Introduction to Weather Derivatives - hu- · PDF fileIntroduction to Weather Derivatives ......

Introduction to Weather Derivatives

Brenda Lopez CabreraWolfgang K. Hardle

CASE-Center for Applied Statistics andEconomicsHumboldt-Universitat zu Berlin

1

�

Motivation 1-1

Weather:

� Influences our daily lives and choices

� Impact on corporate revenues and earnings

� According to Metereological institutions: 80% of the businessactivity in the world is weather dependent

I British Met Office: daily beer consumption increases by 10% iftemperature decreases by 3◦C


Motivation 1-2

� Climate changes related ”El Nino”

� Global climate changes the volatility of weather

� The occurrence of extreme weather events increases

� Emergence of international weather markets new opportunitiesarise to handle these risks

I Weather Derivatives (WD)


Motivation 1-3

Examples

� 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 insea temperatures


Outline 2-1

Outline

1. Motivation X

2. What are Weather Derivatives?

3. Weather Indices

4. Valuation of Weather Derivatives

5. Application


What are Weather Derivatives (WD)? 3-1

What are WD?

� Hedge weather related risk exposures

� WD returns uncorrelated with other financial instruments

� Formal exchanges: Chicago Mercantile Exchange (CME)I Monthly and seasonal temperature future contractsI 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 intrinsicfinancial value



Anatomy of a WD

A weather derivative is defined by several elements:

� Reference Weather Station

� Index

� Term

� Structure: puts, calls, swaps, collars, straddles, and strangles.

� Premium



Example: A beverage company hedge itself against a cold summerwith the following option:

Weather Stations: Munich, Berlin and FrankfurtRisk Period: 01 Jul - 31 AugustWeather Index: Average temperature for the 3 Weather StationsPut Strike: 19◦CTick: EUR 30,000 per 0.01◦CMaximum Payout: EUR 3,000,000Payout if: Weather Index < Put Strike:

Min {(Strike −WeatherIndex) xTick;MaximumPayout}



Weather indices: temperature

Heating degree day (HDD): over a period [τ1, τ2]∫ τ2

τ1

max(T0 − Tu, 0)du (1)

Cooling degree day (CDD): over a period [τ1, τ2]∫ τ2

τ1

max(Tu − T0, 0)du (2)

T0 is some baseline temperature (typically 18◦C or 65◦F), Tu isthe average temperature on day u.



Average of average temperature (PRIM): measure the”excess” or deficit of temperature. The average of averagetemperatures over [τ1, τ2] days is:

1

τ1 − τ2

∫ τ2

τ1

Tudu (3)

Cumulative averages (CAT): The accumulated averagetemperature over [τ1, τ2] days is:∫ τ2

τ1

Tudu (4)

Event indices: number of times a certain meteorological eventoccurs in the contract period

� Frost days: temperature at 7:00-10.00 local time is less thanor equal to −3.5◦C



Weather indices: rain

Cumulative rainfall index I c : sum of daily rainfall amount in acertain accumulation period

I c =x∑

t=1

yt (5)

x : length of the accumulation period

Rainfall deficit index I d : measures the shortfall accumulated overz periods of the rainfall sum in an s-days period relative to areference level ymin

I d =z∑

τ=1

min

0,τ ·s∑

t=(τ−1)·s+1

yt − ymin

(6)


Valuation of WD 4-7

Valuation of WD

The price of a contingent claim F can be calculated as:

F = e−rT E [ψ(I )] (7)

I : stochastic variable that expires at time T , weather indexψ(I ): payoff of the derivative at expirationr : risk free interest rateE [·]: expectation by means of risk neutral probabilities


Valuation of WD 4-8

but...

� weather cannot be traded → no-arbitrage models to WD areimpractical!

