The In uence of Weather Forecasts on the Pricing of ...sfb649.wiwi.hu-berlin.de/fedc/events/Motzen10/Presentation_Ritter.pdf · Weather Derivatives Temperature Indices Pricing of

IntroductionMethods

ResultsConclusion

The Influence of Weather Forecasts on the Pricingof Weather Derivatives

Matthias Ritter Martin Odening Oliver Mußhoff

Georg-August-Universitat GottingenHumboldt-Universitat zu Berlin

5 June 2010

Matthias Ritter 5 June 2010 1/37

IntroductionMethods

ResultsConclusion

Weather DerivativesTemperature IndicesPricing of Weather Derivatives

Contents

1 IntroductionWeather DerivativesTemperature IndicesPricing of Weather Derivatives

2 Methods

3 Results

4 Conclusion


IntroductionMethods

ResultsConclusion


Weather Derivatives

Weather strongly affects the economy.

Agriculture

Beverage industry

Energy sector

”About 80% of the global economyis directly or indirectly affected by weather irregularities.”(Josef Auer, Deutsche Bank Research, 2003)

”Everybodytalks about the weather, but nobody does anything about it.”(Mark Twain, 1835-1910)

⇒ Weather derivatives


IntroductionMethods

ResultsConclusion


Weather derivatives

Weather index

Weather station

Tick size

Contract period

Strike

Cap

Premium


IntroductionMethods

ResultsConclusion


Example: Host at Oktoberfest 2002

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

50000

100000

150000

200000

250000

Rainy days

EU

R

PayoffPremium

Figure: Payoff of a weather derivative on rainy days in Munich


IntroductionMethods

ResultsConclusion


Temperature Indices

Daily average temperature (DAT)

The daily average temperature Tt is defined as the mean of theminimal and the maximal temperature on a day t, t ∈ N.

J F M A M J J A S O N D−15

−10

−5

0

5

10

15

20

25

2009

°C

DAT Berlin 2009

Figure: Daily average temperature in 2009 in Berlin-Tempelhof(Data: Deutscher Wetterdienst, www.dwd.de)


www.dwd.de

IntroductionMethods

ResultsConclusion


Heating degree days (HDD)

The (cumulated) heating degree days over a period[τ1, τ2], τ1, τ2 ∈ N, τ1 ≤ τ2, with threshold K (usually 18 ◦C/65 ◦F) aredefined as

HDD(τ1, τ2) =τ2∑

t=τ1

max(0,K − Tt).

Cooling degree days (CDD)

The (cumulated) cooling degree days over a period[τ1, τ2], τ1, τ2 ∈ N, τ1 ≤ τ2, with threshold K (usually 18 ◦C/65 ◦F) aredefined as

CDD(τ1, τ2) =τ2∑

t=τ1

max(0,Tt − K ).


IntroductionMethods

ResultsConclusion


J F M A M J J A S O N D0

5

10

15

20

25

30

35

40

2009

°C

HDDs Berlin 2009


5

10

15

20

25

30

35

40

2009

°C

CDDs Berlin 2009

Figure: Daily HDDs and CDDs in 2009 in Berlin-Tempelhof


IntroductionMethods

ResultsConclusion


Pricing of Weather Derivatives

The arbitrage free price at time t of a derivative with payoff YM(Tt) attime M > t is given by

F(t;τ1,τ2) = EQθ[YM(Tt)|Ft ]

with an equivalent measure Qθ, the market price of risk θ, the contractperiod [τ1, τ2] and Ft containing the historical temperature valuesavailable at time t.

Much effort has been put on the calculation of this market price of risk,Hardle and Lopez Cabrera (2009) describe it as a deterministic timedependent function.


IntroductionMethods

ResultsConclusion


Problem(Meyer-Brandis 2010)

Ft contains only historical temperature values, but more information isusually available on the market.

⇒ include weather forecasts in temperature model

Usually:

Ft = FNFt = {T0, . . . ,Tt−1}

Now:

FFkt = Ft ∪ {Tt+0, . . . , Tt+k}

with Tt+k being the temperature forecast k days in advance, k ∈ N0.


