Debt Instruments Set 9 Interest-Rate Options 0. Overview

Debt Instruments Set 9

Backus/Novemb er 30, 1998

Interest-Rate Options

0. Overview

Fixed Income Options

Option Fundamentals

Caps and Flo ors

Options on Bonds

Options on Futures

Swaptions

Debt Instruments 9-2

1. Fixed Income Options

Options imb edded in b onds:

{ Callable b onds

{ Putable b onds

{ Convertible b onds

Options on futures

{ Bond futures

{ Euro currency futures

OTC options

{ Caps, o ors, and collars

{ Swaptions

Debt Instruments 9-3

2. Option Basics

Big picture

{ Options are everywhere

Sto ck options for CEOs and others

Corp orate equity: call option on a rm

Mortgages: the option to re nance

{ Options are like insurance

Premiums cover the down side, keep the up side

Customers like this combination

Insurer b ears risk or shares it

diversi cation or reinsurance

{ Managing cost of insurance

Out-of-the-money options are cheap er

insurance with a big deductable

Collars: sell some of the up side

Aggregate: basket option cheap er than basket of op-

tions

{ Managing option b o oks

Customer demands may result in exp osed p osition

Particular exp osure to volatility:

puts and calls b oth rise with volatility

Hedging through replication is another route

Debt Instruments 9-4

2. Option Basics continued

Option terminology

{ Basic terms

Options are the right to buy a cal l or sell a put

at a xed price strike price

The thing b eing b ought or sold is the underlying

This righttypically has a xed expiration date

A short p osition is said to have written an option

{ Kinds of options

European options can b e exercised only at expira-

tion

American options can b e exercised any time

Bermuda options can b e exercised at sp eci c dates eg, b onds callable only on coup on dates

Debt Instruments 9-5

2. Option Basics continued

Features of options

{ Leverage cheap source of exp osure

{ Nonlinear payo s

Payo s vary with underlying

in- and out-of-the money

Translates into variable duration

convexity rears its ugly head

Creates risk management hazards

{ Volatility has p ositive e ect on b oth puts and calls

Another risk management hazard!

{ They're state-contingent claims

no way around it, but nothing new either

Debt Instruments 9-6

3. Approaches to Valuation

Why use a pricing mo del? No choice | the instruments de-

mand that wevalue uncertain cash ows state-contingent

claims.

What pricing mo del? Go o d question.

Interest rate trees

{ Been there... and it hasn't changed

{ We'll return to them shortly

The Black-Scholes b enchmark Black's formula

{ Underlying: an arbitrary b ond with say maturity m

{ Parameters: n-p erio d Europ ean call with strike price k

{ Formula:

1=2

Call Price = pN x d kN x n

with

p = current price of underlying

f = forward price of underlying

d = n-p erio d discount factor

N = normal cdf

log f=k+n =2

x =

1=2

Debt Instruments 9-7

3. Approaches to Valuation continued

Remarks on Black-Scholes for xed income

{ Formula based on log-normal price of underlying

normal continuously comp ounded sp ot rates

p ossibility of negative sp ot rates

{ Volatility varies systematically with maturities of op-

tion and underlying \term structure of volatility"

Sample swaption volatility matrix :

Option Swap Maturity

Maturity 1yr 2yr 5yr 10 yr

1m 15.50 16.00 16.75 15.25

3m 17.50 18.50 18.25 18.25

1yr 21.50 21.25 19.25 16.50

5yr 20.00 19.00 17.50 15.50

Source: Tradition, Inc, global swap broker, Jan 2, 1996.

Remark: \hump" is typical

{ Despite problems, a common b enchmark dealers often

quote volatility instead of price

Debt Instruments 9-8

3. Approaches to Valuation continued

Prop erties of Black-Scholes option prices

most of these generalize to other settings

{ The Delta:

Call Price

Delta = = N x;

whichvaries b etween zero and one nonlinear.

