options 1 (1)

8/8/2019 Options 1 (1)

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Rates University 2007

Options 1

Greg KnudsenInternational Rate Structuring & MTNs

8/8/2019 Options 1 (1)


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Contents

This module is an introduction to the theory and practice of Options.

We will look at the following theoretical concepts ...

• Option definitions• Call and Put payoffs

• Volatility

• Black Scholes pricing

• Put / Call Parity

• Option sensitivities (the Greeks)

We will then consider at some specific option examples ...

• FX Options

• Caps & Floors

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Option Definitions

An Option is the right, but not the obligation, to buy (Call) or sell (Put) an asset at a specified date

at a specified price.

Note the difference to a Forward which is the obligation to buy (or sell) an asset at a specified future date

at a specified price.

A Forward obliges the participants to transact the agreed exchange on the agreed date at the agreed price.

An Option allows the Option Buyer to choose to exercise the exchange (a purchase or a sale) or not,

depending upon the real or perceived value of that purchase or sale.

We define some common terms …

Call Option the buyer of a Call Option has the right to buy the underlying asset

on a specified date at a specified price (the strike)

Put Option the buyer of a Put Option has the right to sell the underlying asseton a specified date at a specified price (the strike)

Strike the price at which the asset is bought or sold if the option is exercised

Premium the price of the Option contract

the buyer of the option pays the seller of the option this price at inception

Expiry the Maturity Date of the Option

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Option Definitions

European Option a European Option can only be exercised on the specified Option Maturity Date

American Option an American Option can be exercised at any time prior to the Option

Maturity Date

Bermudan Option a Bermudan Option can be exercised (once) on one of a specified set

of exercise dates up to and including the Option Maturity Date

These often co-incide with coupon payment dates on structured bonds and swaps

Option Value the value for which the option can be bought or sold in the market

At inception this is the premium paid to buy the option

Intrinsic Value the value (if positive) of immediate exercise of the Option

For a Call this is Max (0, Asset Value – Strike)

For a Put this is Max (0, Strike – Asset Value)

Time Value the residual value of the option after accounting for the Intrinsic Value

It captures the potential additional payoff from the underlying moving(deeper) into the money before expiry

By definition … Time Value = Option Value – Intrinsic Value

In-the-money an option is In-the-money if the Intrinsic Value is positive

At-the-money an option is At-the-money if the current Asset Value = Strike

Out-of-the-money an option is Out-of-the-money if immediate exercise would result in a loss

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European Options

European Options have a well-established analytical valuation methodology.

This theory is called Black-Scholes Theory

Black-Scholes Theory was developed in the 1970s and was the first truly comprehensive solution to the

problem of European option valuation.

Under a set of assumptions about the distribution of the underlying asset (interest rates, FX rates etc), about

the volatility of the asset, etc, the theory uses arbitrage arguments to derive analytical formulas for calls and

puts.

In the Black-Scholes world, Geometric Brownian Motion is assumed as the dynamic model for the evolution

of the underlying asset.

The variable could be for example

• a short term interest rate

• a spot FX rate

• an observable Index

• a commodity price, etc.

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5

Payoffs and Volatilities

The Payoffs for Calls and Puts are given by

Volatility a measure of the variability of the price of the underlying Asset or IndexIn Black-Scholes, Lognormal Volatility is technically the standard deviation of the

Log of the Asset Price in 1 year’s time.

Lognormal Vol is also effectively the standard deviation of the percentage change

in the asset price over 1 year. For interest rate options, Basis Point Vol is the

standard deviation of the absolute change in the interest rate Index over 1 year.

Historical Vol the historical vol is the observed volatility of the asset price over a specified period.

Implied Vol the implied vol is the volatility implied by the option price traded in the market.

The implied vol may be higher or lower than the historical vol.

