Hedging crop yields against weather uncertainties -- a weather derivative perspective
HHedging Crop Yields Against Weather Uncertainties–A Weather Derivative Perspective
Samuel Asante Gyamerah , Philip Ngare ,Dennis Ikpe , Pan African University, Institute of Basic Sciences Technology and Innovation, Kenya. University ofNairobi, Kenya African Institute for Mathematical Sciences, South Africa. Michigan State University,USA.
Abstract
The effects of weather on agriculture in recent years have become a major global concern.Hence, the need for an effective weather risk management tool (i.e., weather derivatives)that can hedge crop yields against weather uncertainties. However, most smallholder farm-ers and agricultural stakeholders are unwilling to pay for the price of weather derivatives(WD) because of the presence of basis risks (product-design and geographical) in the pricingmodels. To eliminate product-design basis risks, a machine learning ensemble technique wasused to determine the relationship between maize yield and weather variables. The resultsrevealed that the most significant weather variable that affected the yield of maize was av-erage temperature. A mean-reverting model with a time-varying speed of mean reversion,seasonal mean, and local volatility that depended on the local average temperature was thenproposed. The model was extended to a multi-dimensional model for different but correlatedlocations. Based on these average temperature models, pricing models for futures, optionson futures, and basket futures for cumulative average temperature and growing degree-daysare presented. Pricing futures on baskets reduces geographical basis risk, as buyers havethe opportunity to select the most appropriate weather stations with their desired weightpreference. With these pricing models, farmers and agricultural stakeholders can hedge theircrops against the perils of extreme weather.
Keywords basis risk, agricultural risk management, weather derivatives, cumulative average tempera-ture, growing degree-days • Published as:Samuel Asante Gyamerah, Philip Ngare and Dennis Ikpe.Mathematical and Computational Applications, Vol. 24, No. 3, 2019 https://doi.org/10.3390/mca24030071 *Corresponding author: E-mail: [email protected] : a r X i v : . [ q -f i n . M F ] A ug . Introduction Agriculture continues to be an important sector that contributes to Ghana’s exports earn-ings, inputs for most manufacturing sectors, and revenue generation for majority of thepopulation. The agriculture and agribusiness sector account for a significant share of themajor economic activities in Ghana and a major source of income for most smallholderfarmers. It is reported to account for about 70% of the labour force and more than 25%of the gross domestic product in Africa (UNECA, 2009). This makes agriculture one ofthe most important and largest sector in the development of the economies in Africa, ofwhich Ghana is a member state. However, in Ghana, the sector continues to be controlledby primary production as a result of high weather variability and hydrological flows, espe-cially in the Northern savannas (Ibn Musah et al., 2018). The Northern savannas of Ghanahave experienced perennial extreme flooding and droughts, both linked to extreme heat andtemperatures (Ibn Musah et al., 2018). These have contributed to crop failures and haveconsequently led to extensive impacts on the economic activities of most rural farmers.Weather variables are difficult to mitigate, especially for smallholder farmers in most devel-oping and under-developed countries, and have great effects on the farming activities of thesefarmers. For this reason, an effective and reliable risk management tool, weather derivative(WD), is needed to hedge farmers and stakeholders from the peril of weather uncertain-ties. By linking the payoffs to a fairly measured weather index (e.g. temperature, rainfall,humidity, and sunshine), a WD reduces or eliminates the disadvantages of traditional insur-ance, such as moral hazards and information asymmetry. Turvey Turvey (2001) examinedthe pricing of weather derivatives in Ontario and contended that weather derivatives andweather insurance can be used as a form of agricultural risk management tools. Zong andEnder Zong and Ender (2016) developed a novel type of weather derivatives contract calleda climatic zone-based growth degree-day contract. Their aim was to mitigate weather risk inthe agricultural sector of mainland China by introducing new types of temperature indices.Even though several weather risk management tools have been recently introduced into theWD market for smallholder farmers in most countries around the world, their purchaseshave been lower than expected. Among the reasons causing the low purchase of WD andthe unwillingness of farmers to pay for WD in most communities are the lack of capacity todesign and determine the value of this insurance product and, most importantly, high basisrisk (product-design and geographical) in the contract design and implementation Woodardand Garcia (2008). Basis risk in WD market is defined as the difference between the actualloss and the actual payout of WD Rohrer (2004). Using a different index rather than theactual index that affects a specific crop yield at a location can lead to a larger gap betweenthe real exposure and the payoff (product-design basis risk). Product design basis risk canbe mitigated if the appropriate weather observation is used to construct the index for WD.Hence, a complete evaluation of the relationship between historical weather and crop yielddata is significant for an effective and reliable design of WD. Forecasting of crop yields andfeature importance of different weather indices for crop yields are, therefore, principal com-ponents for an effective rate-making process in the insurance/derivatives market. Anotherform of basis risk–geographical basis risk occurs when there is a deviation of weather con-ditions at the measurement station of the weather derivative and the weather conditions atthe location of the buyer Ritter et al. (2014). Spicka and Hnilica Spicka and Hnilica (2013)evaluated the effectiveness of weather derivatives as a revenue risk management tool by con-sidering crop growing conditions in the Czech Republic. The authors concluded that highbasis risk can significantly misrepresent the payoff of the contract. Musshoff et al. Musshoffet al. (2011), in their research, concluded that hedging effectiveness for the agricultural sec-2or using weather derivatives is controlled by the contract design. They categorized basisrisk into local and geographical basis risk, and asserted that basis risk has a greater influenceon the hedging effectiveness of weather derivatives.In this study, geographical basis risk is mitigated by pricing futures on a temperature bas-ket rather than a single index contract. This requires the determination of the correlationbetween the locations under study.In the literature of weather derivative pricing, different methods have been proposed forpricing temperature-based weather derivatives. Among these methods are the indifferencepricing approach, actuarial pricing methods, the equilibrium model approach, and incom-plete market pricing models. The indifference pricing approach is a valuation method forweather derivatives which is founded on the arguments of expected utility. It is also referredto as the incomplete market pricing model, reservation price, or private valuation. This ap-proach for pricing has been used in most traditional financial derivatives where the marketis incomplete (see Henderson (2002); Zariphopoulou (2001); ? ). In the weather derivativemarket, Davis Davis (2001) used the marginal utility approach or ”shadow price” method ofmathematical economics to price weather derivatives. The basis for their pricing approachwas centered on the concept that investors in the weather derivative market are not rep-resentative but experience distinct risks that are linked to the effect of weather on theirbusiness. Brockett et al. Brockett et al. (2009) used the indifference pricing approach tovalue weather derivative futures and options. The relationships between the indifferencepricing and actuarial pricing approaches for weather derivatives were studied in a mean-variance context, where they provided instances that the actuarial pricing method does notgive a distinct valuation for weather derivatives. From the concept of utility maximization,Barrieu and Karoui Barrieu and Karoui (2002) calculated the optimal profile (and its value)of derivatives written on an illiquid asset; for example, the weather or a catastrophic event.Cao and Wei Cao and Wei (2000) generalized the equilibrium model of Lucas Jr (1978) byincorporating weather as a basic variable in the economy. Based on this, they proposed anequilibrium valuation framework for temperature-based weather derivatives. Using a modelthat captured the daily temperature dynamics of five cities in the United States of America(Atlanta, Chicago, Dallas, New York, and Philadelphia), they performed numerical analysisfor forward and option contracts on heating degree-days (HDDs) and cooling degree-days(CDDs). Their analysis revealed that the market price of risk is mostly trivial when linkedto the temperature variable, particularly when the aggregate dividend process is mean-reverting. They showed that unrealistic assumptions in historical simulation methods resultin inaccurate pricing of temperature-based weather derivatives. As stated by Alaton et al.