{ Volatility is the only unobservable

we use call prices to \imply" it

{ If volatility rises, so do es the call price puts, to o

Debt Instruments 9-9

4. Caps, Flo ors, and Collars

Terminology:

{ A cap pays the di erence b etween a reference rate and

the cap rate, if p ositive. Series of call options on an

interest rate

{ A oor pays the di erence b etween the o or rate and

a reference rate, if p ositive. Series of put options on

an interest rate

{ A col lar is a long p osition in a cap plus a short p osition

in a o or.

Contract terms:

{ Cap and/or o or rate

{ Reference rate typically LIBOR

{ Frequency of payment

{ Notional principal amount on whichinterest is paid

Approaches:

{ Apply Black's formula

{ Interest rate tree we did this earlier

{ An uncountable numb er of other mo dels

Debt Instruments 9-10

4. Caps, Flo ors, and Collars continued

Example 1: two-year semiannual 7 cap on 6-m LIBOR

Payments shifted back one p erio d:

r 7 =2

Notional Principal

1+r=2

three such semi-annual payments, excluding the rst

Short rate tree same as b efore:

8.913

7.869

6.913 6.587

P P

5.869 6.036

P P

4.913 4.587

3.869

2.913

Price path for cap for 100 notional:

0.916

0.858

0.416 0.000

P P

0.000 0.202

P P

0.000 0.000

0.000

P 0.000

Debt Instruments 9-11

4. Caps, Flo ors, and Collars continued

Example 1 continued: the e ects of volatility

Short rate tree = 2, same implied sp ot rates:

12.001

9.912

8.001 7.602

P P

5.912 6.036

P P

4.001 3.602

1.912

0.001

Price path for cap for 100 notional:

2.359

2.740

0.481 1.712

P P

0.234 0.892

P P

0.000 0.115

0.000

Remarks:

{ Volatility increase cap prices they're b ets on extreme

events, and higher makes them more likely

{ Similar in this resp ect to Black-Scholes

Debt Instruments 9-12

4. Caps, Flo ors, and Collars continued

Example 2: 2-Year FRN with collar 7 cap, 4 o or

E ectiveinterest rates b oxes indicate cap/ o or binds:

7.000

6.587 6.913

P P

6.036 5.869

P P

4.587 4.913

4.000

Price path for note:

99.08

99.14

100.00 99.58

P P

100.00 99.88

P P

100.00 100.16

100.33

100.54

Remarks:

{ Di erences from 100 indicate impact of collar

{ Giving up low rates partially o sets cost of cap

0:12 = 100 99:88 < 0:20

{ Issuers would generally adjust cap and o or to get a price of 100

Debt Instruments 9-13

5. Options on Bonds

Earliest and most common interest-rate option?

Examples of callable corp orate b onds

{ Apple Computer Corp oration's 6-1/2s, issued February

10, 1994, due February 15, 2004. Callable at \make

whole."

{ Ford Motor Company's 6.11 p ercent b onds, issued Septem-

b er 22, 1993, due January 1, 2001. Callable at \make

whole under sp ecial circumstances."

{ Intel Overseas Corp oration's 8-1/8s, issued April 1, 1987

really, due March 15, 1997, callable at par. Par in this

situation means par plus accrued interest: the rm pays

the relevantinterest as well as the face value. The b onds

were called March 15, 1994, at 100.

{ Texas Instruments' 9s of July 99, issued July 18, 1989,

due July 15, 1999. Callable on or after July 15, 1996, at par.

Debt Instruments 9-14

5. Options on Bonds continued

Example: call option on 2-year 5 b ond

the usual rate tree

Price path of b ond:

102.50

100.63

102.50 99.77

P P

101.58 100.00

P P

102.50 101.65 100.94

P P

102.54 102.81

P P

102.50 103.58

103.53

102.50

18-month Europ ean option callable at 102.5 | \par"

Price path is:

0.000

0.000 0.000

P P

0.010 0.000

P P

0.134 0.021 0.000

P P

0.227 0.042

P P

0.525 0.000

1.028

0.000

No des in boxes indicate cash ows from exercise, other

no des indicate value in earlier p erio ds.

