Options

1
Options

• Fin 284
• Fixed Income Analysis

2
Option Terminology

• Call Option the right to buy an asset at some
  point in the future for a designated price.
• Put Option the right to sell an asset at some
  point in the future at a given price

3
Review of Option Terminology

• Expiration Date The last day the option can be
  exercised (American Option) also called
  the strike date, maturity, and exercise date
• Exercise Price The price specified in the
  contract
• American Option Can be exercised at any time up
  to the expiration date
• European Option Can be exercised only on the
  expiration date

4
Review of Option Terminology

• Long position Buying an option
• Long Call Bought the right to buy the asset
• Long Put Bought the right to sell the asset
• Short Position Writing or Selling the option
• Short Call Agreed to sell the other party the
  right to buy the underlying asset, if the other
  party exercises the option you deliver the asset.
  
• Short Put - Agreed to buy the underlying asset
  from the other party if they decide to exercise
  the option.

5
Review of Terminology

• In - the - money options
• when the spot price of the underlying asset for a
  call (put) is greater (less) than the exercise
  price
• Out - of - the - money options
• when the spot price of the underlying asset for a
  call (put) is less (greater) than the exercise
  price
• At the money options
• when the exercise price and spot price are equal.

6
Interest Rate Options

• Traded on Chicago Board of Options Exchange
  (CBOE)
• Interest rate Options are traded on 13 Week,
• 5 year, 10 year and 30 year treasury securities

7
Options on Futures

• Options on futures are as popular or even more
  popular than on the actual asset.
• Options on futures do not require payments for
  accrued interest.
• The likelihood of delivery squeezes is less.
• Current prices for futures are readily available,
  they are more difficult to find for bonds.

8
Futures Options

• Call option holder will own a long futures
  position if the option is exercised.
• The writer of the call option accepts the
  corresponding short position at the exercise
  price.

9
Mechanics of Options on Futures

• Call Option Example
• Exercise price 85 Current Futures price 95
• Upon exercise both the long position and the
  short owned by the writer of the short option is
  set to 85.
• When marked to market the holder of the long
  makes 10, the holder of the short looses 10.
• The holders of the short and long position then
  face the same risks as any other holder of the
  futures contract.

10
Buyer Margin Requirements on Futures Options

• The buyer of the call option is not required to
  place any margin deposits. The most that could
  be lost is the cost of the option.

11
Call Option Writer Margin Requirements

• The writer of the call option accepts all of the
  risk since the buyer will not exercise if there
  would be a loss.
• The writer is required to deposit the original
  margin that would be required on the futures
  contract and the option price that is received
  for writing the option. The writer is also
  required to deposit variation margin as the
  contract is marked to market.

12
Call Option Profit

• Call option as the price of the asset increases
  the option is more profitable.
• Once the price is above the exercise price
  (strike price) the option will be exercised
• If the price of the underlying asset is below the
  exercise price it wont be exercised you only
  loose the cost of the option.
• The Profit earned is equal to the gain or loss on
  the option minus the initial cost.

13
Profit Diagram Call Option(Long Call Position)

• Profit
• Spot Price
• Cost

S-X-C
S
X
14
Call Option Intrinsic Value

• The intrinsic value of a call option is equal to
  the current value of the underlying asset minus
  the exercise price if exercised or 0 if not
  exercised.
• In other words, it is the payoff to the investor
  at that point in time (ignoring the initial cost)
  
• the intrinsic value is equal to
• max(0, S-X)

15
Payoff Diagram Call Option

• Payoff
• Spot
• Price

S-X
X
S
16
Example Naked Call Option

• Assume that you can purchase a call option on an
  8 coupon bond with a par value of 100 and 20
  years to maturity. The option expires in one
  month and has an exercise price of 100.
• Assume that the option is currently at the money
  (the bond is selling at par) and selling for 3.
• What are the possible payoffs if you bought the
  bond and held it until maturity of the option?

