Delta-Hedging with Tracking Risk

1
WEMBA 2000 Real Options 77
Delta-Hedging with Tracking Risk
Basis Risk When the cashflows from the project
are not perfectly correlated with the asset we
are using to delta-hedge. Example Suppose 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 the price of oil in the
market. Even if the delta-hedging had been
continuous and with zero transactions costs,
however, this will cost us 1.2
million. Summary If there are any risks in the
project that are not perfectly correlated with
the price risk in the underlying asset, then the
delta-hedging may result in a shortfall.
2
WEMBA 2000 Real Options 78
Sell delta 1.2 barrels 26.88 less trans.
0.5 -0.134
Reserve 14.8 (Call option value)
Date Price Delta T0 28
0.8 T1 41.77 0.91 T2
62.32 0.98 T3 92.96
1.00 T4 138.68 1.00
Invest remainder 11.95 at 6.3
Re-hedge sell further 0.13 barrels for
5.51 less trans. 0.5 -0.0276
12.70 at year end
18.19 Invest at 6.3

Re-hedge sell further 0.084 barrels for
5.23 less trans. 0.5 -0.026
24.54 Invest at 6.3
19.33 at year end

28.30 Invest at 6.3
Re-hedge sell further 0.024 barrels for
2.23 less trans. 0.5 -0.011
26.08 at year end

Buy back 1.2 barrels for - 166.42 less trans.
0.5 -0.83
30.08 at year end
- 137.17

Exercise Option (138.68 - 25)1.2 136.41
3
WEMBA 2000 Real Options 79
Delta-Hedging with Transactions Costs
Rigby Oil
Without Transactions Costs With Transactions
Costs 320,000 excess from delta-hedging 762,000
shortfall from delta-hedging
How do we minimize the transactions costs?
Hedge "within a band"!
With transactions costs and hedging within a
band 9,560,000 excess!
4
WEMBA 2000 Real Options 80
Sell delta 1.2 barrels 26.88 less trans
0.5 -0.134
Reserve 14.8 (Call option value)
"Real" Hedge Date Price Delta
Position T0 28 0.8 0.8 T1
41.77 0.91 0.81 T2 62.32
0.98 0.88 T3 92.96 1.00
0.90 T4 138.68 1.00 0.90
Invest remainder 11.95 at 6.3
Re-hedge sell further 0.01 1.2 barrels for
0.5 less trans. 0.5 -0.003
12.7 at year end
13.20 Invest at 6.3

Re-hedge sell further 0.07 1.2 barrels for
5.23 less trans. 0.5 -0.026
19.23 Invest at 6.3
14.02 at year end

22.29 Invest at 6.3
Re-hedge sell further 0.02 1.2 barrels for
1.86 less trans. 0.5 -0.009
20.44 at year end

23.70 at year end
Buy back 0.91.2 -149.8 less trans. 0.5
-0.749
- 126.85

Exercise Option (138.68 - 25)1.2 136.41
5
WEMBA 2000 Real Options 81
"Within-Band" Delta-Hedging with Transactions
Costs
Alternative Price movement scenarios
Excess/Shortfall no trans.costs trans. costs
costs band 320,000 -758,000
9,660,000 - 1,720,000 -2,060,000
-4,503,000
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
OOPS!!
So, should we forget "hedging within the
band"? Or is there something we could do
differently?