Managing Fixed-Income
Positions with OTC Derivatives
1
1. Hedging with OTC Derivatives
2. Hedging a Series of Cash Flows—OTC
Caps and Floors
3. Financing Caps and Floors: Collars and
Corridors
4. Other Interest Rate Derivatives
5. Hedging Currency Positions with
Currency Options
2
Hedging with OTC Derivatives
3
Forward Rate Agreements (FRA)
 A forward rate agreement, FRA, requires a cash
payment or provides a cash receipt based on the
difference between a realized spot rate such as the
LIBOR and a pre-specified rate.
 For example, the contract could be based on a
specified rate of Rk = 6% (annual) and the 3-month
LIBOR (annual) in 5 months and a notional principal,
NP (principal used only for calculation purposes) of
$10,000,000.
4
Forward Contracts and Forward
Rate Agreements (FRA)
 In five months the payoff would be
Payoff  ($10,000,000)
LIBOR
 .06(91 / 365)
1  LIBOR (91 / 365)
 If the LIBOR at the end of five months exceeds the
specified rate of 6%, the buyer of the FRA (or long
position holder) receives the payoff from the seller.
 If the LIBOR is less than 6%, the seller (or short
position holder) receives the payoff from the buyer.
5
Forward Contracts and Forward
Rate Agreements (FRA)
Payoff  ($10,000,000)
LIBOR
 .06(91 / 365)
1  LIBOR (91 / 365)
 If the LIBOR were at 6.5%, the buyer
would be entitled to a payoff of $12,267
from the seller;
 If the LIBOR were at 5.5%, the buyer
would be required to pay the seller
$12,297.
6
Forward Contracts and Forward
Rate Agreements (FRA)
 In general, a FRA that matures in T months and is
written on a M-month LIBOR rate is referred to as a
T x (T+M) agreement.
 Thus, in this example the FRA is a 5 x 8 agreement.
 At the maturity of the contract (T), the value of the
contract, VT is
VT 

LIBOR  R k (M / 365)
NP
1  LIBOR (M / 365)
7
Forward Contracts and Forward
Rate Agreements (FRA)
 FRAs originated in 1981 amongst large London
Eurodollar banks that used these forward agreements
to hedge their interest rate exposure.
 Today, FRAs are offered by banks and financial
institutions in major financial centers and are often
written for the bank’s corporate customers.
 They are customized contracts designed to meet the
needs of the corporation or financial institution.
8
Forward Contracts and Forward
Rate Agreements (FRA)
 FRAs are used by corporations and financial institutions to manage
interest rate risk in the same way as financial futures are used.
 Different from financial futures, FRAs are contracts between two
parties and therefore are subject to the credit risk of either party
defaulting.
 The customized FRAs are also less liquid than standardized futures
contracts.
 The banks that write FRAs often takes a position in the futures
market to hedge their position or a long and short position in spot
money market securities to lock in a forward rate.
 As a result, in writing the FRA, the specified rate Rk is often set
equal to the rate implied on a futures contract.
9
Forward Contracts and Forward
Rate Agreements (FRA)
Example:
 Suppose Kendall Manufacturing forecast a cash
inflow of $10,000,000 in 2 months that it is
considering investing in a Sun National Bank CD for
90 days.
 Sun National Bank’s jumbo CD pays a rate equal to
the LIBOR.
 Currently such rates are yielding 5.5%.
 Kendall is concerned that short-term interest rates
could decrease in the next 2 months and would like to
lock in a rate now.
10
Forward Contracts and Forward
Rate Agreements (FRA)
Example:

As an alternative to hedging its investment with Eurodollar futures, Sun
National suggests that Kendall hedge with a Forward Rate Agreement
with the following terms:
1. FRA would mature in 2 months (T) and would be written on a 90-day
(3-month) LIBOR (T x (T+M) = 2 x 5 agreement
2. NP = $10,000,000
3. Contract rate = Rk = 5.5%
4. Day count convention = 90/365
5. Cagle would take the short position on the FRA, receiving the payoff
from Sun National if the LIBOR were less than Rk = 5.5%
6. Sun National would take the long position on the FRA, receiving the
payoff from Cagle if the LIBOR were greater than Rk = 5.5%
11
Forward Contracts and Forward
Rate Agreements (FRA)
 The exhibit slide shows Kendall’s FRA receipts or
payments and cash flows from investing the
$10,000,000 cash inflow plus or minus the FRA
receipts or payments at possible LIBORs of 5%,
5.25%, 5%, 5.75%, and 6%.
 As shown, Kendall is able to earn a hedged rate of
return of 5.5% from its $10,000,000 investment.
