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**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) Dr. Michael Moreno

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**Dynamic Hedging Temperature Simulation process used**

Swap hedging and cap effects Greeks neutral hedging Dr. Michael Moreno

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**1. Temperature Simulation process used**

Dr. Michael Moreno

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**Temperature simulation**

Part 1 Temperature Simulation process used Temperature simulation Short Memory Heteroskedasticity GARCH ARFIMA FBM ARFIMA-FIGARCH Bootstrapp Long Memory Homoskedasticity Heteroskedasticity & Long Memory Dr. Michael Moreno

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**ARFIMA-FIGARCH model Seasonality Trend ARFIMA-FIGARCH**

Part 1 Temperature Simulation process used ARFIMA-FIGARCH model Seasonality Trend ARFIMA-FIGARCH Seasonal volatility Dr. Michael Moreno

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**ARFIMA-FIGARCH definition**

Part 1 Temperature Simulation process used ARFIMA-FIGARCH definition 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 innovationst are white noises with the variance σ2. Dr. Michael Moreno

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**FIGARCH noise Given the conditional variance We suppose that**

Part 1 Temperature Simulation process used FIGARCH noise Given the conditional variance We suppose that Long term memory Cf Baillie, Bollerslev and Mikkelsen 96 or Chung 03 for full specification Dr. Michael Moreno

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**Distributions of London winter HDD**

Part 1 Temperature Simulation process used Distributions of London winter HDD Histo Sim Average St Dev 128.52 119.26 Skewness 0.42 -0.01 Kurtosis 3.63 3.13 Minimum Maximum With similar detrending methods The slight differences come mainly from the year 1963 Dr. Michael Moreno

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**2. Swap hedging and cap effects**

Dr. Michael Moreno

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**Swap Hedging Long HDD Call and optcall HDD Swap Dynamic values**

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 Dr. Michael Moreno

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**Deltas of a capped call Part 2 Swap hedging and cap effects**

Dr. Michael Moreno

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**Deltas of capped swaps Part 2 Swap hedging and cap effects**

Dr. Michael Moreno

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**Call optimal delta hedge**

Part 2 Swap hedging and cap effects Call optimal delta hedge optcall= call/ swap NOT = 1 Dr. Michael Moreno

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**Put optimal delta hedge**

Part 2 Swap hedging and cap effects Put optimal delta hedge optput= put/ swap NOT = 1 Dr. Michael Moreno

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**3. Greeks neutral hedging**

Dr. Michael Moreno

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**Traded swap levels THE DATA USED IS MOST CERTAINLY INCOMPLETE**

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 Dr. Michael Moreno

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**Historical swap levels LONDON HDD December**

Part 3 Greeks Neutral Hedging Historical swap levels LONDON HDD December Forward 380 Before the period started: swap level below Then swap level above like the partial index Dr. Michael Moreno

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**Historical swap levels LONDON HDD January**

Part 3 Delta Vega Neutral Hedging Historical swap levels LONDON HDD January 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 Dr. Michael Moreno

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**Historical swap levels LONDON HDD February**

Part 3 Greeks Neutral Hedging Historical swap levels LONDON HDD February Forward 350 Before the start of the period, the swap level is well below the forward Then swap level converges toward with forward Dr. Michael Moreno

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**Historical swap levels LONDON HDD March**

Part 3 Greeks Neutral Hedging Historical swap levels LONDON HDD March Forward 340 Before the period started: swap level below the forward Then swap level converges toward final swap level Dr. Michael Moreno

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**There can be very erratic levels**

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 Dr. Michael Moreno

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**Consequences on Option Hedging**

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 Dr. Michael Moreno

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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) Dr. Michael Moreno

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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. Dr. Michael Moreno

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**Trajectory example 1 2 3 4 Part 3 Greeks Neutral Hedging**

1: decrease in vol (15%) implies a higher gamma and theta => rehedge 2: increase in vol => less sensitive to gamma and theta but forward down by 25% of vol => rehedge 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 1 2 3 4 Dr. Michael Moreno

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**Simulation results summary**

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. Dr. Michael Moreno

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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. Dr. Michael Moreno

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**References Spectron Ltd for swap levels**

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, pp D.C. Brody, J. Syroka, M. Zervos, “Dynamical pricing of weather derivatives”, Quantitative Finance volume 2 (2002) pp , 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 Dr. Michael Moreno

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