Bm Fi6051 Wk11 Lecture

Derivative Instruments FI6051 Finbarr Murphy Dept. Accounting & Finance University of Limerick Autumn 2009

Week 11 – Interest Rate Derivatives

Black’s Model 

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The Black-Scholes Model was an “overnight” success It was quickly adapted for Interest Rate (IR) derivatives But, IR Derivatives are more complex than equity/currency options because:    

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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 Black’s (Black’s) model

Black’s Model 

Consider a european call option, on an underlying variable, value V   

T – time to maturity F – forward price of V with maturity at T F0 – Value of F at time = 0 (now)

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K – The option strike price P(t,T) – The price at t, of a zero coupon bond paying $1 at time T VT – Value of V at T

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σ – Volatility of F

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VT has a lognormal distribution, with SD of lnVt =σ T

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E[VT] = F0

Black’s Model

V=VT E[VT ]=F0

F=F0 $1 P(t,T)

t=t t=0 F = F0 = forward price of V (maturity T)

t=T V = VT

Black’s Model 

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

max(VT − K ,0 )

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The lognormal assumption implies a payoff

E (VT ) N ( d1 ) − KN ( d 2 )

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where

ln[ E (VT ) K ] + σ 2T 2 d1 = σ T d2

ln[ E (VT ) =

K ] −σ T 2 = d1 − σ T σ T 2

Black’s Model  

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

c = P( 0, T ) [ F0 N ( d1 ) − KN ( d 2 ) ] 

similarly p

= P( 0, T ) [ KN ( − d 2 ) − F0 N ( − d1 ) ]

Black’s Model 

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Blacks (1976) model is very similar to the BlackScholes (1973) model. The two main differences are: 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.

Bond Options 

Embedded Options  

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Callable Bonds Puttable Bonds

European Bond Options Recall:c = P 0, T F0 N d1

( )[ ( ) − KN ( d 2 ) ] p = P( 0, T ) [ KN ( − d 2 ) − F0 N ( − d1 ) ] 2 [ ] ln F K + σ T 2 0 Where d = 1

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and

σ T d 2 = d1 − σ T

Bond Options 

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

F0 = ( S 0 − I ) e rT 

Substituting from previous slides:

B0 − I F0 = P (0, t )

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

Bond Options 

Take the example from Hull p649: Item

Description

B0

$960

Current Bond (cash price)

K

$1,000

Strike Price

T10m

0.8333 (=10/12 years)

Option Time to Maturity

r10m

10%

Risk free rate, 10months

r9m

9.5%

Risk free rate, 9months

r3m

9.0%

Risk free rate, 3months

C

10%

Annual coupon (paid semi-annually)

σ

9%

Forward Bond Price volatility

Bond Options t=

s th 3m

s s th mth 9m 10 = t t=

t=0

t=9.75yrs

9years, 9months – Bond Maturity

10months Option Maturity

A coupon of 5% is paid ($50) 

B0 = $960

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I = 50e-r x0.25 +50e-r P(0,T10m ) = e-r x(10/12)

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F0 = (B0 – I)/P(0, T10m )

3m

10m

9m

= $95.45 = e-0.1x(10/12) = 0.9200

x0.75

= (960-95.45)/0.92 = $939.68

Floating Rate Notes   

AKA, “floaters” A note with a variable interest rate The adjustments to the interest rate are usually made every six months and are tied to a certain money-market index such as  

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Issued by corporations or agencies such as   

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3-month Treasury bill or 3-month LIBOR The Federal Home Loan Bank Fannie Mae Freddie Mac

Can have a spread above the benchmark

Floating Rate Notes  

Example terms Corporation XYZ issues a seven-year floating-rate note with the following features:     

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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 2.16000 2.15000 2.14000 2.13000 2.12000 2.11000 2.10000 EUR LIBOR

2.09000

1-Month FRN 2.08000 2.07000

03 /0 9/ 05

03 /0 8/ 05

03 /0 7/ 05

03 /0 6/ 05

03 /0 5/ 05

03 /0 4/ 05

03 /0 3/ 05

03 /0 2/ 05

03 /0 1/ 05

2.06000

Data Source: BBA

Interest Rate Caps OTC Instruments Provide insurance against the rate of interest on a FRN exceeding a certain level

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2.8 2.7

FRN Rate CAP

2.6 2.5 2.4 2.3 2.2 2.1

F

Ja

n0 eb 5 M -05 ar A 05 p r M -0 5 ay Ju - 0 5 n Ju 05 l A - 05 u g S - 05 ep O 05 ct N -05 o v D -05 ec Ja 05 n F -0 6 eb M -06 ar A 06 p r M -0 6 ay Ju - 06 n Ju 06 l A -06 u g S - 06 ep O 06 ct N -06 o v D -06 ec -0 6

