Impied Market Price of Weather Risk

Wolfgang Karl Hardle, Brenda Lopez Cabrera

CASE - Center for Applied Statistics and EconomicsHumboldt-Universitat zu Berlin

Abstract

Weather influences our daily lives and choices and has an enormous im-pact on cooperate revenues and earnings. Weather derivatives differ frommost derivatives in that the underlying weather cannot be traded and theirmarket is relatively illiquid. This paper implements a pricing methodol-ogy for weather derivatives that can increase the precision of measuringweather risk, which is an important issue for financial institutions and en-ergy companies. We applied continous autoregressive models (CAR) tomodel the temperature in Berlin and with that to get explicite nature ofnon-arbitrage prices for temperature derivatives. A clear seasonal variationin the regression residuals of the temperature is observed and the volatil-ity term structure of cumulative average temperature futures presents a aSamuelson effect. We infer the implied market price of temperature riskfor Berlin futures traded at the Chicago Mercantile Exchange (CME).

Keywords: Weather derivatives, weather risk, weather forecasting, seasonality,continuous autoregressive model, stochastic variance

JEL classification: G19, G29, N26, N56, Q29, Q54

Acknowledgements: The financial support from the Deutsche Forschungsgemein-schaft via SFB 649 ”Okonomisches Risiko”, Humboldt-Universitat zu Berlin isgratefully acknowledged.

1 Introduction

Weather influences our daily lives and has an enormous impact on corporaterevenues and earnings. The global climate changes the volatility of weather andthe occurrence of extreme weather events increases. In particular, disfavoured ex-treme natural events like earthquakes, hurricanes, long cold winter, heat, drought,freeze, etc. may cause substantial financial losses. The traditional way of protec-tion against unpredictable weather conditions is the insurance, which covers thelosses in exchange for the payment of a premium. However, recently one observesan inception of new financial instruments linked to weather conditions: CATbonds, sidecars and weather derivatives. It is therefore of scientific importance to

1

study Weather Derivatives (WD), especially when they are used to hedge againstweather related exposures.

The key factor in efficient usage of weather derivatives is a reliable valuationprocedure. However, due to their specific nature one suffers several difficulties.First, weather derivatives are different from most financial derivatives because theunderlying weather cannot be traded. Second, the weather derivatives market isrelatively illiquid, i.e. the weather derivatives cannot be cost-efficiently replicatedwith other weather derivatives. One of the consequences of this is that the valua-tion of weather derivatives is in spirit and methodolgy closer to insurance pricingthan to derivative pricing (arbitrage pricing).

It is therefore important to concentrate on pricing methodologies that can increasethe accuracy of measuring weather risk. A precise validation is a question and iscentral under new insurance and climate change regulations and for the emergenceof a liquid weather derivative market.

The pricing based of weather derivatives attracted the attention of many re-searchers. ? fitted an Ornstein-Uhlenbeck stochastic process with constant vari-ance to temperature observations at Chicago O’Hare airport and started to in-vestigate future prices on temperature indices. Later ? applied the Ornstein-Uhlenbeck model with a monthly variation in the variance to temperature dataof Bromma airport (Stockholm). They applied their model to get prices for dif-ferent temperature prices. ? modelled temperature in several US cities witha high order autoregressive model. They observed seasonality behaviour in theautocorrelation function (ACF) of the squared residuals. However, they did notprice temperature derivatives. ? studied the temperature in Casablanca, Mo-rocco using a mean reverting model with stochastic volatility and a temperatureswap was considered. ? calculates an arbitrage free price for different tempera-ture derivatives prices by using the fractional Brownian motion model of ?, whichdrives the noise in an Ornstein-Uhlenbeck process.

In the temperature derivative market, ? proposed to use a marginal utility tech-nique to price temperature derivatives based on the HDD- index. ? present anoptimal design of weather derivatives in an illiquid framework, arguing that thestandard risk neutral point of view is not applicable to valuate them. ? and? apply an extended version of Lucas’ (1978) equilibrium pricing model wheredirect estimation of weather risk’s market price is avoided. Instead, pricing isbased on the stochastic processes of the weather index, an aggregated dividendand an assumption about the utility function of a representative investor. ? usedthe world stock index as the numeraire to price temperature derivatives. ? and(2007) propose the continuous time autoregressive model with seasonality for thetemperature evolution in time and fit this model to data observed in Stockholm,Sweden. They derive future and option prices for contracts on CDD and CATindices. They also discuss hedging strategies for the options and volatility termstructure.