Note that implied vol is in a sense forward looking, historical vol backwards looking

A key component of Options pricing is the assumed volatility of the underlying

variable or asset. We look at the definitions …

Call Payoff = Max (0, ST – X) where …

ST = the asset price at Maturity (time T)

X = the strike

Put Payoff = Max (0, X – ST)

Call Payoff X

X ST

XPut Payoff

X ST

Both Puts and Calls become more valuable when the implied volatility increases.

This is because higher volatility implies a higher probability of finishing deep in the money

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Calls and Puts … Long & Short

There are 4 basic payoff diagrams for Calls and Puts …

Payoff

X

Payoff

X Asset X Asset

Long Call Long Put

L = Libor Payoff Payoff

XX

Short Put

Asset

Short Call

Asset

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Black-Scholes Methodology

Black and Scholes proved that, under certain assumptions, an underlying asset could perfectly hedge the

profits and losses of a standard European option by following a self-financing dynamic replication portfolio

strategy.

The Black-Scholes European option price is the discounted expected payoff of the option at Maturity and is

derivable analytically since the distribution of the underlying variable at the sole exercise date is available.

This is of course intuitively appealing. Under Geometric Brownian motion the distribution of the underlying at

Maturity is Lognormal. This distribution can be used to calculate the expected value of the option payoff.

Since this payoff occurs at Maturity it must be discounted in order to calculate the option value (premium)

Call Price vs Maturity Distribution

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

3.00% 3.40% 3.80% 4.20% 4.60% 5.00% 5.40% 5.80% 6.20% 6.60% 7.00%

Forward

P r i c e

The graph shows the Call Price and Payoff for

a Call with as functions of the Forward.

Also shown are the Lognormal Distributions

corresponding to 10% and 20% volatilities.

The Call is priced using a volatility of 20%

If the Call were repriced using only 10% vol

then the Call price would fall.

This is because the payoff is asymmetric.

A higher vol implies a greater chance of extreme positive payoffs.

Lognormal Dist

Vol = 10%

Lognormal Dist

Vol = 20%Call Price

Payoff

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Intrinsic Value and Time Value

Intrinsic Value is the gain that could be realised through immediate exercise of the option.

For a Call this is Max (0, Asset – Strike)

Time Value is the remaining value of the option.

It captures the potential additional payoff from the underlying moving (deeper) into the money before expiry.

Call Price - Intrinsic Value & Time Value

Forward

P r i c e

Call Price

Payoff

Time Value

Intrinsic Value

X

Time Value

The graph shows both the Call Price

and Payoff for a Call with Strike X, as

functions of the Forward Rate.

The Call Price is divided into …

- Intrinsic Value

- Time Value

For a Call …

• if the Call is Out-of-the-money(Forward less than Strike)

there is only Time Value

(Intrinsic = 0)

• if the Call is In-the-money

(Forward greater than Strike)there is both Time Value and

Intrinsic ValueOut-of-the-money In-the-money

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Puts: Intrinsic Value and Time Value

The above graph shows the Price and Payoff for

a Put with Strike X.

The Put Price is again divided into …

- Intrinsic Value

- Time Value

X

O p t i o n

V a l u e

Forward

Intrinsic Value

Time Value

In-the-money Out-of-the-moneySource: Citigroup

For a Put …

• if the Put is Out-of-the-money

(Forward greater than Strike)

there is only Time Value

(Intrinsic = 0)

• if the Put is In-the-money

(Forward less than Strike)

there is both Time Value and

Intrinsic Value

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Black-Scholes Call Price

Suppose the distribution of an Index at time T in the future is assumed to be lognormal.

