Ch. 13: 1

1
Chapter 13 Interest Rate Forwards and Options

• Well, it helps to look at derivatives like atoms.
  Split them one way and you have heat and energy
  - useful stuff. Split them another way and you
  have a bomb. You have to understand the
  subtleties.
• Kate Jennings
• Moral Hazard, 2002, p. 8

2
Important Concepts

• The notion of a derivative on an interest rate
• Pricing, valuation, and use of forward rate
  agreements (FRAs), interest rate options,
  swaptions, and forward swaps

3

• A derivative on an interest rate
• The payoff of a derivative on a bond is based on
  the price of the bond relative to a fixed price.
• The payoff of a derivative on an interest rate is
  based on the interest rate relative to a fixed
  interest rate.
• In some cases these can be shown to be the same,
  particularly in the case of a discount
  instrument. In most other cases, however, a
  derivative on an interest rate is a different
  instrument than a derivative on a bond.
• See Figure 13.1, p. 446 for notional principal of
  FRAs and interest rate options over time.

4
Forward Rate Agreements (FRA)

• Definition
• A forward contract in which the underlying is an
  interest rate
• A forward rate agreement (FRA) is an agreement
  that a certain rate will apply to a certain
  principal during a certain future time period
• An FRA can work better than a forward or futures
  on a bond, because its payoff is tied directly to
  the source of risk, the interest rate.

5
Forward Rate Agreements (continued)

• The Structure and Use of a Typical FRA
• Underlying is usually LIBOR
• Payoff is made at expiration (contrast with
  swaps) and discounted. For FRA on m-day LIBOR,
  the payoff is
• Example Long an FRA on 90-day LIBOR expiring in
  30 days. Notional principal of 20 million.
  Agreed upon rate is 10 percent. Payoff will be

6
Forward Rate Agreements (continued)

• Some possible payoffs. If LIBOR at expiration is
  8 percent,
• So the long has to pay 98,039. If LIBOR at
  expiration is 12 percent, the payoff is
• Note the terminology of FRAs A ? B means FRA
  expires in A months and underlying matures in B
  months.

7
Zero Rates

• A zero rate (or spot rate), for maturity T is the
  rate of interest earned on an investment that
  provides a payoff only at time T

8
Example

9
Forward Rates

• The forward rate is the future zero rate implied
  by todays term structure of interest rates

10
Formula for Forward Rates

• Suppose that the zero rates for time periods T1
  and T2 are R1 and R2 with both rates continuously
  compounded.
• The forward rate for the period between times T1
  and T2 is

11
Calculation of Forward Rates

Zero Rate for
Forward Rate
an
n
-year Investment
for
n
th Year
Year (
n
)
( per annum)
( per annum)
1
3.0
2
4.0
5.0
3
4.6
5.8
4
5.0
6.2
5
5.3
6.5
12
FRA Valuation

• An FRA is equivalent to an agreement where
  interest at a predetermined rate, RK is exchanged
  for interest at the market rate
• An FRA can be valued by assuming that the forward
  interest rate is certain to be realized

13
Valuation Formulas

• Value of FRA where a fixed rate RK will be
  received on a principal L between times T1 and T2
  is
• Value of FRA where a fixed rate is paid is
• RF is the forward rate for the period and R2 is
  the zero rate for maturity T2

14
Example FRA Valuation

• Suppose that the three-month LIBOR rate is 5 and
  the six-month LIBOR rate is 5.5 with continuous
  compounding. Consider an FRA where you will
  receive a rate 7 measured with quarterly
  compounding, on a principal of 1 million between
  the end of month 3 and the end of month 6. The
  forward rate is 6 percent with continuous
  compounding or 6.0452 with quarterly compounding.
  The value of the FRA is
• 1,000,000 x (0.07 0.0652) x 0.25 x e-0.055 x
  0.5 2,322

15
Forward Rate Agreements (continued)

• Applications of FRAs
• FRA users are typically borrowers or lenders with
  a single future date on which they are exposed to
  interest rate risk.
• See Table 13.3, p. 452 and Figure 13.2, p. 453
  for an example.
• Note that a series of FRAs is similar to a swap
  however, in a swap all payments are at the same
  rate. Each FRA in a series would be priced at
  different rates (unless the term structure is
  flat). You could, however, set the fixed rate at
  a different rate (called an off-market FRA).
  Then a swap would be a series of off-market FRAs.

