weather derivative pricing and risk management applications · much less common than...
TRANSCRIPT
Weather Derivative Pricing and Risk Management Applications
Jon Tindall
1. Introduction2. Pricing Principles3. Temperature Derivatives4. Rainfall Derivatives
Overview
1. Introduction
First formal recorded transaction in 1996 – Enron and Energy-Koch .
• HDD swap – Milwaukee, winter 1997
• De-regulation of energy industries – mainly in US and Europe.
• Initially used as a hedge against variability in electricity supply.
US Department of Commerce estimates that weather adversely affects:
• 70% of all US companies;
• 22% of total GDP.
Introduction
� Mature Over-the-Counter (OTC) market:
• Existed since early 1990’s.
• Specifically ‘tailored’ products.
• Large European banks and insurance brokers.
� Chicago Mercantile Exchange (CME):
• operates electronic exchange for weather derivatives.
• futures and option contracts over US and Canadian cities.
• 55% of total global turnover in 2005.
� L.I.F.F.E – Closed in 2004
• series of contracts based on daily average temperatures in London, Paris and Berlin.
Weather Markets
Market Size
* Source: PwC 2005 Market Survey
$0
$1,000
$2,000
$3,000
$4,000
$5,000
$6,000
$7,000
$8,000
$9,000
2001 2002 2003 2004 2005 OTC Summer OTC Winter CME Summer CME Winter
Market size 2001 - 2005
�Stagnant after Enron collapse.
�2005 shows strong growth may be returning.
Contract Types:• Futures - CME, OTC.
• Options – Majority of transactions to date.
• Swaps - increasing in popularity.
Underlying Variables:• Temperature
• Rainfall
• Wind Speed
• Snow Fall
• Barometric Pressure
Contracts
Contract Types
* Source: PwC 2005 Market Survey
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
2001 2002 2003 2004 2005
Other Temp. HDD CDD Rain Snow Wind Other
Break-up of Transaction Types
�Large increases in rainfall contracts.
�CDD now equal with HDD contracts.
� Average daily temperature
� The most popular derivative contracts are over Heating Degree Days (HDD) and Cooling Degree Days (CDD).
• Where the reference level, T, is usually 18º.
• Heating is generally required below the reference temperature and cooling above it.
• Cumulative number of degrees the average temperature was below the reference level
Temperature Derivatives
)}(,0max{ i
month
TTHDD −= ∑
)}(,0max{∑ −=month
i TTCDD
2
minmax TTT i
+=
Much less common than temperature-based derivatives.
� Market was born out of temperature exposure.
� ‘Discreetness’ of Rainfall
• Basis risk - greatest barrier to expansion.
• Modelling difficulties.
� Lack of counter-parties – only water supply companies as a possible partner.
Rainfall Derivatives
Energy / Utility Companies
� Sales are highly correlated with temperatures.
� Definition of HDD and CDD contracts is based on an energy companies exposure.
• Heating/cooling reference level (18º)
• Cumulative based underlying variable.
� Enron
• Oil and gas pipeline manager.
• Used weather derivatives to reduce exposure to weather.
• Soon became a ‘market-maker’ on CME and others.
Risk Management Applications
Hedging Temperature
Construction
� Temperature:
• Concrete curing (setting) is temperature dependent.
• Productivity reduces at unusually high and low temperatures.
• ‘Stop work’ laws.
� Rainfall:
• Precipitation delays can often represent 10% of contract.
• Subsidence.
� Other exposures:
• Snow fall.
• Wind speed – cranes and other heavy equipment.
Risk Management Applications
Some key differences:
� Identifiable Loss: There is no need to prove that a loss has occurred. Reduces costs – claims assessors, lawyers etc.
� Moral Risk: Nearly entirely removed – referenced to a transparent index
� Minimal Underwriting: Only counter-party risk requires investigation.
� Immediate Payout – known magnitude.
� Basis risk
Weather Derivatives vs. Insurance
2. Pricing Principles
Traditional Black-Scholes assumptions:
• A traded underlying asset that can be used to create a hedge, i.e. sold short.
• Log-normal distribution.
Other methods must be found for the pricing of these contracts:
• Alternative BS framework.
• Martingale approach.
• Numerical simulation.
Pricing
• Weather variables do not rise or fall without bound
• Mean reversion strength depends on several factors – most significantly latitude.
Mean-reversion component:
Ornstein-Uhlenbeck (OU) process:
Modified OU process:
Mean Reversion
).( XXdt
dXt
t −−= γ
ttt dWdtXXdX .).( σγ +−=
t
t
t dWdtdt
XdXXdX .)( σγ +
+−=
� Futures Price:
� Process s.d.e:
� Modified Black-Scholes p.d.e:
� Solution:
Alternative Black-Scholes
)(. tTr
tt eXY−=
2
222
2
1
dy
VdyrV
dt
dV t σ−=
),0,,(.