17
Five possible results

• The price of the bond at maturity of the option
  is 100. The buyer looses the entire purchase
  price, no reason to exercise.
• The price of the bond at maturity is less than
  100 (the YTM is gt 8). The buyer looses the 3
  option price and does not exercise the option.

18
Five Possible Results continued

• The price of the bond at maturity is greater than
  100, but less than 103. The buyer will
  exercise the option and recover a portion of the
  option cost.
• The price of the bond is equal to 103. The
  buyer will exercise the option and recover the
  cost of the option.
• The price of the bond is greater than 103. The
  buyer will make a profit of S-100-3.

19
Profit Diagram Call Option(Long Call Position)

• Profit
• Spot Price
• -3

S-100-3
S
103
100
20
Price vs. Rate

• Note buying a call on the price of the bond is
  equivalent to buying a put on the interest rate
  paid by the bond.
• As the rate decreases, the price increases
  because of the time value of money.

21
Profit Diagram Call Option(Short Call Position)

• Profit
• Spot Price

S
X
CX-S
22
Put option payoffs

• The writer of the put option will profit if the
  option is not exercised or if it is exercised and
  the spot price is less than the exercise price
  plus cost of the option.
• In the previous example the writer will profit as
  long as the spot price is less than 103.
• What if the spot price is equal to 103?

23
Put Option Profits

• Put option as the price of the asset decreases
  the option is more profitable.
• Once the price is below the exercise price
  (strike price) the option will be exercised
• If the price of the underlying asset is above the
  exercise price it wont be exercised you only
  loose the cost of the option.

24
Profit Diagram Put Option

• Profit
• Spot Price
• Cost

X-S-C
S
X
25
Put Option Intrinsic Value

• The intrinsic value of a put option is equal to
  exercise price minus the current value of the
  underlying asset if exercised or 0 if not
  exercised.
• In other words, it is the payoff to the investor
  at that point in time (ignoring the initial cost)
  
• the intrinsic value is equal to
• max(X-S, 0)

26
Payoff Diagram Put Option

• Profit
• Spot Price
• Cost

X-S
X
S
27
Profit Diagram Put OptionShort Put

• Profit
• Spot Price

S
X
S-XC
28
Pricing an Option

• Arbitrage arguments
• Black Scholes
• Binomial Tree Models

29
PV and FV in continuous time

• e 2.71828 y lnx x ey
• FV PV (1k)n for yearly compounding
• FV PV(1k/m)nm for m compounding periods per
  year
• As m increases this becomes
• FV PVern PVert let t n
• rearranging for PV PV FVe-rt

30
Lower Bound of Call Option Price

• Assume that you have an asset that does not pay a
  cash income (A non dividend paying stock for
  example)
• Consider the case of an option as it expires.
• In this case, regardless of whether it is an
  American or European option it will be worth its
  intrinsic value (max(S-X,0)).
• Assuming a positive value the lower bound is
  given by
• S - X

31
Formal Argument

• Consider two portfolios
• A One European call option on the stock of
  Widget Inc. plus cash equal to Xe-rT
• B One share of stock in Widget Inc.
• Note If the cash in portfolio A is invested at
  r, it will grow to be worth X at time T.

32
Portfolio A

• There are two possible outcomes at time T
  depending upon the value of S at time T
• ST gt X Exercise the option and purchase the
  asset with a current value of ST (The value of
  portfolio A at time T is ST ).
• ST lt X Do not exercise the option, The portfolio
  is then worth the value of the cash, X.
• Therefore the portfolio is worth
• max(ST,X)

33
Portfolio B

• The value of portfolio B is simply the value of
  the stock at time T, ST.

34
Comparing A to B

• Combining the two results it is easy to see that
  portfolio A (the option and the cash) is always
  worth at least as much as portfolio B (owning the
  stock), and sometimes it is worth more than
  B.Without arbitrage, the same relationship
  should be true today as well as at time T in the
  future.

