Interest Rate Options:

1
Chapter 15

• Interest Rate Options
• Hedging Applications and Pricing

2
Topics

• Hedging fixed income and debt positions with
  interest rate options
• Hedging a series of cash flows with OTC caps,
  floors, and other interest-rate derivatives
• Valuation of interest-rate options with the
  binomial interest rate tree
• Pricing interest rate options with the Black
  Futures Model

3
Hedging

• By hedging with either exchange-traded futures
  call options on Treasury or Eurodollar contracts
  or with an OTC spot call option on a debt
  security or an interest rate put (floorlet), a
  fixed-income manager planning to invest a future
  inflow of cash, can obtain protection against
  adverse price increases while still realizing
  lower costs if security prices decrease.

4
Hedging

• For cases in which bond or money market managers
  are planning to sell some of their securities in
  the future, a manager, for the costs of buying an
  interest rate option, can obtain downside
  protection if bond prices decrease while earning
  greater revenues if security prices increase.
• Similar downside protection from hedging
  positions using interest rate put options can be
  obtained by bond issuers, borrowers, and
  underwriters.

5
Hedging Risk

• Like futures-hedged position, option-hedged
  positions are subject to quality, quantity, and
  timing risk.
• For OTC options, some of the hedging risk can be
  minimized, if not eliminated, by customizing the
  contract.
• For exchange-traded option-hedged positions the
  objective is to minimize hedging risk by
  determining the appropriate number of option
  contracts.

6
Hedging Risk

• For direct hedging cases (cases in which the
  future value of the asset or liability to be
  hedged, VT, is the same as the one underlying the
  option contract) the number of options can be
  determined by using the naïve hedge where n
  VT/X.
• For cross hedging cases (cases in which the asset
  or liability to be hedge is not the same as the
  one underlying the option contract), the number
  of options can be determined by using the
  price-sensitivity model defined in Chapter 13.

7
Long Hedging Cases
8
Future CD Purchase

• Consider the case of a money market manager who
  is expecting a cash inflow of approximately
  985,000 in September that he plans to invest in
  a 90-day jumbo CD with a face value of 1M and
  yield tied to the LIBOR.
• Assume
• CDs are currently trading at a spot index of
  94.5 (S0 98.625 per 100 face value), for YTM
  (100/98.625)(365/90) - 1 .057757).

9
Future CD Purchase

• Suppose the manager would like to earn a minimum
  rate on his September investment that is near the
  current rate, with the possibility of a higher
  yield if short-term rates increase.
• To achieve these objectives, the manager could
  take a long position in a September Eurodollar
  futures call.
• Suppose the manager buys one September 94
  Eurodollar futures call option priced at 4
• RD 6
• X 985,000
• C0 4(250) 1,000
• nc VT/X 985,000/985,000 1 call contract

10
Future CD Purchase

• Assume the manager will close the futures call
  at expiration at its intrinsic value (fT X) if
  the option is in the money and then buy the 1M,
  90-day CD at the spot price.
• The exhibit shows the effective investment
  expenditures at expiration (costs of the CD minus
  the profit from the Eurodollar futures call) and
  the hedged YTM earned from the hedged investment

YTM 1,000,000/Effective Investment(365/90)
1)
11
Future CD Purchase
Profit Max(fT- 985,000), 0 - 1,000 YTM
1M/Column 4365/90 - 1
12
Future CD Purchase

• If spot rates (LIBOR) are lower than RD 6 at
  expiration, the Eurodollar futures call will be
  in the money and the manager will be able to
  profit from the call position, offsetting the
  higher costs of buying the 90-day, 1M CD.
• As a result, the manager will be able to lock in
  a maximum purchase price of 986,000 and a
  minimum yield on his 90-day CD investment of
  5.885 when RD is 6 or less.

13
Future CD Purchase

• If spot discount rates (LIBOR) are 6 or higher,
  the call will be worthless. In these cases,
  though, the managers option losses are limited
  to just the 1,000 premium he paid, while the
  prices of buying CDs decrease, the higher the
  rates.
• As a result, for rates higher than 6, the
  manager is able to obtain lower CD prices and
  therefore higher yields on his CD investment as
  rates increase.

14
Future CD Purchase

• Thus, by hedging with the Eurodollar futures
  call, the manager is able to obtain at least a
  5.885 YTM, with the potential to earn higher
  returns if rates on CDs increase.

15
Future 91-Day T-Bill Investment

• In Chapter 13, we examined the case of a
  corporate treasurer who was expecting a 5
  million cash inflow in June, which she was
  planning to invest in T-bills for 91 days.
• In that example, the treasurer locked in the
  yield on the T-bill investment by going long in
  June T-bill futures contracts.
• Suppose the treasurer expected higher short-term
  rates in June but was still concerned about the
  possibility of lower rates.
• To be able to gain from the higher rates and yet
  still hedge against lower rates, the treasurer
  could alternatively buy June call options on
  T-bill futures.

16
Future 91-Day T-Bill Investment

• Suppose there is a June T-bill futures call with
• Exercise price 987,500 (index 95, RD 5)
• Price 1,000 (quote 4 C (4)(250)
  1,000)
• June expiration (on both the underlying futures
  and futures option) occurring at the same time
  the 5M cash inflow is to be received.

17
Future 91-Day T-Bill Investment

• To hedge the 91-day investment with this call,
  the treasurer would need to buy 5 calls at a cost
  of 5,000

18
Future 91-Day T-Bill Investment

• If T-bill rates were lower at the June
  expiration, then the treasurer would profit from
  the calls and could use the profit to defray part
  of the cost of the higher priced T-bills.
• As shown in the exhibit, if the spot discount
  rate on T-bills is 5 or less, the treasurer
  would be able to buy 5.058 spot T-bills (assume
  perfect divisibility) with the 5M cash inflow
  and profit from the futures calls, locking in a
  YTM for the next 91 days of approximately 4.75
  on the 5M investment.

19
Future 91-Day T-Bill Investment
Profit 5Max(fT- 987,500), 0 - 5,000
YTM (Number of
Bill)(1M)/5M365/91 - 1
20
Future 91-Day T-Bill Investment

• If T-bill rates are higher, then the treasurer
  would benefit from lower spot prices while her
  losses on the call would be limited to just the
  5,000 costs of the calls.
• For spot discount rates above 5, the treasurer
  would be able to buy more T-bills, the higher the
  rates, resulting in higher yields as rates
  increase.

