Delta-Hedging with Tracking Risk

WEMBA 2000 Real Options 77

Basis Risk When the cashflows from the project are not perfectly correlated with the assetwe are using to delta-hedge.

ExampleSuppose that, in Rigby Oil, we extract crude oil, while the market price of oil is indexed to refined oil. Normally the cost difference between crude and refined is $1/barrel (say). This cost has been included in the $25/barrel extraction costs.

Now suppose that, right before extraction at the end of year 4, our usual (outsourced)refinery goes out of business. The only alternative is to ship the crude to a more distant refinery, at an additional cost of $1/barrel.

This change will have no impact on any of the hedging activity; nor will it affect theprice of oil in the market. Even if the delta-hedging had been continuous and withzero transactions costs, however, this will cost us $1.2 million.

SummaryIf there are any risks in the project that are not perfectly correlated with the price risk inthe underlying asset, then the delta-hedging may result in a shortfall.

WEMBA 2000 Real Options 78

Date Price Delta

T=0 28 0.8

T=1 41.77 0.91

T=2 62.32 0.98

T=3 92.96 1.00

T=4 138.68 1.00

Sell delta * 1.2 barrels = $26.88less trans. 0.5% = -0.134

Reserve $14.8(Call option value)

Invest remainder: $11.95 at 6.3%

Re-hedge: sell further0.13 barrels for $5.51less trans. 0.5% = -0.0276

$12.70 at year end +

= $18.19 Invest at 6.3%

$19.33 at year end

Re-hedge: sell further0.084 barrels for $5.23less trans. 0.5% = -0.026

+= $24.54 Invest at 6.3%

$26.08 at year end

Re-hedge: sell further0.024 barrels for $2.23less trans. 0.5% = -0.011

+= $28.30 Invest at 6.3%

$30.08 at year end

Buy back 1.2 barrelsfor - $166.42less trans. 0.5% = -0.83+

= - $137.17

Exercise Option: (138.68 - 25)*1.2 = $136.41

WEMBA 2000 Real Options 79

Without Transactions Costs With Transactions Costs

$320,000 excess from delta-hedging $762,000 shortfall from delta-hedging

Delta-Hedging with Transactions Costs

How do we minimize the transactions costs?

Hedge "withina band"!

With transactions costs and hedging within a band

$9,560,000 excess!

Rigby Oil

"Real" HedgeDate Price Delta Position

T=0 28 0.8 0.8

T=1 41.77 0.91 0.81

T=2 62.32 0.98 0.88

T=3 92.96 1.00 0.90

T=4 138.68 1.00 0.90

Sell delta * 1.2 barrels = $26.88less trans 0.5% = -0.134

Reserve $14.8(Call option value)

Invest remainder: $11.95 at 6.3%

Re-hedge: sell further0.01 * 1.2 barrels for $0.5less trans. 0.5% = -0.003

$12.7 at year end +

= $13.20 Invest at 6.3%

$14.02 at year end

Re-hedge: sell further0.07 * 1.2 barrels for $5.23less trans. 0.5% = -0.026

+= $19.23 Invest at 6.3%

$20.44 at year end

Re-hedge: sell further 0.02 * 1.2 barrels for $1.86less trans. 0.5% = -0.009

+= $22.29 Invest at 6.3%

$23.70 at year end

Buy back 0.9*1.2 = -$149.8less trans. 0.5% = -0.749+

= - $126.85

WEMBA 2000 Real Options 80

Exercise Option: (138.68 - 25)*1.2 = $136.41

"Within-Band" Delta-Hedging with Transactions Costs

Alternative Price movement scenarios:

Scenario 1: S = 28 : 41.77 : 62.32 : 92.96 : 138.68 = 0.8 : 0.91 : 0.98 : 1.00 : 1.00

'band-hedge' = 0.8 : 0.81 : 0.88 : 0.90 : 0.90

Scenario 2: S = 28 : 41.77 : 28.00 : 18.77 : 28.00 = 0.8 : 0.91 : 0.76 : 0.36 : 1.00

'band-hedge' = 0.8 : 0.81 : 0.81 : 0.46 : 0.46

Excess/Shortfallno trans.costs trans. costs costs & band

$320,000 -$758,000 $9,660,000

- $1,720,000 -$2,060,000 -$4,503,000

OOPS!!

So, should we forget "hedging within the band"?Or is there something we could do differently?

WEMBA 2000 Real Options 81