35
Equal value of the portfolios today

• Let c be the call price (value of option) today.
• Then the value of portfolio A is c Xe-rT
• The value of portfolio B is S
• Since the value of A is always worth as much as B
  and sometimes it is worth more
• c Xe-rT gtS
• or rearranging
• c gt S - Xe-rT

36
Final result

• The worst outcome to buying a call option is that
  it expires worthless, so the option is worth
  either nothing or S-Xe-rT
• Therefore c gt max(S - Xe-rT,0)

37
Put Option

• Similar to a call option the put option should
  always have a positive value.
• Considering the case of an option as it expires
  (either an American or European Option), the
  value of the option should be equal to its
  intrinsic value. The lower bound is therefore
• X - S

38
European Put Option

• Again, in the case of a European option prior to
  maturity this equation will not hold and it is
  necessary to account for the time value of money.
  In this case the lower bound for the option is
  given by
• Xe-rT - S

39
Formal Argument

• Consider two Portfolios
• C One European put option plus one share
• and
• D An amount of cash equal to Xe-rT

40
Portfolio C

• There are two possibilities
• ST lt X Exercise the option at time T and the
  portfolio is worth X.
• ST gt X The option expires and the portfolio is
  worth STPortfolio C is therefore worth
  max(ST,X)

41
Portfolio D

• Investing the amount at a rate equal to r, the
  portfolio will be worth X at Time T.
• Combining the two arguments it is easy to see
  that portfolio C is always worth at least the
  same amount as portfolio D and sometimes it is
  worth more.

42
Comparing the two

• Let p the value (price) of the put option
• Without arbitrage opportunitiesp S gt Xe-rT
• or rearranging
• p gt Xe-rT - S the value of the put option is
  then given as p gt max(Xe-rT-S,0)

43
Put Call Parity

• Consider portfolio A and C above A One European
  call option plus an amount of cash equal to
  Xe-rTC One European put option plus one share
• Both portfolios are have a value of max(X,ST) at
  the expiration of the options. If no arbitrage
  opportunities exist, they should also have the
  same value today which implies
• c Xe-rT p S

44
Put Call Parity

• In other words, the value of a European call with
  a given exercise date can be deduced from the
  value of a European put with the same exercise
  date and exercise price.

45
Put call Parity

• Without this condition arbitrage opportunities
  exist
• Put-Call Parity specifies that
• c Xe-rT p S
• which rearranges to
• p c Xe-rT - S

46
A Fixed Income Example

• Previously we discussed had a call option on an
  8 coupon bond with a par value of 100 and 20
  years to maturity. The option expires in one
  month and has an exercise price of 100. The
  option is currently at the money (the bond is
  selling at par) and selling for 3.
• Assume we also have a put option on the same bond
  and the put option is selling for 2.

47
A Fixed Income Example

• For now, ignore the coupon payments on the bond.
• Consider three possible strategies
  simultaneously
• Buy the bond in the spot market for 100
• Enter into a short call position (sell the call)
  for 3
• Buy a put at a price of 2

48
Possible Outcomes at expiration

• Bond Price gt 100
• The call option is exercised so you are forced to
  sell the bond at a price equal to 100. The put
  option expires. You make 1 profit from the
  difference in the call and put prices
• Bond Price lt100
• You exercise the Put option and sell the Bond for
  100, which is the same price you paid. The Call
  expires worthless . You make a 1 from the
  difference in the price.

49
Arbitrage

• Regardless of the price of the bond at
  expiration, there was a 1 profit.
• Three possible things would eliminate arbitrage.
• An increase in the price of the bond today.
• A decrease in the call option price.
• An increase in the put option price.

50
Arbitrage continued

• Assume that the price of the bond doesnt change
• There would be an increase in market participants
  attempting to short call options. To compete
  with each other they lower the price and the call
  price will decrease.
• There will be an increase in the number of market
  participants wanting to purchase long put
  options. To compete with each other they will
  offer higher prices increasing the price.

51
Put Call Parity Revisited p c Xe-rT - S

• We have ignored the Time value of money so the
  relationship becomes
• pcX-S
• In our example XS which implies that p should
  equal c.
• If both the put and call price equaled each other
  there would be no arbitrage profits regardless of
  what happened to the bond price at maturity.