A Call on an Index with strike X, maturing at time T, is defined to have the Payoff …

X

IndexX

Call Payoff

Normal Distribution

N (x)

x

Standard Normal Distribution

Call Payoff = Max (0, Index – X)

The Black-Scholes equation states that the value of the Call is given by

}d(NXd(NF{e=icePr Call 21)tT(r )-)--

where …

F = Index Fwd Rate to time T

X = Strike

σ = Volatility of the Index

N(x) = Prob (Z<x)

Z is a N(0,1) variable

(standard Normal distribution)

)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ )tT(d=d 12 -- σ

∫∞-

-

x

z)21( dze21=)x(N

2

π

Ttimefor factor discounttheise )tT(r --

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Call Price – evolution over time

We graph the Call price as a function of the Forward Rate F for a selection of Terms to Maturity

Note that the graph of the Call Price approaches the payoff Max (0, Index – X) as the Term to Maturity falls.

The time value falls to 0 and the Call Price converges to the Payoff (intrinsic value)

Call Price

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

Forward

P r i c e

24 months

12 months

6 months

Maturity

X

Terms to Maturity



X = Strikeσ = Volatility of the Index

)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ

)tT(d=d 12 -- σ


N(x) = Prob (Z<x)Z is a N(0,1) variable


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12

Black-Scholes Put Price

Again we suppose the distribution of an Index at time T in the future is lognormal.

A Put on an Index with strike X, maturing at time T, is defined to have the Payoff …

Normal Distribution

N (x)

x

IndexX

Put Payoff


Call Payoff = Max (0, X – Index )X

The Black-Scholes equation states that the value of the Put is given by

}d(NFd(NX{e=icePr Put 12)tT(r )--)---

where …


X = Strike

σ = Volatility of the Index

N(x) = Prob (Z<x)



)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ )tT(d=d 12 -- σ


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Put Price – evolution over time

We also graph the Put price as a function of the Forward Rate F for a selection of Terms to Maturity

Note that the graph of the Put Price approaches the payoff Max (0, X – Index) as the Term to Maturity falls.

The time value falls to 0 and the Call Price converges to the Payoff (intrinsic value)

Put Price

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

Forward

P r i c e

24 months

12 months

6 months

Maturity

Terms to Maturity

X



X = Strikeσ = Volatility of the Index

)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ

)tT(d=d 12 -- σ


N(x) = Prob (Z<x)Z is a N(0,1) variable


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Put - Call Parity

Put - Call Parity for European options implies that for identical Calls and Puts …

Call – Put = Long ForwardPayoff

Since Max (0, Index – X) – Max (0, X – Index) = (Index – X)

it follows that …

and

To see this we look at the Payoffs at expiry …

Call Payoff = Max (0, Index – X)

Put Payoff = Max (0, X– Index )

Put – Call = Short Forward

Call Payoff – Put Payoff = Long Forward Payoff

Put Payoff – Call Payoff = Short Forward Payoff

Long Call

X Index

+

X

Payoff

Short Put

Index

=

Payoff

Long

Forward

IndexX

15

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Put - Call Parity

We can also check Put-Call Parity against the prices of European Calls and Puts

}d(NXd(NF{e=icePr Call 21)tT(r )-)-- (1)

Normal Distributionwhere …

F = Index Fwd Rate at time T

X = Strike

σ = Volatility of the swap rate

N(x) = Prob (Z<x)



Since Z is symmetric distribution around 0, we have for any x

Prob (Z< -x) = 1 – Prob (Z> -x) = 1 – Prob (Z< x)

N (–x)


0 –x x

1 – N (x)


)tT()tT()2/1(+)X/Fln(=d

21

--

σ

σ )tT(d=d 12 -- σ

(2)

ie N(–x) = 1 – N(x)

}d(NFd(NX[d(NXd(NF[{e=icePr PuticePr Call 1221)tT(r )]--)--)]-)- --Then …

}d(N+d(N[Xd(N+d(N[F{e= 2211)tT(r )]-)-)]-)--

}XF{e= )tT(r ---

= Price of entering a long Forward on the Index struck at X

16

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16

Put - Call Parity

Similarly, (2) – (1) gives …

}FX{e=icePr CallicePr Put )tT(r -- --

= Price of entering a short Forward on the Index struck at X

Now, If the Call and Put are at-the-money (ATM), so that X = F = Index Forward Price

then Call (F) – Put (F) = Buy Index forward at F

or

0=}FF{e= )tT(r ---Call (F) – Put (F)

and

Price (at the money Call) = Price (at the money Put)

17

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The Greeks

In pricing and trading options, it is important to consider one or more risk parameters, collectively calledthe Greeks.