21
Future 91-Day T-Bill Investment

• Thus, for the cost of the call options, the
  treasurer is able to establish a floor by locking
  in a minimum YTM on the 5M June investment of
  approximately 4.75, with the chance to earn a
  higher rate if short-term rates increase.

22
Hedging a CD Rate with an OTC Interest Rate Put

• Suppose the ABC manufacturing company was
  expecting a net cash inflow of 10M in 60 days
  from its operations and was planning to invest
  the excess funds in a 90-day CD from Sun Bank
  paying the LIBOR.
• To hedge against any interest rate decreases
  occurring 60 days from the now, suppose the
  company purchases an interest rate put
  (corresponding to the bank's CD it plans to buy)
  from Sun Bank for 10,000.

23
Hedging a CD Rate with an OTC Interest Rate Put

• Suppose the put has the following terms
• Exercise rate 7
• Reference rate LIBOR
• Time period applied to the payoff 90/360
• Day Count Convention 30/360
• Notional principal 10M
• Payoff made at the maturity date on the CD (90
  days from the options expiration)
• Interest rate puts expiration T 60 days
  (time of CD purchase)
• Interest rate put premium of 10,000 to be paid
  at the options expiration with a 7 interest
  Cost 10,000(1(.07)(60/360)) 10,117

24
Hedging a CD Rate with an OTC Interest Rate Put

• As shown in the exhibit, the purchase of the
  interest rate put makes it possible for the ABC
  company to earn higher rates if the LIBOR is
  greater than 7 and to lock in a minimum rate of
  6.993 if the LIBOR is 7 or less.

25
Hedging a CD Rate with an OTC Interest Rate Put
26
Hedging a CD Rate with an OTC Interest Rate Put

• For example, if 60 days later the LIBOR is at
  6.5, then the company would receive a payoff
  (90 day later at the maturity of its CD) on the
  interest rate put of 12,500
• The 12,500 payoff would offset the lower (than
  7) interest paid on the companys CD of
  162,500
• At the maturity of the CD, the company would
  therefore receive CD interest and an interest
  rate put payoff equal to 175,000

12,500 (10M).07-.065(90/360)
162,500 (10M)(.065)(90/360)
175,000 162,500 12,500
27
Hedging a CD Rate with an OTC Interest Rate Put

• With the interest-rate puts payoffs increasing
  the lower the LIBOR, the company would be able to
  hedge any lower CD interest and lock in a hedged
  dollar return of 175,000.
• Based on an investment of 10M plus the 10,117
  costs of the put, the hedged return equates to an
  effective annualized yield of 6.993
• On the other hand, if the LIBOR exceeds 7, the
  company benefits from the higher CD rates, while
  its losses are limited to the 10,117 costs of
  the puts.

6.993 (4)(175,000)/10M 10,117
28
Short Hedging Cases
29
Hedging Future T-Bond Sale with an OTC T-Bond Put

• Consider the case of a trust-fund manager who
  plans to sell ten 100,000 face value T-bonds
  from her fixed income portfolio in September to
  meet an anticipated liquidity need.
• The T-bonds the manager plans to sell pay a 6
  interest and are currently priced at 94 (per 100
  face value), and at their anticipated selling
  date in September, they will have exactly 15
  years to maturity and no accrued interest.
• Suppose the manager expects long-term rates in
  September to be lower and therefore expects to
  benefit from higher T-bond prices when she sells
  her bonds, but she is also concerned that rates
  could increase and does not want to risk selling
  the bonds at prices lower than 94.

30
Hedging Future T-Bond Sale with an OTC T-Bond Put

• As a strategy to lock in minimum revenue from the
  September bond sale if rates increase, while
  obtaining higher revenues if rates decrease,
  suppose the manager decides to buy spot T-bond
  puts from an OTC Treasury security dealer who is
  making a market in spot T-bond options.
• Suppose the manger pays the dealer 10,000 for a
  put option on ten 15-year, 6 T-bonds with an
  exercise price of 94 per 100 face value and
  expiration coinciding with the managers
  September sales date.
• The exhibit shows the manager's revenue from
  either selling the T-bonds on the put if T-bond
  prices are less than 94 or on the spot market if
  prices are equal to or greater than 94.

31
Hedging Future T-Bond Sale with an OTC T-Bond Put
32
Hedging Future T-Bond Sale with an OTC T-Bond Put

• If the price on a 15-year T-bond is less than 94
  at expiration (or rates are approximately 6.60
  or more) the manager would be able to realize a
  minimum net revenue of 930,000 by selling her
  T-bonds to the dealer on the put contract at X
  940,000 and paying the 10,000 cost for the put
  
• If T-bond prices are greater than 94 (below
  approximately 6.60), her put option would be
  worthless, but her revenue from selling the
  T-bond would be greater, the higher T-bond
  prices, while the loss on her put position would
  be limited to the 10,000 cost of the option.

33
Hedging Future T-Bond Sale with an OTC T-Bond Put

• Thus, by buying the put option, the trust-fund
  manager has attained insurance against decreases
  in bond prices. Such a strategy represents a bond
  insurance strategy.

34
Hedging Future T-Bond Sale with an OTC T-Bond Put

• Note If the portfolio manager were planning to
  buy long-term bonds in the future and was worried
  about higher bond prices (lower rates), she could
  hedge the future investment by buying T-bond spot
  or futures calls.

35
Managing the Maturity Gap with a Eurodollar
Futures Put

• In Chapter 13, we examined the case of a small
  bank with a maturity gap problem resulting from
  making 1M loans in June with maturities of 180
  days, financed by selling 1M worth of 90-day CDs
  at the current LIBOR of 5 and then 90 days later
  selling new 90-day CDs to finance its June CD
  debt of 1,012,103
• To minimize its exposure to market risk, the bank
  hedged its 1,012,103 CD sale in September by
  going short in 1.02491 (1,012,103/987,500)
  September Eurodollar futures contract trading at
  quoted index of 95 (987,500).

1,012,103 1M(1.05)90/365
36
Managing the Maturity Gap with a Eurodollar
Futures Put

• Instead of hedging its future CD sale with
  Eurodollar futures, the bank could alternatively
  buy put options on Eurodollar futures.
• By hedging with puts, the bank would be able to
  lock in or cap the maximum rate it pays on it
  September CD.

