Ch. 13: 1

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Chapter 13 Interest Rate Forwards and Options

• If a deal was mathematically complex in 1993 and
  1994, that was considered innovation. But this
  year, what took you forward with clients wasnt
  the math it was bringing them the most
  efficient application of a product.
• Mark Wells
• Risk, January, 1996, p. R15

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Important Concepts in Chapter 13

• 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

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• 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 different on a bond.
• See Figure 13.1, p. 467 for notional principal of
  FRAs and interest rate options over time.

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Forward Rate Agreements

• Definition
• A forward contract in which the underlying is an
  interest rate
• 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.

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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

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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.

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Forward Rate Agreements (continued)

• The Pricing and Valuation of FRAs
• Let F be the rate the parties agree on, h be the
  expiration day, and the underlying be an m-day
  rate. L0(h) is spot rate on day 0 for h days,
  L0(hm) is spot rate on day 0 for h m days.
  Assume notional principal of 1.
• To find the fixed rate, we must replicate an FRA
• Short a Eurodollar maturing in hm days that pays
  1 F(m/360). This is a loan that can be paid
  off early or transferred to another party
• Long a Eurodollar maturing in h days that pays 1

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Forward Rate Agreements (continued)

• The Pricing and Valuation of FRAs (continued)
• On day h,
• Loan we owe has a market value of
• Pay if off early. Collect 1 on the ED we hold.
  So total cash flow is

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Forward Rate Agreements (continued)

• The Pricing and Valuation of FRAs (continued)
• This can be rearranged to get
• This is the payoff of an FRA so this strategy is
  equivalent to an FRA. With no initial cash flow,
  we set this to zero and solve for F
• This is just the forward rate in the LIBOR term
  structure. See Table 13.1, p. 471 for an example.

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Forward Rate Agreements (continued)

• The Pricing and Valuation of FRAs (continued)
• Now we determine the market value of the FRA
  during its life, day g. If we value the two
  replicating transactions, we get the value of the
  FRA. The ED we hold pays 1 in h g days. For
  the ED loan we took out, we will pay 1 F(m/360)
  in h m g days. Thus, the value is
• See Table 13.2, p. 472 for example.

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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. 473 and Figure 13.2, p. 474
  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.

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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.

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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.

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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

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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. 478 for illustration.

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Interest Rate Options (continued)

• Interest Rate Option Strategies
• See Table 13.5, p. 479 and Figure 13.3, p. 480
  for an example of the use of an interest rate
  call by a borrower to hedge an anticipated loan.
• See Table 13.6, p. 481 and Figure 13.4, p. 483
  for an example of the use of an interest rate put
  by a lender to hedge an anticipated loan.

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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 floors 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.

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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. 485 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.

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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. 486 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.

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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. 487 for example.

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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 gt X
• The payoff of a short put is
• - (X LIBOR) if LIBOR ? X
• 0 if LIBOR gt 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.

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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

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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)

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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?

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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

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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

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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 - .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.

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Interest Rate Swaptions and Forward Swaps
(continued)

• Pricing Swaptions
• We do not cover this advanced topic here, but
  note that based on the previous result, we would
  price swaptions using models for pricing options
  on bonds.

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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).

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Interest Rate Swaptions and Forward Swaps
(continued)

• Forward Swaps (continued)
• 360 days .09 (0.9174)
• 720 days .1006 (0.8325)
• 1080 days .1103 (0.7514)
• 1440 days .12 (0.6757)
• 1800 days .1295 (0.6070)
• We need the forward rates two years ahead for
  periods of one, two, and three years.

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Interest Rate Swaptions and Forward Swaps
(continued)

• Forward Swaps (continued)

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Interest Rate Swaptions and Forward Swaps
(continued)

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

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Interest Rate Swaptions and Forward Swaps
(continued)

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

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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.

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Summary
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