2

Currently weather derivatives are priced using detrended historical data, ?. Theimportant issues here are the methods of detrending, the choice of models forresiduals and model validation. The temperature dynamics are modelled withautoregressive processes (continuous or discrete) with lag p and seasonal variation.The seasonality is observed watching at the autocorrelation function (ACF) forthe (squared) residuals.

In this paper, we applied continous autoregressive models (CAR) to model thetemperature in Berlin and with that to get explicite nature of non-arbitrage pricesfor temperature derivatives, as ? did for Stockholm Temperature data. In fact,his model works also for Berlin Temperature data. Our paper is structured asfollows. In the next section we discuss fundamentals of WD, weather dynamicsexplained by a continous autoregressive model (CAR), definitions of temperaturederivatives (future and options) and their indices, and the obtained explicit pricesof WD. Section 3 is devoted to an application to Berlin temperature data. Section4 presents the explicit prices of WD for Berlin and its comparison to ChicagoMercantile Exchange (CME) data prices. In section 5, we propose a benchmarkmodel to calibrate the market price of risk for temperature derivatives of Berlintraded at the Chicago Mercantile Exchange (CME). In section 6, similary toBerlin, the assumptions of weather and prices dynamics hold for other Germancities. Section 7 concludes.

2 Weather dynamics for Berlin temperature Data

In this section, we compute the weather dynamics for Berlin daily temperaturedata by means of the model used in ?. The temperature data considers daily av-erage temperatures from 1950/1/1-2006/7/24 recorded at the Tempelhof AirportStation. We worked with 20,645 recordings after removing 29 February’s, thiswith the end of working with 360 days per year.

We check first the presence of a linear trend and investigate the seasonal patternof the data. The upper panel in Figure ?? shows 57 years of daily averagetemperatures from Berlin and the fitted seasonal function with trend

Λt = a0 + a1t+ a2 sin

(2πt

365− a3

)where a0 = 91.51, a1 = 0.00, a2 = 97.95, a3 = −180.34 are significant at 10%and the mean squared error (RMSE) is equal to 38.20. The estimation is madeby means of the least squares minimization. The lower panel in Figure ??, itis more clear to observe the fitness of the seasonal function to the daily averagetemperatures. We observe low temperatures in the winter and high temperaturesin the summer.

3

After removing the seasonality from the daily average temperatures,

Xt = Tt − Λt (1)

we check whether the process Xi is a stationary process I(0). In order to do that,we apply the Augmented Dickey-Fuller test (ADF):

[(1− L)x = c1 + c2trend + τLy + α1(1− L)Lx+ . . . αp(1− L)Lpx+ u]

where the test statistic for a unit root in a time series τ = −39.81, with 1%critical value equal to -2.56. Therefore, we reject the null hypothesis H0 (τ = 0)and hence Yi is a stationary process I(0).

Figure 1: Upper Panel: Berlin daily temperature data from 1950/1/1-2006/7/24, weatherstation: Tempelhof Airport Station. The line in red color denotes the daily average temper-atures and the blue line is the fitted seasonal function. Lower Panel: Temperature in Berlinduring 1990-2000, weather station: Tempelhof Airport Station. The line in red color denotesthe daily average temperatures and the blue line is the fitted seasonal function

We plot the Partial Autocorrelation Function (PACF) in Figure ?? to study thetime behaviour of the residuals of Yt. Due to its periodic behaviour, temperature

4

reverts back to its mean over time. This is well explained by the PACF, whichsuggests that the AR(3) model suggested by ? also holds for Berlin Temperaturedata. In this case,

Xi+3 = β1Xi+2 + β2Xi+1 + β3Xi + σiεi

with β1 = 0.91, β2 = −0.20, β3 = 0.07, σ2i = 510.63, the AIC estimator is equal to

6.23 and the BIC estimator equals to 6.23.

Figure 2: Partial autocorrelation function (PACF)

The residuals and squared residuals of the Berlin Temperature data, after trendand seasonal component were removed, are plotted in Figure ??. We reject at1% significance level the null hypothesis H0 that the residuals are uncorrelated.

