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Saunders & Cornett, Financial Institutions Management, 4th ed. 1 “One can no more ban derivatives than the Luddites could ban power looms in the early nineteenth century.” R. Bliss

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Saunders & Cornett, Financial Institutions Management, 4th ed. 2 Financial Options The right, not the obligation, to buy (for a call) or to sell (for a put) some underlying financial security, at a predetermined price (exercise or strike price) at or before a preset expiration date. Hedge by buying (not selling) options. Short hedge: buy puts. Long hedge: buy calls. Can reduce the cost of options hedge using compound options such as caps, collars & floors.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 3 Pricing Default-free Bond Options Using the Binomial Model Spot Rates: 1 yr 0 R 1 = 5% 2 yr 0 R 2 = 6.5% p.a. Expectations hypothesis implied forward rate 1 R 1 = 8% p.a. Suppose that there is a 50-50 chance that 1 yr rates 1 yr from now will be either 7% or 9% p.a. Bond Valuation using the Binomial Model: Time period 0 Time period 1 Time period 2 ----50%----100/1.09=$91.74----25%----$100 $88.17=100/1.065 2 ----25%----$100 ----50%----100/1.07=$93.46-----25%---$100 ----25%----$100

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Saunders & Cornett, Financial Institutions Management, 4th ed. 4 Pricing a Call Option on the Bond Using the Binomial Model Exercise Price = $92.60 =.5(91.74) +.5(93.46) Time period 0 Time period 1 Time period 2 ----50%----------------0--------------25%----$7.40 $6.93=.5(0)+.5(.86)/1.05 ------25%----$7.40 +7.40/(1.065) 2 ----50%---$0.86= $93.46-92.60-----25%---$7.40 ----25%----$7.40= max (0,100-92.60)

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Saunders & Cornett, Financial Institutions Management, 4th ed. 5 Delta Hedging = P O / P S If the option’s underlying instrument is a bond then = P O / B Delta of call >0 Delta of put <0 Out of the money is between 0 - 0.5 At the money is around 0.5 In the money is between 0.5 - 1

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Saunders & Cornett, Financial Institutions Management, 4th ed. 6 Example of Delta Neutral Option Microhedge FI intends to sell its T-bond portfolio in 60 days to underwrite an $11.168m investment project. The T-bonds are 15 yr 8% p.a. coupon with FV=$10m and yield of 6.75% p.a. T-bond MV = $11.168m. Step 1: Analyze the risk of cash position. Calculate duration = 9.33 yrs. Risk that price (interest rates) will decline (increase) over the next 60 days. Assume a 50 bp unanticipated increase in T-bond spot interest rates: E -D S P S R S /(1+R S ) = -9.33($11.168m)(.0050) 1.03375 = - $504,000 Step 2: Loss of $504,000 on position when T-bond spot rates increase 50 bp.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 7 Microhedge Example (contd.) Step 3: Perfect hedge would generate cash flows of $504,000 whenever interest rates go up 50 bp. Short hedge: buy put options. Step 4: On the day that the hedge is implemented, the 112 strike price T-bond put option premium is priced at 1-52 = 1 52/64 = $1,812.50 per $100,000 contract. The spot T- bond price is 111,687.50, so the 112 put is at the money and has a delta=0.5. D O =9.33 yrs. 6.75% pa yield. Calculate impact on a T-bond put option value of a 50 bp increase in T-bond rates:. O - D O P O R /(1+R) = -(.5)9.33($111,687.50)(.0050) 1.03375 = $2,520 gain per put option bought The number of put options bought is: N O O = E N O = -$504,000/2,520 = -200 puts

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Saunders & Cornett, Financial Institutions Management, 4th ed. 8 Microhedge Example (cont.) But, the delta neutral options hedge neglects the cost of the premium. 200 puts would cost 200(1,812.50) = $362,500. To fully immunize against loss, would have to buy $504,000/(2520-1812.50) = 712 puts. There may be basis risk such that rates on the underlying securities do not have the same volatility as the cash instruments. So: if br = (rates on options underlying)/(rates on cash instrument) is not equal to 1: N O = E/(br O). So if br = 1.15 then the number of puts bought (without considering the premium) would be: $504,000/(1.15)(2520) = 173 put options rather than 200.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 9 Example of Macrohedge Against Interest Rate Risk Step 1: D A = 7.5 yrs. D L =2.9 yrs. A=$750m L=$650m. DG = 5 yrs. Assume a 25 bp increase in interest rates such that R S /(1+R S ) = + 25bp E -D G A R S /(1+R S ) = -5($750m)(.0025) = - $9.375m Step 2: Loss of $9.375million in the market value of equity when interest rates unexpectedly increase by 25 bp.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 10 Macrohedge Example (cont.) Step 3: Perfect hedge would generate positive cash flows of $9.375 million whenever spot rates increase 25 bp. Short hedge: buy T-bill futures put options. Step 4: T-bill future IMM Index price = 97.25. Implies T-bill futures rate = 2.75% p.a. T-bill futures are 91day pure discount instruments. T-bill futures price P F =$1m(1-.0275(91)/360)=$993,048.61. Options on T-bill futures are at the money with delta=0.5. O - D F P F R F /(1+R F ) = -(0.5)0.25($993,048.61)(.0025) = $310.33 gain per futures contract sold The number of put options bought is: N O O = E N O = -$9.375m/310.33 = -30,208 puts on T-bill futures bought to implement macrohedge to immunize against ALL interest rate risk