12
Forward Contracts and Forward
Rate Agreements (FRA)
LIBOR
0.0500
0.0525
0.0550
0.0575
0.0600
Sun National
Payoff
-$12,178.62
-$6,085.60
$0.00
$6,078.21
$12,149.03
Payoff  ($10,000,000)
Kendall
Payoff
$12,178.62
$6,085.60
$0.00
-$6,078.21
-$12,149.03
Cagle CD Investment
$10m + FRA Payoff
$10,012,178.62
$10,006,085.60
$10,000,000.00
$9,993,921.79
$9,987,850.97
CF at CD Maturity
Hedged Rate
$10,135,616
$10,135,616
$10,135,616
$10,135,616
$10,135,616
0.0550
0.0550
0.0550
0.0550
0.0550
LIBOR
 .055(90 / 365)
1  LIBOR (90 / 365)
13
Interest Rate Call
 An interest rate call, also called a caplet, gives the buyer a payoff on a
specified payoff date if a designated interest rate, R, such as the
LIBOR, rises above a certain exercise rate, Rx.
 On the payoff date:
 If the designated rate is less than Rx, the interest rate call expires
worthless.
 If the rate exceeds Rx, the call pays off the difference between the
actual rate and Rx, times a notional principal, NP, times the
fraction of the year specified in the contract, θ.
Payoff  Max[R  R x ,0]()( NP)
14
Interest Rate Call
Example:
 Given an interest rate call with a designated rate of
LIBOR, Rx = 6%, NP = $1,000,000, time period of
180 days, and day-count convention of actual/360, the
buyer would receive a $5,000 payoff on the payoff
date if the LIBOR were 7%:
Payoff = Max[.07−.06, 0](180/360)($1,000,000)
Payoff = $5,000
15
Interest Rate Call
Hedging Use
 Interest rate call options are often written by
commercial banks in conjunction with futures
loans they plan to provide to their customers.
 The exercise rate on the option usually is set
near the current spot rate, with that rate often
being tied to the LIBOR.
16
Hedging a Future Loan Rate
with an OTC Interest Rate Call
Example:
 Suppose a construction company plans to finance one of
its project with a $10,000,000 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.
17
Hedging a Future Loan Rate
with an OTC Interest Rate Call
Example:

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:
1. Exercise rate = 7%
2. Reference rate = LIBOR
3. Time period applied to the payoff = 90/360
4. Notional principal = $10,000,000
5. Payoff made at the maturity date on the loan (90 days after
the option’s expiration)
6. Interest rate call’s expiration = T = 60 days (time of the
loan)
7. Interest rate call premium of $20,000 to be paid at the
option’s expiration with a 7% interest: Cost = $20,000(1 +
(.07)(60/360)) = $20,233
18
Hedging a Future Loan Rate
with an OTC Interest Rate Call
Example:
 The exhibit slide 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 slide, the company is
able to lock in a maximum interest cost of 8.016% if
the LIBOR is 7% or greater at expiration, and still
benefit with lower rates if the LIBOR is less than 7%.
19
Hedging a Future Loan Rate
with an OTC Interest Rate Call
Company's Loan: $10m at LIBOR+100BP for 90 days (.25 per year)
Interest Rate Call Option: Exercise Rate = 7%, Reference Rate = LIBOR, NP = $10m, Time Period = .25, Option Expiration = T= 60 days
Cost of Option = $20,000, payable at T plus 7% interest
1
2
3
4
5
6
LIBOR
Interest Rate Call
Cost of the Option at T
Interest Paid on
Cost at Maturity
Annualized Hedged Rate
Payoff: $10M[Max[LIBOR-.07,0]](.25)
$20,000(1+.07(60/360))
Loan at its Maturity
Col (4) - Col (2)
4[Col (5)/($10m-Col (3))]
(LIBOR+100bp)(.25)($10,000,000)
0.0550
$0
$20,233
$162,500
$162,500
0.06513
0.0575
$0
$20,233
$168,750
$168,750
0.06764
0.0600
$0
$20,233
$175,000
$175,000
0.07014
0.0625
$0
$20,233
$181,250
$181,250
0.07265
0.0650
$0
$20,233
$187,500
$187,500
0.07515
0.0675
$0
$20,233
$193,750
$193,750
0.07766
0.0700
$0
$20,233
$200,000
$200,000
0.08016
0.0725
$6,250
$20,233
$206,250
$200,000
0.08016
0.0750
$12,500
$20,233
$212,500
$200,000
0.08016
0.0775
$18,750
$20,233
$218,750
$200,000
0.08016
0.0800
$25,000
$20,233
$225,000
$200,000
0.08016
0.0825
$31,250
$20,233
$231,250
$200,000
0.08016
0.0850
$37,500
$20,233
$237,500
$200,000
0.08016
20
Interest Rate Put
 An interest rate put, also called a floorlet, gives the
buyer a payoff on a specified payoff date if a designated
interest rate, R, is below the exercise rate, Rx.