37
Managing the Maturity Gap with a Eurodollar
Futures Put

• For example, suppose the bank decides to hedge
  its September CD sale by buying a September
  Eurodollar futures put with
• Expiration coinciding with the maturity of its
  September CD
• Exercise price of 95 (X 987,500)
• Quoted premium of 2 (P 500)

38
Managing the Maturity Gap with a Eurodollar
Futures Put

• With the September debt from the June CD of
  1,012,103, the bank would need to buy 1.02491
  September Eurodollar futures puts (assume perfect
  divisibility) at a total cost of 512.46 to cap
  the rate it pays on its September CD

39
Managing the Maturity Gap with a Eurodollar
Futures Put

• If the LIBOR at the September expiration is
  greater than 5, the bank will have to pay a
  higher rate on its September CD, but it will
  profit from its Eurodollar futures put position,
  with the put profits being greater, the higher
  the rate.
• The put profit would serve to reduce part of the
  1,012,103 funds the bank would need to pay on
  the maturing June CD, in turn, offsetting the
  higher rate it would have to pay on its September
  CD.

40
Managing the Maturity Gap with a Eurodollar
Futures Put

• As shown in the exhibit, if the LIBOR is at
  discount yield of 5 or higher, then the bank
  would be able to lock in a debt obligation 90
  days later of 1,025,435 (allow for slight
  rounding differences), for an effective 180-day
  rate of 5.225.
• If the rate is less than or equal to 5, then the
  bank would be able to finance its 1,012,615.46
  debt (June CD of 1,012,103 and put cost of
  512.46) at lower rates, while its losses on its
  futures puts would be limited to the premium of
  512.46.
• As a result, for lower rates the bank would
  realize a lower debt obligation 90 days later and
  therefore a lower rate paid over the 180-day
  period.

41
Managing the Maturity Gap with a Eurodollar
Futures Put
42
Managing the Maturity Gap with a Eurodollar
Futures Put

• Thus, for the cost of the puts, hedging the
  maturity gap with puts allows the bank to lock in
  a maximum rate on its debt obligation, with the
  possibility of paying lower rates if interest
  rates decrease.

43
Hedging a Future Loan Rate with an OTC Interest
Rate Call

• Suppose a construction company plans to finance
  one of its project with a 10M, 90-day loan from
  Sun Bank, with the loan rate to be set equal to
  the LIBOR 100 BP when the project commences 60
  day from now.
• Furthermore, suppose that the company expects
  rates to decrease in the future, but is concerned
  that they could increase.

44
Hedging a Future Loan Rate with an OTC Interest
Rate Call

• To obtain protection against higher rates,
  suppose the company buys an interest rate call
  option from Sun Bank for 20,000 with the
  following terms
• Exercise rate 7
• Reference rate LIBOR
• Time period applied to the payoff 90/360
• Notional principal 10M
• Payoff made at the maturity date on the loan (90
  days after the options expiration)
• Interest rate calls expiration T 60 days
  (time of the loan)
• Interest rate call premium of 20,000 to be paid
  at the options expiration with a 7 interest
  Cost 20,000(1(.07)(60/360)) 20,233

45
Hedging a Future Loan Rate with an OTC Interest
Rate Call

• The exhibit shows the company's cash flows from
  the call, interest paid on the loan, and
  effective interest costs that would result given
  different LIBORs at the starting date on the loan
  and the expiration date on the option.
• As shown in Column 6 of the table, the company is
  able to lock in a maximum interest cost of 8.016
  if the LIBOR is 7 or greater at expiration,
  while still benefiting with lower rates if the
  LIBOR is less than 7.

46
Hedging a Future Loan Rate with an OTC Interest
Rate Call
47
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• In Chapter 13, we defined the price sensitivity
  model for hedging debt positions in which the
  underlying futures contract was not the same as
  the debt position to be hedge.
• The model determines the number of futures
  contracts that will make the value of a portfolio
  consisting of a fixed-income security and an
  interest rate futures contract invariant to small
  changes in interest rates.

48
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• The model also can be extended to hedging with
  put or call options. The number of options
  (calls for hedging long positions and puts for
  short positions) using the price-sensitivity
  model is

where Durs duration of the bond being
hedged. Duroption duration of the bond
underlying the option contract. V0 current
value of bond to be hedged. YTMs yield to
maturity on the bond being hedged. YTMoption
yield to maturity on the options underlying bond.
49
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• Suppose a bond portfolio manager is planning to
  liquidate part of his portfolio in September.
• The portfolio he plans to sell consist of a mix
  of A to AAA quality bonds with a weighted average
  maturity of 15.25 years, face value of 10M,
  weighted average yield of 8, portfolio duration
  of 10, and current value of 10M.
• Suppose the manager would like to benefit from
  lower long-term rates that he expects to occur in
  the future but would also like to protect the
  portfolio sale against the possibility of a rate
  increase.

50
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• To achieve this dual objective, the manager could
  buy a spot or futures put on a T-bond.
• Suppose there is a September 95 (X 95,000)
  T-bond futures put option trading at 1,156 with
  the cheapest-to-deliver T-bond on the puts
  underlying futures being a bond with a current
  maturity of 15.25 years, duration of 9.818, and
  currently priced to yield 6.0.

51
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• Using the price-sensitivity model, the manager
  would need to buy 81 puts at a cost of 93,636 to
  hedge his bond portfolio

52
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• Suppose that in September, long-term rates were
  higher, causing the value of the bond portfolio
  to decrease from 10M to 9.1M and the prices on
  September T-bond futures contracts to decrease
  from 95 to 86.
• In this case, the bond portfolios 900,000 loss
  in value would be partially offset by a 635,364
  profit on the T-bond futures puts ?
  81(95,000 - 86,000) - 93,636 635,364.
• The managers hedged portfolio value would
  therefore be 9,735,364 a loss of 2.6 in value
  (this loss includes the cost of the puts)
  compared to a 9 loss in value if the portfolio
  were not hedged.

