trading Co, TradeWeather.com, Natsource, Southern Co. TradeWeather.com is a New York based company servicing the global weather derivatives market. It is an Internet system that includes automated order placement and execution, real-time quotes 24 hours per day. In Europe, the market is also gaining amplitude with an increase in the number of 10
deals executed by firms such as Swiss Re New Markets, Soci´et´e G´en´erale. Swiss re-insurance has recently launched the sale of weather derivatives via its ELRiX platform. ELRiX stands for Electronic Risk Exchange and forms the Swiss Re’s electronic trading of standardised risk transfer products. Weather derivatives structures commonly used are: i) cap - a call option; ii) floor - a put option; iii) collar - a put and a call option, usually with little or no premium; iv) swap - a derivative with a profit and loss profile of a futures contract and v) digital option - an option that pays either a predetermined amount if a certain temperature or degree day level is reached, or nothing at all in other case. A business with weather exposure may choose to buy or sell a futures contract, which is equivalently to a swap such that one counterparty gets paid if the degree day over a specific period are greater than the strike level, and the other party gets paid if the degree day over that period are less than the strike. A business may also choose to write an option. A heating oil retailer may feel that if the winter is very cold they will have high revenues - so they might sell an HDD call. If the winter is very cold, the retailer can afford to write the option and pay out with higher than normal revenues. The weather risk market has a huge potential and the growing number of deals has sparked interest among some of the world’s biggest corporation, banks, brokers and insurers. However the market’s rise has not evolved as participants hopped. Several issues have stopped the way of this recent market such as lack of end-user demand and liquidity. A recent conference of the Weather Risk Management Association (WRMA) tried to shed some light on the future of the weather derivatives market. An important issue mentioned there was the fact that to create an effective commodity market to operate the main aspect is whether the data is reasonably consistent and the market participants agree upon.
1.3
Impact of weather conditions on the economy
Almost all businesses are in certain way affected or exposed to weather conditions, sometimes in a cyclical way like the energy, gas, heating oil sector or in a irregular way such as entertainment or leasure businesses and, as a consequence the providers of those. Nonetheless, several businesses have performed their activities without taking in consideration meteorological conditions nor have built a team of weather experts in the running of their deals. So, the question posed here is why companies should care about weather events throughout the time. To answer this question first, let us have a look at the kinds of weather events that might 11
affect firms and then how weather derivatives can provide the right solutions for the consequences of some weather events.
1.3.1 Extreme weather conditions and natural catastrophes Natural catastrophes such as earthquakes, hurricanes, floods, and large scale fires3 have increased in terms of frequency and losses caused during the past twenty years. One answer for this phenomenon might be attributed to climate changes, specifically global warming. The expansion of cities with the consequent growth of gas emission by industries and vehicles together with an increase of building areas, which avoid the solar rays to be be absorbed, have certainly contributed to shift upward the temperature level throughout this century. However, there is still little agreement on the effects of the overall warming. Global warming trend is also displayed by temperature data, principally in cities that have had an increased population growth. Although this warming trend is very small in terms of absolute values, it produces significant changes in the temperature as one may observe today. Additionally, seasonality is a feature that one observes when depicts temperature data over years. This seasonality is not completely the same for different samples of past data even when we observe very similar characteristics for all years. Another reason for this is the climatic response to El Ni˜no. El Ni˜no, the oceanic phenomenon of warming sea surface temperature in the Eastern Pacific, is widely known to alter the patterns around the world. According to the National Oceanic and Atmospheric Administration, the 1997-1998 El Ni˜no coincided with the highest land and ocean temperatures during this century. In addition to El Ni˜no, other recent weather phenomena have had a deep impact in the US economy. Though not defined as an El Ni˜no, a less pronounced warm ocean phase even led to 500 year floods in the Midwest during the summer of 1993. 1993, the year of the March “Superstorm”, dumped feet of snow in the eastern US disrupting utility services cancelling schools and shutting down businesses on the coast. A 1996 cold ocean phase led to record cold temperatures in the upper Midwest and included massive snow-melt flooding in the North Dakota the following spring. These are disasters that have occurred since 1988 with at least one disaster each year according to survey conducted in the U.S. by the National Climatic Data 3 The insurance industry defines a catastrophe as “an event which causes in excess of $5 million in insured property damage and effects a significant number of insurers and insurers”. See Louberg´e and Schlesinger (1999)
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Centre (NCDC), an agency responsible for monitoring and assessing the climate4 .