 On the payoff date:
 If the designated rate (or reference rate) is more than Rx, the
interest rate put expires worthless.
 If the reference rate is less than Rx, the put pays the difference
between Rx and the actual rate times a notional principal, NP,
times the fraction of the year, θ, specified in the contract.
Payoff  Max[R x  R,0]()( NP)
21
Interest Rate Put
Hedging Use
 A financial or non-financial corporation that is planning
to make an investment at some future date could hedge
that investment against interest rate decreases by
purchasing an interest rate put from a commercial bank,
investment banking firm, or dealer.
22
Hedging a CD Rate with
an OTC Interest Rate Put
Example:
 Suppose the ABC manufacturing company was
expecting a net cash inflow of $10,000,000 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 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

Example:
Suppose the put has the following terms:
1.
2.
3.
4.
5.
6.
7.
8.
Exercise rate = 7%
Reference rate = LIBOR
Time period applied to the payoff = θ = 90/360
Day Count Convention = 30/360
Notional principal = $10 million
Payoff made at the maturity date on the CD (90 days from
the option’s expiration)
Interest rate put’s expiration = T = 60 days (time of CD
purchase)
Interest rate put premium of $10,000 to be paid at the
option’s 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
Example:
 As shown in the exhibit slide, 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
Company's Investment: $10m at LIBOR for 90 days (.25 per year)
Interest Rate Put Option: Exercise Rate = 7%, Reference Rate = LIBOR, NP = $10m, time period = .25, option expiration = T = 60 days
Cost of Option = $10,000, payable at T plus 7% interest
1
LIBOR
2
3
4
5
6
Interest Rate Put
Cost of the Option at T
Interest Received on
Revenues at Maturity
Annualized Hedged Rate
Payoff: $10M[Max[.07−LIBOR,0](.25)
$10,000(1+.07(60/360))
CD at its Maturity
Col (2) + Col 4
4[Col (5)/($10m+Col (3))]
0.0550
$37,500
$10,117
$137,500
$175,000
0.06993
0.0575
$31,250
$10,117
$143,750
$175,000
0.06993
0.0600
$25,000
$10,117
$150,000
$175,000
0.06993
0.0625
$18,750
$10,117
$156,250
$175,000
0.06993
0.0650
$12,500
$10,117
$162,500
$175,000
0.06993
0.0675
$6,250
$10,117
$168,750
$175,000
0.06993
0.0700
$0
$10,117
$175,000
$175,000
0.06993
0.0725
$0
$10,117
$181,250
$181,250
0.07243
0.0750
$0
$10,117
$187,500
$187,500
0.07492
0.0775
$0
$10,117
$193,750
$193,750
0.07742
0.0800
$0
$10,117
$200,000
$200,000
0.07992
0.0825
$0
$10,117
$206,250
$206,250
0.08242
0.0850
$0
$10,117
$212,500
$212,500
0.08491
(LIBOR)(.25)($10,000,000)
26
Hedging a CD Rate with
an OTC Interest Rate Put
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:
$12,500 = ($10,000,000)[.07 − .065](90/360)
 The $12,500 payoff would offset the lower (than 7%) interest
paid on the company’s CD of $162,500:
$162,500 = ($10,000,000)(.065)(90/360)
 At the maturity of the CD, the company would therefore receive
CD interest and an interest rate put payoff equal to $175,000:
$175,000 = $162,500 + $12,500
27
Hedging a CD Rate with
an OTC Interest Rate Put
Example:
 With the interest-rate put’s 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 $10,000,000 plus the
$10,117 costs of the put, the hedged return equates to
an effective annualized yield of 6.993%:
6.993% = [(4)($175,000)]/[$10,000,000 + $10,117]
 On the other hand, if the LIBOR exceeds 7%, the
company benefits from the higher CD rates.
28
Cap
 A popular option offered by financial
institutions in the OTC market is the cap.
 A plain-vanilla cap is a series of European
interest rate call options—a portfolio of
caplets.
29
Cap
Example:
 A 7%, 2-year cap on a 3-month LIBOR, with a NP of
$100,000,000, provides, for the next 2 years, a payoff
every 3 months of (LIBOR − .07)(.25)($100M) if the
LIBOR on the reset date exceeds 7% and nothing if the
LIBOR equals or is less than 7%.
 Note: Typically, the payoff does not occur on the reset
date, but rather on the next reset date.
