Weather derivative hedging
& Swap illiquidity
Dr. Michael Moreno
Call/Put Hedging
• Diversification or Static hedging
(portfolio oriented)
– PCA
– Markowitz
– SD
• Dynamic hedging (Index hedging)
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Dr. Michael Moreno
2
Dynamic Hedging
1. Temperature Simulation process used
2. Swap hedging and cap effects
3. Greeks neutral hedging
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3
1. Temperature Simulation process used
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4
Part 1 Temperature Simulation process used
Temperature simulation
• GARCH
• ARFIMA
• FBM
Short Memory
Heteroskedasticity
Long Memory
Homoskedasticity
• ARFIMA-FIGARCH
• Bootstrapp
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Heteroskedasticity
& Long Memory
5
Part 1 Temperature Simulation process used
ARFIMA-FIGARCH model
Ti  Si  mi   i yi
Seasonality
Trend
ARFIMA-FIGARCH
Seasonal volatility
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Part 1 Temperature Simulation process used
ARFIMA-FIGARCH definition
We consider first the ARFIMA process:
 L1  Ld  yt      L t
0
Where, as in the ARMA model,  is the unconditional mean
a
of yt while the autoregressive operator
 L   1    j L j
j 1
m
and the moving average operator
 L   1    j L j
j 1
are polynomials of order a and m, respectively, in the lag
operator L, and the innovationst are white noises with the
variance σ2.
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Part 1 Temperature Simulation process used
FIGARCH noise
Given the conditional variance
ht  Var  t t 1 
We suppose that
1   Lht    [1   L]
2
t
 L1  L 
d
2
t
Long term memory
Cf Baillie, Bollerslev and Mikkelsen 96 or Chung 03 for full specification
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Part 1 Temperature Simulation process used
Distributions of London winter HDD
Densities
0.003
Histo
Sim
0.003
0.003
0.002
0.002
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0
1,000
1,200
1,400
1,600
1,800
2,000
Histo
Sim
1700.79
1704.54
128.52
119.26
Skewness
0.42
-0.01
Kurtosis
3.63
3.13
Minimum
1474.39
1375.13
Maximum
2118.64
2118.92
Average
St Dev
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Dr. Michael Moreno
2,200
2,400
With similar detrending methods
The slight differences come mainly
from the year 1963
9
2. Swap hedging and cap effects
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Part 2 Swap hedging and cap effects
Swap Hedging
Long HDD Call and optcall HDD Swap
Dynamic values
Long HDD Put and optput HDD Swap
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Part 2 Swap hedging and cap effects
Deltas of a capped call
Delta of Capped Calls
0.9
0.8
0.7
0.6
Delta 0.5
0.4
0.3
0.2
0.1
1 300
140
130
120
110
1 400
1 500
1 600
1 700
M ean
b
c
d
e
f
g
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Vol
100
1 800
cap 200 g
b
c
d
e
f
1 900
2 000
cap 400 g
b
c
d
e
f
Dr. Michael Moreno
90
2 100
cap 800
12
Part 2 Swap hedging and cap effects
Deltas of capped swaps
Delta of Capped Swaps
1
0.8
0.6
Delta
0.4
140
130
0.2
120
110
1 300
1 400
1 500
100
1 600
1 700
Strike
g
b
c
d
e
f
b
c
d
e
f
g
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Vol
1 800
Delta Sw ap cap 200
b
c
d
e
f
g
Delta of Sw ap cap 800
Dr. Michael Moreno
1 900
2 000
90
Delta of Sw ap cap 400
13
Part 2 Swap hedging and cap effects
Call optimal delta hedge
optcall= call/ swap
NOT = 1
Delta of Capped Call & Swap
Prices of Capped Call & Swap
Fair Values
0.9
150
0.8
100
0.7
50
0.6
Delta
0
0.5
0.4
-50
0.3
-100
0.2
-150
0.1
1 300 1 400 1 500 1 600 1 700 1 800 1 900 2 000 2 100
1 300
1 400
1 500
1 600
b
c
d
e
f
g
sw ap cap 200 g
b
c
d
e
f
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1 700
1 800
1 900
2 000
Mean
Mean
b
c
d
e
f
g
call cap 200
Dr. Michael Moreno
call cap 200
b
c
d
e
f
g
sw ap cap 200
14
2 100
Part 2 Swap hedging and cap effects
Put optimal delta hedge
optput= put/ swap
Prices of Capped Put & Swap
Delta of Capped Put & Swap
0.8
150
0.6
100
Fair Values
NOT = 1
0.4
50
-50
Delta 0.2
0
-100
-0.2
-150
-0.4
0
-0.6
1 300 1 400 1 500 1 600 1 700 1 800 1 900 2 000 2 100
1 300
1 400
1 500
Mean
b
c
d
e
f
g
sw ap cap 200 g
b
c
d
e
f
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1 600
1 700
1 800
1 900
2 000
Mean
b
c
d
e
f
g
put cap 200
Dr. Michael Moreno
sw ap cap 200 g
b
c
d
e
f
put cap 200
15
2 100
3. Greeks neutral hedging
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Part 3 Greeks Neutral Hedging
Traded swap levels
• THE DATA USED IS MOST CERTAINLY
INCOMPLETE
• We would like to thank Spectron Group plc
for providing the weather market swap data
