bm fi6051 wk11 lecture

8/14/2019 Bm Fi6051 Wk11 Lecture

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Derivative InstrumentsFI6051

Finbarr MurphyDept. Accounting & FinanceUniversity of LimerickAutumn 2009

Week 11 Interest Rate Derivatives


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The Black-Scholes Model was an overnightsuccess

It was quickly adapted for Interest Rate (IR)derivatives

But, IR Derivatives are more complex thanequity/currency options because: IRs behave differently

The entire zero-curve must be modeled

Volatilities across the curve are different IRs are used for discounting and payoff

We start with Fischer Blacks (Blacks) model

Blacks Model


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Blacks Model

P(t,T)

$1

t=0 t=t t=T

F = F0= forward price of V (maturity T)

V = VT

F=F0V=VTE[VT ]=F0


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At maturity (time = T), the payoff from the option isgiven by

The lognormal assumption implies a payoff

where

Blacks Model

( )0,max KVT

( ) ( ) ( )21 dKNdNVE T

( )[ ]TTKVEd T

2ln

2

1

+

=

( )[ ]Td

T

TKVEd T

=

= 1

2

2

2ln


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Discounting at the risk-free rate and Assuming E[VT] = F0

similarly

Blacks Model

( ) ( ) ( )[ ]210,0 dKNdNFTPc =

( ) ( ) ( )[ ]102,0 dNFdKNTPp =


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Blacks (1976) model is very similar to the Black-Scholes (1973) model. The two main differencesare:

Blacks Model uses the forward bond price instead of

the spot price There is no drift, we only assume that the forward

bond price is lognormally distributed.

Blacks Model


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Embedded Options Callable Bonds

Puttable Bonds

European Bond Options

Recall:

Where

and

Bond Options

( ) ( ) ( )[ ]210,0 dKNdNFTPc =

( ) ( ) ( )[ ]102,0 dNFdKNTPp =[ ]

T

TKFd

2ln2

01

+=

Tdd = 12


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Recall that the forward contract on an investmentasset providing an income with PV = I

Substituting from previous slides:

All prices are assumed to be cash prices (not quotedprices)

Bond Options

( ) rTeISF = 00

),0(

00

tP

IBF

=


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B0 = $960

I = 50e-r3m x0.25 +50e-r

9m x0.75 = $95.45 P(0,T10m ) = e

-r 10m x(10/12) = e-0.1x(10/12) = 0.9200

F0 = (B0 I)/P(0, T10m )

= (960-95.45)/0.92

= $939.68

Bond Options

t=3m

ths

t=9m

ths

t=0t=9.75yrs

t=10

mths

9years, 9months Bond Maturity

10monthsOption Maturity

A coupon of 5% is paid ($50)


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AKA, floaters A note with a variable interest rate

The adjustments to the interest rate are usuallymade every six months and are tied to a certain

money-market index such as 3-month Treasury bill or

3-month LIBOR

Issued by corporations or agencies such as The Federal Home Loan Bank Fannie Mae

Freddie Mac

Can have a spread above the benchmark

Floating Rate Notes


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Example terms Corporation XYZ issues a seven-year floating-rate

note with the following features: Maturity date: September 1, 2010

Benchmark rate: Three-month U.S. Treasury bill Spread: 75 basis points

Interest frequency: Quarterly

Initial interest rate: 1.69% (based on an initial Treasury bill

rate of 0.94% on September 9, 2003) If three-month Treasury bill rates increase 0.5% to

1.44%, the coupon would reset to 2.19%.

If three-month Treasury bill rates decline 0.5% to

0.44%, the coupon would reset to 1.19%.

Floating Rate Notes


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Floating Rate Notes

2.06000

2.07000

2.08000

2.09000

2.10000

2.11000

2.12000

2.13000

2.14000

2.15000

2.16000

03/0

1/05

03/02/05

03/03/05

03/0

4/05

03/05/05

03/06/05

03/07/05

03/08/05

03/09/05

EUR LIBOR

1-Month FRN

Data Source: BBA


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OTC Instruments Provide insurance against the rate of interest on a

FRN exceeding a certain level

Interest Rate Caps

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Jan-05

Feb-05

Mar-05

Apr-0

5

May-05

Jun-05

Jul-0

5

Aug-05

Sep-05

Oct-05

Nov-05

Dec-05

Jan-06

Feb-06

Mar-06

Apr-0

6

May-06

Jun-06

Jul-0

6

Aug-06

Sep-06

Oct-06

Nov-06

Dec-06

FRN Rate

CAP

(FRNRate-Cap)*Nominal

----------------------

No. of Payments per Year


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From the previous slide, assume: A 2-year FRN