52
Put Call Extensions

• We ignored the time value of money and the coupon
  payments paid by the bond.
• The coupon payment can be treated similar to the
  price. If you own the bond you will receive the
  cash payment in the future.
• The put call parity relationship for a coupon
  bond is simply
• pcXe-rTCPe-rT-S
• where CP is the coupon payment received at time T
  

53
Put Call Parity and American Options

• Put-Call Parity holds only for European Options
  but it is possible to use the relationship to
  specify some generalizations concerning the
  relationship between American Puts and Calls.

54
American call Option vs. European Call Option

• Should an American call option on a non dividend
  paying stock be exercised prior to maturity?
• NO (assuming that the investor plans to hold the
  stock past the maturity date of the option.)

55
Should an American call Option be Exercised
Early?

• Assume that the option currently is deep in the
  money, The following possibilities exist
• 1) S gt X The investor can earn interest amount
  of cash equal to X and then still pay X for the
  stock upon expiration of the option.
• 2) S lt X The investor can then purchase the stock
  at the spot price and let the option expire.
• 3) SX Again there is no reason to exercise the
  option, and the investor will let the option
  expire.

56
Exercising Call Options

• Since it is never optimal to exercise the call
  early, the value of the American Call (C) should
  be equal to the value of the European Call (c).

57
Exercising Put Options

• Should an American put option on a non dividend
  paying stock be exercised prior to maturity?
• Yes (if it is sufficiently in the money)
• The general argument is that the Put option
  serves as insurance and that early exercise is a
  good idea if the investor realizes a significant
  gain from the exercise of the option

58
Exercising Put Options

• The price of an American put option should be
  above that of an Equivalent European option (Pgtp)
• The value of an American Call should equal the
  value of an European Call.
• Using the put call parity relationship and
  substituting generalizations can be made about
  American Options

59

• P gt p c Xe-rT - S
• P gt C Xe-rT - S
• Which rearranges to
• C - P lt S - Xe-rT It can also be shown that C -
  P gt S-XWhich combines with the above equation to
  proveS - X lt C - P lt S - Xe-rT

60
Black Scholes

• The basic starting point for the actual pricing
  of an European option is the model developed by
  Fisher Black, Myron Scholes, and Robert Merton.

61
Black Scholes Assumptions

1. Stock prices follow a lognormal distribution with
   m and s constant.
2. There are no transaction costs or taxes and all
   securities are perfectly divisible
3. There are no dividend on the asset during the
   life of the option
4. There are no riskless arbitrage opportunities
5. Security trading is continuous
6. Investors can borrow and lend at the same risk
   free rate
7. The short term risk free rate is constant

62
Inputs you will need

• S Current value of underlying asset
• X Exercise price
• t life until expiration of option
• r riskless rate
• s2 variance

63
Black Scholes

• Value of Call Option SN(d1)-Xe-rtN(d2)
• S Current value of underlying asset
• X Exercise price
• t life until expiration of option
• r riskless rate
• s2 variance
• N(d ) the cumulative normal distribution (the
  probability that a variable with a standard
  normal distribution will be less than d)

64
Black Scholes (Intuition)

• Value of Call Option
• SN(d1) - Xe-rt N(d2)
• The expected PV of cost Risk Neutral
• Value of S of investment Probability of
• if S gt X S gt X

65
Black Scholes

• Value of Call Option SN(d1)-Xe-rtN(d2)
• Where

66
Extending Black Scholes to Futures Options

• Black extended the original model to price
  options on futures.

67
Time Value of an Option

• The time value of an option is the difference in
  the theoretical price of the option and the
  intrinsic value.
• It represents the the possibility that the value
  of the option will increase over the time it is
  owned.

68
Time Value of Call Option

• Payoff
• Spot
• Price

Time value of option
S-X
X
S
69
Delta of an option

• The delta of the option shows how the theoretical
  price of the option will change with a small
  change in the underlying asset.