53
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• On the other hand, if rates in September were
  lower, causing the value of the bond portfolio to
  increase from 10M to 10.5M and the prices on
  the September T-bond futures contracts to
  increase from 95 to 100, then the puts would be
  out of the money and the loss of the options
  would be limited to the 93,636.
• In this case, the hedged portfolio value would be
  10.406365M a 4.06 gain in value compared to
  the 5 gain for an unhedged position.

54
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge

• The exhibit shows the put-hedged bond portfolio
  values for a number of pairs of T-bond futures
  and bond portfolio values (and their associate
  yields).
• As shown, for increasing interest rates cases in
  which the pairs of the T-bond and bond portfolio
  values are less than 95 (the exercise price) and
  10,000,000, respectively, the hedge portfolio
  losses are between approximately 1 and 2.6,
  while for increasing interest rate cases in which
  the pairs of T-bond prices and bond values are
  greater than 95 and 10,000,000, the portfolio
  increases as the bond value increases.

55
Hedging a Bond Portfolio with T-Bond Puts
Cross Hedge
56
Hedging a Series of Cash Flows
57
Hedging a Series of Cash Flows Using
Exchange-Traded Options

• When there is series of cash flows, such as a
  floating-rate loan or an investment in a
  floating-rate note, a series or strip of interest
  rate options can be used.
• For example, a company with a one-year
  floating-rate loan starting in September at a
  specified rate and then reset in December, March,
  and June to equal the spot LIBOR plus BP, could
  obtain a maximum rate or cap on the loan by
  buying a series of Eurodollar futures puts
  expiring in December, March, and June.

58
Hedging a Series of Cash Flows Using
Exchange-Traded Options

• At each reset date, if the LIBOR exceeds the
  discount yield on the put, the higher LIBOR
  applied to the loan will be offset by a profit on
  the nearest expiring put, with the profit
  increasing the greater the LIBOR.
• If the LIBOR is equal to or less than the
  discount yield on the put, the lower LIBOR
  applied to the loan will only be offset by the
  limited cost of the put.
• Thus, a strip of Eurodollar futures puts used to
  hedge a floating-rate loan places a ceiling on
  the effective rate paid on the loan.

59
Hedging a Series of Cash Flows Using
Exchange-Traded Options

• In the case of a floating-rate investment, such
  as a floating-rate note tied to the LIBOR or a
  banks floating rate loan portfolio, a minimum
  rate or floor can be obtained by buying a series
  of Eurodollar futures calls, with each call
  having an expiration near the reset date on the
  investment.
• If rates decrease, the lower investment return
  will be offset by profits on the calls.
• If rates increase, the only offset will be the
  limited cost of the calls.

60
Hedging a Series of Cash Flows Using OTC Caps
And Floors

• Using exchange-traded options to establish
  interest rate floors and ceiling on floating rate
  assets and liabilities is subject to hedging
  risk.
• As a result, many financial and non-financial
  companies looking for such interest rate
  insurance prefer to buy OTC caps or floors that
  can be customized to meet their specific needs.

61
Floating Rate Loan Hedged with an OTC Cap

• Example Suppose the Diamond Development Company
  borrows 50M from Commerce Bank to finance a
  two-year construction project.
• Suppose the loan is for two years, starting on
  March 1 at a known rate of 8, then resets every
  three months -- 6/1, 9/1, 12/1, and 3/1 -- at
  the prevailing LIBOR plus 150 BP.

62
Floating Rate Loan Hedged with an OTC Cap

• In entering this loan agreement, suppose the
  company is uncertain of future interest rates and
  therefore would like to lock in a maximum rate,
  while still benefiting from lower rates if the
  LIBOR decreases.

63
Floating Rate Loan Hedged with an OTC Cap

• To achieve this, suppose the company buys a cap
  corresponding to its loan from Commerce Bank for
  150,000, with the following terms
• The cap consist of seven caplets with the first
  expiring on 6/1/2003 and the others coinciding
  with the loans reset dates.
• Exercise rate on each caplet 8.
• NP on each caplet 50M.
• Reference Rate LIBOR.
• Time period to apply to payoff on each caplet
  90/360. (Typically the day count convention is
  defined by the actual number of days between
  reset date.)
• Payment date on each caplet is at the loans
  interest payment date, 90 days after the reset
  date.
• The cost of the cap 150,000 it is paid at
  beginning of the loan, 3/1/2003.

64
Floating Rate Loan Hedged with an OTC Cap

• On each reset date, the payoff on the
  corresponding caplet would be
• With the 8 exercise rate (sometimes called the
  cap rate), the Diamond Company would be able to
  lock in a maximum rate each quarter equal to the
  cap rate plus the basis points on the loan, 9.5,
  while still benefiting with lower interest costs
  if rates decrease.
• This can be seen in the exhibit, where the
  quarterly interests on the loan, the cap payoffs,
  and the hedged and unhedged rates are shown for
  different assumed LIBORs at each reset date on
  the loan.

Payoff (50M) (MaxLIBOR-.08, 0)(90/360)
65
Floating Rate Loan Hedged with an OTC Cap
66
Floating Rate Loan Hedged with an OTC Cap

• For the five reset dates from 12/1/2003 to the
  end of the loan, the LIBOR is at 8 or higher.
• In each of these cases, the higher interest on
  the loan is offset by the payoff on the cap,
  yielding a hedged rate on the loan of 9.5 (the
  9.5 rate excludes the 150,000 cost of the cap
  the rate is 9.53 with the cost included).
• For the first two reset dates on the loan,
  6/1/2003 and 9/1/2003, the LIBOR is less than the
  cap rate. At these rates, there is no payoff on
  the cap, but the rates on the loan are lower with
  the lower LIBORs.

67
Floating Rate Asset Hedged with an OTC Floor

• As noted, floors are purchased to create a
  minimum rate on a floating-rate asset.
• As an example, suppose the Commerce Bank in the
  above example wanted to establish a minimum rate
  or floor on the rates it was to receive on the
  two-year floating-rate loan it made to the
  Diamond Company.

68
Floating Rate Asset Hedged with an OTC Floor

• To this end, suppose the bank purchased from
  another financial institution a floor for
  100,000 with the following terms corresponding
  to its floating-rate asset
• The floor consist of seven floorlets with the
  first expiring on 6/1/2003 and the others
  coinciding with the reset dates on the banks
  floating-rate loan to the Diamond Company.
• Exercise rate on each floorlet 8.
• NP on each floorlet 50M.
• Reference Rate LIBOR.
• Time period to apply to payoff on each floorlet
  90/360. Payment date on each floorlet is at the
  loans interest payment date, 90 days after the
  reset date.
• The cost of the floor 100,000 it is paid at
  beginning of the loan, 3/1/2003.