Cap Rate = 2.5%

Principle = 100,000,000

Tenor () = 1/12 (time between resets) On June 1st , the FRN rate set to 2.55%

So the payment to the cap holder on July 1st (end of

the period)

Interest Rate Caps

66166412

100502552.,

*)..(=

=

MMPayoff


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Each reset date is an option

Interest Rate Caps

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Jan-05

Feb-05

Mar

-05

Apr-0

5

May

-05

Jun-05

Jul-0

5

Aug-05

Sep-05

Oct-0

5

Nov-05

Dec-05

= Rk

= RK

It is in-the-money if Rk

>RK

Look more closely at the IR Cap from before

time=tk time=tk+1


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Each option is known as a caplet

Where L = the principal amount

Interest Rate Caps

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Jan-05

Feb-05

Mar

-05

Apr-0

5

May

-05

Jun-05

Jul-0

5

Aug-05

Sep-05

Oct-0

5

Nov-05

Dec-05

= Rk

= RK

)0,max( Kk RRLcaplet =


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Recall that for a european bond option:

Setting K = RK and F0 = Fk

where

Interest Rate Caps

( ) ( ) ( )[ ]210,0 dKNdNFTPc =

( ) ( ) ( )[ ]211,0. dNRdNFtPLcaplet Kkk = +

[ ]

kk

kkKk

t

tRFd

2ln2

1

+

=

[ ]kk

kk

kkKk tdt

tRFd

=

= 1

2

2

2ln


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The value of the IR Cap is the sum of the caplets A CAP trader is interested in the k series, I.e. the

volatility of the forward rate for each caplet.

A Floor, is similar to a Cap as a put option is similarto a call option.

A floorlet, is a series of put options

A Floor is the sum of a series of floorlets

A Collar, is a long Cap plus a short Floor position

A collar guarantees against a FRN exceeding floor

and ceiling limits

Interest Rate Caps


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A Swaption gives the holder the right (but not theobligation) to enter into an interest rate swap ata certain price at a certain date in the future

These are popular OTC derivative products

Remember, a swap allows a company to swapfixed for floating rate

This can be used for cashflow management E.g. Lock-in a fixed rate

Or for speculative reasons Bullish on 3-months rate so buys repo

Swap Options (Swap Options)


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A swap can be considered a short position on afixed income bond and a long positon in a FRN

Or visa versa



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You can replicate a long swap position by issuing(selling) a fixed income paying bond and buying aFRN (I.e. you pay fixed and receive floating)

Therefore, a swaption can be viewed as theoption to simultaneously sell a fixed income bondand buy a FRN at a specific rate at a specific timein the future.

Or A swaption gives you the right to pay fixed rate

and receive floating rate



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You can replicate a short swap position by buyinga fixed income paying bond and issuing (selling)

a FRN (I.e. you receive fixed and pay floating)

Therefore, a swaption can (also) be viewed as theoption to simultaneously buy a fixed income bondand sell a FRN at a specific rate at a specific timein the future.

Or A swaption gives you the right to receive fixed

rate and pay floating rate



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A Receiver Swaption is the right but not theobligation to enter into an Interest Rate Swapwhere the buyer RECEIVES fixed rate and paysFLOATING.

The buyer will therefore benefit if rates FALL.

A Payer Swaption is the right but not theobligation to enter into an Interest Rate Swap

where the buyer PAYS fixed rate and receivesFLOATING.

The buyer will therefore benefit if rates RISE.



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Arent swaptions a bit obscure? Interest rate swaps are the most widely held

single product type among all over-the-counter(OTC) derivatives (around 55 per cent of total

notional outstandings worldwide). The global interest rate swaps market has

experienced significant growth in recent years.

Total notional outstandings reached

approximately $347,093,635,353,043 in June2007

Average daily swaps trade volumes rose to$611bn


Source:International Swaps and Derivatives Association (ISDA)

And Bank for International Settlements (BIS)


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How do we value swaptions?

We assume that the swap rate at option maturityis lognormal



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Assume we purchase a swaption with thefollowing terms: At option maturity we can pay Sk fixed

At option maturity we can received LIBOR floating

The swap agreement lasts for n years The option matures in Tyears


NyearsTyears


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Look at what happens at option maturity


Sk ST

The actual swap rate = ST

But the strike swap rate = Sk You would not enter into a swap agreement at a

rate higher than the prevailing market swap rate

So this swaption expires out-of-the-money


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In general, the payoff on a swaption is:

Where L is the notional principal and m is thenumber of payments per year.


)0,max( kT SSm

L


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Hull, J.C, Options, Futures & Other Derivatives,2009, 7thth Ed. Chapter 28

Further reading

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