CHAPTER11 Interest Rate Forwards and Options (《衍生工具与风险管理》PPT课件)

With an exercise rate of X, the payoff of an interest
rat(eNocatilol nisal Principal)Max(0, LIBOR X)(m/360)
The(Npoatyiofnfaol fParinninctpearle)sMt raaxt(e0p, Xut-isLIBOR)(m/3 60)
(.12 - .10)(90/360) 1 .12(90/360)


$97,087
Note the terminology of FRAs: A B means FRA
expires in A months and underlying matures in B
months.
FORWARD RATE AGREEMENTS
g)/360)

See Table 13.2, p. 472 for example.
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.
1 F(m/360) Pay if off early. C1olleLcth$(1mo)n(mth/3e6E0D)we hold. So total
cash flow is
1 1 F(m/360) 1 Lh (m)(m/360)
FORWARD RATE AGREEMENTS
(CONTINUED)
The fixed rate is called the exercise rate.
Most are European-style.
INTEREST RATE OPTIONS (CONTINUED)
The Structure and Use of a Typical Interest Rate Option
took out, we will pay 1 + F(m/360) in h + m – g days.
Thus, the value is
VFRA

1
Lg (h

1 g)((h
g)/360)



1

Lg (h

1 F(m/360) m g)((h m
payoff is
(Notional
Principal)
(LIBOR
- Agreed upon rate)(m/360) 1 LIBOR(m/36 0)

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
($20,000,000)
(LIBOR - .10)(90/360) 1 LIBOR(90/360)

FORWARD RATE AGREEMENTS (CONTINUED)
Some possible payoffs. If LIBOR at expiration is 8 percent,
INTEREST RATE OPTIONS (CONTINUED)
The Structure and Use of a Typical Interest Rate Option (continued)
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.
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.
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.
(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(h+m) is spot rate on day 0 for h + m days. Assume notional principal of $1.
(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
The Pricing and Valuation of FRAs (continued)
This can be rearranged to get
(Lh (m) F)(m/360)

This
is
the
1 L
payoff of
ha(nmF)(RmA/3s6o0t)his
strategy
See Figure 13.1, p. 467 for notional principal of FRAs and interest rate options over time.
FORWARD RATE AGREEMENTS
Definition
A forward contract in which the underlying is an interest rate
($20,000,000)
(.08 - .10)(90/360) 1 .08(90/360)


$98,039
So the long has to pay $98,039. If LIBOR at
expiration is 12 percent, the payoff is
($20,000,000)
To find the fixed rate, we must replicate an FRA:
Short a Eurodollar maturing in h+m days that pays 1 + F(m/360). This is a loan that can be paid off early or transferred to another party
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.
FORWARD RATE AGREEMENTS
Long a Eurodollar maturing in h days that pays $1
FORWARD RATE AGREEMENTS (CONTINUED)
The Pricing and Valuation of FRAs (continued)
On day h,
Loan we owe has a market value of
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
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

structure. See Table 13.1, p. 471 for an example.
FORWARD RATE AGREEMENTS
(CONTINUED)
The Pricing and Valuation of FRAs (continued)
Now we determine the market value of the FRA
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is
equivalent to an FRA. With no initial cash flow, we
set this to zero and solve for F:

This
is
juFstth1efoLr0w(1hardLmr0)a((ht(e/h3i6n0m)th)/e36L0IB) OR1te3rm6m0
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.