These parameters measure the sensitivity of the option price to a variety of variables.

The most common greeks are

• Delta sensitivity of price to change in the underlying Index (or the Forward)

• Vega sensitivity of price to change in the volatility

• Theta sensitivity of price to change in time

• Gamma sensitivity of Delta to change in the underlying Index (a 2nd order effect)

where …

T = Option Maturity

t = today (valuation date)


X = Strike

σ = Volatility of the Index )tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ

For example, we know the Black-Scholes value of a European Call is …

N(x) = Prob (Z<x)



Normal Distribution


N (x)

x


)tT(d=d 12 -- σ

18

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18

The Greeks

We can use Calculus to derive the Greeks. To do this we differentiate the Call Price with respect to thevariable for which we require sensitivity (keeping other variables constant)

Sparing you the details (see the Appendix) we have for Calls ...

)--

1

)tT(r d(Ne=F

c=Delta

∂

∂

)tT(e2

1Fe=

c=Vega

21d)21()tT(r -

---

πσ∂

∂

)tT(F

1e

2

1e=

F

c=Gamma

21d)21()tT(r

2

2

-

---

σπ∂

∂

Looking now at Puts we differentiate the Put Price with respect to the variable for which we requiresensitivity (keeping other variables constant)

Thus for Puts …Note that the Vega and Gamma for a Put are identicalto the Vega and Gamma for a Call.

This follows from Put – Call Parity which states …Call – Put = Long Forward

1d(Ne=F

p=Delta 1

)tT(r -)--

∂

∂

)tT(e

2

1Fe=

p=Vega

21d)21()tT(r -

---

πσ∂

∂

)tT(F

1e

2

1e=

F

p=Gamma

21d)21()tT(r

2

2

-

---

σπ∂

∂

19

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19

Call Delta

The Delta of a Call measures the sensitivity of the Call Price with respect to the underlying.

Delta is the rate of change of the Call Price

with respect to the underlying forward rate FF

c=Delta

∂

∂

The Delta will therefore be the slope of the graph of Call Price as a function of the forward rate F

Call … Price, Payoff and Delta

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00% 3.40% 3.80% 4.20% 4.60% 5.00% 5.40% 5.80% 6.20% 6.60% 7.00%

Forward

P r i c e

out of the money in the money

Call Price

Call Payoff

Delta is the

rate of change

(slope) of Price

The graph on the right shows …

• Call Price as a function of F

• Call Payoff as a function of F

• Call Delta as the slope of the Call Price

where F = Forward Rate

The Strike on this Call is 5.00%.

At the Money (when the Forward Rate = 5.00%)

the Delta is approximately 0.50

Deep In the Money (Forward Rate >> 5.00%)

the Delta approaches 1 as the Call behaveslike an outright long Forward Strike = 5.00%

20

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Call Delta

The Delta of a Call measures the sensitivity of the Call Price with respect to the underlying.

Here we show the Delta of an at-the-money 2yr forward Caplet with strike X = 4.84%

The Delta is calculated and graphed as a function of the forward rate F

Caplet Delta

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%

Forward Rate

at the money

Delta ~ 0.50

in the money

Delta approaches 1out of the money

Delta approaches 0


Gamma is therate of change

(slope) of DeltaF

c=Delta

∂

∂

)--1

)tT(r d(Ne=Delta

N(x) = Prob (Z<x)



)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ


21

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Put Delta

The Delta of a Put measures the sensitivity of the Put Price with respect to the underlying.

