Options, Caps, Floors and Options, Caps, Floors and CollarsCollars

Chapter 24

© 2008 The McGraw-Hill Companies, Inc., All Rights Reserved.McGraw-Hill/Irwin

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Overview

Derivative securities as a whole have become increasingly important in the management of risk and this chapter details the use of options in that vein. A review of basic options –puts and calls– is followed by a discussion of fixed-income, or interest rate options. The chapter also explains options that address foreign exchange risk, credit risks, and catastrophe risk. Caps, floors, and collars are also discussed.

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Option Terms

Long position in an option is synonymous with: Holder, buyer, purchaser, the long Holder of an option has the right, but not the

obligation to exercise the option

Short position in an option is synonymous with: Writer, seller, the short Obliged to fulfill terms of the option if the option

holder chooses to exercise.

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Call option

A call provides the holder (or long position) with the right, but not the obligation, to purchase an underlying security at a prespecified exercise or strike price. Expiration date: American and European

options The purchaser of a call pays the writer of

the call (or the short position) a fee, or call premium in exchange.

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Payoff to Buyer of a Call Option

If the price of the bond underlying the call option rises above the exercise price, by more than the amount of the premium, then exercising the call generates a profit for the holder of the call.

Since bond prices and interest rates move in opposite directions, the purchaser of a call profits if interest rates fall.

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The Short Call Position

Zero-sum game: The writer of a call (short call position) profits

when the call is not exercised (or if the bond price is not far enough above the exercise price to erode the entire call premium).

Gains for the short call position are losses for the long call position.

Gains for the long call position are losses for the short call position.

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Writing a Call

Since the price of the bond could rise to equal the sum of the principal and interest payments (zero rate of interest), the writer of a call is exposed to the risk of very large losses.

Recall that losses to the writer are gains to the purchaser of the call. Therefore, potential profit to call purchaser could be very large. (Note that call options on stocks have no theoretical payoff limit at all).

Maximum gain for the writer occurs if bond price falls below exercise price.

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Call Options on Bonds

Buy a call Write a call

X

X

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Put Option

A put provides the holder (or long position) with the right, but not the obligation, to sell an underlying security at a prespecified exercise or strike price. Expiration date: American and European

options The purchaser of a put pays the writer of the

put (or the short position) a fee, or put premium in exchange.

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Payoff to Buyer of a Put Option

If the price of the bond underlying the put option falls below the exercise price, by more than the amount of the premium, then exercising the put generates a profit for the holder of the put.

Since bond prices and interest rates move in opposite directions, the purchaser of a put profits if interest rates rise.

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The Short Put Position

Zero-sum game: The writer of a put (short put position) profits

when the put is not exercised (or if the bond price is not far enough below the exercise price to erode the entire put premium).

Gains for the short position are losses for the long position. Gains for the long position are losses for the short position.

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Writing a Put

Since the bond price cannot be negative, the maximum loss for the writer of a put occurs when the bond price falls to zero. Maximum loss = exercise price minus the

premium

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Put Options on Bonds

Buy a Put Write a Put

(Long Put) (Short Put)

X

X

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Writing versus Buying Options

Many smaller FIs constrained to buying rather than writing options. Economic reasons

Potentially large downside losses for calls. Potentially large losses for puts Gains can be no greater than the premiums so less

satisfactory as a hedge against losses in bond positions

Regulatory reasons Risk associated with writing naked options.

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Combining Long and Short Option Positions

The overall cost of hedging can be custom tailored by combining long and short option positions in combination with (or alternative to) adjusting the exercise price.

Example: Suppose the necessary hedge requires a long call option but the hedger wishes to lower the cost. A higher exercise price would lower the premium but provides less protection. Alternatively, the hedger could buy the desired call and simultaneously sell a put (with a lower exercise price). The put premium offsets the call premium. Presumably any losses on the short put would be offset by gains in the bond portfolio being hedged.

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Hedging Downside with Long Put

Payoffs to Bond + Put

X

X

Put

Bond

Net

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Tips for plotting payoffs

Students often find it helpful to tabulate the payoffs at critical values of the underlying security: Value of the position when bond price equals

zero Value of the position when bond price equals X Value of position when bond price exceeds X Value of net position equals sum of individual

payoffs

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Tips for plotting payoffs

B=0 B < X B=X B > X

Long call Short call

- C + C

- C + C

-C +C

B-X-C X+C-B

Long put Short put

X - P P - X

X-B-P P+B-X

-P +P

-P +P

Long Bond

B = 0

B

B=X

B

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Futures versus Options Hedging

Hedging with futures eliminates both upside and downside

Hedging with options eliminates risk in one direction only

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Hedging with Futures

Bond Portfolio

Bond Price

Purchased Futures Contract

X

0

Gain

Loss

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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-par.

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Example: FI purchases zero-coupon bond with 2 years to maturity, at BP0 = $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%, BP1= 100/1.1382 = $87.86

If r1=12.18%, BP1= 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(BP1) = 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-the-counter. Open interest on CBOE relatively small

Preferred method of hedging is an option on an interest rate futures contract. Combines best features of futures contracts

with asymmetric payoff features of options.

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Web Resources

Visit:

Chicago Board Options Exchange www.cboe.com

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Hedging with Put Options

To hedge net worth exposure,

P = - E

Np = [(DA-kDL)A] [ D B] Adjustment for basis risk:

Np = [(DA-kDL)A] [ D B br]

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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

Credit spread call option 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 loss-ratios. Example: Increasing payoff if loss-ratio 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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Fair Cap Premium

Two period cap:

Fair premium = P

= PV of year 1 option + PV of year 2 option Cost of a cap (C)

Cost = Notional Value of cap × fair cap premium (as percent of notional face value)

C = NVc pc

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Collars: Buy a Cap and Sell a Floor

Net cost of long cap and short floor:

Cost = (NVc × pc) - (NVf × pf )

= Cost of cap - Revenue from floor Counterparty credit risk is an issue

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Pertinent websites

Chicago Board of Trade www.cbot.com

CBOE www.cboe.com

Chicago Mercantile Exchange www.cme.com

Wall Street Journal www.wsj.com