IntroductionMethods

ResultsConclusion

Temperature ModelTemperature ForecastsExtended Model

Contents

1 Introduction

2 MethodsTemperature ModelTemperature ForecastsExtended Model

3 Results

4 Conclusion


IntroductionMethods

ResultsConclusion


Temperature Model

Temperature as an AR-process

The daily average temperature Tt on day t, t ∈ N, consists of:

trend (linear),

seasonality (Fourier series),

autoregression (AR(L)-process),

stochastic component (seasonal variance).


IntroductionMethods

ResultsConclusion


Temperature as an AR-process(Benth et. al. 2007, Hardle/Lopez Cabrera 2009)

The daily average temperature Tt on day t, t ∈ N, is given by:

Tt = a + bt︸︷︷︸trend

+P∑

p=1

[ap cos

(2πpt

365

)+ bp sin

(2πpt

365

)]︸︷︷︸

seasonality

+L∑

l=1

ρt−lTt−l︸︷︷︸autoregression

+ σtεt︸︷︷︸stochastic

,

σ2t =

Q∑q=1

[cq cos

(2πqt

365

)+ dq sin

(2πqt

365

)], εt ∼ N (0, 1).


IntroductionMethods

ResultsConclusion


Temperature data

CME: Daily temperature data for New York since 01/01/97

Trend and seasonality (P = 3)

97 98 99 00 01 02 03 04 05 06 07 08 090

10

20

30

40

50

60

70

80

90

100

° F

ahre

nhei

t

Daily average temperatureEstimated seasonality

Figure: Historical DAT and estimated seasonality in New York, 01/01/97–31/12/09


IntroductionMethods

ResultsConclusion


Autoregression (L = 3)

0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Par

tial A

utoc

orre

latio

ns

Sample Partial Autocorrelation Function

Figure: Partial correlation function of residuals without trend and seasonality


IntroductionMethods

ResultsConclusion


Seasonal variance (Q = 1)


10

20

30

40

50

60

70

80

90

100

° F

ahre

nhei

t

Empirical daily varianceFitted daily variance

Figure: Empirical and fitted daily variance


IntroductionMethods

ResultsConclusion


Normal distribution (µ = −0.00014, σ = 0.99947)

−5 −4 −3 −2 −1 0 1 2 3 40

50

100

150

200

250

Figure: Histogram of the residuals without trend, seasonality, autoregression andseasonal variance


IntroductionMethods

ResultsConclusion


Temperature Forecasts

Forecast data

WeatherOnline: Daily forecast data from 29/12/2008 to 12/02/2010(from 0 to 13 days in advance)

Calculation Date Forecast Date Tmin in ◦C Tmax in ◦C01 January 2010 Friday, 01 January 2010 1 6

Saturday, 02 January 2010 -2 0Sunday, 03 January 2010 -5 -2

Monday, 04 January 2010 -3 3Tuesday, 05 January 2010 -2 1

Wednesday, 06 January 2010 -2 2Thursday, 07 January 2010 -3 1

Friday, 08 January 2010 -3 -2Saturday, 09 January 2010 -7 -5

Sunday, 10 January 2010 -7 -4Monday, 11 January 2010 -7 -2Tuesday, 12 January 2010 -4 0

Wednesday, 13 January 2010 -3 1Thursday, 14 January 2010 -3 0

Figure: Forecast data for 1 January 2010 in New York


IntroductionMethods

ResultsConclusion


0 1 2 3 4 5 6 7 8 9 10 11 12 130

10

20

30

40

50

60

° F

ahre

nhei

t

Days in advance

Forecasts

Figure: Quadratic difference of the forecasts compared with the observed temperaturein New York in 2009


IntroductionMethods

ResultsConclusion


0 1 2 3 4 5 6 7 8 9 10 11 12 130

10

20

30

40

50

60

° F

ahre

nhei

t

Days in advance

Simulated temperatures NFForecasts

Figure: Quadratic difference of the forecasts and the temperature model comparedwith the observed temperature in New York in 2009


IntroductionMethods

ResultsConclusion


Extended Model

Procedure

1 Extension of the time series with forecast data (1–14 values)

2 Daily estimation of the parameters of the temperature model

3 Simulation of the temperature in the contract period and theexpected payoff

Monte Carlo simulation with 10 000 repetitionssimplifying assumptions: θ = 0, r = 0

4 Comparison with observed market prices


IntroductionMethods

ResultsConclusion


Temperature model fitting(AR-model)

Historical temperature data[t0, ... , t-1]

Temperature forecast[t, ... , t+k], k[0,13]