),,,(),(
σ
στ
τ
tyBSe
rtyeBStyV
r
r
−
−
=
=
])[( tt dWdtrydY σµ +−=
‘Burn’ Analysis:
• No assumptions needed re: the process dynamics;
• No parameters to be estimated;
• Agreement on price.
Monte Carlo Simulations:
• Model dependant;
• Data intensive.
Numerical Methods
)),((.1
)]([1
i
N
i
t tXfN
Xf ψ∑=
=E
3. Temperature Modellingand Derivative Pricing
Australian Bureau of Meteorology (BOM)
� Sydney Airport. (Jan 1940 – Dec 2005)
� Observatory Hill. (Jan 1940 – Dec 2005)
� Prospect Dam. (Jan 1965 – Dec 2005)
Missing data:
�Temperature – Backup stations.
�Rainfall – much more difficult. No accurate measure due to discreetness of rainfall.
Data
� Bi-modal Distribution
Temperature Distributions
AveT emp - SydAirport
0
200
400
600
800
1000
1200
1400
1600
1800
2000
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Ave. T emp (degrees C)
Co
un
t
AveTemp
Average D aily T em p. - S yd ney Airp ort
0
2 0 0
4 0 0
6 0 0
8 0 0
1 0 0 0
1 2 0 0
1 4 0 0
1 6 0 0
1 8 0 0
6 7 8 9 1 0 11 1 2 13 14 1 5 16 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 29 3 0 31 32 3 3 34 3 5
Av e. T em pe rature (d eg C )
Co
un
t
W inte r M onths (M ay-O ct)
S um m er M o nths (Nov-A p r)
Steps:
• De-trend data;
• Choose functional form for seasonal fluctuations;
• Estimate the parameters, including mean-reversion;
• Simulate the process;
• Analyse residuals.
Modelling Temperature
All temperature data sets revealed a significant positive slope
• Time series over 70 years should de-trend with a quadratic function.
• Natural geological based heating + human induced global warming
Long-term Trends
2.. tctbaTlong ++=
tbaT Long .+=
Fourier series to model seasonal component:
�A first order series is sufficient to capture seasonal pattern.
Combining this with the linear trend we obtain:
Seasonal Trends
∑∑ ++++=i
i
i
iSeasonal tCostSinT )()(.0 θλβφγαεα
)()(.. θλβφγα +++++= tCostSintbaT
Model Fit
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
0 500 1000 1500 2000 2500 3000 3500
Days
Te
mp
. (d
eg
C)
Actual Ave. Temp
Modelled Temp.
Actual vs Expected – Sydney Airport (10 years)
Temperature Volatility
Daily Temperature Volatility – Sydney Airport (10 years)
0
2
4
6
8
10
12
1
10
19
28
37
46
55
64
73
82
91
10
0
10
9
11
8
12
7
13
6
14
5
15
4
16
3
17
2
18
1
19
0
19
9
20
8
21
7
22
6
23
5
24
4
25
3
26
2
27
1
28
0
28
9
29
8
30
7
31
6
32
5
33
4
34
3
35
2
36
1
Day of Year
%
Raw Mod
� Degree-4 polynomial fitted to volatility distribution.
• Parameter estimation – least squares
• Residuals – Sydney Airport.
Parameter estimation
Syd. Airport Observatory Prospect
a 16.925 17.434 17.26 b 6.30*10
-5 5.16*10
-5 4.91*10
-5
α 5.14 4.91 5.194 β 0.69 -0.20 0.986 φ 1.097 1.25 1.100 θ 0.97 1.10 0.675
Residual Distribution
0
500
1000
1500
2000
2500
3000
3500
-5.0
0
-4.6
0
-4.2
0
-3.8
0
-3.4
0
-3.0
0
-2.6
0
-2.2
0
-1.8
0
-1.4
0
-1.0
0
-0.6
0
-0.2
0
0.2
0
0.6
0
1.0
0
1.4
0
1.8
0
2.2
0
2.6
0
3.0
0
3.4
0
3.8
0
4.2
0
4.6
0
5.0
0
Residual
Co
un
t
• CDD option - January
• Pricing via:
1. Normal approximation.
2. ‘Burn’ analysis – 66 years of data.