30
Cap
Uses
 Caps are often written by financial
institutions in conjunction with a floatingrate loan and are used by buyers as a hedge
against interest rate risk.
31
Cap
 A company with a floating-rate loan tied to the LIBOR
could lock in a maximum rate on the loan by buying a
cap corresponding to its loan.
 At each reset date, the company would receive a payoff
from the caplet if the LIBOR exceeded the cap rate,
offsetting the higher interest paid on the floating-rate
loan; on the other hand, if rates decrease, the company
would pay a lower rate on its loan whereas its losses on
the caplet would be limited to the cost of the option.
 Thus, with a cap, the company is able to lock in a
maximum rate each quarter, and yet still benefit with
lower interest costs if rates decrease.
32
Floor
 A plain-vanilla floor is a series of European
interest rate put options—a portfolio of
floorlets.
33
Floor
Example:
 A 7%, 2-year floor on a 3-month LIBOR, with a NP of
$100,000,000, provides for the next 2 years a payoff
every 3 months of (.07 − LIBOR)(.25)($100M) if the
LIBOR on the reset date is less than 7% and nothing if
the LIBOR equals or exceeds 7%.
34
Floor
Uses
 Floors are often purchased by investors as a tool to
hedge their floating-rate investment against interest
rate declines.
 Thus, with a floor, an investor with a floating-rate
security is able to lock in a minimum rate each
period, and yet still benefit with higher yields if rates
increase.
35
Hedging a Series of Cash Flows:
OTC Caps and Floors
36
Hedging a Series of Cash Flows:
OTC Caps and Floors
 We have examined how a strip of Eurodollar futures
puts can be used to cap the rate paid on a floating-rate
loan, and how a strip of Eurodollar futures calls can be
used to set a floor on a floating-rate investment.
 Using such exchange-traded options to establish
interest rate floors and ceiling on floating rate assets
and liabilities, though, 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.
37
Hedging a Series of Cash Flows:
OTC Caps and Floors
 Financial institutions typically provide caps
and floors with:
1.
2.
3.
4.
5.
Terms that range from 1 to 5 years
Monthly, quarterly, or semiannual reset dates
LIBOR as the reference rate
Notional principal and the reset dates that often
match the specific investment or loan
Settlement dates that usually come after the reset
dates
38
Hedging a Series of Cash Flows:
OTC Caps and Floors
 In cases where a floating-rate loan (or investment)
and cap (or floor) come from the same financial
institution, the loan and cap (or investment and
floor) are usually treated as a single instrument so
that when there is a payoff, it occurs at an interest
payment (receipt) date, lowering (increasing) the
payment (receipt).
 The exercise rate is often set so that the cap or floor
is initially out of the money, and the payments for
these interest rate products are usually made up
front, although some are amortized.
39
Floating Rate Loan
Hedged with an OTC Cap
Example:
 Suppose the Diamond Development Company borrows
$50 million from Commerce Bank to finance a 2-year
construction project.
 Suppose:
 The loan is for 2 years
 The loan starts on March 1 at a known rate of 8%
 The loan rate resets every three months—6/1, 9/1,
12/1, and 3/1—at the prevailing LIBOR plus 150 bp.
40
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,
but still benefit from lower rates if the LIBOR
decreases.
41
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:
1.
2.
3.
4.
5.
6.
7.
The cap consist of seven caplets with the first expiring on
6/1/Y1 and the others coinciding with the loan’s reset dates.
Exercise rate on each caplet = 8%
NP on each caplet = $50,000,000
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 loan’s 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/Y1.
42
Floating Rate Loan
Hedged with an OTC Cap
 On each reset date, the payoff on the corresponding caplet would
be
Payoff = ($50,000,000) (Max[LIBOR − .08, 0])(90/360)
 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%, but still benefit with lower interest costs if rates decrease.
 This can be seen in the exhibit slide, 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.
43
Floating Rate Loan
Hedged with an OTC Cap
Loan: Floating Rate Loan; Term = 2 years; Reset dates: 3/1, 6/1, 9/1, 12/1; Time frequency = .25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date
Cap: Cost of cap =$150,000; Cap Rate = 8%; Reference Rate = LIBOR; Time frequency = .25; Caplets' Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date.