Temperature simulation[t+k+1, ... , 1, ... ,2]

Payoff simulation[1, ... ,2]

Monte Carlo simulation

Figure: Model for temperature simulations on day t


IntroductionMethods

ResultsConclusion


Example

10−Dec−09 20−Dec−09 01−Jan−10 10−Jan−10 20−Jan−100

10

20

30

40

50

60°

F

Historical Temperature

Figure: Example for temperature simulation in New York on 1 January 2010


IntroductionMethods

ResultsConclusion


Example


10

20

30

40

50

60°

F

Historical TemperatureSimulated Temperature NF



IntroductionMethods

ResultsConclusion


Example


10

20

30

40

50

60°

F

Historical TemperatureSimulated Temperature NFForecasted Temperature



IntroductionMethods

ResultsConclusion


Example


10

20

30

40

50

60°

F

Historical TemperatureSimulated Temperature NFForecasted TemperatureSimulated Temperature F13



IntroductionMethods

ResultsConclusion


Market price data

CME market prices for weather futures from Bloomberg: Daily price andtraded volume on trading days

Futures Trading days Traded volume (days)HDD Feb09 38/247 4225/4809 (22/26)HDD Mar09 61/217 5496/6216 (26/30)CDD Apr09 82/143 0 (0)CDD May09 102/249 4550 (12)CDD Jun09 124/185 3635 (23)CDD Jul09 145/206 2100 (20)CDD Aug09 166/228 5878 (20)CDD Sep09 187/249 4575 (15)HDD Okt09 66/68 1895 (14)HDD Nov09 172/177 2555 (14)HDD Dec09 185/189 2646/3211 (21/23)HDD Jan10 205/209 2829 (14)

Figure: New York contracts used in this study overlapping with forecast data period


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Contents

1 Introduction

2 Methods

3 ResultsNew YorkUSAEurope

4 Conclusion


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

New York

A M J J A S O N D J F0

200

400

600

800

1000

1200

Inde

x po

ints

Price at CME (lin. interpolated)Simulated price F13Simulated price NFHDD−Index January 2010

Figure: Observed and simulated prices of an HDD contract for January 2010 in NewYork


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

New York

A M J J A S O N D J F0

200

400

600

800

1000

1200

Inde

x po

ints

Price at CME (lin. interpolated)Simulated price F13Simulated price NFHDD−Index January 2010

D J F900

920

940

960

980

1000

1020

1040

1060

1080

1100

Inde

x po

ints

Figure: Observed and simulated prices of an HDD contract for January 2010 in NewYork

&%'$


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

How to measure the goodness of the prediction?

Root mean squared prediction error:

RMSPE(FFk ,F ) =

√√√√ 1

N·

N∑i=1

(FFkti− Fti

)2

ModelRMSPE NF F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13Feb09 28.8 28.2 29.4 30.5 28.1 27.3 26.1 28.9 28.0 27.1 26.1 24.0 20.3 17.9 18.8Mar09 22.9 22.6 22.3 21.8 19.8 20.5 20.1 21.0 20.2 19.7 20.9 20.6 22.3 25.0 25.2Apr09 6.8 6.8 6.7 6.4 6.2 6.4 6.7 6.7 6.8 7.0 7.2 7.3 7.5 7.8 7.7May09 21.2 21.3 21.2 21.0 21.0 20.7 20.5 20.5 20.6 20.3 20.4 20.5 20.5 20.7 20.8Jun09 17.2 16.8 15.9 15.4 15.0 14.4 14.2 14.6 14.8 15.5 16.4 17.3 18.4 19.2 19.9Jul09 27.0 26.0 25.1 24.0 23.1 22.4 21.4 21.1 20.2 20.2 19.7 19.5 18.8 18.8 18.7Aug09 20.2 20.6 20.5 20.6 20.3 20.0 19.9 19.6 18.2 17.5 16.7 16.1 16.0 16.2 15.5Sep09 18.5 17.8 17.5 17.3 16.9 16.7 16.6 16.6 16.6 16.6 16.2 16.3 16.2 16.3 16.4Okt09 24.1 23.3 22.4 21.9 21.3 21.6 19.5 17.4 14.9 13.5 9.9 9.6 8.3 7.6 8.0Nov09 26.8 26.8 26.7 26.4 25.9 25.5 25.3 24.9 24.8 25.0 25.6 25.7 26.2 26.7 27.1Dec09 38.3 37.6 36.4 35.3 34.4 34.3 34.1 33.1 32.5 32.3 32.1 31.7 31.7 31.8 31.7Jan10 38.0 37.7 37.4 36.8 35.9 35.2 34.8 34.4 33.6 33.1 32.2 31.7 31.3 30.8 30.5Mean 24.1 23.8 23.4 23.1 22.3 22.1 21.6 21.6 20.9 20.7 20.3 20.0 19.8 19.9 20.0