Delta is the rate of change of the Put Price

with respect to the underlying forward rate FF

p=Delta

∂

∂

The Delta will therefore be the slope of the graph of Put Price as a function of the forward rate F

Put … Price, Payoff and Delta

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00% 3.40% 3.80% 4.20% 4.60% 5.00% 5.40% 5.80% 6.20% 6.60% 7.00%

Forward

P r i c e

in the money out of the money

Put Price

Put Payoff

Delta is the

rate of change

(slope) of Price

The graph on the right shows …

• Put Price as a function of F

• Put Payoff as a function of F

• Put Delta as the slope of the Put Price

where F = Forward Rate

The Strike on this Put is 5.00%.

At the Money (when the Forward Rate = 5.00%)

the Delta is approximately 0.50

Deep In the Money (Forward Rate << 5.00%)

the Delta approaches 1 as the Put behaveslike an outright short Forward Strike = 5.00%

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Put Delta

The Delta of a Put measures the sensitivity of the Put Price with respect to the underlying.

Here I show the Delta of an at-the-money 2yr forward Floorlet with strike X = 4.84%

The Delta is calculated and graphed as a function of the forward rate F

Floorlet Delta

-1.00

-0.90

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%

Forward Rate

at the money

Delta ~ - 0.50

out of the money

Delta approaches 0

in the money

Delta approaches -1

out of the moneyin the money

Gamma is therate of change

(slope) of DeltaF

p=Delta

∂

∂

1d(Ne=Delta 1)tT(r -)--

N(x) = Prob (Z<x)



)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ


23

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Vega

The Vega of an option measures the sensitivity of the option with respect to the volatility.

Here I show the Vega of an at-the-money 2yr forward Caplet with strike X = 4.84%

The Vega is calculated and graphed as a function of the Forward Rate F

Caplet Vega

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%

Forward Rate

Vega is largest

at the strike

Vega smaller

as call goes into

the money


Vega smaller

out of the money

)tT(e2

1Fe=Vega

21d)21()tT(r -

---

π

)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ

Ttimefor factor discounttheise

)tT(r --

(T - t) = term to Maturity of the Option

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Gamma

The Gamma of an option measures sensitivity of Delta to change in the underlying Index

Here I show the Gamma of an at-the-money 2yr forward Caplet with strike X = 4.84%

The Gamma is calculated and graphed as a function of the Forward Rate F

)tT(

)tT()2/1(+)X/Fln(=d

2

1-

-

σ

σ

Ttimefor factor discounttheise

)tT(r --

Caplet Gamma

0.00

5.00

10.00

15.00

20.00

25.00

30.00

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%

Forward Rate

at the money

Delta changing

most quickly

Delta changes

less as call goes

into the money


)tT(F

1e

2

1e=Gamma

21d)21()tT(r

-

---

σπ

(T - t) = term to Maturity of the Option

25

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Call Spread

A Call Spread is a combination of a bought Call and a sold Call at different strikes.

The Long Call is at the Low Strike, the Short Call at the High strike

Payoff = Max (0, IT – X1) – Max (0, I

T – X2)

0 if IT

< X1 < X2

= (IT – X1) if X1 < I

T< X2

(X2 – X1) if X1 < X2 < IT

X

X1 = Low Strike

X2 = High Strike

X2 – X1

X2 Index

where …

IT

= Index fixing at Maturity (time T)

X1 = Low Strike

X2 = High Strike X1

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Put Spread

A Put spread is a combination of a bought Put and a sold Put at different strikes.

The Long Put is at the High Strike, the Short Put at the Low strike

Payoff = Max (0, X2 – IT) – Max (0, X1 – I

T)

(X2 – X1) x DayCount if L < X1 < X2

= (X2 – L) x DayCount if X1 < L < X2

0 if X1 < X2 < L

X L = Libor X2 – X1

X1 X2 Index

where …

IT

= Index fixing at Maturity (time T)

X1 = Low Strike

X2 = High Strike

X1 = Low Strike

X2 = High Strike

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FX Options

An FX Option maturing at time T gives the option holder the right to exchange say M units of CCY 1 for N units of CCY2.