(2002), the weather derivative market is a typical example of an incomplete market. Hence,pricing models based on incomplete markets are the most applicable valuation method forweather derivatives. These models consider the hedgeable and unhedgeable components ofrisk. The price of a weather derivative is usually dependent on different weather indices, suchas HDD, CDD, Pacific Rim (PRIM), cumulative average temperature (CAT), and growingdegree-days (GDD). Different authors (Alaton et al., 2002; Mraoua, 2007) have used theHDD and CDD indices as the major indices for pricing weather derivatives for the energyindustry.The contributions made in this study are: (1) We are able to empirically determine the mainunderlying weather variable (average temperature) that affects the yield of the selected cropusing machine learning ensemble techniques and feature selection through crop yield fore-casting, rather than the usual assumption of using temperature as the underlying withoutproper empirical studies. This will eliminate product-design basis risk during pricing of the3eather derivatives. (2) Previous studies have either used a piecewise constant volatilityfunction or a seasonal volatility (e.g., Alaton et al. (2002); Benth and Benth (2007); Benthand ˇSaltyt˙e-Benth (2005)). However, our proposed model includes a local volatility whichis able to capture the local variations of the daily average temperature. (3) Futures, optionson futures, and basket futures (futures for multi-dimensional locations) on CAT and GDDare priced using the constructed daily average temperature model. These pricing models forCAT and GDD are the first of their kind in the literature. (4) The basket futures pricingwill help in mitigating geographical basis risks in the weather derivative market.
2. Crop Yield–Weather Model and Feature Importancefor Weather Derivatives
Different factors, such as condition of the soil, the variety of seedling, and type and amountof fertilizer applied to the soil affect the yield and productivity of crops. These factorscan be controlled by the farmer and industry players. Weather variables, especially surfacetemperature, rainfall, and humidity, are the principal drivers of the differences in crop yields(Hu and Buyanovsky, 2003). These weather variables directly affect the moisture content ofthe soil and the level of nutrients in the soil. The effect of the uncertainties in the patternof weather, both between and within planting seasons, can affect the crop production andyield significantly. This has significantly affected the yield of most crops, causing economicand food security risks in most developing and under-developed countries. Maize is the mostwidely cultivated staple crop in the Northern savanna, and is the major source of incomefor about 45% of households there (Wood, 2013). For this reason, the yield of maize wasused as a proxy to determine the effect of the selected weather variables on crop yield. Thisweather variable can, then, be used as the underlying for weather derivatives pricing.
Machine learning (ML) is used in regression and classification to solve most of the problemsthat arise as a result of nonlinearity in the features and the response variable. In this study,ML ensemble classification algorithms are used to predict the possibility of an improved cropyield harvest or crop yield loss (a two-class (binary) problem). The choice of an optimalML algorithm for prediction is a major factor to consider in any forecasting problem. In ourcase, the chosen ML technique should be able to predict whether there will be an increaseor decrease in crop yields for a period of years with a small margin of error. The targetvariable for the ensemble classifier is whether there will be a loss or increase in crop yieldsfor the year-ahead harvest; that is, • If the long-term average crop yield ( ¯ Y ) is smaller than or equal to the present yearscrop yield ( Y t ), then there is an increase in crop yield; else: • If the long-term average crop yield ( ¯ Y ) is greater than the present years crop yield( Y t ), then there is a decrease or loss in crop yield.We assign labels to this classification: “0” for a decrease/loss in crop yields and “1” for anincrease in crop yields, y = (cid:40) Y ≤ Y t , Y > Y t . (2.1)4umidity, sunlight, rainfall, minimum temperature (minT), maximum temperature (maxT),and average temperature (aveT) are the features ( X ) used in predicting the target variable( Y ). Now, suppose a set of features X = { x i ∈ R n } with associated labels Y = { y i ∈ Y, y i ∈ (0 , } , are given for a training data set T = { ( x i , y i ) } . In this way, we solve the supervisedclassification problem where the learning model N depends on D .The yearly datasets (2000–2016) for maize yield Bole, Tamale, and Yendi were taken from theStatistics, Research, and Information Directorate (SRID) of the Ministry of Food and Agri-culture, Ghana. Daily historical data for sunlight, humidity, rainfall, maximum and mini-mum temperature from 2000–2016 were obtained from the Ghana Meteorological Agency,Ghana. K-nearest neighbors (KNN) algorithm was used to compute the missing data pointsin the dataset. The yearly data points were calculated from the daily data of the selectedweather variables using the arithmetic average. The yearly average temperature was com-puted from the arithmetic average of the yearly maximum and minimum temperature. Datafor the weather variables was also taken from Bole, Tamale, and Yendi. These towns arepart of the Northern Savanna and they are considered to be the food basket of Ghana. Dueto the sensitivity of the ML algorithms used and the unequal weight of the data sets, thesample data sets were set into an identical scale–the min-max normalization scale in the in-terval [0 , Stacking is an ensemble learning technique that combines multiple classification or regressionmodels through a meta-classifier/meta-regressor to a single predictive model to reduce bias(boosting), variance (bagging), and improve the accuracy of predictions (stacking). Usingthe training data set, the base level classifiers were trained. The meta-model was trained onthe outputs of the base level algorithm as features. Stacking ensemble learning algorithms areheterogenous because the base level classifiers are made of different classification algorithms.In this study, Adapted boosting (AdaBoost) and artificial neural network (ANN), were usedas the base classifier and gradient boosting machine (GBM) was used as the meta-classifier.Stacking ensemble algorithm is outlined in Algorithm 1,5 lgorithm 1: Algorithm of Stacking EnsembleInput: D : { ( x i , y i ) } | x i ∈ X, y i ∈ Y Output:
An ensemble classifier H Step 1. Learn base-level classifiers for t ← N do Learn a base classifier h t based on T , h t = H t ( T ) end for Step 2. Construct new data set from T , T (cid:48) = φ for i ← n do Construct a new data set that contains { x newi , y i } where { x newi , y i } = { h t ( x i ) for t = 1 to N } end for Step 3. Learn a second-level classifier Learn a new classifier h new based on the newly constructed data set. Return H ( x ) = h new ( h ( x ) , h ( x ) , · · · , h N ( x ))The performance of the evaluation metrics of the proposed stacking ensemble classifieris presented in Table 1. The table gives the performance of the ensemble classifier on thetesting data set. For binary classification, an AUC value of 50% or less is as good asrandomly selecting the labels. An AUC value closer to 1 and an accuracy value closer to100% indicate the superiority of the proposed model. In general, the classification modelwas very optimal in predicting the class of the target variable (crop yield) using the selectedfeatures (weather variables). Bole Tamale YendiAccuracy
AUC . . . eature Bole Tamale Yendi ImportanceValue
Rank
ImportanceValue
Rank
ImportanceValue
RankminT maxT aveT
Rainfall
Sunlight
Humidity
Table 2: Performance of evaluation metrics of the proposed stacking ensemble classification.