Figure: RMSPE for New York contracts for different models


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

NF F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F130.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPE

Figure: Average nRMSPE


IntroductionMethods

ResultsConclusion

New YorkUSAEurope


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPEnormalized RMSPE (2 months)

Figure: Average nRMSPE (whole period, last 2 months)


IntroductionMethods

ResultsConclusion

New YorkUSAEurope


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPEnormalized RMSPE (2 months)normalized RMSPE (VOL>0)

Figure: Average nRMSPE (whole period, last 2 months, days with volume > 0)


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Correlation coefficient

CC(FFk ,F ) =Cov(FFk ,F )√

Var(FFk ) ·√

Var(F )

ModelCC NF F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13Feb09 0.41 0.39 0.31 0.25 0.31 0.31 0.32 0.05 0.13 0.24 0.32 0.41 0.56 0.62 0.60Mar09 0.53 0.58 0.60 0.62 0.68 0.56 0.52 0.48 0.54 0.62 0.56 0.58 0.57 0.55 0.62Apr09 0.56 0.56 0.58 0.61 0.64 0.63 0.62 0.61 0.59 0.58 0.56 0.54 0.52 0.51 0.53May09 0.65 0.63 0.63 0.67 0.65 0.65 0.68 0.69 0.67 0.69 0.68 0.68 0.68 0.67 0.67Jun09 0.94 0.95 0.96 0.96 0.97 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.96 0.96 0.96Jul09 0.88 0.90 0.91 0.92 0.93 0.94 0.95 0.95 0.95 0.95 0.96 0.96 0.97 0.97 0.97Aug09 0.26 0.20 0.22 0.20 0.20 0.24 0.25 0.29 0.47 0.52 0.55 0.58 0.57 0.56 0.61Sep09 0.71 0.75 0.77 0.78 0.79 0.80 0.81 0.80 0.80 0.80 0.82 0.83 0.83 0.84 0.84Oct09 0.62 0.64 0.67 0.69 0.72 0.72 0.77 0.81 0.86 0.88 0.94 0.94 0.96 0.96 0.96Nov09 0.85 0.84 0.85 0.87 0.88 0.89 0.90 0.92 0.92 0.92 0.90 0.91 0.90 0.89 0.89Dec09 0.63 0.68 0.74 0.79 0.82 0.81 0.82 0.85 0.85 0.86 0.84 0.84 0.83 0.81 0.81Jan10 0.40 0.41 0.42 0.47 0.54 0.59 0.63 0.67 0.70 0.74 0.80 0.84 0.87 0.89 0.90Mean 0.62 0.63 0.64 0.65 0.68 0.68 0.69 0.67 0.71 0.73 0.74 0.76 0.77 0.77 0.78

Figure: Correlation coefficient for New York contracts for different models


IntroductionMethods

ResultsConclusion

New YorkUSAEurope


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPECorrelation Coefficient

Figure: Average nRMSPE and correlation coefficient


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Better Forecast

”Prediction is very difficult, especially about the future.”(Niels Bohr, 1885–1962)

Would we get better results with better forecasts?

Best forecast = perfect forecast: FF13perfectt = Ft ∪ {Tt , · · · ,Tt+13}

(= Ft+14

)Model

nRMSPE NF F13 F13perfectFeb09 1.00 0.66 1.16Mar09 1.00 1.10 1.08Apr09 1.00 1.13 0.86May09 1.00 0.98 0.99Jun09 1.00 1.16 1.25Jul09 1.00 0.69 0.63Aug09 1.00 0.77 0.96Sep09 1.00 0.89 0.93Okt09 1.00 0.33 0.47Nov09 1.00 1.01 0.94Dec09 1.00 0.83 0.85Jan10 1.00 0.80 0.83Mean 1.00 0.86 0.91