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Saunders & Cornett, Financial Institutions Management, 4th ed. 11 Options Hedge for Interest Rate Risk Immunization - Summary No. of Options Contracts to Immunize Using: Microhedge: N O = (D S P S )/( D O P O ) Macrohedge: N O = (DG)A/( D O P O ) With Basis Risk: No. of Options Contracts to Immunize Using: For microhedge: N O = (D S P S )/( D O P O br) For macrohedge: N O = (DG)A/( D O P O br)

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Saunders & Cornett, Financial Institutions Management, 4th ed. 12 Options vs. Futures Hedging Advantages of Options Hedging –Options keep upside gain potential. Disadvantages of Options Hedging –Options premium is an upfront cost. –Premium reduces options gains. –To reduce the cost of options hedge, use compound options strategies.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 13 Pricing a Cap Using the Binomial Model Current spot rates=6% p.a. 50-50 chance of interest rates increase to 7% or down to 5% in 1 year. In 2 years, 25% chance of 4%, 50% of 6%, 25% of 8% pa interest rate. Price a cap with a $100m notional value & strike P = 6%pa Cap premium = $849,000 Time period 0 Time period 1 Time period 2 ----50%---($100m)(.07-.06)=$1m---25% (8%)--$2m 6% pa $849,000=.5(0)+.5($1m)/(1.06)(1.07) -----25% (6%)----$0 +.25($2m)/(1.06)(1.07)(1.08) ----50%---0 if 5% pa -----25% (6%)---$0 ----25% (4%)----$0

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Saunders & Cornett, Financial Institutions Management, 4th ed. 14 Pricing the Floor Using the Binomial Model Current spot rates=6% p.a. 50-50 chance of interest rates increase to 7% or down to 5% in 1 year. In 2 years, 25% chance of 4%, 50% of 6%, 25% of 8% pa interest rate. Price a cap with a $100m notional value & strike P = 5% Cap premium = $215,979 Time period 0 Time period 1 Time period 2 ----50%---0 if 7% pa--------------25% (8%)--$2m 6% pa $215,979 = -----25% (6%)----$0 +.25($1m)/(1.06)(1.05)(1.04) ----50%---0 if 5% pa -----25% (6%)---$0 ----25% (4%)----$100(.05-.04)=$1m

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Saunders & Cornett, Financial Institutions Management, 4th ed. 15 Constructing a Collar By Buying the Cap and Selling the Floor Buy 6% cap with notional value (NV)=$100m. Premium=$849,000 Sell 5% floor with NV=$100m. Premium=$215,979 Collar premium NV=100m = 849,000-215,979 = (NV c x pc) - (NV f x pf) = (.00849 x $100m) - (.00215979 x $100m) = $633,302 If out-of-money cap bough and in-the-money floor then collar premium is negative (money making product). Zero cost collar: set floor NV so that collar premium=0 C = (NV c x pc) – (NV f x pf) = 0 so that: NV f = (.00849x$100m)/0.00215979 =$393 million Buy $100m 6% cap and sell a $393 million 5% floor to get a costless collar.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 16 Hedging Credit Risk: Credit Spread Call Option Payoff increases as the credit spread (CS) on a benchmark bond increases above some exercise spread S T. Payoff on option = Modified duration x FV of option x (current CS – S T ) Basis risk if CS on benchmark bond is not closely related to borrower’s nontraded credit risk. Figure 15.3 shows the payoff structure.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 17

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Saunders & Cornett, Financial Institutions Management, 4th ed. 18

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Saunders & Cornett, Financial Institutions Management, 4th ed. 19 Default Option Pays a stated amount in the event of default. Usually specifies physical delivery in the event of default. Figure 15.4 shows the payoff structure. Variation: “barrier” option – if CS fall below some amount, then the option ceases to exist. Lowers the option premium.

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Saunders & Cornett, Financial Institutions Management, 4th ed. 20

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Saunders & Cornett, Financial Institutions Management, 4th ed. 21 Breakdown of Credit Derivatives: Rule (2001) British Bankers Assoc Survey 50% of notional value are credit swaps 23% are Collateralized Loan Obligations (CLOs) 8% are baskets (credit derivatives based on a small portfolio of loans each listed individually. A first- to-default basket credit default swap is triggered by the default of any security in the portfolio). 6% are credit spread options

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