This is best explained by looking at an example.

Consider an option that allows the option holder at Maturity to …

sell P € EUR in return for receiving 1.30 x P € USD.

We say that this option is a EUR Put / USD Call struck at X = 1.30

Conversely a EUR Call / USD Put struck at 1.30 allows the holder

the right to …

buy P € EUR in return for paying 1.30 x P € USD

Exercise of EUR Put / USD Call

X

Option

Buyer

USD 1.30 x P €

EUR P €

EUR P €

Option

Buyer

Option

Seller

USD 1.30 x P €

Exercise of EUR Call / USD Put

Option

Seller

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FX Options

Note that in an FX Option there are by definition 2 currencies involved and potentially exchanged.This can create confusion when comparing to regular calls and puts which are settled in a single currency.

Consequently, in considering FX Options, it is very useful to divide the 2 currencies involved into aCommodity Currency and a Terms currency.

It is supposed then that the Spot Rate between the 2 currencies equates 1 unit of the Commodity Currencyto S units of the Terms Currency.

Spot 1 unit of Comm CCY = S units of Terms CCY S = Spot Rate

The option holder then has the right to buy (or sell) 1 unit of the Commodity Currency in return for paying Xunits of the Terms CCY.

In our example … EUR is the Commodity CurrencyUSD is the Terms Currency

Exercise of EUR Put / USD Call Exercise of EUR Call / USD Put

EUR 1 EUR 1

Option

Buyer

Option

Seller

Option

Buyer

Option

Seller

USD X USD X

Option Buyer has the right to

sell EUR 1 against USD X

Option Buyer has the right to

buy EUR 1 against USD X

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FX Option Price … Commodity Call / Terms Put

We assume as usual that the distribution of the Spot FX rate at time T in the future is lognormal.

A Commodity CCY Call / Terms CCY Put maturing at time T will have the payoff (in Terms CCY) …

Payoff = Max (0, ST – X) value in Terms CCY Comm Call / Terms Put

Note .. at time T, 1 Comm CCY = ST Terms CCY Comm 1

Option

Buyer

Option

Seller The Black-Scholes equation states that the price in Terms CCY of the

Comm Call / Terms Put is given by Terms X

where …

FT = Fwd FX Rate to time T

X = Strike

σ = Spot Volatility )tT(d=d 12 -- σ

)tT(

)tT()2/1(+)X/Fln(=d

2T

1-

-

σ

σ

Ttimefor factor discount=e )tT(r --

Normal Distribution}d(NXd(NF{e=icePr CallComm 21T

)tT(r

)-)--

N (x)

x

Standard Normal Distributionprice in Terms CCY

N(x) = Prob (Z<x)


(std Normal distribution)Note that FT is given by

)T(d

)T(d×S=F

t

cT ∫

∞-

-x

z)21( dze2

1=)x(N

2

π

dc(T) is the Comm discount factor for time T

dt(T) is the Terms discount factor for time T

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FX Option Price … Commodity Put / Terms Call

We assume again that the distribution of the Spot FX rate at time T in the future is lognormal.

A Commodity CCY Put / Terms CCY Call maturing at time T will have the payoff (in Terms CCY) …

Payoff = Max (0, X – ST) value in Terms CCY Comm Put / Terms Call

Note .. at time T, 1 Comm CCY = ST Terms CCY Comm 1

Option

Buyer

Option

Seller The Black-Scholes equation states that the price in Terms CCY of the

Comm Put / Terms Call is given by Terms X

where …

FT = Fwd FX Rate to time T

X = Strike

σ = Spot Volatility )tT(d=d 12 -- σ

)tT(

)tT()2/1(+)X/Fln(=d

2T

1-

-

σ

σ

Ttimefor factor discount=e )tT(r --

Normal Distribution}d(NFd(NX{e=icePr PutComm 1T2

)tT(r

)--)---

N (x)

x

Standard Normal Distributionprice in Terms CCY

N(x) = Prob (Z<x)


(std Normal distribution)Note that FT is given by

)T(d

)T(d×S=F

t

cT ∫

∞-

-x

z)21( dze2

1=)x(N

2

π

dc(T) is the Comm discount factor for time T

dt(T) is the Terms discount factor for time T

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Caps

A Cap is a basic Call on a floating Index, for example Libor or Euribor.