3. Monte Carlo simulations
Pricing Example
Period: January Measure: Cumulative CDD
Exercise Prices: 170 / 180 / 190 / 200 CDD’s Tick:: $100,000 /CDD Location: Sydney Airport (Kingsford Smith)
• Stochastic form:
• Euler approximation - discrete
Monte Carlo
∫∆−∆− +−+=
t
s
tt
ott dWeeTTTT ττγγ σ.).( 0
Z.).(1 σγ ++−=−+dt
TdTTTT
t
ytt
8.00
10.00
12.00
14.00
16.00
18.00
20.00
22.00
24.00
26.00
28.00
30-S
ep
14-O
ct
28-O
ct
11-N
ov
25-N
ov
9-D
ec
23-D
ec
6-J
an
20-J
an
3-F
eb
17-F
eb
3-M
ar
17-M
ar
31-M
ar
14-A
pr
28-A
pr
12-M
ay
26-M
ay
9-J
un
23-J
un
7-J
ul
21-J
ul
4-A
ug
18-A
ug
1-S
ep
15-S
ep
29-S
ep
de
g C
Simulation
Pricing Example
Pricin g C om p ariso n - Janu ary
$0
$100,000
$200,000
$300,000
$400,000
$500,000
$600,000
170 180 190 200Exe rcise (CDD)
Op
tio
n P
ric
e
'B urn' A nalys is M onte Carlo S im ulation Norm al A pprox im ation
Sydney Airport
January
Method 170 180 190 200
'Burn' Analysis $473,306 $268,763 $137,865 $59,582
Monte Carlo Simulation $489,044 $230,479 $90,848 $13,990
Normal Approximation $463,670 $288,627 $167,993 $91,012
Exercise (CDD)
4. Rainfall Modelling
• Annual
• Monthly
Rainfall Correlations Ye a rly R a in fa l l C o rre la tio n
y = 1 .0088x + 128 .21
R 2 = 0 .8846
0
500
1000
1500
2000
2500
500 700 900 1100 1300 1500 1700 1900 2100
S yd . A irp o rt
Ob
serv
ato
ry
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
Jan Feb Mar Apr May June July Aug Sep Oct Nov Dec
Co
rrela
tio
n
Monthly Rainfall Correlation
Compound Model – size & frequency.
�Frequency: Markov Chain.
�Size: Gamma distribution (4 segments).
Transition probabilities:
Modelling Rainfall
Rain No rain
Rain 0.55 0.45
No rain 0.28 0.72
Actual vs Expected Frequency – Sydney Airport
Frequency Simulation
� Errors are greatest in winter - i.e. errors not proportional.
� Clearly defined seasonal patterns.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
15
29
43
57
71
85
99
11
3
12
7
14
1
15
5
16
9
18
3
19
7
21
1
22
5
23
9
25
3
26
7
28
1
29
5
30
9
32
3
33
7
35
1
36
5
Da y of Year
Ra
in F
req
ue
nc
y
f_rainf_norainPoly. (f_ra in)
Magnitude
Rain t-1 No Yes
Average 3.74 7.20
StdDev 6.21 12.23
Max 79.0 182.1
Min 0.1 0.2
Average 6.43 13.89
StdDev 11.22 21.46
Max 132.6 216.2
Min 0.2 0.2
No
Yes
Rain t+1
Gamma distribution parameter values
NRN RRN NRR RRR
alpha 0.58 0.45 0.47 0.49
beta 19.32 12.09 13.07 6.39
�4 segments conditional on t-1 and t+1.
�Fit 4 Gamma distributions
Gamma Distributions
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Magnitude (m m )
Pro
po
rtio
n
NRN RRNNRR RRR
Actual vs Expected Magnitude – Sydney Airport
Magnitude
� Close Gaussian fit.
� Clearly defined seasonal patterns.
Actual vs Expected: No rain / Rain / No rain
0
50
100
150
200
250
300
350
400
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54
M agnitude (mm)
Fre
qu
en
cy
Actual Model
Actual vs Expected Magnitude – Sydney Airport
Simulation
Feb .
0
1
2
3
4
5
6
7
8
0
10
20
30
40
50
60
70
80
90
10
0
11
0
12
0
13
0
14
0
15
0
16
0
17
0
18
0
19
0
20
0
21
0
22
0
23
0
24
0
25
0
26
0
27
0
28
0
29
0
30
0M odelA c tual
� New Markets:
• Australian market practically non-existent – agricultural based economy.
• Must promote to seek out suitable counter-parties.
• Improve product design – reduce basis risk.
• Centralised data recording methodologies – Europe in particular.
� New Interest:
• Hedge funds – attracted to immature market.
• Diversification tool – minimal correlation to debt and equity markets.
• Weather-based indexed insurance contracts.
Where to from here?