70
Time Value of Call Option

• Payoff
• Spot
• Price

Time value of option
S-X
X
S
Delta is the slope of the tangent line at the
given stock price
71
Delta of an option

• Intuitively a higher stock price should lead to a
  higher call price. The relationship between the
  call price and the stock price is expressed by a
  single variable, delta.
• The delta is the change in the call price for a
  very small change it the price of the underlying
  asset.

72
Delta

• Delta can be found from the call price equation
  as
• Using delta hedging for a short position in a
  European call option would require keeping a long
  position of N(d1) shares at any given time. (and
  vice versa).

73
Delta explanation

• Delta will be between 0 and 1.
• A 1 cent change in the price of the underlying
  asset leads to a change of delta cents in the
  price of the option.

74
Delta

• For deep in the money call options the delta will
  be close to 1.
• For deep out of the money call options the delta
  will be close to zero.

75
Gamma

• Gamma measures the curvature of the theoretical
  call option price line.

76
Gamma of an Option

• The change in delta for a small change in the
  stock price is called the options gamma
• Call gamma

77
Other Measures
78
Hedging

• If a firm is worried about an increase in
  borrowing costs it could buy a call option on the
  relevant interest rate. Any gains on the call
  options will offset the increased borrowing.
• Similarly if the firm is worried about a decline
  in rates decreasing income it could buy a put
  option on the interest rate. Any decline in
  income would be offset by the change in rates.

79
A Short Hedge

• Agree to sell 10 Eurodollar future contracts
  (each with an underlying value of 1 Million).
• We want to look at two results the spot market
  and the futures market. Assume you close out the
  futures position and that the futures price will
  converge to the spot at the end of the three
  months.

80
Rates increase to 8

• Spot position
• Need to pay 8 1 9 on 10 Million 10
  Million(.09/4) 225,000
• Futures Position
• Fut Price 92 interest rates increased by .9
• Close out futures position
• profit (10 million)(.009/4) 22,500

81
Rates Increase to 8

• Net interest paid
• 225,000 - 22,500 202,500
• 10 million(.0810/4) 202,500

82
Rates decrease to 6

• Spot position
• Need to pay 6 1 7 on 10 Million 10
  Million(.07/4) 175,000
• Futures Position
• Fut Price 94 interest rates decreased by 1.1
• Close out futures position
• loss (10 million)(.011/4) 27,500

83
Rates Decrease to 8

• Net interest paid
• 175,000 27,500 202,500
• 10 million(.0810/4) 202,500

84
Results of Hedge

• Either way the final interest rate expense was
  equal to 8.10 or 100 basis points above the
  initial futures rate of 7.10
• Should the position be hedged?
• It locks in the interest rate, but if rates had
  declined you were better off without the hedge.

85
Hedging with options

• Assume that you can purchase a put option on the
  futures contract with a strike price of 93 (7)
  and a cost of .40
• If interest rates rates increase to 8 the put
  guarantees that the worst case would be a rte of
  7 1 .4 8.4 which implies a total
  interest cost of 210,000
• If rates decrease to 7 you only have the added
  cost of the option resulting in a total interest
  expense of 7.4 or 185,000

86
(No Transcript)
87
Hedging Fixed Income Securities with Options

• Finance 284
• Analysis of Fixed Income Securities

88
Hedging Interest Rate Risk

• Previously (in the last class) we purchased a put
  option on a eurodollars futures contract to hedge
  against a change in interest rates.
• The result was that it limited the upper rate
  that might be paid and allowed a decrease in
  rates to decease the actual rate paid (the option
  wasnt exercised).
• There is a (sometimes substantial) cost to
  entering into the option contracts to accomplish
  this.

89
Option position

• In the example the option profited as the level
  of interest rates increased (the price of bonds
  decreased)

90
Profit Diagram Put Option

• Profit
• Spot Price
• Cost

X-S-C
S
X
91
Spot Position

• The investor was attempting to hedge against an
  increase in the level of interest rates, in other
  words, they paid a higher borrowing cost as rates
  increased.
• This is the same idea as saying the they lost
  money as the price of the bonds declined.
• This was offset by the profit from the option,
  but you incur the cost of the option.