3. Temperature-Based Weather Derivatives
Even though the literature on weather derivatives has evolved rapidly over the past fewdecades, a consensus theoretical framework for evaluating weather derivatives has not beenreached. This can be attributed to the fact that the underlying factors of WD are nottradable in the financial market and, hence, traditional pricing approaches cannot be usedfor the valuation of this product. Further, weather indices do not correlate strongly withthe prices of other financial products in the financial market and, as a result, the underlyingindices cannot be substituted for a linked exchange security.
Alaton et al. Alaton et al. (2002) proposed the following mean-reverting model for thedynamics of temperature variations: dT ( t ) = dS ( t ) + β ( T ( t ) − S ( t )) dt + σ ( t ) dB ( t ) , (3.1)where T ( t ) represents the daily average temperature, S ( t ) is the deterministic seasonalcomponent, B ( t ) is a Brownian motion, β is a constant mean-reversion rate, and σ ( t ) isthe volatility. They assumed that the volatility is a piecewise constant function whichcharacterizes the monthly variation in volatility. From the proposed model of Alaton et al.(2002), Benth and Benth Benth and ˇSaltyt˙e-Benth (2005) suggested the following mean-reverting model for the time-dynamics of Norwegian daily average temperature: dT ( t ) = dS ( t ) + β ( T ( t ) − S ( t )) dt + σ ( t ) dL ( t ) . (3.2)As indicated by Benth and ˇSaltyt˙e-Benth (2005), the only innovation from Equation (3.1) isthe introduction of a L´evy noise, L ( t ), instead of the Brownian motion B ( t ). They used thegeneralized hyperbolic distribution, which is a special class of L´evy process, to capture theskewness and (semi-) heavy tails of their temperature data. However, to allow analyticalpricing using the temperature dynamics model, Benth and Benth Benth and Benth (2007)later proposed to use a Brownian motion, instead of the L´evy process, in Equation (3.2).Clearly, References Alaton et al. (2002); Benth and Benth (2007); Benth and ˇSaltyt˙e-Benth(2005) used a constant mean-reversion rate, a volatility which is piecewise constant function,and they did not consider multi-dimensional locations in their models.The introduction of a time-varying speed of mean-reversion, a local volatility that is ableto capture the local variations of the daily average temperature at the selected locations,7nd a multi-dimensional daily average temperature model for multi-locations gave rise tothe innovations in our proposed model and pricing. Our proposed models capture all thestylized facts of the selected locations and, to the best of our knowledge, it is the first of itskind in the literature. The daily average temperature data (degree Celsius) for Bole and Tamale in the Northernsavanna of Ghana, over a measurement period from 01/01/1992 to 31/08/2017, were takenfrom the Ghana Meteorological Service. The data consisted of the daily maximum andminimum temperatures. Missing data points were computed using KNN. The daily averagetemperature was computed from the arithmetic average of the daily minimum and maximumtemperatures. For consistency in days (365 days) per year, February 29 was removed fromthe dataset for each leap year. As a result, the total daily average temperature had 9368data points.The seasonal and seasonally-adjusted (de-seasonalized) plot of the daily average temper-ature data for Bole and Tamale are presented in Figures 1 and 2, respectively. The spatialcorrelation between Bole and Tamale was estimated, in order to develop a basket tempera-ture for weather derivatives. The de-seasonalized daily average temperature (as suggestedby Alexandridis and Zapranis (2012)) was used to estimate the correlation matrix betweenthe two towns (Bole and Tamale). From Table 3, it is clear that the average temperaturecorrelation between the selected locations was very high. This indicates that a maize farm atBole will encounter the same weather-related risk as a maize farm in Tamale or Yendi, andvice versa. Hence, a weather station in any of these locations can be used for the contractwithout introducing geographical basis risk in the contract. A farmer with two or morefarms at these locations can buy a single temperature basket derivatives contract.Figure 1: Seasonal and de-seasonalized daily average temperature of Bole.8igure 2: Seasonal and de-seasonalized daily average temperature of Tamale.
Bole Tamale YendiBole
Tamale
Yendi
Motivated by Alaton et al. (2002); Benth and ˇSaltyt˙e-Benth (2005); Jones (2003), we proposea new average temperature dynamics model which is able to capture major stylized factsof daily average temperature, such as locality features, seasonality, mean-reversion, andvolatility. These stylized facts were consistent with the daily average temperature of thechosen location for this study. The proposed one-dimensional model was extended to a multi-dimensional temperature model to cover the selected locations under study. For convenienceand analytical tractability, we assumed that the residuals of the daily average temperaturewere independently and identically distributed (i.i.d.) standard normal. The proposed dailyaverage temperature model is given as dT ( t ) = dS ( t ) + β ( t ) (cid:0) T ( t ) − S ( t ) (cid:1) dt + σT ( t ) dB ( t ) , (3.3)where T ( t ) represents the daily average temperature, S ( t ) is the deterministic seasonalcomponent, β ( t ) is the time-varying speed of mean-reversion, and σT ( t ) is the daily averagetemperature volatility through time.Following Alaton et al. (2002), the seasonality component is defined as S ( t ) = A + Bt + C sin (cid:18) πt
365 + ϑ (cid:19) , (3.4)9hich is made up of a seasonal component ( C sin 2 πt/
365 + ϑ ) and a trend component( A + Bt ), where A and B denote the constant and the coefficient of the linearity of theseasonal trend, respectively; C denotes the daily average temperature amplitude, ϑ ; and t is the time, measured in days.Equation (3.4) can be transformed to S ( t ) = a + bt + c sin (cid:18) πt (cid:19) + d cos (cid:18) πt (cid:19) . (3.5)By comparing (3.4) to (3.5), the relationship of the parameters is given below A = a ; B = b ; C = √ c + d ; ϑ = arctan (cid:18) dc (cid:19) . The numerical values of the constant in Equation (3.5) are estimated by fitting the functionto the historical daily average temperature data using the method of least squares. Theseasonal component for Bole and Tamale are given in the following function, respectively, S ( t ) = 22 .