1.3.2 Abnormal weather Less dramatic climate events might also generate important losses. Several industries are closely related to weather and even not so extreme events can cause huge losses if they are abnormal and persistent during certain period of time. Viticulture industry is extremely sensitive to the weather. Lack of sunshine and cool temperatures during the stages between pre-bloom and maturation do significantly affect the quality of grapes, and consequently the vintage of the resulting wine. In 1998, California’s production of wine grapes felt almost 30%. This was due to a rainy and cold spring, followed by a very hot July and August. Higher-than-average rainfall during the summer months can also be very expensive for wine makers as this leads to the grapes rotting on the vines and delays the harvest. Brewing industry is also affected by changes of weather. Sales of beer drop during colder-than normal summers, and although it is possible to estimate seasonal trading patterns, long term forecasts are still notoriously unreliable. Without considering reduced sales, brewers are also affected by colder summers due to the fact that the beer not consumed has to be stored, increasing the overall expenses. Another important industry that is directly related with weather is the construction industry and here the size of the contracts are generally high. In this sector, heavy penalties can be imposed for works that are not concluded in the schedule period and delays can soar the costs. Concrete needs to set at a certain temperature to obtain its maximum strength, but if it is too hot or too cold, it sets too quickly or too slowly respectively. High winds mean that workers cannot work at heights, and crane use is banned due to safety regulations. Nevertheless, the biggest forewarning to the construction industry is the rainfall followed by freezing temperature. Risk Professional Magazine5 has underlined the successfully effects in the US entertainment-to-drinks conglomerates in smoothing their earnings by actively managing weather exposures. The lucrative but volatile world of entertainment is exposed to weather changes that can severely decrease the revenues. A long than normal rainfall during the production of a movie would make the costs immense. The same principle is applicable to theme parks where cloudy days could lead 4 A complete survey of extreme weather and climate events can be obtained on the following web site : http://www.ncdc.noaa.gov/extremes.html 5 Risk Professional issue 2/6 July/August 2000
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to a significant decline in volume of visitants during the holidays’ season. Some studies have been conducted in this field trying to find some evidence about the relationship between weather and businesses. Ross (1984) presents the influences of weather changes in the frozen orange juice concentrated production in the US. More than 98 per cent of the production took place in central Florida region surrounding Orlando. The study suggests the interaction between prices and a truly exogenous determinant of value, the weather. His empirical results show that cold weather is bad for orange production. Orange trees cannot withstand freezing temperatures that last for more than a few hours. Florida occasionally has freezing weather and the history of citrus production in the state has been marked by famous freezes. In 1895, almost every orange tree in Florida was killed to the ground, production declined by 97 per cent and 16 years passed before it had recovered to its previous level. Even a mild freeze will prompt the trees to drop significant amounts of fruit.