92
Diagramming the spot

• The spot position could be represented by a
  straight line that represents the corresponding
  savings in interest rates.
• The line will also slope up to the right. As the
  price increases (rate decreases) there is a
  relative improvement since the rate decrease
  saves the investor money.

93
Profit Diagram Spot

• Profit
• Spot Price
• Cost

94

• Assume that the current interest rate is just
  below the rate implied by the strike price.
• The two positions could be represented on a
  single graph which explains the results of the
  hedge.
• At rates above the strike price the profit on the
  option cancels out the loss from the increased
  rates. At rates below the strike price you gain,
  but the gain is reduced by the cost of the option.

95
Profit Diagram Put Option

X
Combined Position
96
Selling off benefits

• It is possible to decrease the impact of the
  options cost
• One approach would be buying a cap as before, but
  also selling at cap at a higher rate (lower
  price).
• The money received from selling the option
  offsets the initial cost of the other option
  position.
• The downside is that if rates increased above the
  second level, you are exposed to the interest
  rate change.

97
Selling off benefits options position

• However if the level of interest rates increased
  too high the option that was sold would
  experience a loss offsetting the gains from the
  original position.
• Therefore the original position is no longer
  hedged.
• Assume that the two options are for the same
  expiration date and are both European

98
Profit Diagram Put Options
Short Put
Spot Price
Long Put
99
Profit Diagram Bear Spread

Spot price
Bear Spread
100
Bear Spread

• The previous example was essentially buying an
  interest rate cap (buying put on price of bonds)
  and selling an interest rate cap (selling a put
  on the price of bonds). The position could also
  be thought of as buying and selling call options
  on the level of interest rates.
• Below the lower price (above the higher yield)
  the two options cancel each other out so the
  increased cost associated with the spot position
  is unhedged.

101
Adding the spot position

• Again assume that the current level of interest
  rates is slightly below the lower of the two
  strike prices.

102
Profit Diagram Bear Spread

Spot price
Hedged Position
Bear Spread
103
Another Strategy

• To avoid the downside of the previous example you
  could buy the same interest rate cap, but sell an
  interest rate floor at a higher price (lower
  yield).

104
Interest Rate Floor

• A call option on the price of the bond can be
  represented as a floor on the level of interest
  rates.
• The option will be profitable if the price of the
  bond increases above the strike price (the
  interest rate decreases below the strike).
• This would offset a loss on an asset that is rate
  sensitive and effectively limit the loss.

105
Profit Diagram Call Option(Long Call Position)

• Profit
• Spot Price
• Cost

S-X-C
S
X
106
Spot position

• Since the option profits as the rates decrease
  (the price of a bond increases) this would offset
  lost income on an asset that is rate sensitive.
• In our new position we want to sell the option.

107
Profit Diagram Call Option(Short Call Position)

• Profit
• Spot Price

S
X
CX-S
108
Another Strategy

• The new strategy is to avoid the downside of the
  previous example you could buy the same interest
  rate cap, while selling an interest rate floor at
  a higher price (lower yield).

109
Profit Diagram Call Option(Short Call Position)

• Profit

Long Put
Short Call
110
Profit DiagramCostless Collar

• Profit

Long Put
Short Call
Combined Profit
111
Combined with Spot

• By using options selling at the same price the
  net cost is zero.
• At prices above the higher strike price (below
  the lower yield) the gain is offset by a loss in
  the option position.
• At prices below the lower strike price (above the
  higher yield) the loss is offset by gains in the
  option.
• You have limited both the gain and loss.
• Assume that the current interest rate is exactly
  between the two strike prices

112
Profit DiagramCostless Collar (Fence)

• Profit

Costless Collar
113
Complications

• In the previous examples we ignored many real
  world complications.
• This is especially true if you are buying options
  on futures (treasury bond futures for example).
• Many of these complications arise from the ideas
  of basis risk presented earlier.