1
2
3
4
5
6
7
Reset Date
Assumed LIBOR
Loan Interest on Payment Date
Cap Payoff on Payment Date
Hedged Interest Payment
Hedged Rate
Unhedged Rate
(LIBOR + 150bp)(.25)($50m)
(Max[LIBOR−.08,0])(.25)($50m)
Col. (3) − Col. (4)
4[Col (5)/$50m]
LIBOR + 150bp
3/1/Y1n
0.065
6/1/Y1
0.070
$1,000,000
$0
$1,000,000
0.080
0.080
9/1/Y1
0.075
$1,062,500
$0
$1,062,500
0.085
0.085
12/1/Y1
0.080
$1,125,000
$0
$1,125,000
0.090
0.090
3/1/Y2
0.085
$1,187,500
$0
$1,187,500
0.095
0.095
6/1/Y2
0.090
$1,250,000
$62,500
$1,187,500
0.095
0.100
9/1/Y2
0.095
$1,312,500
$125,000
$1,187,500
0.095
0.105
12/1/Y2
0.100
$1,375,000
$187,500
$1,187,500
0.095
0.110
$1,437,500
$250,000
$1,187,500
0.095
0.115
3/1/Y3
n
There is no caplet for this date
44
Floating Rate Loan
Hedged with an OTC Cap
 For the 5 reset dates from 12/1/Y1 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 2 reset dates on the loan, 6/1/Y1 and
9/1/Y1, 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.
45
Floating Rate Asset
Hedged with an OTC Floor
Example:
 As noted, floors are purchased to create a minimum
rate on a floating-rate asset.
 As an example, suppose the Commerce Bank in the
preceding example wanted to establish a minimum
rate or floor on the rates it was to receive on the 2year floating-rate loan it made to the Diamond
Company.
46
Floating Rate Asset
Hedged with an OTC Floor

Suppose the bank purchased from another financial
institution a floor for $100,000 with the following
terms corresponding to its floating-rate asset:
1.
2.
3.
4.
5.
6.
The floor consist of 7 floorlets with the first expiring on
6/1/Y1 and the others coinciding with the reset dates on the
bank’s floating-rate loan to the Diamond Company
Exercise rate on each floorlet = 8%
NP on each floorlet = $50,000,000
Reference Rate = LIBOR
Time period to apply to payoff on each floorlet = 90/360
Payment date on each floorlet is at the loan’s 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/Y1
47
Floating Rate Asset
Hedged with an OTC Floor
 On each reset date, the payoff on the corresponding
floorlet would be
Payoff = ($50,000,000) (Max[.08 − LIBOR, 0])(90/360)
 With the 8% exercise rate, Commerce Bank would be
able to lock in a minimum rate each quarter equal to the
floor rate plus the basis points on the floating-rate asset,
9.5%, but still benefit with higher returns if rates
increase.
48
Floating Rate Asset
Hedged with an OTC Floor
 In the exhibit slide, Commerce Bank’s 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.
49
Floating Rate Asset
Hedged with an OTC Floor
Asset: Floating-rate loan made by bank; Term = 2 years; Reset dates: 3/1, 6/1, 9/1, 12/1; Time frequency = .25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date
Floor: Cost of floor =$100,000; Floor Rate = 8%; Reference Rate = LIBOR; Time frequency = .25; Floorlets' expirations: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date.
1
2
Reset Date
3
4
5
6
7
Hedged Interest Income
Hedged Rate
Unhedged Rate
(Max[.08−LIBOR,0})(.25)($50m)
Col. (3) + Col. (4)
4[Col (5)/$50m]
LIBOR + 150BP
Assumed LIBOR Interest Received on Payment Date
Floor Payoff on Payment Date
(LIBOR + 150bp)(.25)($50m)
3/1/Y1n
0.065
6/1/Y1
0.070
$1,000,000
$0
$1,000,000
0.080
0.080
9/1/Y1
0.075
$1,062,500
$125,000
$1,187,500
0.095
0.085
12/1/Y1
0.080
$1,125,000
$62,500
$1,187,500
0.095
0.090
3/1/Y2
0.085
$1,187,500
$0
$1,187,500
0.095
0.095
6/1/Y2
0.090
$1,250,000
$0
$1,250,000
0.100
0.100
9/1/Y2
0.095
$1,312,500
$0
$1,312,500
0.105
0.105
12/1/Y2
0.100
$1,375,000
$0
$1,375,000
0.110
0.110
$1,437,500
$0
$1,437,500
0.115
0.115
3/1/Y3
n
There is no floorlet for this date
50
Floating Rate Asset
Hedged with an OTC Floor
 For the first two reset dates on the loan, 6/1/Y1 and 9/1/Y1,
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 bank’s 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/Y1 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.
51
Financing Caps and Floors:
Collars and Corridors
52
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 the preceding 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.
53
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:
1. Effective dates on floorlet = reset date on loan
2. Reference rate = LIBOR
3. NP on floorlets = $50,000,000
4. Time period for rates = .25
54
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 slide, it still locked 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/Y1, Diamond has to pay
$125,000 ninety days later on its short floor position.