69
Floating Rate Asset Hedged with an OTC Floor

• On each reset date, the payoff on the
  corresponding floorlet would be
• With the 8 exercise rate, Commerce Bank would be
  able to lock in a minumum rate each quarter equal
  to the floor rate plus the basis points on the
  floating-rate asset, 9.5, while still benefiting
  with higher returns if rates increase.

Payoff (50M) (Max.08 - LIBOR, 0)(90/360)
70
Floating Rate Asset Hedged with an OTC Floor

• In the exhibit, Commerce Banks quarterly
  interests received on its loan to Diamond, its
  floor payoffs, and its hedged and unhedged yields
  on its loan are shown for different assumed
  LIBORs at each reset date.

71
Floating Rate Asset Hedged with an OTC Floor
72
Floating Rate Asset Hedged with an OTC Floor

• For the first two reset dates on the loan,
  6/1/2003 and 9/1/2003, the LIBOR is less than the
  floor rate of 8. At theses rates, there is a
  payoff on the floor that compensates for the
  lower interest Commerce receives on the loan
  this results in a hedged rate of return on the
  banks loan asset of 9.5 (the rate is 9.52 with
  the 100,000 cost of the floor included).
• For the five reset dates from 12/1/2003 to the
  end of the loan, the LIBOR equals or exceeds the
  floor rate. At these rates, there is no payoff
  on the floor, but the rates the bank earns on its
  loan are greater, given the greater LIBORs.

73
Collars

• A collar is combination of a long position in a
  cap and a short position in a floor with
  different exercise rates.
• The sale of the floor is used to defray the cost
  of the cap.
• For example, the Diamond Company in our above
  case could reduce the cost of the cap it
  purchased to hedge its floating rate loan by
  selling a floor.
• By forming a collar to hedge its floating-rate
  debt, the Diamond Company, for a lower net
  hedging cost, would still have protection against
  a rate movement against the cap rate, but it
  would have to give up potential interest savings
  from rate decreases below the floor rate.

74
Collars

• Example suppose the Diamond Company decided to
  defray the 150,000 cost of its 8 cap by selling
  a 7 floor for 70,000, with the floor having
  similar terms to the cap
• Effective dates on floorlet reset date on loan
• Reference rate LIBOR
• NP on floorlets 50M
• Time period for rates .25

75
Collars

• By using the collar instead of the cap, the
  company reduces its hedging cost from 150,000 to
  80,000, and as shown in the exhibit can still
  lock in a maximum rate on its loan of 9.5.
• However, when the LIBOR is less than 7, the
  company has to pay on the 7 floor, offsetting
  the lower interest costs it would pay on its
  loan. For example
• When the LIBOR is at 6 on 6/1/2003, Diamond has
  to pay 125,000 ninety days later on its short
  floor position.
• When the LIBOR is at 6.5 on 9/1/2003, the
  company has to pay 62,500.
• These payments, in turn, offset the benefits of
  the respective lower interest of 7.5 and 8
  (LIBOR 150) it pays on its floating rate loan.

76
Collars
77
Collars

• Thus, for LIBORs less than 7, Diamond has a
  floor in which it pays an effective rate of 8.5
  (losing the benefits of lower interest payments
  on its loan) and for rates above 8 it has a cap
  in which it pays an effective 9.5 on its loan.

78
Corridor

• An alternative financial structure to a collar is
  a corridor.
• A corridor is a long position in a cap and a
  short position in a similar cap with a higher
  exercise rate.
• The sale of the higher exercise-rate cap is used
  to partially offset the cost of purchasing the
  cap with the lower strike rate.

79
Corridor

• For example, the Diamond company, instead of
  selling a 7 floor for 70,000 to partially
  finance the 150,000 cost of its 8 cap, could
  sell a 9 cap for say 70,000.
• If cap purchasers believe there was a greater
  chance of rates increasing than decreasing, they
  would prefer the collar to the corridor as a tool
  for financing the cap.

80
Reverse Collar

• A reverse collar is combination of a long
  position in a floor and a short position in a cap
  with different exercise rates. The sale of the
  cap is used to defray the cost of the floor.
• For example, the Commerce Bank in our above floor
  example could reduce the 100,000 cost of the 8
  floor it purchased to hedge the floating-rate
  loan it made to the Diamond company by selling a
  cap.
• By forming a reverse collar to hedge its
  floating-rate asset, the bank would still have
  protection against rates decreasing against the
  floor rate, but it would have to give up
  potential higher interest returns if rates
  increase above the cap rate.

81
Reverse Collar

• Example Suppose Commerce sold a 9 cap for
  70,000, with the cap having similar terms to the
  floor.
• By using the reverse collar instead of the floor,
  the company would reduce its hedging cost from
  100,000 to 30,000,
• As shown in the exhibit, Commerce would lock in
  an effective minimum rate on its a asset of 9.5
  and an effective maximum rate of 10.5.

82
Reverse Collar
83
Reverse Corridor

• Instead of financing a floor with a cap, an
  investor could form a reverse corridor by selling
  another floor with a lower exercise rate.

84
Barrier Options

• Barrier options are options in which the payoff
  depends on whether an underlying security price
  or reference rate reaches a certain level.
• They can be classified as either knock-out or
  knock-in options
• Knock-out option is one that ceases to exist once
  the specified barrier rate or price is reached.
• Knock-in option is one that comes into existence
  when the reference rate or price hits the barrier
  level.

85
Barrier Options

• Knock-out and knock-in options can be formed with
  either a call or put and the barrier level can be
  either above or below the current reference rate
  or price when the contract is established
• Down-and-out or up-and-out knock out options
• Up-and-in or down-and-in knock in options

86
Barrier Options

• Barrier caps and floors with termination or
  creation feature are offered in the OTC market at
  a premium above comparable caps and floors
  without such features.

87
Barrier Options

• Down-and-out caps and floors are options that
  ceases to exist once rates hit a certain level.
• Example A two-year, 8 cap that ceases when the
  LIBOR hits 6.5, or a two-year, 8 floor that
  ceases once the LIBOR hits 9.