� Black and Scholes does not work: volatility changes

� Price of the derivative must account the market price ofweather risk (λ)

� Benth (2004): many arbitrage-free prices exist

� Hull (2006): Weather risk is not a systematic risk, λ = 0


Valuation of WD 4-9

Pricing Weather Derivatives

� 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


Valuation of WD 4-10

Burn analysis

� Evaluates how a contract would have performed in previousyears

� Uses the empirical distribution of the underlying index

� Implementation of the pricing formula

F = e−rT

[1

n

n∑t=1

ψ(It)

](8)

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



Call option price:

F = e−rT

[1

n

n∑t=1

{It − K , 0}+

]

Put option price:

F = e−rT

[1

n

n∑t=1

{K − It , 0}+

]

r : risk free interest rateK : strikeIt : accumulated weather index for t-th yearn: number of years



Index Value simulation

� Utilizes a statistical model for the weather index or thederivative payoff

� Supported by goodness of fit tests

Steps:

1. Given an appropriate distribution, the values of the index arerandomly drawn and the payoffs are determined




Daily simulation

� Describes the dynamics of the weather variable over time

� Yield smaller confidence intervals than Burn and indexsimulation: smaller for rainfall than for temperature

Steps:

1. Derive the weather index from the simulated sample pathI summing up daily precipitation/daily average temperature

2. Determined pay offs




Stochastic Model for temperature

Define the Ornstein-Uhlenbleck process Xt ∈ Rp:

dXt = Atdt +pt σtdBt

ekp: k’th unit vector in Rp

σt : temperature volatilityA: p × p-matrixBt : Wiener Process

A =

(0 1

−αp ... −α1

)Solution of Xt :

Xs = exp (A(s−t))x +

∫ s

texp (A(s − u))epσudBu



Temperature dynamics:

Tt = Λt + X1t

� X1t : continuous-time AR(p) (CAR(p)) model

� Λt : seasonality function

� Temperature will be normally distributed each time

Let p = 1, implying Xt = X1t :

dX1t = −α1X1tdt + σtdBt



Berlin temperature data

Daily average temperatures: 1950/1/1-2006/7/24

� Station: BERLIN-TEMP.(FLUGWEWA)� 29 February removed� 20645 recordings



Seasonal function

Suppose seasonal function with trend:

Λt = a0 + a1t + a2sin

(2π(t)

365− a3

)Estimates: a0 = 86.0543, a1 = 0.0008, a2 = 98.0582, a3 = 174.058Squared 2-norm of the residual at x:

minx{1

2‖Λt − Tt‖2

2} =1

2

m∑t=1

(Λt − Tt)2 = 3.02 + e07



Fitting an autoregressive model

Remove effect of Λt from the data: Yi = Ti − Λi

ADF-Test statistic:

(1−L)y = c1 +c2trend+ τLy +α1(1−L)Ly + . . . αp(1−L)Lpy +u

� ADF-Test Result of returns: τ = −51.53, where the 1%critical value is -2.5659 we can reject H0 that τ = 0 ⇒ Yi is astationary process I (0) ⇒ no unit roots d = 0



Partial autocorrelation function suggest AR(3):Yi+3 = βiYi+2 + β2Yi+1 + β3Yi + σiεiEstimates:β1 = 0.91748, β2 = −0.20122, β3 = 0.077464, σ2

i = 510.63



Residuals

Reject H0 for zero expected innovations: stat= 2.1013.



Seasonal volatility

Close to zero ACF for residuals of AR(3) and highly seasonal ACFfor squared residuals of AR(3)

Figure 1: ACFIntroduction to Weather Derivatives


Figure 2: PACF



Calculate the daily variances of residuals for 56 years and calibrateit to the next truncated Fourier series:

σ2t = c +

4∑i=1

ci sin

(2iπt

365

)+

4∑j=1

djcos

(2jπt

365

)

Figure 3: Seasonal varianceIntroduction to Weather Derivatives


Dividing out the seasonal volatility from the regression residuals:

� ACF for residuals unchanged, residuals become normallydistributed

� ACF for squared residuals non-seasonal

Figure 4: Residual/seasonal variance



Temperature futures

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

FHDD(t,τ1,τ2) = EQ

[∫ τ2

τ1

max(18◦C − Tu, 0)du|Ft

]No trade in settlement periodConstant interest rate rQ is a risk neutral probability

� Not unique since market is incomplete

� Temperature is not tradable

Analogously: CDD and CAT future prices



Risk neutral probabilities

Defined by the Girsanov transformation of Bt : dBθt = dBt − θtdt

θt : time-dependent market price of risk.