16
Interest Rate Options

• Definition an option in which the underlying is
  an interest rate it provides the right to make a
  fixed interest payment and receive a floating
  interest payment or the right to make a floating
  interest payment and receive a fixed interest
  payment.
• The fixed rate is called the exercise rate.
• Most are European-style.

17
Interest Rate Options (continued)

• The Structure and Use of a Typical Interest Rate
  Option
• With an exercise rate of X, the payoff of an
  interest rate call is
• The payoff of an interest rate put is
• The payoff occurs m days after expiration.
• Example notional principal of 20 million,
  expiration in 30 days, underlying of 90-day
  LIBOR, exercise rate of 10 percent.

18
Interest Rate Options (continued)

• The Structure and Use of a Typical Interest Rate
  Option (continued)
• If LIBOR is 6 percent at expiration, payoff of a
  call is
• The payoff of a put is
• If LIBOR is 14 percent at expiration, payoff of a
  call is
• The payoff of a put is

19
Interest Rate Options (continued)

• Pricing and Valuation of Interest Rate Options
• A difficult task binomial models are preferred,
  but the Black model is sometimes used with the
  forward rate as the underlying.
• When the result is obtained from the Black model,
  you must discount at the forward rate over m days
  to reflect the deferred payoff.
• Then to convert to the premium, multiply by
  (notional principal)(days/360).
• See Table 13.4, p. 456 for illustration.

20
Interest Rate Options (continued)

• Interest Rate Option Strategies
• See Table 13.5, p. 458 and Figure 13.3, p. 459
  for an example of the use of an interest rate
  call by a borrower to hedge an anticipated loan.
• See Table 13.6, p. 460 and Figure 13.4, p. 461
  for an example of the use of an interest rate put
  by a lender to hedge an anticipated loan.

21
Interest Rate Options (continued)

• Interest Rate Caps, Floors, and Collars
• A combination of interest rate calls used by a
  borrower to hedge a floating-rate loan is called
  an interest rate cap. The component calls are
  referred to as caplets.
• A combination of interest rate puts used by a
  lender to hedge a floating-rate loan is called an
  interest rate floor. The component puts are
  referred to as floorlets.
• A combination of a long cap and short floor at
  different exercise prices is called an interest
  rate collar.

22
Interest Rate Options (continued)

• Interest Rate Caps, Floors, and Collars
  (continued)
• Interest Rate Cap
• Each component caplet pays off independently of
  the others.
• See Table 13.7, p. 463 for an example of a
  borrower using an interest rate cap.
• To price caps, price each component caplet
  individually and add up the prices of the caplets.

23
Interest Rate Options (continued)

• Interest Rate Caps, Floors, and Collars
  (continued)
• Interest Rate Floor
• Each component floorlet pays off independently of
  the others
• See Table 13.8, p. 464 for an example of a lender
  using an interest rate floor.
• To price floors, price each component floorlet
  individually and add up the prices of the
  floorlets.

24
Interest Rate Options (continued)

• Interest Rate Caps, Floors, and Collars
  (continued)
• Interest Rate Collars
• A borrower using a long cap can combine it with a
  short floor so that the floor premium offsets the
  cap premium. If the floor premium precisely
  equals the cap premium, there is no cash cost up
  front. This is called a zero-cost collar.
• The exercise rate on the floor is set so that the
  premium on the floor offsets the premium on the
  cap.
• By selling the floor, however, the borrower gives
  up gains from falling interest rates below the
  floor exercise rate.
• See Table 13.9, p. 466 for example.

25
Interest Rate Options (continued)

• Interest Rate Options, FRAs, and Swaps
• Recall that a swap is like a series of off-market
  FRAs.
• Now compare a swap to interest rate options. On
  a settlement date, the payoff of a long call is
• 0 if LIBOR ? X
• LIBOR X if LIBOR X
• The payoff of a short put is
• - (X LIBOR) if LIBOR ? X
• 0 if LIBOR X
• These combine to equal LIBOR X. If X is set at
  R, which is the swap fixed rate, the long cap and
  short floor replicate the swap.