Debt Instruments 9-15

5. Options on Bonds continued

American option has greater value

can exercise either at expiration, or earlier if b etter

Approach:

{ Start at expiration, work backwards

{ At each no de, cho ose b etter of \exercise" or \hold"

Cash ows from immediate exercise:

0.000

0.000 0.000

P P

0.000 0.000

P P

0.000 0.000 0.000

P P

0.314 0.042

P P

0.000 1.080

1.028

0.000

No de with b ox:

{ If hold:

0:5

Value = 0:042 + 1:028 = 0:525

1+0:03869=2

{ If exercise:

Value = 1:080

This is b etter: we take it.

Debt Instruments 9-16

5. Options on Bonds continued

American option continued

No de 0,1 one down move from start:

{ If hold:

0:5

Value = 0:000 + 1:080 = 0:538

1+0:04587=2

{ If exercise:

Value = 0:314

Hold is b etter in this case, so we write 0.538 here.

Complete price path:

0.000

0.000 0.000

P P

0.000 0.010

P P

0.000 0.021 0.266

P P

0.538 0.042

P P

0.000 1.080

1.028

0.000

Boxes indicate no des where option is exercised.

Summary

{ Worth more than Europ ean call

{ Valued recursively as usual

Debt Instruments 9-17

6. Callable Bonds

Example: Callable b ond based on previous example:

2-year 5 b ond with 18-month American call

Price path:

102.50

100.63

102.50 99.77

P P

101.58 99.99

P P

102.50 101.63 100.67

P P

102.50 102.28

P P

102.50 102.50

102.50

Interest-sensitivity1: replication with x ;x units, resp,

a b

of underlying b ond and one-p erio d zero

1.00,.00

.99,.10

.96,.04 .81,.19

.56,.45

0,1.025

Eg, the callable b ond is equivalent, in the initial no de, to

0.81 units of the underlying noncallable b ond and 0.19 units

of a one-p erio d zero.

Remarks:

{ The callable has shorter duration than the noncallable

{ Howmuch shorter varies throughout the tree

Debt Instruments 9-18

6. Callable Bonds continued

Interest-sensitivity 2: price-yield relation

{ How do es price vary if we shift the whole short rate tree

up and down?

{ Belowwe graph price against initial short rate

{ Slop e used to compute \e ective duration"

115

110

105 Noncallable bond

Bond Prices 100 Callable bond

90 0 2 4 6 8 10 12 Current Short Rate

Debt Instruments 9-19

6. Callable Bonds continued

Dumb ideas

{ Yield to rst call date for b onds in the money

{ Yield to worst: nd call date with highest yield

{ Remarks:

These approaches ignore the intrinsic diculties of

valuing uncertain cash ows

They're dumb for exactly that reason

Our approach: call decision varies through the tree

Option-Adjusted Spread OAS

{ Consider the valuation of a callable b ond

{ Supp ose market price is p

{ Compute spread s added to the short rate tree required

to repro duce the market price

{ Positive spread means the market values the b ond more highly than the mo del

Debt Instruments 9-20

7. Options on Futures

Options available on ma jor futures contracts

{ Government b ond contracts

{ Euro currency contracts

{ Brady b ond futures

Same strengths as the underlying futures

{ Highly liquid markets

{ Low transaction costs

Debt Instruments 9-21

8. Swaptions

Swaptions: options on swaps

{ Option to enter a swap

{ Option to extend a swap

{ Option to terminate a swap

{ Europ ean, American, and Bermuda

Prop erties

{ Similar to b ond options swap = b ond - FRN, or re-

verse

{ Exp osure to long-dated volatility

{ Currently the OTC option standard

Debt Instruments 9-22

Summary

Options are ubiquitous.

Their nonlinear payo s p ose challenges to valuation and

risk management.

Nonlinearity translates in this context into nonlinear price-

yield relations | convexity, in other words.

Black-Scholes is less well suited for xed income than other

securities, but remains a common b enchmark nonetheless.

American options are valued recursively: at each no de, we

decide whether to exercise or hold.

Shortcuts don't work: yield-to-call is meaningless.