Density of Qθ:

Z θt = exp

(∫ t

0θsdBs −

1

2

∫ t

0θ2s ds

)Dynamics Xt under Qθ:

dXt = (AXt + epσtθt)dt + epσtdBθt

with solution:

Xs = exp (A(s−t))x +

∫ s

texp (A(s−u))epσuθudu

+

∫ s

texp (A(s−u))epσudBθ

u (9)Introduction to Weather Derivatives


CDD futures

FCDD(t,τ1,τ2) =

∫ τ2

τ1

v(t,s)ψ

(m(t, s, e

′1 exp(A(s−t))Xt))

v(t,s)

)ds (10)

where m(t, s, x) = Λs − c +∫ τ2

τ1σuθue

′1 exp(A(s−t))epdu + x

v2(t,s) =

∫ st σ

2u

(e

′1 exp(A(s−t))ep

)2du

ψx = xφx + φ′x

φ is the cumulative standard normal distribution fct.

� The future price is a function of the lagged temperaturesTt ,Tt−1, ...,Tt−p



From the martingale property and Ito ′s formula, the timedynamics of the CDD-futures prices:

dFCDD(t,τ1,τ2) = σt

∫ τ2

τ1

(e

′1 exp(A(s−t))ep

)×φ

(m(t, s, e

′1 exp(A(s−t))Xt))

v(t,s)

)dsdBθ

t

Benth (2005): No analytical solution, do Monte Carlo simulation



CAT futures price

FCAT (t,τ1,τ2) =

∫ τ2

τ1

Λudu + a(t,τ1,τ2)Xt

+

∫ τ2

τ1

θuσua(t,τ1,τ2)epdu

+

∫ τ2

τ1

θuσue′1A

−1(exp(A(τ2−u))− Ip)epdu

where Ip : p × p identity matrix anda(t,τ1,τ2) = e

′1A

−1(exp(A(τ2−u))− exp(A(τ1−t))

Time dynamics of FCAT :

dFCAT (t,τ1,τ2) = σta(t,τ1,τ2)epdBθt



Data adjustments

Station changes

� Instrumentation

� Location

Trends: to capture weather surprise

� Global Climate Cycles

� Urban heat

Forecasting


Research 5-31

Questions for Research

� Memory in AR(3): volatility with monthly/weeklymeasurement period

� Role of the strike value

� Longterm (interannual) variability of parameters - capturevolatility due to climate changes

� Relationship between weather variables and production

� Formula for options on CAT/CDD/HDD futures

� WD related to catastrophes


Conclusion 6-1

Conclusion

� CAR(p) model for the temperature dynamicsI Autoregressive process with seasonal mean and seasonal

volatility

� Analytical future prices for the traded contract on CMEI HDD/CDD/CAT/ futures with delivery over months or seasons


Conclusion 6-2

Reference

F.E. BenthOption Theory with Stochastic Analysis: An Introduction toMathematical Finance(Berlin: Springer), 2004.

Benth and S. BenthStochastic Modelling of temperature variations with a viewtowards weather derivatives(Appl.Math. Finance), 2005.

K. BurneckiWeather derivatives(Warsaw), 2004.


Conclusion 6-3

S. Cambell, F. DieboldWeather Forecasting for weather derivativesJ.American Stat. Assoc. , 2006.

J.C. HullOption, Futures and other Derivatives(6th ed. New Jersey: Prentice Hall International), 2006.

M. OdeningAnalysis of Rainfall Derivatives using Daily precipitationmodels: opportunities and pitfalls


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