CASE-Center for Applied Statistics and
Economics
1

Motivation
1-1
Weather:
Influences our daily lives and choices
Impact on corporate revenues and earnings
According to Metereological institutions: 80% of the business activity in the world is weather dependent
I British Met Office: daily beer consumption increases by 10% if temperature decreases by 3
◦
C
Introduction to Weather Derivatives

Motivation
1-2
Global climate changes the volatility of weather
The occurrence of extreme weather events increases
Emergence of international weather markets new opportunities arise to handle these risks
I Weather Derivatives (WD)
Introduction to Weather Derivatives

Motivation
1-3
Natural gas company suffer negative impact in mild winter
Construction companies can buy WD during rain period
Cloth retailers sell fewer clothes in hot summer
Salmon fishery suffer losses with a single degree of warming in sea temperatures
Introduction to Weather Derivatives

Outline
1.
Motivation X
2.
What are Weather Derivatives?
3.
Weather Indices
4.
Valuation of Weather Derivatives
5.
Application
Introduction to Weather Derivatives
2-1

What are Weather Derivatives (WD)?
3-1
Hedge weather related risk exposures
WD returns uncorrelated with other financial instruments
Formal exchanges: Chicago Mercantile Exchange (CME)
I
I
Monthly and seasonal temperature future contracts
European call and put options on these futures
Common underlying weather variables: temperature, rainfall, wind, snow, frost
I No physical markets in weather: the underlying has no intrinsic financial value
Introduction to Weather Derivatives

What are Weather Derivatives (WD)?
3-2
A weather derivative is defined by several elements:
Reference Weather Station
Index
Term
Structure: puts, calls, swaps, collars, straddles, and strangles.
Premium
Introduction to Weather Derivatives

What are Weather Derivatives (WD)?
3-3
Example: A beverage company hedge itself against a cold summer with the following option:
Weather Stations: Munich, Berlin and Frankfurt
Risk Period: 01 Jul - 31 August
Weather Index: Average temperature for the 3 Weather Stations
Put Strike: 19
◦
C
Tick: EUR 30,000 per 0.01
◦
C
Maximum Payout: EUR 3,000,000
Payout if: Weather Index < Put Strike:
Min { ( Strike − WeatherIndex ) xTick ; MaximumPayout }
Introduction to Weather Derivatives

What are Weather Derivatives (WD)?
3-4
Heating degree day (HDD): over a period [ τ
1
, τ
2
]
Z
τ
2 max( T
0
− T u
, 0) du
τ
1
(1)
Cooling degree day (CDD): over a period [ τ
1
, τ
2
]
Z
τ
2 max( T u
− T
0
, 0) du
τ
1
T
0 is some baseline temperature (typically 18
◦
C or 65
◦
F), T u is the average temperature on day u .
(2)
Introduction to Weather Derivatives

What are Weather Derivatives (WD)?
Average of average temperature (PRIM): measure the
” excess ” or deficit of temperature. The average of average temperatures over [ τ
1
, τ
2
] days is:
1
τ
1
− τ
2
Z
τ
2
T u du
τ
1
3-5
(3)
Cumulative averages (CAT): The accumulated average temperature over [ τ
1
, τ
2
] days is:
Z
τ
2
T u du
τ
1
(4)
Event indices: number of times a certain meteorological event occurs in the contract period
Frost days: temperature at 7:00-10.00 local time is less than or equal to − 3 .
5
◦
C
Introduction to Weather Derivatives

What are Weather Derivatives (WD)?
3-6
Cumulative rainfall index I c
: sum of daily rainfall amount in a certain accumulation period
I c
= x
X y t t =1
(5) x : length of the accumulation period
Rainfall deficit index I d
: measures the shortfall accumulated over z periods of the rainfall sum in an s -days period relative to a reference level y min
I d
=
 z
X min

0 ,
τ =1
τ · s
X t =( τ − 1) · s +1
 y t
− y min

(6)
Introduction to Weather Derivatives

Valuation of WD
4-7
The price of a contingent claim F can be calculated as:
F = e
− rT
E [ ψ ( I )]
I : stochastic variable that expires at time T , weather index
ψ ( I ): payoff of the derivative at expiration r : risk free interest rate
E [ · ]: expectation by means of risk neutral probabilities
(7)
Introduction to Weather Derivatives

Valuation of WD
4-8 but...
weather cannot be traded → no-arbitrage models to WD are impractical!
Black and Scholes does not work: volatility changes
Price of the derivative must account the market price of weather risk ( λ )
Benth (2004): many arbitrage-free prices exist
Hull (2006): Weather risk is not a systematic risk, λ = 0
Introduction to Weather Derivatives

Valuation of WD
Temperature derivatives: Campbell and Diebold (2005),
Jewson and Brix (2005), Benth (2005)
Rainfall: Cao, Li and Wei (2004)
Approaches:
Burn analysis
Index Value Simulation
Daily simulation
Stochastic Pricing Model
Introduction to Weather Derivatives
4-9