15 + (4 . · − ) t + 1 .
98 sin (cid:18) πt − . (cid:19) ,S ( t ) = 18 .
38 + (7 . · − ) t + 2 .
06 sin (cid:18) πt − . (cid:19) . (3.6)Using additive seasonal decomposition by moving averages, the daily average temperaturewas decomposed into a seasonal, linear, and a random (residual) component as shown in Fig-ures 3–5 respectively. The trend component (Figure 4) was smoother than the actual dailyaverage temperature data plot (Figures 1 and 2, seasonalized) and captured the main move-ment of the daily average temperature data without the minor variations. The estimatedseasonal component of the daily average temperature is presented in figure 6.Figure 3: Seasonal trend of the daily average temperature for Bole and Tamale.10igure 4: Linear trend of the daily average temperature for Bole and Tamale.Figure 5: Residuals of the daily average temperature for Bole and Tamale.11igure 6: Estimated seasonal figure of the daily average temperature for Bole and Tamale. Theorem 3.1 (Girsanov Theorem) . Let B t be a Brownian motion on a probability space(Ω , F , P ) and λ = { λ t : 0 ≤ t ≤ T } be an adaptive process satisfying the Novikov condition E (cid:20) exp (cid:18) (cid:90) t λ u du (cid:19)(cid:21) < ∞ . (3.7)Let Z ( t ) = exp (cid:18)(cid:90) t λ u dB u − (cid:90) t λ u du (cid:19) . Then, Q ∼ P can be determined by the Radon–Nikodym derivative d Q d P | F t = Z ( T ) . (3.8)Then, the random process W t = B t − (cid:90) t λ s ds,dW t = dB t − λ t dt, (3.9)is a standard Brownian motion under the measure Q λ . Remark 1.
The Novikov condition in the Girsanov theorem ensures that Z is positivemartingale whenever E ( Z ) = 1. This is referred to as the Radon–Nikodym derivative. Remark 2. λ is refered to as the market price of risk (MPR). As there is no real weatherderivative market in Africa from which the prices can be obtained, λ is assumed to be aconstant. For a constant λ , Equation (3.9) can be re-defined as dW t = dB t − λdt. (3.10)12 emma 3.1. If the daily average temperature follows the proposed model in Equation (3.3),then the explicit solution is given by T t = S t + ( T − S ) e (cid:82) t β s ds + e (cid:82) t β s ds (cid:90) t σT u e (cid:82) t β s ds dB u . (3.11) Proof.
We have that dT t = dS t + β t ( T t − S t ) dt + σT t dB t d ˜ T t = β t ˜ T t + σT t dB t , (3.12)where ˜ T = T t − S t . Using the transformation below, d ˜ T t can be evaluated, F ( ˜ T t , t ) = ˜ T t e − (cid:82) t β s ds ∂F∂ ˜ T t = e − (cid:82) t β s ds ; ∂ F∂ ˜ T t = 0; ∂F∂t = − β t ˜ T t e − (cid:82) t β s ds . Applying Itˆo’s Lemma and Equation (3.12), dF t = σT t e − (cid:82) t β s ds dB t . (3.13)Integrating Equation (3.13) over the interval [0 , t ], F t = F + (cid:90) t σT u e − (cid:82) u β s ds dB u ˜ T t e − (cid:82) t β s ds = ˜ T + (cid:90) t σT u e − (cid:82) u β s ds dB u ˜ T t = ˜ T e (cid:82) t β s ds + e (cid:82) t β s ds (cid:90) t σT u e − (cid:82) u β s ds dB u T t = S t + ( T − S ) e (cid:82) t β s ds + e (cid:82) t β s ds (cid:90) t σT u e − (cid:82) u β s ds dB u . Lemma 3.2.
Under the risk-neutral measure Q , the explicit solution of the daily averagetemperature model is given by T t = S t + ( T − S ) e (cid:82) t β s ds + (cid:90) t σλT u e (cid:82) tu β s ds du + (cid:90) t σT u e (cid:82) tu β s ds dW u . (3.14) Proof.
By substituting Equation (3.10) into Equation (3.12) and following the steps of theproof of Lemma 3.11, the lemma can be derived.
Suppose that, for a contract period [ t , t ], the temperature dynamics follow the TML model.Then, there is a price dynamic of futures written on a CAT index with t ≤ t < t . Thefutures price of CAT is given by0 = e − r ( t − t ) E Q (cid:20)(cid:90) t t T x dx − F CAT ( t, t , t ) | F t (cid:21) . (3.15)As the future price F ( t, t , t ) is F t adapted under the measure Q , F CAT ( t, t , t ) = E Q (cid:20)(cid:90) t t T x dx | F t (cid:21) . (3.16)13 roposition 3.1. Suppose the daily average temperature follows Model (3.3). Then, theprice of CAT-futures at time t ≤ t ≤ t for the contract period [ t , t ] is given by: F CAT ( t, t , t ) = (cid:90) t t S x dx + (cid:90) t t ( T t − S t ) e (cid:82) xt β s ds dx + L , where L = (cid:82) t t (cid:82) t t e (cid:82) x β s ds σλT u e (cid:82) u β s ds dxdu + (cid:82) t t (cid:82) t t e (cid:82) x β s ds σλT u e (cid:82) u β s ds dxdu. Proof.
From Equation (3.16), F CAT ( t, t , t ) = E Q (cid:20) (cid:90) t t T x dx | F t (cid:21) . From Lemma 3.2 and for any time x ≥ tT x = S x + ( T t − S t ) e (cid:82) xt β s ds + (cid:90) xt σλT u e (cid:82) xu β s ds du + (cid:90) xt σT u e (cid:82) xu β s ds dW u F CAT ( t, t , t ) = E Q (cid:20) (cid:90) t t (cid:18) S x + ( T t − S t ) e (cid:82) xt β s ds + (cid:90) xt σλT u e (cid:82) xu β s ds du + (cid:90) xt σT u e (cid:82) xu β s ds dW u (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:90) t t S x dx + (cid:90) t t ( T t − S t ) e (cid:82) xt β s ds dx + (cid:90) t t (cid:90) xt σλT u e (cid:82) xu β s ds dudx = (cid:90) t t S x dx + (cid:90) t t ( T t − S t ) e (cid:82) xt β s ds dx + L , where L = (cid:90) t t (cid:90) xt σλT u e (cid:82) xu β s ds dudx = (cid:90) t t (cid:90) t t [ t,x ] ( u ) σλT u e (cid:82) xu β s ds dudx = (cid:90) t t (cid:90) t t [ t,x ] ( u ) σλT u e (cid:82) xu β s ds dxdu = (cid:90) t t (cid:90) t t [ t,x ] ( u ) σλT u e (cid:82) xu β s ds dxdu + (cid:90) t t (cid:90) t t [ t,x ] ( u ) σλT u e (cid:82) xu β s ds dxdu = (cid:90) t t (cid:90) t t e (cid:82) x β s ds σλT u e (cid:82) u β s ds dxdu + (cid:90) t u (cid:90) t t e (cid:82) x β s ds σλT u e (cid:82) u β s ds dxdu. Proposition 3.2.