26
Interest Rate Swaptions and Forward Swaps

• Definition of a swaption an option to enter
  into a swap at a fixed rate.
• Payer swaption an option to enter into a swap
  as a fixed-rate payer
• Receiver swaption an option to enter into a
  swap as a fixed-rate receiver

27
Interest Rate Swaptions and Forward Swaps
(continued)

• The Structure of a Typical Interest Rate Swaption
• Example MPK considers the need to engage in a
  10 million three-year swap in two years.
  Worried about rising rates, it buys a payer
  swaption at an exercise rate of 11.5 percent.
  Swap payments will be annual.
• At expiration, the following rates occur
  (Eurodollar zero coupon bond prices in
  parentheses)
• 360 day rate .12 (0.8929)
• 720 day rate .1328 (0.7901)
• 1080 day rate .1451 (0.6967)

28
Interest Rate Swaptions and Forward Swaps
(continued)

• The Structure of a Typical Interest Rate Swaption
  (continued)
• The rate on 3-year swaps is, therefore,
• So MPK could enter into a swap at 12.75 percent
  in the market or exercise the swaption and enter
  into a swap at 11.5 percent. Obviously it would
  exercise the swaption. What is the swaption
  worth?

29
Interest Rate Swaptions and Forward Swaps
(continued)

• The Structure of a Typical Interest Rate Swaption
  (continued)
• Exercise would create a stream of 11.5 percent
  fixed payments and LIBOR floating receipts. MPK
  could then enter into the opposite swap in the
  market to receive 12.75 fixed and pay LIBOR
  floating. The LIBORs offset leaving a three-year
  annuity of 12.75 11.5 1.25 percent, or
  125,000 on 10 million notional principal. The
  value of this stream of payments is
• 125,000(0.8929 0.7901 0.6967) 297,463

30
Interest Rate Swaptions and Forward Swaps
(continued)

• The Structure of a Typical Interest Rate Swaption
  (continued)
• In general, the value of a payer swaption at
  expiration is
• The value of a receiver swaption at expiration is

31
Interest Rate Swaptions and Forward Swaps
(continued)

• The Equivalence of Swaptions and Options on Bonds
• Using the above example, substituting the formula
  for the swap rate in the market, R, into the
  formula for the payoff of a swaption gives
• Max(0,1 0.6967 - 0.115(0.8929 0.7901
  0.6967))
• This is the formula for the payoff of a put
  option on a bond with 11.5 percent coupon where
  the option has an exercise price of par. So
  payer swaptions are equivalent to puts on bonds.
  Similarly, receiver swaptions are equivalent to
  calls on bonds.

32
Swaption and Callable Bonds

• One application of swaptions relates to callable
  bonds
• Recall callable bond issuer has sold (issued)
  bonds and purchased a call option
• A receiver swaption is comparable to the embedded
  call option of a bond
• See Figure 13.5, p. 474

33
Interest Rate Swaptions and Forward Swaps
(continued)

• Forward Swaps
• Definition a forward contract to enter into a
  swap a forward swap commits the parties to
  entering into a swap at a later date at a rate
  agreed on today.
• Example The MPK situation previously described.
  Let MPK commit to a three-year pay-fixed,
  receive-floating swap in two years. To find the
  fixed rate at the time the forward swap is agreed
  to, we need the term structure of rates for one
  through five years (Eurodollar zero coupon bond
  prices shown in parentheses).

34
Interest Rate Swaptions and Forward Swaps
(continued)

• Forward Swaps (continued)
• 360 days 0.09 (0.9174)
• 720 days 0.1006 (0.8325)
• 1080 days 0.1103 (0.7514)
• 1440 days 0.12 (0.6757)
• 1800 days 0.1295 (0.6070)
• We need the forward rates two years ahead for
  periods of one, two, and three years.

35
Interest Rate Swaptions and Forward Swaps
(continued)

• Forward Swaps (continued)

36
Interest Rate Swaptions and Forward Swaps
(continued)

• Forward Swaps (continued)
• The Eurodollar zero coupon (forward) bond prices

37
Interest Rate Swaptions and Forward Swaps
(continued)

• Forward Swaps (continued)
• The rate on the forward swap would be

38
Interest Rate Swaptions and Forward Swaps
(continued)

• Applications of Swaptions and Forward Swaps
• Anticipation of the need for a swap in the future
• Swaption can be used
• To exit a swap
• As a substitute for an option on a bond
• Creating synthetic callable or puttable debt
• Remember that forward swaps commit the parties to
  a swap but require no cash payment up front.
  Options give one party the choice of entering
  into a swap but require payment of a premium up
  front.

39
(Return to text slide)
40
(Return to text slide)
41
(Return to text slide)
42
(Return to text slide)
43
(Return to text slide)
44
(Return to text slide)
45
(Return to text slide)
46
(Return to text slide)
47
(Return to text slide)
48
(Return to text slide)
49
(Return to text slide)
50
(Return to text slide)