Weather derivative hedging & Swap illiquidity Dr. Michael Moreno

Weather derivative hedging& Swap illiquidity

Dr. Michael Moreno

www.weatherderivs.com Dr. Michael Moreno 2

Call/Put Hedging

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

• Dynamic hedging (Index hedging)


Dynamic Hedging

1. Temperature Simulation process used

2. Swap hedging and cap effects

3. Greeks neutral hedging


1. Temperature Simulation process used


Temperature simulation

• GARCH

• ARFIMA

• FBM

• ARFIMA-FIGARCH

• Bootstrapp

Long MemoryHomoskedasticity

Short MemoryHeteroskedasticity

Heteroskedasticity& Long Memory

Part 1 Temperature Simulation process used


ARFIMA-FIGARCH model

iiiii ymST

Seasonality Trend ARFIMA-FIGARCH


Seasonal volatility


ARFIMA-FIGARCH definition

ttd LyLL 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 innovationst are white noises with the

variance σ2.

a

j

jj LL

1

1

We consider first the ARFIMA process:

m

j

jj LL

1

1



FIGARCH noise

1 ttt Varh


Given the conditional variance

We suppose that

22 1]1[1 td

tt LLLhL

Cf Baillie, Bollerslev and Mikkelsen 96 or Chung 03 for full specification

Long term memory


Distributions of London winter HDDHistoSim

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 mainlyfrom the year 1963



2. Swap hedging and cap effects


Swap Hedging

Long HDD Call and optcall HDD Swap

Long HDD Put and optput HDD Swap

Dynamic values

Part 2 Swap hedging and cap effects


Deltas of a capped call

Delta of Capped Calls

cap 200gfedcb cap 400gfedcb cap 800gfedcb

M ean2 100

2 0001 900

1 8001 7001 600

1 5001 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



Deltas of capped swaps

Delta of Capped Swaps

Delta Sw ap 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



Call optimal delta hedgeoptcall= call/ swap

Delta of Capped Call & Swap

call cap 200gfedcb sw ap cap 200gfedcb

Mean2 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

Mean2 1002 0001 9001 8001 7001 6001 5001 4001 300

Fai

r V

alue

s

150

100

50

0

-50

-100

-150



Put optimal delta hedge

optput= put/ swap NOT = 1

Delta of Capped Put & Swap

sw ap cap 200gfedcb put cap 200gfedcb

Mean2 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

Mean2 1002 0001 9001 8001 7001 6001 5001 4001 300

Fai

r V

alue

s

150

100

50

0

-50

-100

-150



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

Part 3 Greeks Neutral Hedging


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

HD

D

MeanMaxMinCurrent Index

Weather 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 380Before the period started: swap level belowThen swap level above like the partial index



Historical swap levels LONDON HDD January

London HDD January

250

300

350

400

450

500

30-Dec-02 04-Jan-03 09-Jan-03 14-Jan-03 19-Jan-03 24-Jan-03

Date

HD

D


Weather Index Cone - LONDON HDD January 2003

31292725232119171513110907050301

580560540

520500480460

440420400380

360340320300

280260240220

200180160

14012010080

604020

Forward 400Before the period started: swap level belowThen swap level has 2 peaks and does not followthe partial index evolution which is well above the mean

Part 3 Delta Vega Neutral Hedging


Historical swap levels LONDON HDD February


Weather 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

HD

D

Forward 350Before the start of the period, the swap level is well below the forwardThen swap level converges toward with forward



Historical swap levels LONDON HDD March


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

HD

D

Forward 340Before the period started: swap level below the forwardThen swap level converges toward final swap level



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



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



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)



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.



Trajectory exampleForward trajectory - London HDD December 02

330

340

350

360

370

380

390

400

410

25/1

1/20

02

30/1

1/20

02

05/1

2/20

02

10/1

2/20

02

15/1

2/20

02

20/1

2/20

02

25/1

2/20

02

30/1

2/20

02

04/0

1/20

03

date

HD

D

0

10

20

30

40

50

60

StD

ev

1 2 3 4

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



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.



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.


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

Documents

Weather derivative hedging & Swap illiquidity Dr. Michael Moreno