2

(FRNRate-Cap)*Nominal ---------------------No. of Payments per Year

Interest Rate Caps 

From the previous slide, assume:     

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A 2-year FRN Cap Rate = 2.5% Principle = €100,000,000 Tenor (τ) = 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)

( 2.55 − 2.50) * €100MM Payoff = = €4,166.66 12

Interest Rate Caps  

Look more closely at the IR Cap from before… Each reset date is an option = Rk = RK

2.8 2.7 2.6 2.5 2.4 2.3 2.2

time=tk

2.1

time=tk+1

2 J



-0 an

5

05 be F

M

5 -0 ar

r-0 p A

5 M

-0 ay

5

5 -0 n Ju

l- 0 Ju

5

-0 ug A

5

05 pe S

It is in-the-money if Rk>RK

O

-0 ct

5

v No

5 -0

ec D

5 -0

Interest Rate Caps

= Rk = RK

2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2 J

-0 an

5

bFe

05 M

5 -0 ar

-0 pr A

5 M

-0 ay

5

nJu

05

l- 0 Ju

5

0 gu A

5

5 -0 p Se

O

-0 ct

5

5 5 -0 -0 c v De No

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Each “option” is known as a caplet

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Where

caplet = Lτ max(Rk − RK ,0)

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L = the principal amount

Interest Rate Caps 

Recall that for a european bond option:

c = P( 0, T ) [ F0 N ( d1 ) − KN ( d 2 ) ] 

Setting K = RK and F0 = Fk

caplet = Lτ .P( 0, t k +1 ) [ Fk N ( d1 ) − RK N ( d 2 ) ] 

ln[ Fk RK ] + σ k2t k 2 d1 = σ k tk 2 ln[ Fk RK ] − σ k t k 2 d2 = = d1 − σ k t k σ k tk

where

Interest Rate Caps  

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The value of the IR Cap is the sum of the caplets A CAP trader is interested in the σk series, I.e. the volatility of the forward rate for each caplet. A Floor, is similar to a Cap as a put option is similar to 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

Swap Options (Swap Options) 

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A Swaption gives the holder the right (but not the obligation) to enter into an interest rate swap at a certain price at a certain date in the future These are popular OTC derivative products Remember, a swap allows a company to swap fixed for floating rate This can be used for cashflow management 

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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 a fixed income bond and a long positon in a FRN Or visa versa

Swap Options (Swap Options) 

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

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Therefore, a swaption can be viewed as the option to simultaneously sell a fixed income bond and buy a FRN at a specific rate at a specific time in the future. Or A swaption gives you the right to pay fixed rate and receive floating rate

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Swap Options (Swap Options) 

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

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Therefore, a swaption can (also) be viewed as the option to simultaneously buy a fixed income bond and sell a FRN at a specific rate at a specific time in the future. Or A swaption gives you the right to receive fixed rate and pay floating rate

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Swap Options (Swap Options) 

A Receiver Swaption is the right but not the obligation to enter into an Interest Rate Swap where the buyer RECEIVES fixed rate and pays FLOATING.  The buyer will therefore benefit if rates FALL.

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A Payer Swaption is the right but not the obligation to enter into an Interest Rate Swap where the buyer PAYS fixed rate and receives FLOATING. The buyer will therefore benefit if rates RISE.

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Swap Options (Swap Options)  

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Aren’t 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 June 2007† Average daily swaps trade volumes rose to $611bn† †

Source: International Swaps and Derivatives Association (ISDA) And Bank for International Settlements (BIS)

Swap Options (Swap Options)

†

Source: Swapstream.net

Swap Options (Swap Options) 

How do we value swaptions?

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We assume that the swap rate at option maturity is lognormal

Swap Options (Swap Options) 

Assume we purchase a swaption with the following terms: 

At option maturity we can pay Sk fixed

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At option maturity we can received LIBOR floating The swap agreement lasts for n years The option matures in T years

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Tyears

Nyears

Swap Options (Swap Options)

Sk    

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ST

Look at what happens at option maturity… 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

Swap Options (Swap Options) 

In general, the payoff on a swaption is:

L max(ST − S k ,0) m 

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

Further reading 

Hull, J.C, “Options, Futures & Other Derivatives”, 2009, 7thth Ed. 

Chapter 28