1
One can no more ban derivatives than the
Luddites could ban power looms in the early
nineteenth century.

• R. Bliss

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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Pricing Default-free Bond Options Using the
Binomial Model

• Spot Rates 1 yr 0R1 5 2 yr 0R2 6.5 p.a.
  Expectations hypothesis implied forward rate 1R1
  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----100/1.0991.74----25----100
• 88.17100/1.0652 ----25----100
• ----50----100/1.0793.46-----25---100
• ----25----100

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)

5
Delta Hedging

• ? ???PO/?PS
• If the options underlying instrument is a bond
  then ? ???PO/?B
• Delta of call gt0 Delta of put lt0
• 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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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 FV10m 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 ? -DSPS ?RS /(1RS) -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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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 delta0.5. DO9.33 yrs. 6.75 pa
  yield.
• Calculate impact on a T-bond put option value of
  a 50 bp increase in T-bond rates.
• ?O ? -?DOPO ?R /(1R) -(.5)9.33(111,687.50)(.00
  50)
  1.03375
• 2,520 gain per put option bought
• The number of put options bought is
• NO ?O ?E NO -504,000/2,520 -200 puts

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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
• NO ?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.

9
Example of Macrohedge Against Interest Rate Risk

• Step 1 DA 7.5 yrs. DL2.9 yrs. A750m
  L650m. DG 5 yrs. Assume a 25 bp increase in
  interest rates such that ?RS /(1RS) 25bp
• ?E ? -DGA ?RS /(1RS) -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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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 PF1m(1-.0275(91)/360)993,
  048.61. Options on T-bill futures are at the
  money with delta0.5.
• ?O ? -?DFPF ?RF /(1RF) -(0.5)0.25(993,048.61)(
  .0025)
  
• 310.33 gain per futures contract
  sold
• The number of put options bought is
• NO ?O ?E NO -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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Options Hedge for Interest Rate Risk Immunization
- Summary

• No. of Options Contracts to Immunize Using
• Microhedge NO (DSPS)/(? DOPO)
• Macrohedge NO (DG)A/(? DOPO)
• With Basis Risk
• No. of Options Contracts to Immunize Using
• For microhedge NO (DSPS)/(? DOPObr)
• For macrohedge NO (DG)A/(? DOPObr)

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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.

13
Pricing a Cap Using the Binomial Model
Current spot rates6 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 6pa 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)(1.0
8) ----50---0 if 5 pa
-----25 (6)---0 ----25
(4)----0

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Pricing the Floor Using the Binomial Model
Current spot rates6 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


15
Constructing a Collar By Buying the Cap and
Selling the Floor

• Buy 6 cap with notional value (NV)100m.
  Premium849,000
• Sell 5 floor with NV100m. Premium215,979
• Collar premium NV100m 849,000-215,979 (NVc
  x pc) - (NVf 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
  premium0
• C (NVc x pc) (NVf x pf) 0 so that
• NVf (.00849x100m)/0.00215979 393 million
• Buy 100m 6 cap and sell a 393 million 5 floor
  to get a costless collar.

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Hedging Credit Risk Credit Spread Call Option

• Payoff increases as the credit spread (CS) on a
  benchmark bond increases above some exercise
  spread ST.
• Payoff on option Modified duration x FV of
  option x (current CS ST)
• Basis risk if CS on benchmark bond is not closely
  related to borrowers nontraded credit risk.
• Figure 15.3 shows the payoff structure.

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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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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