At time t ≤ t ≤ t , the in-period (in the contract) valuation of the CATfutures is given by F CAT ( t, t , t ) = (cid:90) tt T x dx + (cid:90) t t S x dx + (cid:90) t t ( T t − S t ) e (cid:82) xt β s ds dx + (cid:90) t u (cid:90) t t e (cid:82) x β s ds σλT u e (cid:82) u β s ds dxdu. roof. From the CAT futures price in Equation (3.16), F CAT ( t, t , t ) = E Q (cid:20) (cid:90) t t T x dx | F t (cid:21) = E Q (cid:20)(cid:18) (cid:90) tt T x dx + (cid:90) t t T x dx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:90) tt T x dx + E Q (cid:20) (cid:90) t t T x dx | F t (cid:21) = (cid:90) tt T x dx + F CAT ( t, t, t ) . From Proposition 3.1, F CAT ( t, t , t ) = (cid:90) tt T x dx + (cid:90) t t S x dx + (cid:90) t t ( T t − S t ) e (cid:82) xt β s ds dx + (cid:90) t u (cid:90) t t e (cid:82) x β s ds σλT u e (cid:82) u β s ds dxdu. Lemma 3.3.
The dynamics of the CAT futures under the equivalent probability measure Q and measured over the contract period [ t , t ] is given by dF CAT ( t, t , t ) = Σ CAT ( t, t , t , T t ) dW t , where Σ CAT ( t, t , t , T t ) = σT t (cid:90) t t e (cid:82) xt β s ds dx. Proof.
We have that dF CAT ( t, t , t ) dT t = (cid:90) t t e (cid:82) xt β s ds dxdF CAT ( t, t , t ) = (cid:90) t t e (cid:82) xt β s ds dxdT t dF CAT ( t, t , t ) = σT t (cid:90) t t e (cid:82) xt β s ds dxdW t . Proposition 3.3.
The call option price at exercise time t n and strike price C is given by C CAT ( t, t n , t , t ) = e − r ( t n − t ) (cid:18)(cid:0) F CAT ( t, t , t ) − C (cid:1) Φ(∆( t, t n , t , t , T t ))+Σ t,t n φ (∆( t, t n , t , t , T t )) (cid:19) , where ∆( t, t n , t , t , T t ) = F CAT ( t, t , t ) − C (cid:113) Σ t,t n ; Σ t,t n = (cid:90) t n t Σ CAT ( s, t , t , T t ) ds, Φ is the cumulative standard normal distribution function, φ is the standard normal densityfunction, and φ ( · ) = Φ (cid:48) ( · ). 15 roof. By definition, a call option price at exercise time t n and strike price C is given as C CAT ( t, t n , t , t ) = e − r ( t n − t ) E Q (cid:2) max (cid:0) F CAT ( t n , t , t ) − C, (cid:1)(cid:12)(cid:12) F t (cid:3) . From Lemma 3.3, (cid:90) t n t dF CAT ( s, t , t ) = (cid:90) t n t Σ CAT ( s, t , t , T t ) dW s ,F CAT ( t n , t , t ) = F CAT ( t, t , t ) + (cid:90) t n t Σ CAT ( s, t , t , T t ) dW s .F CAT ( t n , t , t ) is normally distributed under Q λ , with expectation E λ [ F CAT ( t n , t , t )] = F CAT ( t, t , t )and variance V ar λ [ F CAT ( t n , t , t )] = (cid:90) t n t Σ CAT ( s, t , t , T t ) ds = Σ t,t n . Hence, C CAT ( t, t n , t , t ) = e − r ( t n − t ) E Q (cid:2) max (cid:0) F CAT ( t n , t , t ) − C, (cid:1)(cid:12)(cid:12) F t (cid:3) = e − r ( t n − t ) (cid:90) ∞ C ( y − C ) f CAT ( y ) dy = e − r ( t n − t ) (cid:18)(cid:0) F CAT ( t, t , t ) − C (cid:1) Φ(∆( t, t n , t , t , T t )) + Σ t,t n φ (∆( t, t n , t , t , T t )) (cid:19) , where ∆( t, t n , t , t , T t ) = F CAT ( t, t , t ) − C (cid:113) Σ t,t n . Similar to the definition of the CAT future price, the GDD future price is given as0 = e − r ( t − t ) E Q (cid:20)(cid:90) t t T x dx − F GDD ( t, t , t ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . Using the same idea in deriving the CAT futures price, the price of the GDD-futures can bederived as F GDD ( t, t , t ) = E Q (cid:20)(cid:90) t t max (cid:0) T x − T optimal , (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , (3.17)where T optimal is the optimal normal temperature at which a crop will develop. Proposition 3.4.
Suppose the daily average temperature follows Model (3.3). Then, theprice of GDD-futures at time t ≤ t ≤ t for the contract period [ t , t ] is given by: W ehavethatF
GDD ( t, t , t ) = (cid:90) t t Ψ( t, x ) (cid:20) φ (cid:0) ∆( t, x ) (cid:1) + ∆( t, x )Φ (cid:0) ∆( t, x ) (cid:1) , (cid:21) whereΨ ( t, x ) = (cid:90) xt σ T u e (cid:82) xu β s ds du ; ∆( t, x ) = S x + ( T t − S t ) e (cid:82) xt β s ds + (cid:82) xt σλT u e (cid:82) xu β s ds du − C Ψ( t, x ) . roof. By definition.