The Autocorrelation function (ACF) of the residuals of AR(3), upper panel inFigure ??, are close to zero and according to Box-Ljung statistic the first fewlags are insignificant. But, the ACF for the squared residuals in the lower panelFigure ?? shows a high seasonality pattern. We calibrate this seasonal dependenceof variance of residuals of the AR(3) for 57 years with a truncanted Fouriwefunction

σ2t = c+

4∑i=1

ci sin

(2iπt

365

)+

4∑j=1

dj cos

(2jπt

365

)

Figure ?? shows the daily empirical variance and the fitted squared volatilityfunction for the residuals. Here again, we also get similar results to the studyconducted by ? to Stockholm temperature data, high variance in winter - earliersummer and low variance in spring - late summer.

After dividing out the seasonal volatility from the regression residuals, we ob-served closed to normal residuals (zero mean uncorrelated noise). Figure ??

5

Figure 3: Residuals (upper panel) and Squared residuals (lower panel) of the AR(3) process.

6

Figure 4: ACF for residuals (upper panel) and squared residuals (lower panel) of the AR(3)process.

Figure 5: Seasonal variance: daily empirical variance (blue line), fitted squared volatilityfunction (red line) at 10% significance level

7

Figure 6: ACF for residuals (upper panel) and squared residuals (lower panel) after removingseasonal volatility.

8

Lag Qstatres QSIGres Qstatres1 QSIGres1

1 0.03 0.85 0.67 0.412 0.05 0.97 0.74 0.693 3.16 0.36 4.88 0.184 4.70 0.32 6.26 0.185 4.76 0.44 6.67 0.246 5.40 0.49 7.17 0.307 6.54 0.47 7.51 0.378 10.30 0.24 10.34 0.249 14.44 0.10 14.65 0.1010 21.58 0.01 21.95 0.10

Table 1: Q-test (Qstat) using Ljung-Box’s and the corresponding significance levels (QSIG)for residuals with (res) and without seasonality in the variance (res1)

shows that the ACF plot of the residuals remain unchanged and now the ACFplot for squared residuals presents a non-seasonal pattern.

We used the Ljung-Box’s test statistic (Qstat) to check the significance level ofthe lags of the ACF of residuals with and without seasonal volatility. Table ??presents the statistics and the corresponding significance levels of the lags.

We plot a histogram of the residuals together with a normal distribution in Fig-ure ?? to verify if residuals become normal distributed. Under the t-test withp-value equal to 0.96, we accept the null hyptohesis H0 of normality and we rejectit, with p=0.0028, under the Kolmogorov-Smirnov test. The obtained residualshave a skewness equal to -0.0765 and a kurtosis equal to 3.5527.

Figure 7: Pdf for residuals (black line) and a normal pdf (red line).

In order to get explicit derivations of prices dynamics for all different temperaturecontracts available at the CME, ? analyze a stochastic model for the temperaturewith continuous dynamics. To get that, they showed (by further substituingiteratively in the discrete-time dynamics) that a AR(p) time series in continous

9

time can be written as a continuous time autoregressive model stochastic processCAR(p) with seasonal variance.

For this inconvenience, they use a X1t-CAR (p) to model the temperature dy-namics:

Tt = Λt +X1t (2)

where Xq is q’th coordinate of vector X with q = 1, .., p and Λt is a deterministicseasonal function that represents the average temperature. They noticed thatthe temperature residuals possess a positive, continous and bounded seasonalvariance σt and stationarity holds when the variance matrix∫ t

o

σ2( t− s) exp(As)epe

>p exp(A>s )ds (3)

converges as t→∞.

For this convenience, they define a matrix p× p-matrix:

A =

0 1 0 . . . 0 00 0 1 . . . 0 0...

. . . 0...

0 . . . . . . 0 0 1−αp −αp−1 . . . 0 −α1

in the vectorial Ornstein-Uhlenbleck process Xt ∈ Rp for p ≥ 1 as:

dXt = AXtdt+ eptσtdBt (4)

where ek denotes the k’th unit vector in Rp for k = 1, ...p, σt > 0 states thetemperature volatility, Bt is a Wiener Process and αk are positive constants.

By applying the multidimensional Ito Formula, the stochastic process Xt hasthe explicit form

Xs = exp (As−t)x +

∫ s

t

exp (As−u)epσudBu (5)

for s ≥ t ≥ 0 and Xt = x ∈ Rp.