1.4
Forecasting the weather
There are at least three methods of forecasting. In order of increasing complexity and sophistication, they are persistence, statistical and modelling. Persistence leads us to say that tomorrow will be similar to today, that next month will continue the trend of last month (warmer than normal, for example). Persistence, by itself, incorporates little about the dynamics of the environment. It can be compared to charting the behaviour of the stock market. This forecasting method can be relatively accurate for a short period. Statistical forecasting is an effort to match the patterns of the past to the present. When past patterns fit, the inclination is to forecast that the future will be similar to what happened in the pattern of the past. Unfortunately, nature rarely repeats itself exactly and the number of variables is great. Statistical forecasting, as persistence forecasting does neglects the dynamics of the environment. Model forecasts incorporate the dynamics of the environment. They include the current conditions and mathematical representation of the physical environmental process that influence weather. The environmental events are written as differential equations that are then solved by numerical integration. In general, in weather models, there is a correspondence between forecast reliability over time - confidence declines with the increasing time to the forecasted future. There is also an issue of geographic scale over which the model is integrated. Many countries have meteorological services that record temperature, wind, 14
humidity since the beginning of the century and the degree of confidence in the measurement techniques are high enough to accept the data as a reliable source to use for weather derivatives. Relevant questions arise about the impact of the change in data collection in places where the degree of population growth could alter the measure mechanism. When using data to provide a consistent pricing model, one observes a controversy among the market participants in determining the length of data that might incorporate all the significant factors that influence the weather. A period of 10 to 20 years is considered relatively short to capture the characteristics of the weather cycle. A longer period such as 50 years might be more accurate in identifying those patterns. Specialists say that even a century would not incorporate all factors due to the extreme complexity of the weather as a changing phenomenon. Pricing weather derivatives requires an historical database and application of statistical methods for fitting distribution functions to data. In our research, historical data is available from the National Climate Data Centre (NCDC) a subsidiary of the National Oceanic and Atmospheric Administration (NOAA) for the case of US and Suisse M´et´eo for Switzerland (similar offices exist for different European countries). Defining the appropriate mean and standard deviation is the key challenge in simple-option pricing. As mentioned above the length of historical data that should be used is a critical factor. This problem is well known among climate researchers who have struggled to determine the Optimal Climate Normal (OCN), or the optimal average time scale of previous years for determining the expected value for this year. The National Centre for Environmental Prediction (NCEP) runs an operational tool that is a simplified OCN calculation. This product examines whether the previous 10 years are a better climate predictor than the defined “climate normal period”. Where the historical data indicates that the previous 10 years provides an improved estimate, this 10-year average is used. One of the primary drivers that make the previous 10 years a better predictor than the period from 1961-1990 is a large trend in urbanisation. Any city’s temperature that has a strong warming trend will be better approximated using the most recent data 10 years than NCEP’s 30-year normal. Actual sequence of weather near a measurement site will differ from the measurements at the site: weather is that variable and the contractual data requirement is that specific. This leads to what is called basis risk: one or more weather stations do not exactly measure an enterprise’s exposure to weather. It can be that the stations are too far from the exposure site, or that the exposure is integrated over a region. Basis risk in weather options is poorly understood and difficult to quantify. The scale of weather impacts varies with local and regional sensitivities, from 15
place to place in a region, and changes with time. To measure exposures beyond the measurement site, hedgers will create a weighted group or a basket of sites within a region. This provides a more representative dimension of weather exposure over that region. Basis risk arises from the reality that site-specific measurements and revenues shortfalls do not correlate perfectly. Although accumulating values over a season and area averaging reduce a hedger’s basis risk, neither technique eliminates it completely. Basis risk is even more complex when more than two factors affect weather sensitivity that also varies over time. Air temperature and precipitation are not correlated when viewed as a concurrent and contiguous time series of data, unless we apply an understanding of the weather patterns to stratify the time series. All farmers intuitively understand this. However, this is particularly true for the US market where the continental distances may cause such differences in temperature values from place to place. When analysing this in Europe, one concludes that the basis risk is relatively small and might be discarded due to the high correlation the temperature presents within countries. Section 1.6 elaborates on the pricing of weather options. Weather option pricing relies on an accurate weather prediction model. In section 1.6.3 we present a case study where a temperature forecast is done for the city of Geneva.
1.5
An analysis of weather models: A literature survey
The previous section lead us to conclude that business losses arising from extreme weather events cannot be hedged using weather derivatives contracts. Those events are insured today by using a number of financial products that include exchange-traded catastrophe options, OTC swaps and options, CAT bonds and, CatEPuts. The recent issuance of a CAT bond related to an earthquake in Disneyland Japan is an example of these innovations in the field of catastrophe insurance market. The events related in the previous section as abnormal weather match the category of non-catastrophic events. That is when the use of weather derivatives trading is particular useful to hedge variability of revenues due to the influences of a colder or warmer period. Those weather phenomena are in general easily measurable, can be independently verifiable and transparent be use in written contracts. Weather derivatives contracts focus on the use of those characteristics by rely-
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ing on a specific index together with selected locations on where they have been written using reliable information sources. We present in the following sections the stochastic process for weather derivatives as well as the market structure and trading features of such contracts.
