The temperature market A stochastic model for temperature Temperature futures Conclusions The Volatility of Temperature and Pricing of Weather Derivatives Fred Espen Benth Work in collaboration with J. Saltyte Benth and S. Koekebakker Centre of Mathematics for Applications (CMA) University of Oslo, Norway Universit¨ at Ulm, April 2007
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The Volatility of Temperature and Pricing of Weather ... · I Empirical analysis of Chicago, O’Hare airport temperatures I Alaton, Djehiche and Stillberger (2002): I Monthly varying
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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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
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 marketA stochastic model for temperature
Temperature futuresConclusions
The temperature market
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
A stochastic model for temperature
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
]
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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)
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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 ,
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
I Higher-order AR-models did not increase R2 significantly
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
A continuous-time AR(p)-processRelated modelsStockholm temperature data
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
Temperature futures
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT 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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
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
]
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
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
)
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
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.
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
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)
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures
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
The temperature marketA stochastic model for temperature
Temperature futuresConclusions
Some generalitiesRisk neutral probabilitiesCDD futuresCAT futures