They frequently occur in structures where coupons dependent on a floating Index are either …

• capped by a maximum(a coupon cap)

• floored by a minimum (a coupon floor)

The Cap payoff, which is typically paid at the end of the coupon period to which it applies, is defined as …

Cap Payoff = Max (0, L - X) x DayCount = (L - X) x DayCount if L > X

0 otherwise

where …

L = Libor fixing

X = Strike

DayCount = day count of the underlying period.

XCaplet Payoff

XX Libor

In practice, we usually call this option a Caplet, reserving the term Cap

for a series of consecutive caplets.

For example, a 5yr Cap might be defined as the series of 10 semi-annual caplets.This is a series of calls on the Index, each with a common strike.

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Caplets

The Payoff in a Caplet usually occurs at the end of the underlying period. This helps ensure that the capletacts in harmony with floating rate coupons to “cap” the coupon.

For example, suppose we have a floating rate Note bearing a coupon of 3m USD Libor + 0.50% paid in arrears

If we impose the condition that the maximum coupon permitted in a period is 6.00%, then we haveimplicitly added a caplet struck at 5.50% to the coupon.

Then the coupon can be decomposed as ..

Min (3m USD Libor + 0.50%, 6.00%) = (3m USD Libor + 0.50%) + Min (0, 5.50% – 3m USD Libor)= (3m USD Libor + 0.50%) - Max (0, 3m USD Libor – 5.50%)

Long a floating coupon Short a Caplet struck at 5.50%

6.00%

Caplet payoff

Then the Coupon payment at the end of a 91 day period is ..

if Libor <= 5.50% [(3m Libor + 0.50%) x 91/360]

if Libor > 5.50% [(3m Libor + 0.50%) x 91/360]

– [(3m Libor – 5.50%) x 91/360]

= 6.00% x 91/360

Coupon

5.50%X Libor

Thus the addition of a short caplet position has “capped” the coupon at 6.00%

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Cap Pricing

We now look to use the Black-Scholes methodology to price a Caplet.

We assume a lognormal distribution for Libor, and adopt the following time-line

t T T +∆

- Today is denoted by time t

- the caplet matures (is fixed) at time T- the payoff of the caplet is at time T+ ∆

Option Period Underlying

A caplet is a call on Libor, maturing at time T with strike X, and paid at time T+ ∆.

X

Caplet Payoff Caplet Payoff = Max (0, L - X) x ∆ = (L - X) x ∆ if L > X

0 otherwise

Thus the price of the caplet is given by

XX Libor

}d(NXd(NF{×d×)F+1(

=icePr Caplet 21TTT

)-)∆

∆

where …

FT = Libor Forward Rate (T, T+ ∆)

X = Strike

σ = Volatility of Libor dT = discount factor to time T

N(x) = Prob (Z < x), Z~ N(0,1)

)tT(

)tT()2/1(+)X/Fln(=d

2T

1-

-

σ

σ

)tT(d=d12

-- σ

∆

∆)1dd(

=F+TT

T

-

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Floors

A Floor is similarly a basic Put on a floating Index, for example Libor or Euribor.

Again they frequently occur in structures where the coupons dependent on a floating Index are either

capped by a maximum or floored by a minimum.

The Floor payoff, which is typically paid at the end of the coupon period to which it applies, is defined as …

Floor Payoff = Max (0, X - L) x DayCount = (X - L) x DayCount if L < X

0 otherwise

where …

L = Libor fixing

X = Strike

DayCount = day count of the underlying period.