 When the LIBOR is at 6.5% on 9/1/Y1, 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
bp) it pays on its floating rate loan.
55
Collars
Loan: Floating Rate Loan; Term = 2 years; Reset dates: 3/1, 6/1, 9/1, 12/1; Time frequency = .25; Rate = LIBOR + 150BP; Payment Date = 90 days after reset date
Cap Purchase: Cost of cap =$150,000; Cap Rate = 8%; Reference Rate = LIBOR; Time frequency = .25;
Caplets' Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date.
Floor Sale: Sale of floor = $70,000; Floor rate = 7%; Reference rate = LIBOR; Time frequency = .25;
Floorlets' expiration: On loan reset dates, starting at 61/Y1; Payoff date = 90 days after reset date.
1
2
3
4
5
6
7
8
Reset Date
Assumed LIBOR
Loan Interest
Cap Payoff
Floor Payment
Hedged Interest Payment
Hedged Rate
Unhedged Rate
(LIBOR + 150bp)(.25)($50m)
Max[LIBOR-.08,0](.25)($50m)
Max[.07-LIBOR,0](.25)($50m)
Col. (3) - Col. (4) + Col (5)
4[Col (6)/$50m]
LIBOR + 150BP
3/1/Y1
0.050
6/1/Y1
0.060
$812,500
$0
$0
$812,500
0.065
0.065
9/1/Y1
0.065
$937,500
$0
$125,000
$1,062,500
0.085
0.075
12/1/Y1
0.070
$1,000,000
$0
$62,500
$1,062,500
0.085
0.080
3/1/Y2
0.075
$1,062,500
$0
$0
$1,062,500
0.085
0.085
6/1/Y2
0.080
$1,125,000
$0
$0
$1,125,000
0.090
0.090
9/1/Y2
0.085
$1,187,500
$0
$0
$1,187,500
0.095
0.095
12/1/Y2
0.090
$1,250,000
$62,500
$0
$1,187,500
0.095
0.100
$1,312,500
$125,000
$0
$1,187,500
0.095
0.105
3/1/Y3
Loan interest, cap payoff, and floor payment made on payment date
56
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.
57
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.
58
Corridor
 For example, instead of selling a 7% floor for
$70,000 to partially finance the $150,000 cost of its
8% cap, the Diamond company 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.
59
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 the 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.
60
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 slide, Commerce would lock in
an effective minimum rate on its a asset of 9.5% and an
effective maximum rate of 10.5%.
61
Reverse Collar
Asset: Floating rate loan made by bank; Term = 2 years; Reset dates: 3/1, 6/1, 9/1, 12/1; Time frequency = .25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date
Floor Purchase: Cost of floor =$100,000; Floor Rate = 8%; Reference Rate = LIBOR; Time frequency = .25;
Floorlets' expirations: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date.
Cap Sale: revenue from cap =$70,000; Cap Rate = 9%; Reference Rate = LIBOR; Time frequency = .25;
Caplets' Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date.
1
2
3
4
5
6
7
8
Reset Date
Assumed LIBOR
Interest Received
Floor Payoff
Cap Payment
Hedged Interest Income
Hedged Rate
Unhedged Rate
(LIBOR + 150bp)(.25)($50m)
Max[.08-LIBOR,0](.25)($50m)
Max[LIBOR-.09,0](.25)($50m)
Col. (3) + Col. (4) - Col (5)
4[Col (6)/$50m]
LIBOR + 150BP
3/1/Y1
0.065
6/1/Y1
0.070
$1,000,000
$0
$1,000,000
0.080
0.080
9/1/Y1
0.075
$1,062,500
$125,000
$0
$1,187,500
0.095
0.085
12/1/Y1
0.080
$1,125,000
$62,500
$0
$1,187,500
0.095
0.090
3/1/Y2
0.085
$1,187,500
$0
$0
$1,187,500
0.095
0.095
6/1/Y2
0.090
$1,250,000
$0
$0
$1,250,000
0.100
0.100
9/1/Y2
0.095
$1,312,500
$0
$0
$1,312,500
0.105
0.105
12/1/Y2
0.100
$1,375,000
$0
$62,500
$1,312,500
0.105
0.110
$1,437,500
$0
$125,000
$1,312,500
0.105
0.115
3/1/Y3
Interest Received, floor payoff, and cap payment made on payment date
62
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.
63
Other Interest Rate Products
64
Other Interest Rate Products
 Caps and floors are one of the more popular interest
rate products offered by the OTC derivative market.
 In addition to these derivatives, a number of other
interest rate products have been created over the last
decade to meet the many different interest rate hedging
needs.
