The Volatility of Temperature and Pricing of Weather ... · I Empirical analysis of Chicago,...

The temperature marketA stochastic model for temperature

Temperature futuresConclusions

The Volatility of Temperature and Pricing ofWeather Derivatives

Fred Espen BenthWork in collaboration with J. Saltyte Benth and S. Koekebakker

Centre of Mathematics for Applications (CMA)University of Oslo, Norway

Universitat Ulm, April 2007



Overview of the presentation

1. The temperature market

2. A stochastic model for daily temperatureI Continuous-time AR(p) modelI with seasonal volatility

3. Temperature futures

I HDD, CDD and CATI Explicit futures pricesI Options on temperature forwards



The temperature market



The temperature market

I Chicago Mercantile Exchange (CME) organizes trade intemperature derivatives:

I Futures contracts on monthly and seasonal temperaturesI European call and put options on these futures

I Contracts on 18 US and 2 Japanese cities

I 9 European cities

I London, Paris, Amsterdam, Rome, Barcelona, MadridI StockholmI Berlin, Essen



HDD and CDD

I HDD (heating-degree days) over a period [τ1, τ2]∫ τ2

τ1

max (18− T (u), 0) du

I HDD is the accumulated degrees when temperature T (u) isbelow 18

I CDD (cooling-degree days) is correspondingly theaccumulated degrees when temperature T (u) is above 18∫ τ2

τ1

max (T (u)− 18, 0) du



CAT and PRIM

I CAT = cumulative average temperatureI Average temperature here meaning the daily average∫ τ2

τ1

T (u) du

I PRIM = Pacific Rim, the average temperature

1

τ2 − τ1

∫ τ2

τ1

T (u) du



At the CME...

I Futures written on HDD, CDD, CAT and PRIM as index

I HDD and CDD is the index for US temperature futuresI CAT index for European temperature futures, along with HDD

and CDDI PRIM only for Japan

I Discrete (daily) measurement of HDD, CDD, CAT and PRIM

I All futures are cash settled

I 1 trade unit=20 Currency (trade unit being HDD, CDD orCAT)

I Currency equal to USD for US futures and GBP for European

I Call and put options written on the different futures



A continuous-time AR(p)-processRelated modelsStockholm temperature data

A stochastic model for temperature




A continuous-time AR(p)-process

I Define the Ornstein-Uhlenbeck process X(t) ∈ Rp

dX(t) = AX(t) dt + ep(t)σ(t) dB(t) ,

I ek : k’th unit vector in Rp

I σ(t): temperature “volatility”

I A: p × p-matrix

A =

[0 I

−αp · · · −α1

]




I Explicit solution of X(t):

X(s) = exp (A(s − t)) x +

∫ s

texp (A(s − u)) epσ(u) dB(u) ,

I Temperature dynamics T (t) defined as

T (t) = Λ(t) + X1(t)

I X1(t) CAR(p) model, Λ(t) seasonality function

I Temperature will be normally distributed at each time




Why is X1 a CAR(p) process?

I Consider p = 3

I Do an Euler approximation of the X(t)-dynamics with timestep 1

I Substitute iteratively in X1(t)-dynamicsI Use B(t + 1)− B(t) = ε(t)

I Resulting discrete-time dynamics

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

+ (1− α3 + α2 − α1)X1(t) + σ(t)ε(t) .




Related models

I Mean-reverting Ornstein-Uhlenbeck process

I Let p = 1, implying X(t) = X1(t)

dX1(t) = −α1X1(t) dt + σ(t) dB(t)

I Benth and Saltyte-Benth (2005,2006)I Seasonal volatility σ(t)I Empirical analysis of Stockholm and cities in NorwayI Pricing of HDD/CDD/CAT temperature futures and optionsI Spatial model for temperature in Lithuania




I Dornier and Querel (2000):

I Constant temperature volatility σI Empirical analysis of Chicago, O’Hare airport temperatures

I Alaton, Djehiche and Stillberger (2002):

I Monthly varying σI Empirical analysis of Stockholm, Bromma airport temperaturesI Pricing of various temperature futures and options

I Brody, Syroka and Zervos (2002):

I Constant σ, but B is fractional Brownian motionI Empirical analysis of London temperaturesI Pricing of various temperature futures and options




I Campbell and Diebold (2005):

I ARMA time series model for the temperatureI Seasonal ARCH-model for the volatility σI Empirical analysis for temperature in US cities

I The model seems to fit data very wellI However, difficult to do analysis with it

I Futures and option pricing by simulation




Stockholm temperature data

I Daily average temperatures from 1 Jan 1961 till 25 May 2006I 29 February removed in every leap yearI 16,570 recordings

I Last 11 years snapshot with seasonal function

0 500 1000 1500 2000 2500 3000 3500 4000−25

−20

−15

−10

−5

0

5

10

15

20

25




I Fitting of model goes stepwise:

1. Fit seasonal function Λ(t) with least squares2. Fit AR(p)-model on deseasonalized temperatures3. Fit seasonal volatility σ(t) to residuals




1. Seasonal function

I Suppose seasonal function with trend

Λ(t) = a0 + a1 t + a2 cos (2π(t − a3)/365)

I Use least squares to fit parametersI May use higher order truncated Fourier series

I Estimates: a0 = 6.4, a1 = 0.0001, a2 = 10.4, a3 = −166I Average temperature increases over sample period by 1.6◦C




2. Fitting an auto-regressive model

I Remove the effect of Λ(t) from the data

Yi := T (i)− Λ(i) , i = 0, 1, . . .

I Claim that AR(3) is a good model for Yi :

Yi+3 = β1Yi+2 + β2Yi+1 + β3Yi + σiεi ,




I The partial autocorrelation functionI Suggests AR(3)

0 20 40 60 80 100−0.2

0

0.2

0.4

0.6

0.8

lag

auto

corre

lation




I Estimation gives β1 = 0.957, β2 = −0.253, β3 = 0.119

I All parameters are significant at the 1% level

I R2 is 94.1%

I Higher-order AR-models did not increase R2 significantly




3. Seasonal volatility

I Consider the residuals from the auto-regressive modelI Autocorrelation function for residuals and their squares

I Close to zero ACF for residualsI Highly seasonal ACF for squared residuals

0 100 200 300 400 500 600 700 800−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

lag

auto

corr

elat

ion

0 100 200 300 400 500 600 700 800−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

lag

auto

corr

elat

ion




I Suppose the volatility is a truncated Fourier series

σ2(t) = c +4∑

i=1

ci sin(2iπt/365) +4∑

j=1

dj cos(2jπt/365)

I This is calibrated to the daily variancesI 45 years of daily residualsI Line up each year next to each otherI Calculate the variance for each day in the year




I A plot of the daily empirical variance with the fitted squaredvolatility function

I High variance in winter, and early summerI Low variance in spring and late summer/autumn

0 50 100 150 200 250 300 3501

2

3

4

5

6

7

8

9

days

season

al varia

nce




I Same observation for other citiesI Several cities in Norway and LithuaniaI Seasonality in ACF for squared residuals observed in Campbell

and Diebold’s paperI Example below from Alta, northern Norway

I Small city on the coast close to North CapeI Hardle and Lopez (2007) confirm same pattern for Berlin




I Dividing out the seasonal volatility from the regressionresiduals

I ACF for squared residuals non-seasonalI ACF for residuals unchangedI Residuals become normally distributed

0 100 200 300 400 500 600 700 800−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

lag

auto

corr

elat

ion

−5 −4 −3 −2 −1 0 1 2 3 40

200

400

600

800

1000

1200

1400



Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures

Temperature futures




Some generalities on temperature futures

I HDD-futures price FHDD(t, τ1, τ2) at time t ≤ τ1

I No trade in settlement period

0 = e−r(τ2−t)EQ

[∫ τ2

τ1

max(c−T (u), 0) du−FHDD(t, τ1, τ2) | Ft

].

I Constant interest rate r , and settlement at the end of indexperiod, τ2

I Q is a risk-neutral probabilityI Not unique since market is incompleteI Temperature (nor HDD) is not tradeable

I c is equal to 65◦F or 18◦C




I Adaptedness of FHDD(t, τ1, τ2) yields

FHDD(t, τ1, τ2) = EQ

[∫ τ2

τ1

max(c − T (u), 0) du | Ft

]I Analogously, the CDD and CAT futures price is

FCDD(t, τ1, τ2) = EQ

[∫ τ2

τ1

max(T (u)− c , 0) du | Ft

]FCAT(t, τ1, τ2) = EQ

[∫ τ2

τ1

T (u) du | Ft

]




A class of risk neutral probabilities

I Parametric sub-class of risk-neutral probabilities Qθ

I Defined by Girsanov transformation of B(t)

dBθ(t) = dB(t)− θ(t) dt

I θ(t) time-dependent market price of risk

I Density of Qθ

Z θ(t) = exp(∫ t

0θ(s) dB(s)− 1

2

∫ t

0θ2(s)

)ds

)




I Change of dynamics of X(t) under Qθ:

dX(t) = (AX(t) + epσ(t)θ(t)) dt + epσ(t) dBθ(t) .