114
New Example Protective Put(From Fabozzi)

• Assume that you own a corporate bond and you are
  afraid that an increase in interest rates will
  decrease the value of the bond.
• It would be possible to use futures or futures
  options to lock in a future sale price for the
  bonds.
• Assume that the coupon on the bond is 11.75 and
  they mature on April 19, 2023. Today is April
  19, 1985 and you plan to sell the bond in June
  1985.

115
The Hedge

• To hedge against the possible increase in
  interest rates you decide to buy a put option on
  the treasury bond futures contract.
• If interest rates increase, the price of the
  underlying bonds will decrease allowing you to
  own a short futures position with the higher
  futures price (equal to the strike price).

116
Determining the Strike Price

• The strike price will effectively set a cap on
  the level of interest rates since it rates
  increase above the rate corresponding to the
  strike price you profit from the option
  offsetting the loss in bond value.
• Assume that you do not want the price of the bond
  to drop below 87.668.

117
Target Strike Price 87.668

• The problem is that the futures option is not for
  the bond which you own, it is for a treasury
  bond.
• You need to set a strike price for the Treasury
  bond that corresponds with a price of 87.688 for
  the corporate bond.

118
Price Vs. Yield

• Choosing the minimum price is equivalent to
  selecting the maximum yield on the corporate
  bond.
• A price of 87.668 implies that the corporate bond
  will be paying a yield of 13.41 (since it is
  selling at a large discount the yield will be
  above the coupon rate)

119
Yield on CTD treasury

• The futures contract underlying the option has a
  large set of acceptable treasuries that can be
  delivered. You can find the cheapest to deliver
  at the current date.
• Assume that after finding the cheapest to deliver
  bond, you find that is has a current yield 90
  basis points less than current yield on the
  corporate. Assume that the yield spread stays
  fixed.

120
Price of CTD Treasury

• Given that the yield spread stays fixed at 90
  basis points and that the maximum acceptable
  yield on the corporate bond is 13.41 it implies
  a yield of 12.51 on the treasury.
• This implies a price of 63.756 for the treasury
  that is currently CTD.
• The price used in the futures option will not be
  this price, it must be found using the conversion
  factor

121
Finding the Strike Price

• Given the treasuries price of 63.756 and the
  conversion factor for the treasury of .9660, a
  futures price is then found to be
• 63.756/.9660 66
• Therefore a strike price of 66 on the treasury
  futures option contract would be used.

122
Hedge ratio

• Hedging the position with just futures contract
  would have required finding the hedge ratio, this
  still applies.
• Assuming that you found the hedge ratio to be
  1.24, you will need 1.24 put futures options for
  each spot position.

123
Another approachA Covered Call

• The assumption is that the change in interest
  rates will be small. To hedge against a possible
  decline in rates, the holder of a bond (or
  portfolio) sells out of the money calls.
• The income from the sale of the option provides
  income to offset a possible increase in rates
  that lowers the bond value.

124
The Maximum Effective Call Price

• Assume that the maximum effective call price you
  set is 102.66 plus the premium from the call
  option.
• The price of 102.66 corresponds to a yield of
  11.436 on the corporate bonds.
• Keeping the 90 basis point spread the yield on
  the CTD treasury should be 10.536 which implies
  a 75.348 price for the treasury

125
The strike price

• Using the conversion factor, this implies a
  futures price of 75.348/.9960 78 which is also
  the strike price on the call option you sell.

126
Comparing Strategies

• Comparing a basic futures position to the covered
  call and protective put in the previous examples
  shows that each has its own advantages and
  disadvantages.
• The basic futures position sets the price (and
  yield) regardless of what happens to the level of
  interest rates in the economy. However the other
  two provide scenarios where you are better off
  than this ( and scenarios where you would have
  been worse off)

127
Comparing Strategies

• The protective put does better if rates decrease
  and the call option in the covered call option is
  exercised. The protective put also outperforms
  the basic futures option if as rates decline, but
  it is outperformed by the covered call.
• For extreme rate increases, the option strategies
  are both outperformed by the basic futures
  position.