 Many of these products are variations of the generic
OTC caps and floors—exotic options; two of these to
note are barrier options and path-dependent options.
65
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:
1. Knock-out option is one that ceases to exist once
the specified barrier rate or price is reached.
2. Knock-in option is one that comes into existence
when the reference rate or price hits the barrier
level.
66
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
67
Barrier Options
 Barrier caps and floors with termination or
creation features are offered in the OTC
market at a premium above comparable
caps and floors without such features.
68
Barrier Options
 Down-and-out caps and floors are options that
ceases to exist once rates hit a certain level.
 Example:
A 2-year, 8% cap that ceases when the
LIBOR hits 6.5%
A 2-year, 8% floor that ceases once the
LIBOR hits 9%
69
Barrier Options
 Up-and-in cap and florr is one that becomes effective
once rates hit a certain level.
 Examples:
 A 2-year, 8% cap that that becomes effective when
the LIBOR hits 9%
 A 2-year, 8% floor that become effective when
rates hit 6.5%
70
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.
71
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.
72
Path-Dependent Options: Average Cap
Example:
 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.
73
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.
74
Path-Dependent Options: Q-Cap
Example:
 Suppose the Diamond Company in the earlier cap
example decided to hedge its 2-year floating rate loan
(paying LIBOR + 150bp) by buying a Q-Cap from
Commerce Bank with the following terms (next 2
slides):
75
Path-Dependent Options: Average Cap
Q-Cap Terms:
1. The cap consist of seven caplets with the first
expiring on 6/1/Y1 and the others coinciding with
the loan’s reset dates
2.
Exercise rates on each caplet = 8%
3.
NP on each caplet = $50,000,000
4.
Reference Rate = LIBOR
5. Time period to apply to payoff on each caplet =
90/360
76
Path-Dependent Options: Average Cap
Q-Cap Terms:
6.
For the period 3/1/Y1 to 12/1/Y1, the caplet will payoff
when the cumulative interest starting from loan date 3/1/Y1
on the company’s loan hits $3 million.
7.
For the period 3/1/Y2 to 12/1/Y2, the caplet will payoff
when the cumulative interest starting from date 3/1/Y2 on
the company’s loan hits $3 million.
8.
Payment date on each caplet is at the loan’s interest
payment date, 90 days after the reset date.
9.
The cost of the cap = $125,000; it is paid at beginning of
the loan, 3/1/Y1.
77
Path-Dependent Options: Q-Cap
 The exhibit slide shows the quarterly interest, cumulative
interests, Q-cap payments, and effective interests for assumed
LIBORs.
 In the Q-caps first protection period, 3/1/Y1 to 12/1/Y1,
Commerce Bank will pay the Diamond Company on its 8%
caplet when the cumulative interest hits $3 million.
 The cumulative interest hits the $3 million limit on reset date
9/1/Y1, but on that date the 9/1/Y1 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 Diamond’s loan.
78
Path-Dependent Options: Q-Cap
 In the second protection period, 3/1/Y2 to 12/1/Y2,
the assumed LIBOR rates are higher.
 The cumulative interest hits the $3 million limit on
reset date 9/1/Y1. The caplet on that date and the
caplet on the next reset date (12/1/Y1) 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 2
payment periods on its loan.
79
Path-Dependent Options: Q-Cap
Loan: Floating Rate Loan; Term = 2 years; Reset dates: 3/1, 6/1, 9/1, 12/1; Time frequency = .25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date
Q-Cap: Cost of Q-cap =$125,000; Cap Rate = 8%; Reference Rate = LIBOR; Time frequency = .25; Caplets' Expiration: On loan reset dates, starting at 6/1/Y1;
Payoff made 90 days after reset date; Cap become effective once cumulative interest reaches $3M; protection periods: Y1 and Y2.
1
2
3
4
5
6
7
Reset Date
Assumed LIBOR
Interest to be paid at next reset date
Cumulative Interest
Q-Cap Payment
Hedged Interest Payment
Hedged Rate
to be paid at next reset date
at payment date: Col (3) −Col (5)
4[Col (6)/$50m]
(LIBOR + 150bp)(.25)($50m)
3/1/Y1
0.070
$1,062,500
$1,062,500
$0
6/1/Y1
0.075
$1,125,000
$2,187,500
$0
$1,062,500
0.085
9/1/Y1
0.080
$1,187,500
$3,375,000
$0
$1,125,000
0.090
12/1/Y1
0.085
$1,250,000
$4,625,000
$62,500
$1,187,500
0.095
3/1/Y2
0.085
$1,250,000
$1,250,000
$0
$1,187,500
0.095
6/1/Y2
0.090
$1,312,500
$2,562,500
$0
$1,250,000
0.100
9/1/Y2
0.095
$1,375,000
$3,937,500
$187,500
$1,312,500
0.105
12/1/Y2
0.100
$1,437,500
$5,375,000
$250,000
$1,187,500
0.095
$1,187,500
0.095
3/1/Y3
80
Path-Dependent Options: Q-Cap
 When compared to a standard cap, the Q-cap provides protection
for the 1-year protection periods, whereas the standard cap
provides protection for each period (quarter).