I or, explicitly

X(s) = exp (A(s − t)) x +

∫ s

texp (A(s − u)) epσ(u)θ(u) du

+

∫ s

texp (A(s − u)) epσ(u) dBθ(u) ,

I Feasible dynamics for explicit calculations




CDD futures

I CDD-futures price

FCDD(t, τ1, τ2) =

∫ τ2

τ1

v(t, s)Ψ

(m(t, s, e′1 exp(A(s − t))X(t))

v(t, s)

)ds

where

m(t, s, x) = Λ(s)− c +

∫ s

tσ(u)θ(u)e′1 exp(A(s − u))ep du + x

v2(t, s) =

∫ s

tσ2(u)

(e′1 exp(A(s − u))ep

)2du

I Ψ(x) = xΦ(x) + Φ′(x), Φ being the cumulative standardnormal distribution function.




I The futures price is dependent on X(t)

I ... and not only on current temperature T (t)

I In discrete-time, the futures price is a function of

I The lagged temperatures T (t),T (t − 1), . . . ,T (t − p)




I Time-dynamics of the CDD-futures price

dFCDD(t, τ1, τ2) = σ(t)

∫ τ2

τ1

e′1 exp(A(s − t))ep

× Φ

(m(t, s, e′1 exp(A(s − t))X(t)

v(t, s)

)ds dBθ(t)

I Follows from the martingale property and Ito’s Formula




Pricing options on CDD-futures

I CDD-futures dynamics does not allow for analytical optionpricing

I Monte Carlo simulation of FCDD is the natural approach

I Simulate X(τ) at exercise time τI Derive FCDD by numerical integration

I Alternatively, simulate the dynamics of the CDD-futures




CAT futures

I CAT-futures price

FCAT(t, τ1, τ2) =

∫ τ2

τ1

Λ(u) du + a(t, τ1, τ2)X(t)

+

∫ τ1

tθ(u)σ(u)a(t, τ1, τ2)ep du

+

∫ τ2

τ1

θ(u)σ(u)e′1A−1 (exp (A(τ2 − u))− Ip) ep du

with Ip being the p × p identity matrix and

a(t, τ1, τ2) = e′1A−1 (exp (A(τ2 − t))− exp (A(τ1 − t)))




I Time-dynamics of FCAT

dFCAT(t, τ1, τ2) = ΣCAT(t, τ1, τ2) dBθ(t)

where

ΣCAT(t, τ1, τ2) = σ(t)e′1A−1 (exp (A(τ2 − t))− exp (A(τ1 − t))) ep

I ΣCAT is the CAT volatility term structure




I Seasonal volatility, with maturity effectI Plot of the volatility term structure as a function of t up till

deliveryI Monthly contractsI Parameters taken from Stockholm for CAR(3)

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec0

1

2

3

4

5

6

7

8

9

measurement month

CAT

volat

ility

CAT volatility term structure




I The Samuelson effectI The volatility is decreasing with time to deliveryI Typical in mean-reverting markets

I AR(3) has memoryI Implies a modification of this effectI Plot shows volatility of CAT with monthly vs. weekly

measurement period

140 142 144 146 148 1501

2

3

4

5

6

7

Trading days prior to measurement period

CAT

volat

ility




I Call option prices easily calculated (strike K , exercise time τ)

C (t) = e−r(τ−t) ×{

(FCAT(t, τ1, τ2)− K ) Φ(d)

+

∫ τ

tΣ2

CAT(s, τ1, τ2) ds Φ′(d)}

where

d =FCAT(t, τ1, τ2)− K√∫ τ

t Σ2CAT(s, τ1, τ2) ds

I Φ cumulative normal distribution



Conclusions

I CAR(p) model for the temperature dynamicsI Auto-regressive process, withI Seasonal meanI seasonal volatility

I Allows for analytical futures prices for the traded contracts onCME

I HDD/CDD, CAT and PRIM futuresI Futures contracts with delivery over months or seasonsI Seasonal volatility with a modified Samuelson effect

I Options on HDD-futures require numerical pricing

I Black-formula for options on CAT-futures



Coordinates

I [email protected]

I www.math.uio.no/˜fredb

I www.cma.uio.no



ReferencesAlaton, Djehiche, and Stillberger. On modelling and pricing weather derivatives. Appl. Math. Finance, 9,2002

Benth and Saltyte-Benth. Stochastic modelling of temperature variations with a view towards weatherderivatives. Appl. Math. Finance, 12, 2005

Benth, Saltyte-Benth and Jalinskas. A spatial-temporal model for temperature with seasonal variance.Preprint 2005. To appear in J. Appl. Statist.

Benth and Saltyte-Benth. The volatility of temperature and pricing of weather derivatives. Preprint 2005.To appear in Quantit. Finance

Benth, Saltyte Benth and Koekebakker. Putting a price on temperature. Preprint 2006.

Brody, Syroka and Zervos. Dynamical pricing of weather derivatives. Quantit. Finance, 3, 2002

Campbell and Diebold. Weather forecasting for weather derivatives. J. Amer. Stat. Assoc., 100, 2005

Dornier and Querel. Caution to the wind. Energy Power Risk Manag., August, 2000

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