For the discrete version of the CAR(p) process, they iterate the finite differenceapproximations of the time dynamics, when p = 1, 2, 3. Using εt = Bt+1−Bt, werepeat the exercise:

for p = 1, we get that Xt = X1t and dX1t = −α1X1tdt+ σtdBt.for p = 2, we have that

X1(t+2) ≈ (2− α1)X1(t+1) + (α1 − α2 − 1)X1(t) + σt(Bt−1 −Bt)

10

AR(3) CAR(3)β1 0.91 α1 2.09β2 -0.20 α2 1.38β3 0.07 α3 0.22

Table 2: Coefficients

for p = 3, the A matrix is defined as

A =

0 1 00 0 1

−α3 −α2 −α1

andX1(t+1) −X1(t) = X1(t)dt+ σtεtX2(t+1) −X2(t) = X3(t)dt+ σtεtX3(t+1) −X3(t) = −α3X1(t)dt− α2X2(t)dt− α1X3(t)dt+ σtεtX1(t+2) −X1(t+1) = X1(t+1)dt+ σt+1εt+1

X2(t+2) −X2(t+1) = X3(t+1)dt+ σt+1εt+1

X3(t+2) −X3(t+1) = −α3X1(t+1)dt− α2X2(t+1)dt− α1X3(t+1)dt+ σt+1εt+1

X1(t+3) −X1(t+2) = X1(t+2)dt+ σt+2εt+2

X2(t+3) −X2(t+2) = X3(t+2)dt+ σt+2εt+2

X3(t+3) −X3(t+2) = −α3X1(t+2)dt− α2X2(t+2)dt− α1X3(t+2)dt+ σt+2εt+2

substituing iteratively in X1 dynamics, we get that:

X1(t+3) ≈ (3− α1)X1(t+2) + (2α1 − α2 − 3)X1(t+1) + (−α1 + α2 − α3 + 1)X1(t)

+ σt(Bt−1 −Bt) (6)

which coefficients are equal to the ones obtained for the CAR process in ?.

Once working in continous time,

From equation ??, the coefficients of the CAR(3) can be estimated. Table ??summarizes the AR(3) and CAR(3) estimated coefficients for 57 years of BerlinTemperature. Also the stationarity condition for the CAR(3) (equation ??) isfulfilled, since the eigenvalues of the matrix A have negative real parts (λ1 =−0.2317, λ2,3 = −0.9291± 0.2934i).

3 Weather Derivatives

In the 1990’s WD were developed to hedge against volatility caused by weather.WD are financial contracts, which payments are based on weather-related mea-surements. They are formally exchanged in the Chicago Mercantile Exchange(CME), where monthly and seasonal temperature future, call and put options

11

contracts on future prices are traded. The futures and options at CME are cashsettled. WDs cover against extreme changes on temperature, rainfall, wind, snow,frost, but do not cover catastrophic events, such as earthquakes or hurricanes.

4 Temperature Indices

Temperature derivatives are written on a temperature index. The most commonweather indices on temperature are: Heating Degree Day (HDD), Cooling De-gree Day (CDD), Cumulative Averages (CAT), Average of Average Temperature(AAT) and Event Indices (EI), ?.

The HDD measures the temperature over a period [τ1, τ2], usually between Oc-tober to April, and it is defined as:

HDD(τ1, τ2) =

∫ τ2

τ1

max(K − Tu, 0)du (7)

where K is the baseline temperature (typically 18C or 65F) and Tu is the averagetemperature on day u. Similarly, the CDD measures the temperature over aperiod [τ1, τ2], usually between November and March, and it is defined as:

CDD(τ1, τ2) =

∫ τ2

τ1

max(Tu −K, 0)du (8)

The HDD and the CDD index are used to trade future and options in 18 UScities and 9 european cities. The CAT index accounts the accumulated averagetemperature over a period [τ1, τ2] days:

CAT (τ1, τ2) =

∫ τ2

τ1

Tudu (9)

Since max(Tu − k, 0)−max(K − Tu, 0) = Tu − k, we get the HDD-CDD parity

CDD(τ1, τ2)−HDD(τ1, τ2) = CAT (τ1, τ2)−K (10)

Therefore, it is sufficient to analyse only CDD and CAT indices. The AATmeasures the ”excess” or deficit of temperature i.e. the average of average tem-peratures over [τ1, τ2] days:

AAT (τ1, τ2) =1

τ1 − τ2

∫ τ2

τ1

Tudu (11)

This index is just the average of the CAT and it is relevant for the Pacific Rimconsisted of two Japanese cities. The EI considers the number of times a certainmeteorological event occurs in the contract period. For example, a frost day isconsidered when the temperature at 7:00-10:00 local time is less than or equal to-3.5C. To illustrate this, Table ?? shows the expected number of HDDs, CDDs,CATs and AATs.