Valuation of WD
4-10
Evaluates how a contract would have performed in previous years
Uses the empirical distribution of the underlying index
Implementation of the pricing formula
F = e
− rT
"
1 n n
X
ψ ( I t
)
# t =1
Steps:
1.
Historical weather data are cleaned and detrended
2.
Determine Index value - hypothetical payoffs for each year
3.
Calculate payoff average and discounted with r
(8)
Introduction to Weather Derivatives

Valuation of WD
Call option price:
F = e
− rT
"
1 n n
X
{ I t
− K , 0 }
+
# t =1
Put option price:
F = e
− rT
"
1 n n
X
{ K − I t
, 0 }
+
# t =1 r : risk free interest rate
K : strike
I t
: accumulated weather index for t -th year n : number of years
Introduction to Weather Derivatives
4-11

Valuation of WD
4-12
Utilizes a statistical model for the weather index or the derivative payoff
Supported by goodness of fit tests
Steps:
1.
Given an appropriate distribution, the values of the index are randomly drawn and the payoffs are determined
2.
Calculate payoff average and discounted with r
Introduction to Weather Derivatives

Valuation of WD
Describes the dynamics of the weather variable over time
Yield smaller confidence intervals than Burn and index simulation: smaller for rainfall than for temperature
Steps:
1.
Derive the weather index from the simulated sample path
I summing up daily precipitation/daily average temperature
2.
Determined pay offs
3.
Calculate payoff average and discounted with r
Introduction to Weather Derivatives
4-13

Valuation of WD
Define the Ornstein-Uhlenbleck process X t
∈
R p
: dX t
= A t dt + pt
σ t dB t e kp
: k’th unit vector in
R p
σ t
: temperature volatility
A: p × p -matrix
B t
: Wiener Process
A =
0
− α p
1
...
− α
1
Solution of X t
:
X s
= exp ( A
( s − t )) x +
Z s t exp ( A ( s − u )) e p
σ u dB u
Introduction to Weather Derivatives
4-14

Valuation of WD
Temperature dynamics:
T t
= Λ t
+ X
1 t
X
1 t
: continuous-time AR(p) (CAR(p)) model
Λ t
: seasonality function
Temperature will be normally distributed each time
Let p = 1, implying X t
= X
1 t
: dX
1 t
= − α
1
X
1 t dt + σ t dB t
4-15
Introduction to Weather Derivatives

Valuation of WD
Daily average temperatures: 1950/1/1-2006/7/24
Station: BERLIN-TEMP.(FLUGWEWA)
29 February removed
20645 recordings
4-16
Introduction to Weather Derivatives

Valuation of WD
4-17
Suppose seasonal function with trend:
Λ t
= a
0
+ a
1 t + a
2 sin
2 π ( t )
365
− a
3
Estimates: a
0
= 86 .
0543 , a
1
= 0 .
0008 , a
2
= 98 .
0582 , a
3
= 174 .
058
Squared 2-norm of the residual at x: min x
{
1
2 k Λ t
− T t k
2
2
} =
1
2 m
X
(Λ t
− T t
)
2
= 3 .
02 + e 07 t =1
Introduction to Weather Derivatives

Valuation of WD
4-18
Remove effect of Λ t from the data: Y i
= T i
− Λ i
ADF-Test statistic:
(1 − L ) y = c
1
+ c
2 trend + τ Ly + α
1
(1 − L ) Ly + . . . α p
(1 − L ) L p y + u
ADF-Test Result of returns: τ = − 51 .
53, where the 1% critical value is -2.5659 we can reject H
0 that τ = 0 ⇒ Y i is a stationary process I (0) ⇒ no unit roots d = 0
Introduction to Weather Derivatives

Valuation of WD
Partial autocorrelation function suggest AR(3):
Y i +3
= β i
Estimates:
Y i +2
+ β
2
Y i +1
β
1
= 0 .
91748 , β
2
+ β
3
Y i
= − 0 .
20122 , β
3
+ σ i
ε i
= 0 .
077464 , σ i
2
= 510 .
63
4-19
Introduction to Weather Derivatives

Valuation of WD
Reject H
0 for zero expected innovations: stat= 2.1013.
4-20
Introduction to Weather Derivatives

Valuation of WD
4-21
Close to zero ACF for residuals of AR(3) and highly seasonal ACF for squared residuals of AR(3)
Figure 1:
ACF
Introduction to Weather Derivatives

Valuation of WD
Figure 2: PACF
Introduction to Weather Derivatives
4-22

Valuation of WD
4-23
Calculate the daily variances of residuals for 56 years and calibrate it to the next truncated Fourier series:
σ
2 t
= c +
4
X c i sin i =1
2 i π
365 t
+
4
X d j cos j =1
2 j π
365 t
Figure 3: Seasonal variance

Valuation of WD
4-24
Dividing out the seasonal volatility from the regression residuals:
ACF for residuals unchanged, residuals become normally distributed
ACF for squared residuals non-seasonal
Figure 4: Residual/seasonal variance
Introduction to Weather Derivatives