Figure: Normalized RMSPE for New York contracts for different models


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

USA

Volume New York Atlanta Cincinnati Houston Kansas City Minneapolis Portland SacramentoFeb09 HDD 4809 1161 608 739 2542 752 532 328Mar09 HDD 6216 828 1139 4690 3057 651 304 950Apr09 CDD 0 0 0 0 0 0 0 0

May09 CDD 4550 150 1450 200 100 100 200 175Jun09 CDD 3635 0 425 1485 490 1400 550 125Jul09 CDD 2100 0 515 1510 1340 950 550 350

Aug09 CDD 5878 50 325 750 750 2400 500 400Sep09 CDD 4575 100 470 50 400 1500 200 700Oct09 HDD 1895 100 350 0 500 150 0 0Nov09 HDD 2555 275 375 100 300 825 275 0Dec09 HDD 3211 425 875 800 200 2262 50 0Jan10 HDD 2829 1030 250 0 50 200 50 0Sum 42253 4119 6782 10324 9729 11190 3211 3028

Figure: Volume traded at CME for different contracts


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Minneapolis


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPE

Figure: Average nRMSPE


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Minneapolis


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPEnormalized RMSPE (2 months)

Figure: Average nRMSPE (whole period, last 2 months)


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Minneapolis


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPEnormalized RMSPE (2 months)normalized RMSPE (VOL>0)

Figure: Average nRMSPE (whole period, last 2 months, days with volume > 0)


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Minneapolis


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Model

normalized RMSPEnormalized RMSPE (2 months)normalized RMSPE (VOL>0)Correlation coefficient

Figure: Average nRMSPE and correlation coefficient


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Europe

Volume Amsterdam Berlin Essen London Madrid ParisFeb09 HDD 0 0 0 1430 0 1250Mar09 HDD 0 200 0 13800 0 0Apr09 CAT 0 0 0 0 0 0

May09 CAT 0 0 0 200 0 0Jun09 CAT 0 0 0 0 0 0Jul09 CAT 0 0 0 250 0 0

Aug09 CAT 0 0 100 50 0 0Sep09 CAT 0 0 0 0 0 0Oct09 HDD 0 0 50 1270 0 50Nov09 HDD 0 0 0 1650 0 0Dec09 HDD 0 100 0 3250 0 0Jan10 HDD 0 0 0 250 0 0Sum 0 300 150 22150 0 1300

Figure: Volume traded at CME for different contracts


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Berlin

Apr May Jun Jul Aug Sep Oct Nov Dec Jan0

100

200

300

400

500

600

Inde

x po

ints

Price at CME (lin. interpolated)HDD−Index December 2009

Figure: Observed prices of an HDD contract for December 2009 in Berlin


IntroductionMethods

ResultsConclusion

New YorkUSAEurope

Berlin

Apr May Jun Jul Aug Sep Oct Nov Dec Jan0

100

200

300

400

500

600

Inde

x po

ints

Price at CME (lin. interpolated)HDD−Index December 2009

Dec Jan400

420

440

460

480

500

520

540

560

580

600

Inde

x po

ints

Figure: Observed prices of an HDD contract for December 2009 in Berlin

��


IntroductionMethods

ResultsConclusion

Contents

1 Introduction

2 Methods

3 Results

4 Conclusion


IntroductionMethods

ResultsConclusion

Conclusion

Weather forecasts influence the pricing of weather derivatives.

Prediction error decreases.

Correlation increases.

Better results if concentrating on trading period.

Further work

Combining with market price of risk.

Factor models (Meyer-Brandis 2010).


IntroductionMethods

ResultsConclusion

Literature

Fred Espen Benth, Jurate Saltyte-Benth, and Steen Koekebakker.

Putting a price on temperature.

Scandinavian Journal of Statistics, 34(4):746–767, December 2007.

Wolfgang Hardle and Brenda Lopez Cabrera.

Implied market price of weather risk.

SFB 649 Discussion Paper 2009-001, January 2009.

Thilo Meyer-Brandis.

Consistent factor models for temperature markets.

Workshop Weather Derivatives and Risk, C.A.S.E.,Humboldt-Universitat zu Berlin, January 2010.


Documents

The In uence of Weather Forecasts on the Pricing of ...sfb649.wiwi.hu-berlin.de/fedc/events/Motzen10/Presentation_Ritter.pdf · Weather Derivatives Temperature Indices Pricing of