12

Indices Jan Feb March April May Jun Jul Aug Sept Oct Nov DecCDD 0 0 0 0 28.3 42 71 23.3 24.9 0 0 0HDD 472.8 526.4 471.4 241.1 150.2 71.8 24.8 43.9 73.5 199.5 398.2 525.8CAT 103.2 -4.4 104.6 316.9 454.1 528.2 622.2 555.4 509.4 376.5 159.8 50.2AAT 3.32 -0.15 3.37 10.56 14.64 17.60 20.07 17.91 16.98 12.14 5.32 1.61

Table 3: Degree day incides for temperature data (2005) Berlin.

5 Temperature Derivatives

In this section, the continuous weather dynamics will allow for explicit derivationsof Future/Option price dynamics for all different temperature contracts availableat the CME.

Basic principles of Finance tell us that the arbitrage free price of a derivative isgiven by the present expected payoff from the derivative under the risk neutralprobability measure Q. The problem with WD is that weather cannot be tradedand therefore no-arbitrage models to weather derivatives are impractical, sinceone cannot replicate a future contract with the undelying index. In order toderive the future price dynamics and to have a class of pricing measures with themartingale property, ? introduce a parametrized class of equivalent probabilitiesQ via the Girsanov transformation

Bθt = Bt −

∫ t

0

θudu (12)

where θt is a real valued, bounded and piecewise continous function (market priceof risk) on [0, τmax]. By Girsanov theorem, there exist an equivalent probabilitymeasure Qθ so that Bθ

t is a Brownian montion for t ∈ [0, τmax]. Under Qθ,equation (??) becomes:

dXt = (AXt + epσtθt)dt+ epσtdBθt (13)

with explicit dynamics

Xs = exp (As−t)x +

∫ s

t

exp (As−u)epσuθudu

+

∫ s

t

exp (As−u)epσudBθu (14)

for s ≥ t ≥ 0.

When an owner of a future contract enters in a contract, at time t, he agrees topay the price Ft,τ1,τ2 at the end of the period τ2, instead of receiving the payofffrom the CAT/HDD/CDD future defined in equations (??, ??, ??). Then theFt-adapted arbitrage free price of this future under the Q risk neutral probabilityis equal to:

0 = exp {−r(τ2 − t)}EQ[Y − F(t,τ1,τ2)|Ft

](15)

13

where r is the constant risk-free interest rate. Under the Qθ pricing probability,the price of the future is equal to:

F(t,τ1,τ2) = EQθ [Y |Ft] (16)

5.0.1 CAT Futures/Option

Following equation (??), the risk neutral price of a future based on a CAT indexis defined as:

FCAT (t,τ1,τ2) = EQθ[∫ τ2

τ1

Tsds|Ft]

(17)

? calculate the future price explicitly by inserting the temperature model (equa-tion ??) into equation ??:

FCAT (t,τ1,τ2) =

∫ τ2

τ1

Λudu+ at,τ1,τ2Xt +

∫ τ2

τ1

θuσuat,τ1,τ2epdu

+

∫ τ2

τ1

θuσue>1 A−1 {exp (Aτ2−u)− Ip} epdu (18)

with at,τ1,τ2 = e>1 A−1 {exp(Aτ2−t)− exp(Aτ1−t)} and the p × p identity matrix

(Ip). Then the Qθ-dynamics of FCAT (t,τ1,τ2) is

dFCAT (t,τ1,τ2) = σtat,τ1,τ2epdBθt

Similary ?, calculate the value of a CAT call option, with strike K at exercisetime T ≤ τ1 and written on a CAT future during the period [τ1, τ2] is:

CCAT (t,T,τ1,τ2) = exp {−r(T − t)} ×{

(FCAT (t,τ1,τ2) −K)Φ(d (t, T, τ1, τ2))

+

∫ T

t

Σ2CAT (s,τ1,τ2)dsΦ

′(d (t, T, τ1, τ2))

}(19)

where

d (t, T, τ1, τ2) =FCAT (t,τ1,τ2)−K√∫ Tt

Σ2CAT (s,τ1,τ2)ds

andΣ2CAT (s,τ1,τ2) = σtat,τ1,τ2ep

and Φ denotes the standard normal cdf.