88
Barrier Options

• Up-and-in cap is one that become effective once
  rates hit a certain level.
• Example A two-year, 8 cap that that becomes
  effective when the LIBOR hits 9 or a two-year,
  8 floor that become effective when rates hit
  6.5.

89
Path-Dependent Options

• In the generic cap or floor, the underlying
  payoff on the caplet or floorlet depends only on
  the reference rate on the effective date.
• The payoff does not depend on previous rates
  that is, it is independent of the path the LIBOR
  has taken.
• Some caps and floors, though, are structured so
  that their payoff is dependent on the path of the
  reference rate.

90
Path-Dependent Options Average Cap

• An average cap is one in which the payoff depends
  on the average reference rate for each caplet.
• If the average is above the exercise rate, then
  all the caplets will provide a payoff.
• If the average is equal or below, the whole cap
  expires out of the money.

91
Path-Dependent Options Average Cap

• Consider a one-year average cap with an exercise
  rate of 7 with four caplets.
• If the LIBOR settings turned out to be 7.5,
  7.75, 7, and 7.5, for an average of 7.4375,
  then the average cap would be in the money
  (.074375 - .07)(.25)(NP).
• If the rates, though, turned out to be 7, 7.5,
  6.5, and 6, for an average of 6.75, then the
  cap would be out of the money.

92
Path-Dependent Options Q-Cap

• In a cumulative cap (Q-cap), the cap seller pays
  the holder when the periodic interest on the
  accompanying floating-rate loan hits or exceeds a
  specified level.
• Example Suppose the Diamond Company in our
  earlier cap example decided to hedge its two-year
  floating rate loan (paying LIBOR 150BP) by
  buying a Q-Cap from Commerce Bank with the
  following terms

93
Path-Dependent Options Average Cap

• Q-Cap Terms
• The cap consist of seven caplets with the first
  expiring on 6/1/2003 and the others coinciding
  with the loans reset dates.
• Exercise rates on each caplet 8.
• NP on each caplet 50M.
• Reference Rate LIBOR.
• Time period to apply to payoff on each caplet
  90/360.
• For the period 3/1/2003 to 12/1/2003, the caplet
  will payoff when the cumulative interest starting
  from loan date 3/1/2003 on the companys loan
  hits 3M.
• For the period 3/1/2004 to 12/1/2004, the caplet
  will payoff when the cumulative interest starting
  from date 3/1/2004 on the companys loan hits
  3M.
• Payment date on each caplet is at the loans
  interest payment date, 90 days after the reset
  date.
• The cost of the cap 125,000 it is paid at
  beginning of the loan, 3/1/2003.

94
Path-Dependent Options Q-Cap

• The exhibit shows the quarterly interest,
  cumulative interests, Q-cap payments, and
  effective interests for assumed LIBORs.
• In the Q-caps first protection period, 3/1/2003
  to 12/1/2003, Commerce Bank will pay the Diamond
  Company on its 8 caplet when the cumulative
  interest hits 3M.
• The cumulative interest hits the 3M limit on
  reset date 9/1/2003, but on that date the
  9/1/2003 caplet is not in the money.
• On the following reset date, though, the caplet
  is in the money at the LIBOR of 8.5. Commerce
  would, in turn, have to pay Diamond 62,500 (90
  days later) on the caplet, locking in a hedged
  rate of 9.5 on Diamonds loan.

95
Path-Dependent Options Q-Cap

• In the second protection period, 3/1/2004 to
  12/1/2004, the assumed LIBOR rates are higher.
• The cumulative interest hits the 3M limit on
  reset date 9/1/2004. Both the caplet on that date
  and the next reset date (12/1/2004) are in the
  money. As a result, with the caplet payoffs,
  Diamond is able to obtained a hedged rate of 9.5
  for the last two payment periods on its loan.

96
Path-Dependent Options Q-Cap
97
Path-Dependent Options Q-Cap

• When compared to a standard cap, the Q-cap
  provides protection for the one-year protection
  periods, while the standard cap provides
  protection for each period (quarter).
• As shown in the next exhibit, a standard 8 cap
  provides more protection given the assumed
  increasing interest rate scenario than the Q-cap,
  capping the loan at 9.5 from date 12/1/2003 to
  the end of the loan and providing a payoff on 5
  of the 7 caplets for a total payoff of 687,500.
• In contrast, the Q-cap pays on only 3 of the 7
  caplets for a total payoff of only 500,000.
• Because of its lower protection limits, a Q-cap
  cost less than a standard cap.

98
Path-Dependent Options Q-Cap
99
Exotic Options

• Q-caps, average caps, knock-in options, and
  knock-out options are sometimes referred to as
  exotic options.
• Exotic option products are non-generic products
  that are created by financial engineers to meet
  specific hedging needs and return-risk profiles.
• The next two slides define some of the popular
  exotics options used in interest rate management.

100
Exotic Options
101
Exotic Options
102
Currency Options

• When investors purchase and hold foreign
  securities or when corporations and governments
  sell debt securities in external markets or incur
  foreign debt positions, they are subject to
  exchange-rate risk.
• In Chapter 13, we examined how foreign currency
  futures contracts could be used by financial and
  non-financial corporations to hedge their
  international positions.
• These positions can also be hedge with currency
  options.

103
Currency Options Markets

• In 1982, the Philadelphia Stock Exchange (PHLX)
  became the first organized exchange to offer
  trading in foreign currency options.
• Foreign currency options also are traded on a
  number of derivative exchanges outside the U.S.
  
• In addition to offering foreign currency futures,
  the International Monetary Market and other
  futures exchanges also offers options on foreign
  currency futures.

104
Currency Options Markets

• There is also a sophisticated dealer's market.
• This interbank currency options market is part of
  the interbank foreign exchange market.
• In this dealer's market, banks provide
  tailor-made foreign currency option contracts for
  their customers, primarily multinational
  corporations.
• Compared to exchange-traded options, options in
  the interbank market are larger in contract size,
  often European, and are available on more
  currencies.

105
Currency Options Hedging

• With exchange-traded currency options and
  dealer's options, hedgers, for the cost of the
  options, can obtain not only protection against
  adverse exchange rate movements, but (unlike
  forward and futures positions) benefits if the
  exchange rates move in favorable directions.

