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Weather derivative hedging
& Swap illiquidity
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Call/Put Hedging• Diversification or Static hedging
(portfolio oriented)
– PCA
– Markowitz
– SD
• Dynamic hedging (Index hedging)
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Dynamic Hedging1. Temperature Simulation process used
2. Swap hedging and cap effects
3. Greeks neutral hedging
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1. Temperature Simulation process used
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Temperature simulation• GARCH
• ARFIMA
• FBM
• ARFIMA-FIGARCH• Bootstrapp
Long Memory Homoskedasticity
Short Memory
Heteroskedasticity
Heteroskedasticity& Long Memory
Part 1 Temperature Simulation process used
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ARFIMA-FIGARCH model
iiiii ymS T
Seasonality Trend ARFIMA-FIGARCH
Part 1 Temperature Simulation process used
Seasonal volatility
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ARFIMA-FIGARCH definition
t t
d L y L L 01
Where, as in the ARMA model, is the unconditional mean
of yt while the autoregressive operator
and the moving average operator
are polynomials of order a and m, respectively, in the lag
operator L, and the innovations t are white noises with the
variance σ2.
a
j
j
j L L1
1
We consider first the ARFIMA process:
m
j
j
j L L
1
1
Part 1 Temperature Simulation process used
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FIGARCH noise
1 t t t Var h
Part 1 Temperature Simulation process used
Given the conditional variance
We suppose that
221]1[1 t
d
t t L L Lh L
Cf Baillie, Bollerslev and Mikkelsen 96 or Chung 03 for full specification
Long term memory
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Distributions of London winter HDD
Histo
Sim
Densities
2,4002,2002,0001,8001,6001,4001,2001,000
0.003
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
Histo Sim
Average 1700.79 1704.54
St Dev 128.52 119.26
Skewness 0.42 -0.01
Kurtosis 3.63 3.13
Minimum 1474.39 1375.13
Maximum 2118.64 2118.92
With similar detrending methods
The slight differences come mainly
from the year 1963
Part 1 Temperature Simulation process used
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2. Swap hedging and cap effects
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Swap Hedging
Long HDD Call and optcall HDD Swap
Long HDD Put and opt put HDD Swap
Dynamic values
Part 2 Swap hedging and cap effects
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Deltas of a capped call
Delta of Capped Calls
cap 200gfedcb cap 400gfedcb cap 800gfedcb
Mean2 100
2 0001 900
1 8001 700
1 6001 500
1 4001 300
Delta
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Vol
140
130
120
110
100
90
Part 2 Swap hedging and cap effects
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Deltas of capped swaps
Delta of Capped Swaps
Delta Swap cap 200gfedcb Delta of Sw ap cap 400gfedcbDelta of Sw ap cap 800gfedcb
Strike 2 0001 9001 8001 7001 6001 5001 4001 300
Delta
1
0.8
0.6
0.4
0.2
Vol
140
130
120
110
100
90
Part 2 Swap hedging and cap effects
S ff
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Call optimal delta hedge
optcall= call/ swap
Delta of Capped Call & Swap
call cap 200gfedcb sw ap cap 200gfedcb
Mean
2 1002 0001 9001 8001 7001 6001 5001 4001 300
Delta
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
NOT = 1
Prices of Capped Call & Swap
sw ap cap 200gfedcb call cap 200gfedcb
Mean
2 102 0001 9001 8001 7001 6001 5001 4001 300
150
100
50
0
-50
-100
-150
Part 2 Swap hedging and cap effects
P t 2 S h d i d ff t
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Put optimal delta hedge
opt put= put/ swap NOT = 1
Delta of Capped Put & Swap
sw ap cap 200gfedcb put cap 200gfedcb
Mean
2 1002 0001 9001 8001 7001 6001 5001 4001 300
Delta
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Prices of Capped Put & Swap
sw ap cap 200gfedcb put cap 200gfedcb
Mean
2 102 0001 9001 8001 7001 6001 5001 4001 300
150
100
50
0
-50
-100
-150
Part 2 Swap hedging and cap effects
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3. Greeks neutral hedging
P t 3 G k N t l H d i
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Traded swap levels
• THE DATA USED IS MOST CERTAINLYINCOMPLETE
• We would like to thank Spectron Group plcfor providing the weather market swap data
Part 3 Greeks Neutral Hedging
P t 3 G k N t l H d i
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Historical swap levels LONDON HDD December
London HDD December
350
360
370
380
390
400
410
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
Date
H D D
Mean
Max
Min
Current Index
eather Index Cone - LONDON HDD December 2002
28/12/200221/12/200214/12/200207/12/2002
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
Forward 380
Before the period started: swap level below
Then swap level above like the partial index
Part 3 Greeks Neutral Hedging
Part 3 Delta Vega Neutral Hedging
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Historical swap levels LONDON HDD January
London HDD January
250
300
350
400
450
500
30 -Dec -0 2 04 -Ja n- 03 0 9-Ja n- 03 1 4-Jan -03 1 9-Jan -0 3 2 4- Ja n-0 3
Date