Valuation of WD
HDD-futures price F
HDD ( t ,τ
1
,τ
2
) at time t ≤ τ
1
:
F
HDD ( t ,τ
1
,τ
2
)
= E
Q
Z
τ
2 max (18
◦
C − T u
, 0) du |F t
τ
1
No trade in settlement period
Constant interest rate r
Q is a risk neutral probability
Not unique since market is incomplete
Temperature is not tradable
Analogously: CDD and CAT future prices
Introduction to Weather Derivatives
4-25

Valuation of WD
4-26
Defined by the Girsanov transformation of B t
: dB t
θ
θ t
: time-dependent market price of risk.
= dB t
− θ t dt
Density of Q
θ
:
Z t
θ
= exp
Z t
0
θ s dB s
−
1
2
Z t
θ
2 s ds
0
Dynamics X t under Q
θ
: dX t
= ( AX t
+ e p
σ t
θ t
) dt + e p
σ t dB
θ t with solution:
X s
Z s
= exp ( A
( s − t )
) x +
+
exp ( A
( s − u )
) e p
σ u
θ u du t
Z s
exp ( A
( s
− u )
) e p
σ u dB
θ u
(9)

Valuation of WD
4-27
F
CDD ( t ,τ
1
,τ
2
)
=
Z
τ
2
τ
1 v
( t , s )
ψ m ( t , s , e
1
0 exp( A
( s − t )
) X t
))
!
ds (10) v
( t , s ) where m ( t , s , x ) = Λ v
ψ
2
( t , s ) x
=
= x φ
R s t x
σ
2 u
+ φ
0 x e
1
0 s exp(
−
A c +
( s − t )
R
τ
2
) e
τ
1 p
σ u
θ u e
1
0
2 du exp( A
( s − t )
) e
φ is the cumulative standard normal distribution fct.
p du + x
The future price is a function of the lagged temperatures
T t
, T t − 1
, ..., T t − p
Introduction to Weather Derivatives

Valuation of WD
4-28
From the martingale property and It ˆ
0 s formula, the time dynamics of the CDD-futures prices: dF
CDD ( t ,τ
1
,τ
2
)
Z
τ
2
= σ t e
1
0 exp( A
( s − t )
) e p
τ
1
× φ m ( t , s , e
1
0 exp( A
( s − t )
) X t
))
!
v
( t , s ) dsdB t
θ
Benth (2005): No analytical solution, do Monte Carlo simulation
Introduction to Weather Derivatives

Valuation of WD
F
CAT ( t ,τ
1
,τ
2
)
=
Z
τ
2
+
Λ u du + a
( t ,τ
1
τ
1
Z
τ
2
,τ
2
)
X t
θ u
σ u a
( t ,τ
1
,τ
2
) e p du
τ
1
Z
τ
2
+ θ u
σ u e
0
1
A
− 1
(exp( A
( τ
2
− u )
) − I p
) e p du
τ
1 where I p a
( t ,τ
1
,τ
2
)
: p × p identity matrix and
= e
1
0
A
− 1
(exp( A
( τ
2
− u )
) − exp( A
( τ
1
− t )
)
Time dynamics of F
CAT
: dF
CAT ( t ,τ
1
,τ
2
)
= σ t a
( t ,τ
1
,τ
2
) e p dB
θ t
Introduction to Weather Derivatives
4-29

Valuation of WD
Station changes
Instrumentation
Location
Trends: to capture weather surprise
Global Climate Cycles
Urban heat
Forecasting
Introduction to Weather Derivatives
4-30

Research
Memory in AR(3): volatility with monthly/weekly measurement period
Role of the strike value
Longterm (interannual) variability of parameters - capture volatility due to climate changes
Relationship between weather variables and production
Formula for options on CAT/CDD/HDD futures
WD related to catastrophes
Introduction to Weather Derivatives
5-31

Conclusion
6-1
CAR(p) model for the temperature dynamics
I Autoregressive process with seasonal mean and seasonal volatility
Analytical future prices for the traded contract on CME
I HDD/CDD/CAT/ futures with delivery over months or seasons
Introduction to Weather Derivatives

Conclusion
F.E. Benth
Option Theory with Stochastic Analysis: An Introduction to
Mathematical Finance
(Berlin: Springer), 2004.
Benth and S. Benth
Stochastic Modelling of temperature variations with a view towards weather derivatives
(Appl.Math. Finance), 2005.
K. Burnecki
Weather derivatives
(Warsaw), 2004.
Introduction to Weather Derivatives
6-2

Conclusion
S. Cambell, F. Diebold
Weather Forecasting for weather derivatives
J.American Stat. Assoc. , 2006.
J.C. Hull
Option, Futures and other Derivatives
(6th ed. New Jersey: Prentice Hall International), 2006.
M. Odening
Analysis of Rainfall Derivatives using Daily precipitation models: opportunities and pitfalls
6-3
Introduction to Weather Derivatives