 As shown in the next exhibit slide, a standard 8% cap provides
more protection than the Q-cap, capping the loan at 9.5% from
date 12/1/Y1 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.
81
Path-Dependent Options: Q-Cap
Loan: Floating Rate Loan; Term = 2 years; Reset dates: 3/1, 6/1, 9/1, 12/1; Time frequency = .25; Rate = LIBOR + 150bp; Payment Date = 90 days after reset date
Cap: Cost of cap =$150,000; Cap Rate = 8%; Reference Rate = LIBOR; Time frequency = .25;
Cap: Caplets' Expiration: On loan reset dates, starting at 6/1/Y1; Payoff made 90 days after reset date.
Q-Cap: Cost of Q-cap =$125,000; Cap Rate = 8%; Reference Rate = LIBOR; Time frequency = .25; Caplets' Expiration: On loan reset dates, starting at 6/1/Y1;
Payoff made 90 days after reset date; Cap become effective once cumulative interest reaches $3M; protection periods: Y1 and Y2.
1
2
3
4
5
6
Reset Date
Assumed LIBOR
Loan Interest
Unhedged Interest
Cumulative Interest
Unhedged Loan Rate
3/1/Y1
0.070
$1,062,500
6/1/Y1
0.075
$1,125,000
0.085
1062500
2187500
0.085
9/1/Y1
0.080
$1,187,500
0.090
3375000
0.090
12/1/Y1
0.085
$1,250,000
0.095
4625000
0.095
3/1/Y2
0.085
$1,250,000
0.100
1250000
0.100
6/1/Y2
0.090
$1,312,500
0.100
2562500
0.100
9/1/Y2
0.095
$1,375,000
0.105
3937500
0.105
12/1/Y2
0.100
$1,437,500
0.110
5375000
0.110
3/1/Y3
0.115
0.115
1
7
8
9
10
11
Reset Date
Q-Cap Payment
Hedged Interest Payment
Q-Cap Hedged Rate
Cap-Payments
Cap-Hedged Rate
3/1/Y1
$0
6/1/Y1
$0
1062500
0.085
$0
0.085
at payment date: Col (3)- Col (5)
9/1/Y1
$0
1125000
0.090
$0
0.090
12/1/Y1
$62,500
1187500
0.095
$62,500
0.095
3/1/Y2
$0
1187500
0.095
$62,500
0.095
6/1/Y2
$0
1250000
0.100
$125,000
0.095
9/1/Y2
$187,500
1312500
0.105
$187,500
0.095
12/1/Y2
$250,000
1187500
0.095
$250,000
0.095
1187500
0.095
3/1/Y3
0.095
82
Exotic Options
 Q-caps, average caps, knock-in options, and
knock-out options are sometimes referred to
as exotic options.
 Exotic option products are nongeneric
products that are created by financial
engineers to meet specific hedging needs and
return-risk profiles.
83
Exotic Options
 Chooser Option: Option that gives the holder the right to choose
whether the option is a call or a put after a specified period of time.
 Bermudan Option: An option in which early exercise is restricted to
certain dates.
 Forward Start Option: An option that will start at some time in the
future.
 Trigger Option: An option that depends on another index; that is,
whether the option is in the money depends on value of another index.
84
Exotic Options
 Asian Option: An option in which the payoff depends on the average
price of the underlying asset during some part of the option’s life:
Call: IV = Max[Sav – X,0]; put: IV = Max[X - Sav,0].
 Lookback Option: An option in which the payoff depends on the
minimum or maximum price reached during the life of the option.
 Binary Option: An option with a discontinuous payoff such as a
payoff or nothing. For example: If the price is equal or less than X,
the option pays nothing; if the price exceeds X, the option pays a fixed
amount.
 Compound Option is an option on an option: Call on a call, call on
put, put on put, and put on call.
85
Exotic Options
 Caption: An option on a cap.
 Floortion: An option on a floor.
 Yield Curve Option: An option between two points on a yield curve.
For example, a yield curve with a exercise equal to 200 basis point on
the difference between the yields on two-year and 10-year notes:
Payoff = Max[(YTM10 – YTM2) – .02, 0]NP.
86