# Irwin/McGraw-Hill

```Options, Caps, Floors and Collars
Chapter 25
Financial Institutions Management, 3/e
By Anthony Saunders
Irwin/McGraw-Hill
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Call Options on Bonds
Write a call
X
X
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Put Options on Bonds
Write a Put
X
X
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
Many smaller FIs constrained to buying
rather than writing options.
• Economic reasons
» Potentially unlimited downside losses.
• Regulatory reasons
» Risk associated with writing naked options.
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Hedging
• Payoffs to Bond + Put
Bond
X
Put
Net
X
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Hedging Bonds

Weaknesses of Black-Scholes model.
• Assumes short-term interest rate constant
• Assumes constant variance of returns on
underlying asset.
• Behavior of bond prices between issuance and
maturity
» Pull-to-maturity.
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Hedging With Bond Options Using
Binomial Model
• Example: FI purchases zero-coupon bond with
2 years to maturity, at P0 = \$80.45. This means
YTM = 11.5%.
• Assume FI may have to sell at t=1. Current
yield on 1-year bonds is 10% and forecast for
next year’s 1-year rate is that rates will rise to
either 13.82% or 12.18%.
• If r1=13.82%, P1= 100/1.1382 = \$87.86
• If r1=12.18%, P1= 100/1.1218 = \$89.14
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Example (continued)
• If the 1-year rates of 13.82% and 12.18% are
equally likely, expected 1-year rate = 13% and
E(P1) = 100/1.13 = \$88.50.
• To ensure that the FI receives at least \$88.50 at
end of 1 year, buy put with X = \$88.50.
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Value of the Put

At t = 1, equally likely outcomes that bond
with 1 year to maturity trading at \$87.86 or
\$89.14.
• Value of put at t=1:
Max[88.5-87.86, 0] = .64
Or,
Max[88.5-89.14, 0] = 0.
• Value at t=0:
P = [.5(.64) + .5(0)]/1.10 = \$0.29.
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Actual Bond Options
Most pure bond options trade over-thecounter.
 Preferred method of hedging is an option on
an interest rate futures contract.

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Using Options to Hedge FX Risk
• Example: FI is long in 1-month T-bill paying
£100 million. FIs liabilities are in dollars.
Suppose they hedge with put options, with
X=\$1.60 /£1. Contract size = £31,250.
• FI needs to buy £100,000,000/£31,250 = 3,200
contracts. If cost of put = 0.20 cents per £, then
each contract costs \$62.50. Total cost =
\$200,000 = (62.50 × 3,200).
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Hedging Credit Risk With Options

• Payoff increases as (default) yield spread on a
specified benchmark bond on the borrower
increases above some exercise spread S.

Digital default option
• Pays a stated amount in the event of a loan
default.
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Hedging Catastrophe Risk
Catastrophe (CAT) call spread options to
hedge unexpectedly high loss events such as
hurricanes, faced by PC insurers.
 Provides coverage within a bracket of lossratios. Example: Increasing payoff if lossratio between 50% and 80%. No payoff if
below 50%. Capped at 80%.

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Caps, Floors, Collars
• Cap: buy call (or succession of calls) on
interest rates.
• Floor: buy a put on interest rates.
• Collar: Cap + Floor.
• Caps, Floors and Collars create exposure to
counterparty credit risk since they involve
multiple exercise over-the-counter contracts.
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