GDD = (cid:90) t t max( T x − C, dx,F GDD ( t, t , t ) = E Q (cid:20) (cid:90) t t max (cid:0) T x − C, (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . (3.18)Recall, from Lemma 3.2 and for any time x ≥ t , T x = S x + ( T t − S t ) e (cid:82) xt β s ds + (cid:90) xt σλT u e (cid:82) xu β s ds du + (cid:90) xt σT u e (cid:82) xu β s ds dW u , (3.19) T x = D x = A ( t, x ) + B ( t, x ) , (3.20)where D ( t, x ) = S x + ( T t − S t ) e (cid:82) xt β s ds + (cid:90) xt σλT u e (cid:82) xu β s ds du, and B ( t, x ) = (cid:90) xt σT u e (cid:82) xu β s ds dW u . The distribution of D x can be determined: A ( t, x ) is deterministic and, hence, B ( t, x ) ∼ N (cid:18) , (cid:90) xt σ T u e (cid:82) xu β s ds du (cid:19) = N (cid:18) , Ψ ( t, x ) (cid:19) . It follows, from Equation (3.20), that D x ∼ N (cid:18) A ( t, x ) , Ψ ( t, x ) (cid:19) . Thus, D x can be written, in terms of the standard normal variable Z ∼ N (0 , D ( t, x ) = A ( t, x ) + (cid:0) Ψ ( t, x ) (cid:1) Z. (3.21)Consider T x − C > . This requires (cid:0) Ψ ( t, x ) (cid:1) Z > C − A ( t, x ) ,Z > C − A ( t, x ) (cid:0) Ψ ( t, x ) (cid:1) Z = ∆ ( t, x ) . (3.22)From Equation (3.22), C = A ( t, x ) + ∆( t, x ) (cid:0) Ψ ( t, x ) (cid:1) Z. (3.23)From Equations (3.18) and (3.22), E Q (cid:20) (cid:90) t t max (cid:0) T x − C, (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:90) + ∞ ∆( t,x ) (cid:18) D ( t, x ) − C (cid:19) e − z √ π dz. (3.24)17ubstituting Equations (3.21) and (3.23) into Equation (3.24),= (cid:90) + ∞ ∆( t,x ) (cid:18) A ( t, x ) + (cid:0) Ψ ( t, x ) (cid:1) Z − A ( t, x ) − ∆( t, x ) (cid:0) Ψ ( t, x ) (cid:1) Z (cid:19) e − z √ π dz = (cid:90) + ∞ ∆( t,x ) (cid:18)(cid:0) Ψ ( t, x ) (cid:1) z − ∆( t, x ) (cid:0) Ψ ( t, x ) (cid:1) z (cid:19) e − z √ π dz = (cid:0) Ψ ( t, x ) (cid:1) (cid:32)(cid:90) + ∞ ∆ ( t,x ) ze − z √ π dz + ∆ ( t, x )Φ( − ∆ ( t, x )) (cid:33) = (cid:0) Ψ ( t, x ) (cid:1) (cid:32) e − ∆( t,x ) √ π dz + ∆( t, x )Φ(∆( t, x )) (cid:33) = (cid:0) Ψ ( t, x ) (cid:1) (cid:2) φ (∆( t, x )) + ∆( t, x )Φ(∆( t, x )) (cid:3) , , where ∆( t, x ) = − ∆ ( t, x ) = G ( t, x ) − C (Ψ ( t, x )) , E Q (cid:20) (cid:90) t t max (cid:0) T x − C, (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = Ψ( t, x ) (cid:20) φ (cid:0) ∆( t, x ) (cid:1) + ∆( t, x )Φ (cid:0) ∆( t, x ) (cid:1)(cid:21) . (3.25)Substituting Equation (3.25) into Equation (3.18) gives the Proposition. Lemma 3.4.
The dynamics of the GDD futures under the equivalent probability measure Q measured over the period [ t , t ] are given by dF GDD ( t, t , t ) = Π GDD ( t, t , t ) dW t , where Π GDD is called the term structure of the GDD futures volatility, Π GDD = σT t (cid:90) t t e (cid:82) xt β s ds Φ (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) ds ; Ψ ( t, x ) = (cid:90) xt σ T u e (cid:82) xu β s ds du ; h ( t, x, e (cid:82) xt β s ds ( T t − S t )) = S x + ( T t − S t ) e (cid:82) xt β s ds + (cid:90) xt σλT u e (cid:82) xu β s ds du − C. Proof.
Let h ( t, x, e (cid:82) xt β s ds ( T t − S t )) = S x + ( T t − S t ) e (cid:82) xt β s ds + (cid:82) xt σλT u e (cid:82) xu β s ds du − CdhdT t = e (cid:82) xt β s ds From Proposition 3.4, dF GDD dT t = (cid:90) t t Ψ( t, x )Υ (cid:48) (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) h (cid:48) ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ ( t, x ) ds = (cid:90) t t Υ (cid:48) (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) h (cid:48) ( t, x, e (cid:82) xt β s ds ( T t − S t )) ds = (cid:90) t t e (cid:82) xt β s ds Φ (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) dsdF GDD = σT t (cid:90) t t e (cid:82) xt β s ds Φ (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) dsdW t dF GDD = Π GDD ( t, t , t ) dW t , t, x )) = φ (∆( t, x )) + ∆( t, x )Φ(∆( t, x )); ∆( t, x ) = h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) ; Π GDD = σT t (cid:90) t t e (cid:82) xt β s ds Φ (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) ds. Proposition 3.5.
For a strike price C and maturity time t ≤ t n ≤ t , the price of a calloption at time t on a GDD futures contract is given by C GDD ( t, t n , t , t ) = e − r ( t n − t ) E Q (cid:20) max (cid:18) (cid:90) t t Ψ( t, x ) P ( t, x, t n , ( T t − S t )) ds − C, (cid:19)(cid:21) , where P ( t, x, t n , ( T t − S t )) = ˜Υ (cid:18) t, x, e (cid:82) xt β s ds ( T t − S t ) + (cid:90) t n t λσe (cid:82) xu β s ds du + Σ( x, t, t n ) Y (cid:19) ˜Υ( t, x, e (cid:82) xt β s ds ( T t − S t )) = Υ (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) and Σ( x, t, t n ) = (cid:90) t n t σ T u e (cid:82) xu β s ds du. Proof.
