Derivatives Hedging Swaps

8/13/2019 Derivatives Hedging Swaps

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Weather derivative hedging

& Swap illiquidity



www.weatherderivs.com 2

Call/Put Hedging• Diversification or Static hedging

(portfolio oriented)

– PCA

– Markowitz

– SD

• Dynamic hedging (Index hedging)




Dynamic Hedging1. 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 Memory Homoskedasticity

Short Memory

Heteroskedasticity

Heteroskedasticity& Long Memory

Part 1 Temperature Simulation process used




ARFIMA-FIGARCH model

iiiii ymS T

Seasonality Trend ARFIMA-FIGARCH


Seasonal volatility




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





FIGARCH noise

1 t t t Var h


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




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





2. Swap hedging and cap effects




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




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





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


S ff




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


P t 2 S h d i d ff t




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





3. Greeks neutral hedging

P t 3 G k N t l H d i




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




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 Delta Vega Neutral Hedging




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





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






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








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






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)






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 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






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.





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.




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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Derivatives Hedging Swaps

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