1.5.1 Understanding the weather evolution models Everyone knows that to predict weather is a hazardous task because of the existence of multiple variables that govern the characteristics of the weather. However looking to the past, one might obtain precious information about possible behaviour of the weather which is possible to assume as regular behaviour because changes in the weather seem to follow a cyclical pattern although with some variability. One could mention wind, precipitation, humidity, snow, temperature and so on as the variables that constitute ‘the weather’. The scope of this work is more precise and we only analyse the influences of the temperature as a weather risk. Likewise, when using the concept of weather derivatives here we are assuming that we are referring to temperature derivatives, and specifically heating degree days (HDD) when we analyse a winter season. On the other hand, there can also be cooling degree days (CDD) in a summer season. Let us first see important characteristics when modelling weather derivatives. 1. Following the same approach used by Black and Scholes for pricing option contracts one should start by modelling the behaviour of the temperature index HDD or CDD. Let us use a geometric Brownian motion for this purpose. Definition 1 Defining the probability space ( ), where de fines the set of states of nature, the filtration of information avail able at time and the statistical probability measure, the dynamics governing the stochastic process of the average daily temperature can be set up by the following differential equation:
(1.1)
where the drift may be a mean-reverting process to capture seasonal cyclical patterns, and a volatility that might be considered not constant. 17
Special attention should be considered when dealing with the parameter . To obtain the standard deviation, we need to set up a time-window frame in order to calculate it. This could present problems when data is not available in certain regions leading to a bias in the value of . Another point is the consistency that past pattern will be repeated in the future and thus, making it a reliable parameter. As we have shown in previous section about the behaviour of the weather, there seems to be a certain volatility over time due to an intrinsic feature as well as due to human-made. Nonetheless, in principle it may be feasible admit that is either a deterministic function of the time or a stochastic pattern but depending only on the current value of the temperature index, i.e. .
The following step should be to construct a riskless portfolio by using the Fundamental Theorem of Finance 6 which would yield a risk-free rate. This gives us the partial differential equation (PDE) for a call option: '
"!
"!$#&%
!
!$#)# (
%
+*
*-,
%
*.,
,
(1.2)
with the corresponding boundary condition given by !
/
+0214365 %
87
*"9
(1.3)
In the case of stock option and even interest-sensitive securities the construction of such portfolio is feasible since the underlying is traded. However, for the case of weather derivatives, the same argument is no more valid because the weather is not traded meaning that one does not have the underlying security such as stock, Treasure bills and so forth to build a riskless portfolio. 2. Having in mind that weather options are written on cumulative HDD or CDD we can conclude that in fact we are working with a type of exotic option namely Asian-type derivatives. 6 The arbitrage theorem gives the formal conditions under which“arbitrage” profits can or cannot exist.
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Definition 2 Asian option, or average rate option, is an option whose payoff at maturity depends on the average price of the underlying instrument during all or part of the option life, rather than the price of the underlying asset on maturity date. Assuming that the price driven under the risk-adjusted probability : by the dynamics described below: ;
over[0, ] by the integral, ? ' @
%
A
CB
B
(1.5)
the value of an Asian call option at time t is expressed, by arbitrage argu? D ments, as EGF 5IH"JLK MONPJ 021436 * 9 S ! RQ (1.6)
T
S
where stands for the strike price of the option. Although the PDE for Asian options is the same as Black and Scholes, the boundary conditions are different. Let U denote the value of an Asian option and % the price of the underlying asset. We introduce > as, ? >
(1.7)
TV
where denotes the maturity of the option. So, we need to solve the following PDE, W W
W U
XU ( %
W
W XU %