To replicate the call option with CAT-futures, one should compute the numberof CAT-futures held in the portfolio, which is simply computed by the option’sdelta:

Φ(d (t, T, τ1, τ2)) =∂CCAT (t,T,τ1,τ2)

∂FCAT (t,τ1,τ2)

(20)

14

5.0.2 CDD Futures/Options

Analogously, ? derived explicit CDD future price dynamics. Following equa-tion (??), the risk neutral price of a CDD future is defined as:

FCDD(t,τ1,τ2) = EQθ[∫ τ2

τ1

max(Tu −K, 0)du|Ft]

(21)

and by inserting the temperature model (equation ??) into equation (??), wehave

FCDD(t,τ1,τ2) =

∫ τ2

τ1

υt,sψ

[m(t,s,e>1 exp (As−t)Xt)

υt,s

]ds (22)

wherem(t,s,x) = Λs−c+∫ sτ1σuθue

>1 exp (As−t)epdu+x, υ2

t,s =∫ stσ2u

{e>1 exp (A(s− t))ep

}2du,

ψ(x) = xΦ(x)+Φ>(x) with x = e>1 exp (A(s− t))Xt and Φ is the standard normalcdf.

From the martingale property and Ito′s formula, the time dynamics of the CDD-futures prices are defined by

dFCDD(t,τ1,τ2) = σt

∫ τ2

τ1

{e>1 exp(As−t)ep

}×Φ

[m(t,s,e>1 exp (As−t)Xt)

vt,s

]dsdBθ

t (23)

For the call option written CDD-future, ? found no analytical solution, but anexpression suitable for Monte Carlo simulation. The value of a call option, withstrike K at exercise time T ≤ τ1 and written on a CDD future during the periodt ≤ T is given as

CCDD(t,T,τ1,τ2) = exp {−r(T − t)}E

[max

(∫ τ2

τ1

υT,s

ψ

(m(T,s,e>1 exp (As−t)x+

∫ Tt e>1 exp (As−u)epσuθudu+Σs,t,TY )

1

υT,s

)ds−K, 0

)]x=Xt

(24)

where Y is a standard normal variable and Σ2s,t,T =

∫ Tt

{e>1 exp (As−u)ep

}2σ2udu.

Moreover, after numerous calculation, the hedging strategy HCDD(t,τ1,τ2), witht ≤ T that replicates de CDD-call with CDD-futures is

HCDD(t,τ1,τ2) =σt

Σ2CDD(s,τ1,τ2)

E

[1

(∫ τ2

τ1

υT,sψ

(m(T,s,Z)

υT,s

)ds > K

)×

∫ τ2

τ1

e>1 exp (As−t)epΦ

(m(T,s,Z)

υT,s

)ds

]x=Xt

(25)

15

where Z is a normal distributed with mean e>1 exp (As−t)x+∫ Tt

e>1 exp (As−u)epσ2udu,

and variance∫ Ttσ2u

(e>1 exp (As−u)ep

)2du, and

Σ2CDD(s,τ1,τ2) = σtat,τ1,τ2epΦ

[m{t, s, e>1 exp (As−t)Xt

}υt,s

]

which is similar to Σ2CAT (s,τ1,τ2) except by the last term.

6 A market price risk model

After computing explicit non-arbitrage prices for temperature derivatives, wepropose a benchmark model to calibrate the market risk of price, i.e. the rightprice among possible arbitrage free prices. We do that by calibrating the marketdata (CME data) and thereby pin down the price.

First let us considere the price for the CAT future, which can be written θu is areal valued piecewise linear function:

θ(u) =

{θ1, u ∈ (u1, u2)θ2, u ∈ (u1, u2)

}

7 Conclusion

We apply a higher order continuous-time autoregressive models CAR(3) withseasonal variance for modelling temperature in Berlin for more than 57 years ofdaily observations.

This paper also analyze the weather options/future products for Berlin tradedat the Chicago Mercantile Exchange (CME), written on different temperaturesindices like heating (HDD), cooling degree days (CDD) and cumulative averagetemperature (CAT) measured over different time periods. We computed futureprices for different contracts available at the CME were computed, after calcu-lating the weather dynamics from daily temperature data in Berlin and usingmethods from mathematical finance.

Similar results hold for other German cities.

16

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