106
Currency Options Hedging

• Example Consider the case of a U.S. fund with
  investments in Eurobonds that were to pay a
  principal in British pounds of 10M next
  September.
• For the costs of BP put options, the U.S. fund
  could protect its dollar revenues from possible
  exchange rate decreases when it converts, while
  still benefiting if the exchange rate increases.

107
Currency Options Hedging

• Suppose there is a September BP put with
• Exercise price of X 1.425/
• Price P 0.02/.
• Contract size of 31,250 British pounds
• The U.S. fund would need to buy 320 put contracts
  at a cost of 200,000 to establish a floor for
  the dollar value of its 10,000,000 receipt in
  September.

np 10,000,000/31,250 320 Cost
(320)(31,250)((0.02/) 200,000
108
Currency Options Hedging

• The exhibit shows the dollar cash flows the U.S.
  Fund would receive in September from converting
  its receipts of 10,000,000 to dollars at the
  spot exchange rate (ET) and closing its 320 put
  contracts at a price equal to the put's intrinsic
  value (assume the September payment date and
  option expiration date are the same).

109
Currency Options Hedging
110
Currency Options Hedging

• If the exchange rate is less than X 1.425/,
  the company would receive less than 14,250,000
  when it converts its 10,000,000 to dollars
  these lower revenues, however, would be offset by
  the profits from the put position.
• On the other hand, if the exchange rate at
  expiration exceeds 1.425/, the U.S. fund would
  realize a dollar gain when it converts the
  10,000,000 at the higher spot exchange rate,
  while its losses on the put would be limited to
  the amount of the premium.

111
Currency Options Hedging

• Thus, by hedging with currency put options, the
  company is able to obtain exchange rate risk
  protection in the event the exchange rate
  decreases while still retaining the potential for
  increased dollar revenues if the exchange rate
  rises.

112
Currency Options Hedging

• Suppose that instead of receiving foreign
  currency, a U.S. company had a foreign liability
  requiring a foreign currency payment at some
  future date.
• To protect itself against possible increases in
  the exchange rate while still benefiting if the
  exchange rate decreases, the company could hedge
  the position by taking a long position in a
  currency call option.

113
Currency Options Hedging

• Example Suppose a U.S. company owed 10,000,000,
  with the payment to be made in September.
• To benefit from the lower exchange rates and
  still limit the dollar costs of purchasing
  10,000,000 in the event the / exchange rate
  rises, the company could buy September British
  pound call options.
• The exhibit shows the costs of purchasing
  10,000,000 at different exchange rates and the
  profits and losses from purchasing 320 September
  British pound calls with X 1.425/ at 0.02/
  (contract size 31,250) and closing them at
  expiration at a price equal to the call's
  intrinsic value.

114
Currency Options Hedging
115
Currency Options Hedging

• As shown in the table, for cases in which the
  exchange rate is greater than 1.425/, the
  company has dollar expenditures exceeding
  14,250,000 the expenditures, though, are offset
  by the profits from the calls.
• On the other hand, when the exchange rate is less
  than 1.425/, the dollar costs of purchasing
  10,000,000 decreases as the exchange rate
  decreases, while the losses on the call options
  are limited to the option premium.

116
Pricing Interest Rate Options with a Binomial
Interest Tree
117
Pricing Interest Rate Options with a Binomial
Interest Tree

• In Chapter 9, we examined how the binomial
  interest rate model can be used to price bonds
  with embedded call and put options, sinking fund
  arrangements, and convertible clauses.
• The binomial interest rate tree also can be used
  to price interest rate options.

118
Valuing T-Bill Options with a Binomial Tree

• The exhibit shows
• A two-period binomial tree for an annualized
  risk-free spot rate (S)
• The corresponding prices on a T-bill (B) with a
  maturity of .25 years and face value of 100
• A futures contract (f) on the T-bill, with the
  futures expiring at the end of period 2.

119
Valuing T-Bill Options with a Binomial Tree

• The features of the binomial tree
• The length of each period is six months
  (six-month steps)
• The upward parameter on the spot rate (u) is 1.1
• The downward parameter (d) is 1/1.1 0.9091
• The probability of spot rate increasing in each
  period is .5
• The yield curve is assumed flat.

120
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121
Valuing T-Bill Options with a Binomial Tree

• Spot T-Bill Prices
• At the current spot rate of 5, the price of the
  T-bill is B0 98.79 ( 100/(1.05).25).
• In period 1, the price is 98.67 when the spot
  rate is 5.5 ( 100/(1.055).25) and 98.895 when
  the rate is 4.54545 ( 100/(1.0454545).25).
• In period 2, the T-bill prices are 98.54, 98.79,
  and 99 for spot rates of 6.05, 5, and 4.13223,
  respectively.

122
Valuing T-Bill Options with a Binomial Tree

• Futures Prices
• The futures prices are obtained by assuming a
  risk neutral market.
• That is If the market is risk neutral, then the
  futures price is an unbiased estimator of the
  expected spot price ft E(ST).
• The futures prices at each node in the exhibit
  are therefore equal to their expected price next
  period.

123
Valuing T-Bill Options with a Binomial Tree

• Values of call and put options on spot and
  futures T-bills
• For European options, the methodology for
  determining the price is to start at expiration
  where we know the possible option values are
  equal to their intrinsic values, IVs.
• Given the options IVs at expiration, we then
  move to the preceding period and price the option
  to equal the present value of its expected cash
  flows for next period.
• Given these values, we then roll the tree to the
  next preceding period and again price the option
  to equal the present value of its expected cash
  flows.
• We continue this recursive process to the current
  period.

124
Valuing T-Bill Options with a Binomial Tree

• Values of call and put options on spot and
  futures T-bills
• If the option is American, then its early
  exercise advantage needs to be taken into account
  by determining at each node whether or not it is
  more valuable to hold the option or exercise.

125
Valuing T-Bill Options with a Binomial Tree

• Values of call and put options on spot and
  futures T-bills
• The methodology for valuing American options
• Start one period prior to the options expiration
  and constrain the price of the American option to
  be the maximum of its binomial value (present
  value of next periods expected cash flows) or
  the intrinsic value (i.e., the value from
  exercising).
• Roll those values to the next preceding period,
  and then price the option value as the maximum of
  the binomial value or the IV.
• Continue this process to the current period.