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Dr. Michael Moreno
17
Part 3 Greeks Neutral Hedging
Historical swap levels LONDON HDD December
Weather Index Cone - LONDON HDD December 2002
500
Mean
Max
Min
Current Index
480
London HDD December
460
440
410
420
400
400
380
HDD
390
360
340
380
320
300
370
280
360
260
240
350
05-Nov-02 10-Nov-02 15-Nov-02 20-Nov-02 25-Nov-02 30-Nov-02 05-Dec-02 10-Dec-02 15-Dec-02
220
200
180
Date
160
140
Forward  380
Before the period started: swap level below
Then swap level above like the partial index
120
100
80
60
40
20
07/12/200214/12/200221/12/200228/12/2002
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Part 3 Delta Vega Neutral Hedging
Historical swap levels LONDON HDD January
London HDD January
500
450
HDD
400
350
300
250
30-Dec-02
04-Jan-03
09-Jan-03
14-Jan-03
19-Jan-03
24-Jan-03
Date
Forward  400
Before the period started: swap level below
Then swap level has 2 peaks and does not follow
the partial index evolution which is well above the
mean
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Weather Index Cone - LONDON HDD January 2003
580
560
540
520
500
480
460
440
420
400
380
360
340
320
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
Dr. Michael Moreno
Mean
Max
Min
Current Index
01 03 05 07 09 11 13 15 17 19 21 23 25 27 29 31
19
Part 3 Greeks Neutral Hedging
Historical swap levels LONDON HDD February
London HDD February
Weather Index Cone - LONDON HDD February 2003
390
500
480
370
460
Mean
Max
Min
Current Index
440
HDD
350
420
400
330
380
310
360
340
320
290
270
300
250
04-Jan- 09-Jan- 14-Jan- 19-Jan- 24-Jan- 29-Jan- 03-Feb- 08-Feb- 13-Feb- 18-Feb- 23-Feb03
03
03
03
03
03
03
03
03
03
03
Date
280
260
240
220
200
180
160
140
Forward  350
Before the start of the period,
the swap level is well below the forward
Then swap level converges toward with forward
120
100
80
60
40
20
02 04 06 08 10 12 14 16 18 20 22 24 26 28
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Part 3 Greeks Neutral Hedging
Historical swap levels LONDON HDD March
Weather Index Cone - LONDON HDD March 2003
London HDD March
440
302
300
HDD
Mean
Max
Min
Current Index
420
400
298
380
296
360
294
340
292
320
290
300
288
280
286
260
240
284
282
30-Dec- 09-Jan- 19-Jan- 29-Jan- 08-Feb- 18-Feb- 28-Feb- 10-Mar- 20-Mar- 30-Mar02
03
03
03
03
03
03
03
03
03
Date
220
200
180
160
140
120
Forward  340
Before the period started: swap level below the forward
Then swap level converges toward final swap level
100
80
60
40
20
02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
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Part 3 Greeks Neutral Hedging
Swap level Behaviour
• OF COURSE IT DEPENDS ON THE
MODEL USED TO ESTIMATE THE
FORWARD REFERENCE
• The swap seems to start to trade below its
forward before the start of the period and
remains quite constant prior the start of the
period (or 10 days before)
• The swap level converges quickly to its
final value (10 days in advance)
• There can be very erratic levels
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Part 3 Greeks Neutral Hedging
Consequences on Option Hedging
•
Before the start of the period when the swap level is below the forward (if it
really is!) then the swap has a strong theta, a non zero gamma (if capped) and a
delta away from 1 (if capped)
•
The delta of the traded swap convergences towards 1 slowly
•
10 days before the end of the period, the delta is close to 1, the theta is close to
zero, the gamma is close to zero
•
The vega of the option will be close to zero 10 days before the end of the period
•
Erratic swap levels must not be taken into account
•
Before the start of the period, assuming the swap level is quite constant, it is
easier to sell the option volatility than during the period
•
During the period, the theta of the option will not offset the theta of the swap,
nor will the gamma of the option offset the gamma of the swap
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Part 3 Greeks Neutral Hedging
No neutral hedging
• Due to the cap on the swap and swap illiquidity the
resulting position is likely to be non Delta neutral,
non Gamma neutral, non Theta neutral and non
Vega neutral
• If the swaps are kept (impossible to roll the swaps),
the Gamma and Theta issues are likely to grow
• Solutions:
– Minimise function of Greeks
– Minimise function of payoffs (e.g. SD)
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Part 3 Greeks Neutral Hedging
Market Assumptions
• Bid/Ask spread of Swap is 1% of standard deviation
(London Nov-Mar Stdev 100 => spread = 1 HDD).