By definition , C GDD ( t, t n , t , t ) = e − r ( t n − t ) E Q (cid:20) (cid:90) t t max (cid:0) F GDD ( t n , t , t ) − C, (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) F GDD ( t n , t , t ) = (cid:90) t t Ψ( t, x ) ˜Υ (cid:0) t, x, e (cid:82) xtn β s ds ( T t n − S t n ) (cid:1) ds, = (cid:90) t t Ψ( t, x ) ˜Υ (cid:18) t, x, e (cid:82) xtn β s ds ( T t − S t ) + (cid:90) t n t λσT u e (cid:82) xu β s ds du + (cid:90) t n t σT u e (cid:82) xu β s ds dW u (cid:19) dsC GDD ( t, t n , t , t ) = e − r ( t n − t ) E Q (cid:20) (cid:90) t t max (cid:0) (cid:90) t t Ψ( t, x ) ˜Υ (cid:18) t, x, e (cid:82) xtn β s ds ( T t − S t ) + (cid:90) t n t λσT u e (cid:82) xu β s ds du + (cid:90) t n t σT u e (cid:82) xu β s ds dW u (cid:19) ds − C, (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , where ˜Υ( t, x, e (cid:82) xt β s ds ( T t − S t )) = Υ (cid:18) h ( t, x, e (cid:82) xt β s ds ( T t − S t ))Ψ( t, x ) (cid:19) . Assuming N is the spatial locations in the basket, then ( ω i ) Ni =1 will be the collection ofweights for the spatial locations ( y i ) Ni =1 . The basket of the daily average temperature at the N spatial locations for a given time t is defined as: M ( t ) := N (cid:88) i =1 ω i T i ( t ) , (3.26)19here (cid:80) Ni =1 ω i = 1.Assume the daily average temperature is spatially correlated across the random noise termand the risk-neutral distribution of the daily average temperature for the spatial locationsis normally distributed in Model (3.3). Hence, the weighted sum of a normally distributedbasket is also normally distributed. From the above assumptions, a new daily averagetemperature model for each spatial location y i can be proposed, dT i ( t ) = dS i ( t ) + β i ( t ) (cid:0) T i ( t ) − S i ( t ) (cid:1) dt + σT i ( t ) dB i ( t ) . (3.27)Expressing Equation (3.27), for locations i = 1 , , · · · , N , as an N -dimensional systemleads to the equation below, d T ( t ) = d S ( t ) + β ( t ) (cid:0) T ( t ) − S ( t ) (cid:1) dt + σT ( t ) d B ( t ) , (3.28)where B ( t ) ∼ N (0 , Ω t ) and Ω is a covariance matrix. Using the linear transformation ofmultivariate normal distributions property, Y ∼ N ( µ, Σ) ⇒ DY ∼ N ( Dµ, D Σ D T ) . Suppose Z ∼ N (0 , I t ) and Y = DZ . Then, it follows that Y ∼ N (0 , DD T t ). ApplyingCholesky factorization to Σ, we can derive a lower triangular form for D , and W t will beexpressed as an N -dimensional Brownian motion V t , B ( t ) = LV ( t ) . (3.29)Then, LL T = Ω , L is a lower triangular matrix with non-negative diagonal entries, L T isan upper triangular matrix, and V ( t ) = ( V ( t ) , V ( t ) , V ( t ) , · · · , V N ( t )) T with dV i ( t ) dV j ( t ) = δ ij dt . Equation (3.28) can be reformulated, in terms of V ( t ), as d T ( t ) = d S ( t ) + β ( t ) (cid:0) T ( t ) − S ( t ) (cid:1) dt + σT ( t ) L d V ( t ) . (3.30) R N Let V ( t ) = (cid:0) V ( t ) , V ( t ) , V ( t ) , · · · , V N ( t ) (cid:1) be an N -dimensional Brownian motion on aprobability space (Ω , F , P ) and λ = (cid:0) λ ( t ) , λ ( t ) , λ ( t ) , · · · , λ N ( t ) (cid:1) be an N -dimensionaladapted process on [0 , T ].Define Z λ ( t ) := exp (cid:18) (cid:90) t λ ( s ) d V ( s ) − (cid:90) t || λ ( s ) || ds (cid:19) , (3.31)where || λ ( s ) || = (cid:80) Ni =1 λ i ( s ) .Let ˜ V ( t ) = V ( t ) + (cid:90) t λ ( s ) ds. (3.32)The component process of ˜ B ( t ) is independent under the measure Q .Suppose that E (cid:90) t || λ ( s ) || Z ( s ) ds < ∞ . Then, ˜ B ( t ) is an N -dimensional standard Brownian motion under the measure Q , definedas d Q d P (cid:12)(cid:12)(cid:12)(cid:12) F t = Z ( T ) . (3.33)20et Z λ ( t ) := exp (cid:18) (cid:90) t (cid:0) σT ( t ) L (cid:1) λ ( s ) d V ( s ) − (cid:90) t || σT ( s ) L || − || λ ( s ) || ds (cid:19) . (3.34)For a constant market price of risk at each geographical reference location in equation(3.34), it can be deduced that W λ ( t ) = V ( t ) − (cid:90) t ( σT ( s ) L ) − λ ds, (3.35)where W λ ( t ) is a standard Brownian motion under the measure Q . Hence, we can definethe temperature model under the measure Q as d T ( t ) = d S ( t ) + (cid:2) λ + β ( t ) (cid:0) T ( t ) − S ( t ) (cid:1)(cid:3) dt + σT ( t ) L d W ( t ) . (3.36) For a a specificied contract period, t ≤ t < t and at a spatial location y i ,the CAT futures price is defined as in Equation (3.38): CAT M [ t , t ] := (cid:90) t t M ( t ):= N (cid:88) i =1 ω i (cid:18)(cid:90) t t T i ( x ) dx (cid:19) . (3.37)From Equations (3.16), (3.37), and the linearity of expectation, F CAT ( t, t , t ; M ) = N (cid:88) i =1 ω i E Q (cid:20)(cid:90) t t T i ( x ) dx | F t (cid:21) . (3.38) Definition 3.2.
For a a specificied contract period, t ≤ t < t and at location y i , the CATfutures price is defined as GDD ( τ , τ ) := (cid:90) t t max (cid:110) M ( t ) − C , (cid:111) dx = (cid:90) t t max (cid:26) N (cid:88) i =1 ω i T i ( x ) − C , (cid:27) dx. (3.39)From Equations (3.2), (3.17), and using the linearity of expectation, F GDD ( t, t , t ; D ) = E Q (cid:32) (cid:90) t t max (cid:26) N (cid:88) i =1 ω i T i ( x ) − C , (cid:27) dx (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:33) = (cid:90) t t E Q (cid:32) max (cid:26) N (cid:88) i =1 ω i T i ( x ) − C , (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:33) dx. (3.40) Lemma 3.5.
If the dynamics of the daily average temperature follows Equation (3.36),then the explicit solution for the i th location y i is given as T i ( t ) = S i ( t )+ (cid:0) T i (0) − S i (0) (cid:1) e (cid:82) t β i ( s ) ds + (cid:90) t λ i e (cid:82) tu β i ( s ) ds du + (cid:90) t σ i T i ( u ) e (cid:82) tu β i ( s ) ds i (cid:88) j =1 L ij dW λj ( u ) . roof. The proof follows directly from the proof of Lemma 3.11 and observing the i th location y i . Proposition 3.6.
At a spatial location i , the futures contract price on basket of CAT indexfollowing the mean-reverting regime in Equation (3.27) is calculated as F CAT ( t, t , t ; M ) = N (cid:88) i =1 ω i (cid:20) (cid:90) t t S i ( x ) + (cid:90) t t (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds + (cid:90) t t (cid:90) t t e (cid:82) x β i ( s ) ds λ i e (cid:82) u β i ( s ) ds dxdu + (cid:90) t u (cid:90) t t e (cid:82) x β i ( s ) ds λ i e (cid:82) u β i ( s ) ds dxdu (cid:21) . Proof.