126
Valuing T-Bill Options with a Binomial Tree

• Values of American and European Spot Call
  Options
• The exhibit show the binomial valuation of both a
  European and an American call on the spot T-bill,
  each with an exercise prices of 98.75 per 100
  face value and expiration of one year.
• The price of the European call is 0.0787.
• The price of the American call is 0.08.

127
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128
Valuing T-Bill Options with a Binomial Tree

• Value of European Futures Call Option
• If the call option were on a European T-bill
  futures contract, instead of a spot T-bill, with
  the futures and option having the same
  expiration, then the value of the futures option
  will be the same as the spot option.
• That is, at the expiration spot rates of 6.05,
  5, and 4.13223, the futures prices on the
  expiring contract would be equal to the spot
  prices (98.54, 98.79, and 99), and the
  corresponding IVs of the European futures call
  would be 0, .04, and .25 the same as the spot
  calls IV.
• Thus, when we roll these call values back to the
  present period, we end up with the price on the
  European futures call of .0787 the same as the
  European spot.

129
Valuing T-Bill Options with a Binomial Tree

• Value of American Futures Call Option
• If the futures call option were American, then
  the option prices at each node needs to be
  constrained to be the maximum of the binomial
  value or the futures options IV.
• In this case, the corresponding prices of the
  American futures option are the same as the spot
  option.
• Thus, the price on the American T-bill futures
  call is .08 -- the same price as the American
  spot option.

130
Valuing T-Bill Options with a Binomial Tree

• Value of American Futures Put Option
• The next exhibit shows the binomial valuation of
  both a European and American T-bill futures put
  with an exercise price of 98.75 and expiration of
  one year (two periods).
• The price of the European futures put is .05.
• The price of the American futures put is .05.

131
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132
Valuing T-Bill Options with a Binomial Tree

• Value of American Futures Put Option
• Note The put price of .05 is consistent with the
  put-call futures parity relation

133
Valuing a Caplet and Floorlet with a Binomial
Tree

• The price of a caplet or floorlet can also be
  valued using a binomial tree of the options
  reference rate.
• Consider an interest rate call and put on the
  spot rate defined by our binomial tree, with
• Exercise rate of 5
• Time period applied to the payoff of ? .25
• Notional principal of NP 100.

134
Valuing a Caplet and Floorlet with a Binomial
Tree

• The next exhibit shows the binomial valuation of
  the interest rate call and the interest rate put.
• The value of the caplet is 0.06236.
• The value of the floorlet is 0.05177.

135
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136
Valuing a Cap and Floorwith a Binomial Tree

• Since a cap is a series of caplets, its price is
  simply equal to the sum of the values of the
  individual caplets making up the cap.
• To price a cap, we can use a binomial tree to
  price each caplet and then aggregate the caplet
  values to obtain the value of the cap.
• Similarly, the value of a floor can be found by
  summing the values of the floorlets comprising
  the floor.

137
Valuing T-Bond Options with a Binomial Tree

• The T-bill underlying the spot or futures T-bill
  option is a fixed-deliverable bill that is, the
  features of the bill (maturity of 91 days and
  principal of 1M) do not change during the life
  of the option.
• In contrast, the T-bond or T-note underlying a
  T-bond or T-note option or futures option is a
  specified T-bond or note or the bond from an
  eligible group that is most likely to be
  delivered.
• Because of the specified bond clause on a T-bond
  or note option or futures option, the first step
  in valuing the option is to determine the values
  of the specified T-bond (or bond most likely to
  be delivered) at the various nodes on the
  binomial tree, using the same methodology we used
  in Chapter 9 to value a coupon bond.

138
Valuing T-Bond Options with a Binomial Tree

• Example Consider an OTC spot option on a T-bond
  with
• 6 annual coupon
• Face value of 100
• 3 years left to maturity

139
Valuing T-Bond Options with a Binomial Tree

• In valuing the bond, assume
• 2-period binomial tree of risk-free spot rates
• Length of each period is one year
• Upward parameter u 1.2
• Downward parameter d .8333
• Current spot rate S0 6

140
Valuing T-Bond Options with a Binomial Tree

• Binomial Valuation of T-Bond
• To value the T-bond, we start at the bonds
  maturity (end of period 3) where the bonds value
  is equal to the principal plus the coupon, 106.
• We next determine the three possible values in
  period 2 given the three possible spot rates.
• Given these values, we next roll the tree to the
  first period and determine the two possible
  values.
• Finally, using the bond values in period 1, we
  roll the tree to the current period where we
  determine the value of the T-bond.
• As shown in the exhibit the value of the T-bond
  is 99.78.

141
Valuing T-Bond Options with a Binomial Tree

• Binomial Valuation of T-Bond Futures
• The exhibit also shows the prices on a two-year
  futures contract on the three-year, 6 T-bond.
• The prices are generated by assuming a
  risk-neutral market.
• The futures price is f0 99.83.

142
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143
Valuing T-Bond Options with a Binomial Tree

• Values of American and European Spot Call
• The next exhibit shows the binomial valuation of
  both a European and an American call on the spot
  T-bond, each with an exercise prices of 98 per
  100 face value, and expiration of two year.
• The price of the European call is 1.734.
• The price of the American call is 2.228.

144
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145
Valuing T-Bond Options with a Binomial Tree

• Value of European Futures Call
• If the European call were an option on a futures
  contract on the three-year, 6 T-bond (or if that
  bond were the most likely to-be-deliver bond on
  the futures contract), with the futures contract
  expiring at the same time as the option (end of
  period 2), then the value of the futures option
  will be the same as the spot.
• That is, at expiration the futures prices on the
  expiring contract would be equal to the spot
  prices, and the corresponding IVs of the European
  futures call would be the same as the spot calls
  IV.
• Thus, when we roll these call values back to the
  present period, we end up with the price on the
  European futures call being the same as the
  European spot 1.734.

146
Valuing T-Bond Options with a Binomial Tree

• Value of American Futures Call
• If the futures call were American, then at the
  spot rate of 5 in period 1, its IV would be 2.88
  ( Max100.88-98,0), exceeding the binomial
  value of 2.743.
• Rolling the 2.88 value to the current period
  yields a price on the American futures option of
  1.798 ( .5(.9328) .5(2.88))