XFloorlet Payoff

X Libor

As for caps, we often call this option a Floorlet, reserving the term Floor

for a series of consecutive floorlets.

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Floorlets

The Payoff in a Floorlet usually occurs at the end of the underlying period. This helps ensure that thefloorlet acts in harmony with floating rate coupons to “floor” the coupon.

For example, suppose we have a floating rate Note bearing a coupon of

3m USD Libor + 0.50% paid in arrears

If we impose the condition that the minimum coupon permitted in a period is 2.00%, then we have implicitly

added a floorlet struck at 1.50% to the coupon.

Then the coupon can be decomposed as ..

Max (3m USD Libor + 0.50%, 2.00%) = (3m USD Libor + 0.50%) + Max (0, 1.50% – 3m USD Libor)

Long a floating coupon Long a Floorlet struck at 1.50%

XCoupon

2.00%

Floorlet payoff

Then the Coupon payment at the end of a 91 day period is ..

if Libor >= 1.50% [(3m Libor + 0.50%) x 91/360]

if Libor < 1.50% [(3m Libor + 0.50%) x 91/360]

+ [(1.50% – 3m Libor) x 91/360]

= 2.00% x 91/360

1.50%

Libor

Thus the addition of a long floorlet position has “floored” the coupon at 2.00%

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Floor Pricing

We now use the Black-Scholes methodology to price a Floorlet.

We assume a lognormal distribution for Libor, and adopt the following time-line

t T T + ∆- Today is denoted by time t

- the floorlet matures (is fixed) at time T- the payoff of the floorlet is at time T+ ∆

Option Period Underlying

A floorlet is a put on Libor, maturing at time T with strike X, and paid at time T+ ∆.

X

Floorlet Payoff Caplet Payoff = Max (0, X - L) x ∆ = (X - L) x ∆ if L < X

0 otherwise

Thus the price of the floorlet is given by

XX Libor

}d(NFd(NX{×d×)F+1(

=icePr Floorlet 1T2TT

)--)-∆

∆

where …

FT = Libor Forward Rate (T, T+ ∆)

X = Strike

σ = Volatility of Libor dT = discount factor to time T

N(x) = Prob (Z < x), Z~ N(0,1)

)tT(

)tT()2/1(+)X/Fln(=d

2T

1-

-

σ

σ

)tT(d=d12

-- σ

∆

∆)1dd(

=F+TT

T

-

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Cap Vols and Caplet Vols

As mentioned, a Cap is essentially a portfolio of Caplets.The Caplets have identical strikes and are temporally sequential, covering the life of the Cap.

t1 t2 t3 t4 tn tn+1 - caplets are c1, c2,.., cn

- caplet c j is fixed at time t j and paid at time t j+1

- each caplet c j has an individual volatilityσ

j

To allow for “skew”, caplet vols are functions of not just maturity but strike as well.

Each caplet c j has an individual volatility σ j(X), a function of strike X.

The caplet is priced using its own vol. But we cannot observe these vols directly.

Instead, what we can observe on trader screens are Cap Vols, which are average (or flat) vols.

They are typically quoted on screens for maturities 1yr, 2yr, 3yr, … etc

If each caplet in the Cap were to be priced using the flat Cap Vol σ(X), then the Total Cap price should

match the Total Cap price obtained by pricing each caplet c j at its individual volatility σ j(X)

Thus by observing the flat Cap Vols σ(X) for a series of maturities, we can use a bootstrap

method to obtain caplet vols σ j

(X) for each underlying period and strike X.

cn

c1

c2

c3

Total Period of the Cap

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Cap & Floor Put-Call Parity

The Put-Call Parity is easy to visualize graphically ...

Payoff Payoff Payoff

Libor X

long Caplet

X Libor

short Floorlet

X

pay X, receive Libor

+ =

Libor

Payoff

Libor X

short Caplet

X Libor

long Floorlet

Payoff Payoff

+ =

Libor X

receive X, pay Libor

options 1 (1)

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