Weather derivative hedging

1
Weather derivative hedging Swap illiquidity

• Dr. Michael Moreno

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Call/Put Hedging

• Diversification or Static hedging (portfolio
  oriented)
• PCA
• Markowitz
• SD
• Dynamic hedging (Index hedging)

3
Dynamic Hedging

1. 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
Part 1 Temperature Simulation process used
Short Memory Heteroskedasticity

• GARCH
• ARFIMA
• FBM
• ARFIMA-FIGARCH
• Bootstrapp

Long Memory Homoskedasticity
Heteroskedasticity Long Memory
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ARFIMA-FIGARCH model
Part 1 Temperature Simulation process used
Seasonality
Trend
ARFIMA-FIGARCH
Seasonal volatility
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ARFIMA-FIGARCH definition
Part 1 Temperature Simulation process used
We consider first the ARFIMA process
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 s2.
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FIGARCH noise
Part 1 Temperature Simulation process used
Given the conditional variance We suppose that
Long term memory
Cf Baillie, Bollerslev and Mikkelsen 96 or Chung
03 for full specification
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Distributions of London winter HDD
Part 1 Temperature Simulation process used
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 mainlyfrom the year 1963
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• 2. Swap hedging and cap effects

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Swap Hedging
Part 2 Swap hedging and cap effects
Long HDD Call and ?optcall HDD Swap
Dynamic values
Long HDD Put and ?optput HDD Swap
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Deltas of a capped call
Part 2 Swap hedging and cap effects
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Deltas of capped swaps
Part 2 Swap hedging and cap effects
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Call optimal delta hedge
Part 2 Swap hedging and cap effects
?optcall ?call/ ?swap
NOT 1
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Put optimal delta hedge
Part 2 Swap hedging and cap effects
?optput ?put/ ?swap
NOT 1
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3. Greeks neutral hedging
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Traded swap levels
Part 3 Greeks Neutral Hedging

• 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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Historical swap levels LONDON HDD December
Part 3 Greeks Neutral Hedging
Forward ? 380 Before the period started swap
level below Then swap level above like the
partial index
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Historical swap levels LONDON HDD January
Part 3 Delta Vega Neutral Hedging
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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Historical swap levels LONDON HDD February
Part 3 Greeks Neutral Hedging
Forward ? 350 Before the start of the period,
the swap level is well below the forward Then
swap level converges toward with forward
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Historical swap levels LONDON HDD March
Part 3 Greeks Neutral Hedging
Forward ? 340 Before the period started swap
level below the forward Then swap level converges
toward final swap level
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Swap level Behaviour
Part 3 Greeks Neutral Hedging

• 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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Consequences on Option Hedging
Part 3 Greeks Neutral 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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No neutral hedging
Part 3 Greeks 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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Market Assumptions
Part 3 Greeks Neutral Hedging

• Bid/Ask spread of Swap is 1 of standard
  deviation (London Nov-Mar Stdev 100 gt spread 1
  HDD).
• No market bias (Bid Ask) / 2 Model Forward
• Option Bid/Ask spread is 20 of StDev.

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Trajectory example
Part 3 Greeks Neutral Hedging
1 decrease in vol (15) implies a higher gamma
and theta gt rehedge 2 increase in vol gt less
sensitive to gamma and theta but forward down by
25 of vol gt rehedge 3 forward down, vol still
high and will go down quickly (near the end of
the period) gt rehedge 4 sharp decrease in vol
and forward gt rehedge
1
2
3
4
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Simulation results summary
Part 3 Greeks Neutral Hedging

• 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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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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References

• J.C. Augros, M. Moreno, Book Les dérivés
  financiers et dassurance, 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