For x ≥ t in Lemma 3.5, T i ( x ) = S i ( x )+ (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds + (cid:90) xt λ i e (cid:82) xu β i ( s ) ds du + (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds i (cid:88) j =1 L ij dW λj ( u )(3.41) E Q (cid:2) (cid:90) t t T i ( x ) dx | F t (cid:3) = E Q (cid:20) (cid:90) t t (cid:18) S i ( x ) + (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds + (cid:90) xt λ i e (cid:82) xu β i ( s ) ds du + (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds i (cid:88) j =1 L ij dW λj ( u ) (cid:19) dx (cid:12)(cid:12) F t (cid:21) = (cid:90) t t S i ( x ) dx + (cid:90) t t (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds dx + (cid:90) t t (cid:90) xt λ i e (cid:82) xu β i ( s ) ds dudx = (cid:90) t t S i ( x ) + (cid:90) t t (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds + L , where L = (cid:90) t t (cid:90) xt λ i e (cid:82) xu β i ( s ) ds dudx = (cid:90) t t (cid:90) t t [ t,x ] ( u ) λ i e (cid:82) xu β i ( s ) ds dudx = (cid:90) t t (cid:90) t t e (cid:82) x β i ( s ) ds λ i e (cid:82) u β i ( s ) ds dxdu + (cid:90) t u (cid:90) t t e (cid:82) x β i ( s ) ds λ i e (cid:82) u β i ( s ) ds dxduF CAT ( t, t , t ; M ) = N (cid:88) i =1 ω i (cid:20) (cid:90) t t S i ( x ) + (cid:90) t t (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds + (cid:90) t t (cid:90) t t e (cid:82) x β i ( s ) ds λ i e (cid:82) u β i ( s ) ds dxdu + (cid:90) t u (cid:90) t t e (cid:82) x β i ( s ) ds λ i e (cid:82) u β i ( s ) ds dxdu (cid:21) . roposition 3.7. The GDD futures price for a contract time t ≤ t < t at a given spatiallocation y i for a basket of CAT index following the normal regime is given by F NGDD ( t, t , t ; D ) = (cid:90) t t (cid:0) ξ ( t, x ) + 2 ∆ ( t, x ) (cid:1) (cid:0) φ (cid:0) Λ( t, x ) (cid:1) + Λ( t, x )Φ (cid:0) Λ( t, x ) (cid:1)(cid:1) dx, (3.42)where Φ is the cumulative standard normal distribution function, φ is the standard normaldensity function, Λ( t, x ) = ψ ( t, x ) − C (cid:18) ξ ( t, x ) + 2 ∆ ( t, x ) (cid:19) ,ψ ( t, x ) = N (cid:88) i =1 ω i (cid:18) S i ( x ) + (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds + (cid:90) xt λ i e (cid:82) xu β i ( s ) ds du (cid:19) ,ξ ( t, x ) = N (cid:88) i =1 ω i i (cid:88) j =1 (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds L ij du, and ∆ ( t, x ) = N (cid:88) i =1 N (cid:88) j = i +1 ω i ω j (cid:16) i (cid:88) q =1 L iq L jq (cid:17) (cid:90) xt σ i ( u ) σ j ( u ) T i ( u ) T j ( u ) e (cid:82) xu ( β i ( s )+ β j ( s )) ds du Proof.
Let D ( x ) = N (cid:88) i =1 ω i ˜ T it . (3.43)For convenience, we denote the deterministic and random components of Equation (3.41)as A i ( t, x ) and B i ( t, x ), respectively. That is, A i ( t, x ) = S i ( x ) + (cid:0) T i ( t ) − S i ( t ) (cid:1) e (cid:82) xt β i ( s ) ds + (cid:90) xt λ i e (cid:82) xu β i ( s ) ds du,B i ( t, x ) = (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds i (cid:88) j =1 L ij dW λj ( u ) = i (cid:88) j =1 (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds L ij dW λj ( u ) . Hence, D ( x ) = N (cid:88) i =1 ω i (cid:16) A i ( t, x ) + B i ( t, x ) (cid:17) . (3.44)The distribution of the basket D ( x ) at time t wll be established. A i ( t, x ) is, however,deterministic. By Itˆo isometry, and at t , (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds L ij dW λj ( u ) ∼ N (cid:18) , (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds L ij du (cid:19) . However, the W λj ( u ) are independent for each j . The variances can, therefore, be summedto obtain the variance of B i ( t, x ), B i ( t, x ) ∼ N (cid:18) , i (cid:88) j =1 (cid:90) xt σ i T i ( u ) e (cid:82) xu β i ( s ) ds L ij du (cid:19) = N (cid:18) , Ψ ( t, x ) (cid:19) . (cid:80) Ni =1 ω i B i ( t, x ) is a sum of normally distributed random variables, it is normally dis-tributed with respective mean and variance: E (cid:16) N (cid:88) i =1 ω i B i ( t, x ) (cid:17) = N (cid:88) i =1 ω i E (cid:16) B i ( t, x ) (cid:17) = 0 , and V ar (cid:16) N (cid:88) i =1 ω i B i ( t, x ) (cid:17) = N (cid:88) i =1 V ar (cid:16) ω i B i ( t, x ) (cid:17) + 2 (cid:88) i Cov (cid:0) B i , B j (cid:1) for j > 1. With respect to the standard Brownian motions W λ ( u ),both B and B j are in the same integral form. For j > W λ ( u )and W λj ( u ), the covariance only exists between these two integrals. Therefore, Cov (cid:16) B i , B j (cid:17) = (cid:90) xt σ i ( u ) σ j ( u ) T i ( u ) T j ( u ) e (cid:82) xu ( β i ( s )+ β j ( s )) ds (cid:16) i (cid:88) q =1 L iq L jq (cid:17) du, ∀ j > (cid:16) i (cid:88) q =1 L iq L jq (cid:17) (cid:90) xt σ i ( u ) σ j ( u ) T i ( u ) T j ( u ) e (cid:82) xu ( β i ( s )+ β j ( s )) ds du, ∀ j > . Define Υ ij ( t, x ) := (cid:90) xt σ i ( u ) σ j ( u ) T i ( u ) T j ( u ) e (cid:82) xu ( β i ( s )+ β j ( s )) ds du , V ar (cid:16) N (cid:88) i =1 ω i B i ( t, x ) (cid:17) = N (cid:88) i =1 ω i Ψ ( t, x ) + 2 (cid:88) i Acknowledgements The first author wishes to thank African Union and Pan African University, Instutute forBasic Sciences Technology and Innovation,Kenya for their financial support for this research. Disclosure statement The authors declare that there is no conflict of interest regarding the publication of thispaper 26 eferences Alaton, P., Djehiche, B., and Stillberger, D. (2002). On modelling and pricing weatherderivatives. 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