• No market bias: (Bid + Ask) / 2 = Model Forward
• Option Bid/Ask spread is 20 % of StDev.
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Part 3 Greeks Neutral Hedging
Trajectory example
1: decrease in vol
(15%) implies a
higher gamma and
theta => rehedge
Forward trajectory - London HDD December 02
410
60
400
50
390
HDD
370
30
360
20
350
StDev
40
380
2: increase in vol =>
less sensitive to
gamma and theta
but forward down by
25% of vol =>
rehedge
10
340
date
1
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2
3
Dr. Michael Moreno
4
04/01/2003
30/12/2002
25/12/2002
20/12/2002
15/12/2002
10/12/2002
05/12/2002
30/11/2002
0
25/11/2002
330
3: forward down, vol
still high and will go
down quickly (near
the end of the
period) => rehedge
4: sharp decrease in
vol and forward =>
rehedge
26
Part 3 Greeks Neutral Hedging
Simulation results summary
•
The smaller the caps on the swap the higher the frequency of adjustment
must be and the higher is the hedging cost (transaction/market/back
office cost). Alternately we can keep the swap to hedge extreme
unidirectional events.
•
For out of the money options, if the caps of the option are identical to the
caps of the swap, then the hedging adjustment frequency is reduced
(delta, gamma are close).
•
The combination of swap illiquidity with caps creates a substantial bias
in Greeks Hedging. The higher the caps the more efficient is the hedge.
•
Optimising a portfolio using SD, Markowitz or PCA criterias is still a
favoured solution for hedging but is inappropriate for option volatility
traders.
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Dr. Michael Moreno
27
Conclusion
With the success of CME contracts, other
exchanges and new players may enter into the
weather market.
This may increase liquidity which will make
dynamic hedging of portfolios more practical.
New speculators such as volatility traders may
be attracted. This may give the opportunity to
offer more complex hedging tools that the
primary market needs with lower risk premia.
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Dr. Michael Moreno
28
References
•
J.C. Augros, M. Moreno, Book “Les dérivés financiers et d’assurance”, Ed
Economica, 2002.
•
R. Baillie, T. Bollerslev, H.O. Mikkelsen, “Fractionally integrated generalized
autoregressive condition heteroskedasticity”, Journal of Econometrics, 1996, vol
74, pp 3-30.
•
F.J. Breidt, N. Crato, P. de Lima, “The detection and estimation of long memory in
stochastic volatility”, Journal of econometrics, 1998, vol 83, pp325-348
•
D.C. Brody, J. Syroka, M. Zervos, “Dynamical pricing of weather derivatives”,
Quantitative Finance volume 2 (2002) pp 189-198, Institute of physics publishing
•
R. Caballero, “Stochastic modelling of daily temperature time series for use in
weather derivative pricing”, Department of the Geophysical Sciences, University
of Chicago, 2003.
•
Ching-Fan Chung, “Estimating the FIGARCH Model”, Institute of Economics,
Academia Sinica, 2003.
•
M. Moreno, "Riding the Temp", published in FOW - special supplement for
Weather Derivatives
•
M. Moreno, O. Roustant, “Temperature simulation process”, Book “La
Réassurance”, Ed Economica, Marsh 2003.
•
Spectron Ltd for swap levels
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Dr. Michael Moreno
29