H D D
Mean
Max
Min
Current Index
eather Index Cone - LONDON HDD January 2003
31292725232119171513110907050301
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
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
Part 3 Delta Vega Neutral Hedging
Part 3 Greeks Neutral Hedging
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Historical swap levels LONDON HDD February
Mean
Max
Min
Current Index
eather Index Cone - LONDON HDD February 2003
2826242220181614121008060402
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
London HDD February
250
270
290
310
330
350
370
390
04-Jan-
03
09-Jan-
03
14-Jan-
03
19-Jan-
03
24-Jan-
03
29-Jan-
03
03-Feb-
03
08-Feb-
03
13-Feb-
03
18-Feb-
03
23-Feb-
03
Date
H D D
Forward 350
Before the start of the period,
the swap level is well below the forward
Then swap level converges toward with forward
Part 3 Greeks Neutral Hedging
Part 3 Greeks Neutral Hedging
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Historical swap levels LONDON HDD March
Mean
Max
Min
Current Index
Weather Index Cone - LONDON HDD March 2003
302826242220181614121008060402
440
420
400
380
360
340
320
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
London HDD March
282
284
286
288
290
292
294
296
298
300
302
30-Dec-
02
09-Jan-
03
19-Jan-
03
29-Jan-
03
08-Feb-
03
18-Feb-
03
28-Feb-
03
10-Mar-
03
20-Mar-
03
30-Mar-
03
Date
H D D
Forward 340
Before the period started: swap level below the forward
Then swap level converges toward final swap level
Part 3 Greeks Neutral Hedging
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Part 3 Greeks Neutral Hedging
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Consequences on Option Hedging
• Before the start of the period when the swap level is below the forward (if itreally is!) then the swap has a strong theta, a non zero gamma (if capped) and adelta 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 tozero, 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
Part 3 Greeks Neutral Hedging
Part 3 Greeks Neutral Hedging
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No neutral hedging
• Due to the cap on the swap and swap illiquidity theresulting position is likely to be non Delta neutral,non Gamma neutral, non Theta neutral and nonVega 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)
Part 3 Greeks Neutral Hedging
Part 3 Greeks Neutral Hedging
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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.
Part 3 Greeks Neutral Hedging
Part 3 Greeks Neutral Hedging
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Trajectory example
Forward trajectory - London HDD December 02
330
340
350
360
370
380
390
400
410
2
5 / 1 1 / 2 0 0 2
3
0 / 1 1 / 2 0 0 2
0
5 / 1 2 / 2 0 0 2
1
0 / 1 2 / 2 0 0 2
1
5 / 1 2 / 2 0 0 2
2
0 / 1 2 / 2 0 0 2
2
5 / 1 2 / 2 0 0 2
3
0 / 1 2 / 2 0 0 2
0
4 / 0 1 / 2 0 0 3
date
H D D
0
10
20
30
40
50
60
S t D e v
1 2 3 4
1: decrease in vol
(15%) implies a
higher gamma andtheta => rehedge
2: increase in vol =>
less sensitive to
gamma and theta
but forward down by25% of vol =>
rehedge
3: forward down, vol
still high and will go
down quickly (near
the end of theperiod) => rehedge
4: sharp decrease in
vol and forward =>
rehedge
Part 3 Greeks Neutral Hedging
Part 3 Greeks Neutral Hedging
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Simulation results summary
• The smaller the caps on the swap the higher the frequency of adjustmentmust be and the higher is the hedging cost (transaction/market/backoffice cost). Alternately we can keep the swap to hedge extremeunidirectional events.
• For out of the money options, if the caps of the option are identical to thecaps of the swap, then the hedging adjustment frequency is reduced(delta, gamma are close).
• The combination of swap illiquidity with caps creates a substantial biasin Greeks Hedging. The higher the caps the more efficient is the hedge.
• Optimising a portfolio using SD, Markowitz or PCA criterias is still afavoured solution for hedging but is inappropriate for option volatilitytraders.
Part 3 Greeks Neutral Hedging
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Conclusion
With the success of CME contracts, otherexchanges and new players may enter into theweather market.
This may increase liquidity which will makedynamic hedging of portfolios more practical.
New speculators such as volatility traders may
be attracted. This may give the opportunity tooffer more complex hedging tools that the primary market needs with lower risk premia.
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References
• J.C. Augros, M. Moreno, Book “Les dérivés financiers et d’assurance”, EdEconomica, 2002.
• R. Baillie, T. Bollerslev, H.O. Mikkelsen, “Fractionally integrated generalizedautoregressive condition heteroskedasticity”, Journal of Econometrics, 1996, vol74, pp 3-30.
• F.J. Breidt, N. Crato, P. de Lima, “The detection and estimation of long memory instochastic 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 inweather derivative pricing”, Department of the Geophysical Sciences, Universityof 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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