1. Delta-Hedging with Tracking Risk WEMBA 2000Real Options77 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 2000Real Options78 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 = $26.88 less trans. 0.5% = -0.134 Reserve $14.8 (Call option value) 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% $19.33 at year end Re-hedge: sell further 0.084 barrels for $5.23 less trans. 0.5% = -0.026 + = $24.54 Invest at 6.3% $26.08 at year end Re-hedge: sell further 0.024 barrels for $2.23 less trans. 0.5% = -0.011 + = $28.30 Invest at 6.3% $30.08 at year end Buy back 1.2 barrels for - $166.42 less trans. 0.5% = -0.83 + = - $137.17 Exercise Option: (138.68 - 25)*1.2 = $136.41

3. WEMBA 2000Real Options79 Without Transactions CostsWith 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 "within a band"! With transactions costs and hedging within a band $9,560,000 excess! Rigby Oil

4. "Real"Hedge Date 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 = $26.88 less trans 0.5% = -0.134 Reserve $14.8 (Call option value) 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% $14.02 at year end Re-hedge: sell further 0.07 * 1.2 barrels for $5.23 less 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 $1.86 less trans. 0.5% = -0.009 + = $22.29 Invest at 6.3% $23.70 at year end Buy back 0.9*1.2 = -$149.8 less trans. 0.5% = -0.749 + = - $126.85 WEMBA 2000Real Options80 Exercise Option: (138.68 - 25)*1.2 = $136.41

5. "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/Shortfall no 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 2000Real Options81