options on stock indices, currencies, and futures

3

LEAPS (Long-term equity anticipation securities)

Leaps are options on stock indices that last up to 3 yearsThey have December expiration datesThey are on 10 times the indexLeaps also trade on some individual stocks

4

Index Option Example

Consider a call option on an index with a strike price of 560Suppose 1 contract is exercised when the index level is 580What is the payoff?

5

Using Index Options for Portfolio Insurance

Suppose the value of the index is S0 and the strike price is XIf a portfolio has a of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars heldIf the is not 1.0, the portfolio manager buys put options for each 100S0 dollars heldIn both cases, X is chosen to give the appropriate insurance level

6

Example 1

Portfolio has a beta of 1.0It is currently worth $5 millionThe index currently stands at 1000What trade is necessary to provide insurance against the portfolio value falling below $4.8 million?

7

Example 2

Portfolio has a beta of 2.0It is currently worth $1 million and index stands at 1000The risk-free rate is 12% per annum( 每 3 个月计一次复利）The dividend yield on both the portfolio and the index is 4%( 每 3 个月计一次复利）How many put option contracts should be purchased for portfolio insurance in 3 months?

8

If index rises to 1040, it provides a 40/1000 or 4% return in 3 monthsTotal return (incl. dividends)=5%Excess return over risk-free rate=2%Excess return for portfolio=4%Increase in Portfolio Value=4+3-1=6%Portfolio value=$1.06 million

Calculating Relation Between Index Level and Portfolio Value in 3 months

)( fmfi rrrr

9

Determining the Strike Price

Value of Index in 3months

Expected Portfolio Valuein 3 months ($ millions)

1,080 1.141,040 1.061,000 0.98 960 0.90 920 0.82

An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value

10

Valuing European Index Options

We can use the formula for an option on a stock paying a continuous dividend yieldSet S0 = current index level

Set q = average dividend yield expected during the life of the option

11

Currency OptionsCurrency options trade on the Philadelphia Exchange (PHLX)There also exists an active over-the-counter (OTC) marketCurrency options are used by corporations to buy insurance when they have an FX exposure

Range Forward Contracts

Have the effect of ensuring that the exchange rate paid or received will lie within a certain range

When currency is to be paid it involves selling a put with strike K1 and buying a call with strike K2 (with K2 > K1)

When currency is to be received it involves buying a put with strike K1 and selling a call with strike K2

Normally the price of the put equals the price of the call

12

Range Forward Contract continued Figure 15.1, page 320

13

Payoff

Asset PriceK

1

K2

Payoff

Asset Price

K

1

K

2

Short Position

Long Position

14

European Options on StocksPaying Continuous Dividends

We get the same probability distribution for the stock price at time T in each of the following cases:1. The stock starts at price S0 and provides a continuous dividend yield = q2. The stock starts at price S0e–q T and provides no income

15

European Options on StocksPaying Continuous Dividendscontinued

We can value European options by reducing the stock price to S0e–q T

and then behaving as though there is no dividend

16

Extension of Chapter 9 Results(Equations 15.1 to 15.3)

c S e XeqT rT 0

Lower Bound for calls:

Lower Bound for puts

p Xe S erT qT 0

Put Call Parity

c Xe p S erT qT 0

17

Extension of Chapter 13 Results (Equations 15.4 and 15.5)

c S e N d Xe N d

p Xe N d S e N d

dS X r q T

T

dS X r q T

T

qT rT

rT qT

0 1 2

2 0 1

10

20

2 2

2 2

( ) ( )

( ) ( )

ln( / ) ( / )

ln( / ) ( / )

where

18

The Binomial Model(Risk-neutral world)

S0u ƒu

S0d ƒd

S0

ƒ

p

(1– p )

f=e-rT[pfu+(1-p)fd ]

19

The Binomial Modelcontinued

In a risk-neutral world the stock price grows at r-q rather than at r when there is a dividend yield at rate qThe probability, p, of an up movement must therefore satisfy

pS0u+(1-p)S0d=S0e (r-q)T

so thatp

e d

u d

r q T

( )

20

The Foreign Interest Rate

We denote the foreign interest rate by rf

When a U.S. company buys one unit of the foreign currency it has an investment of S0 dollarsA unit foreign currency will become

units This shows that the foreign currency provides a “dividend yield” at rate rf

( )fr T te

21

Valuing European Currency Options

A foreign currency is an asset that provides a continuous “dividend yield” equal to rf

We can use the formula for an option on a stock paying a continuous dividend yield :

Set S0 = current exchange rate

Set q = rƒ

22

Formulas for European Currency Options

c S e N d Xe N d

p Xe N d S e N d

dS X r r

fT

T

dS X r r

fT

T

r T rT

rT r T

f

f

0 1 2

2 0 1

1

0

2

0

2 2

2 2

( ) ( )

( ) ( )

ln( / ) ( / )

ln( / ) ( / )

where

23

Alternative Formulas

F S e r r Tf

0 0 ( )Using

c e F N d XN d

p e XN d F N d

dF X T

T

d d T

rT

rT

[ ( ) ( )]

[ ( ) ( )]

ln( / ) /

0 1 2

2 0 1

10

2

2 1

2

24

Mechanics of Call Futures OptionsMost of Futures options are American.The maturity date is usually on, or a few

days before, the earliest delivery date of the underlying futures contract.

When a call futures option is exercised the holder acquires

1. A long position in the futures 2. A cash amount equal to the excess of

the futures price over the strike price

25

Mechanics of Put Futures Option

When a put futures option is exercised the holder acquires

1. A short position in the futures 2. A cash amount equal to the

excess of the strike price over the futures price

26

The Payoffs

If the futures position is closed out immediately:Payoff from call = Ft-X

Payoff from put = X-Ft

where Ft is futures price at time of exercise

27

Why Futures Option instead of Spot Option?

Futures is more liquid and easier to get the price informationCan be settled in cashLower transaction cost

28

Put-Call Parity for Futures Option

Consider the following two portfolios:1. European call plus Xe-rT of cash

2. European put plus long futures plus cash equal to F0e-rT

They must be worth the same at time T so that

c+Xe-rT=p+F0 e-rT

29

Futures Price = $33Option Price = $4

Futures Price = $28Option Price = $0

Futures price = $30Option Price=?

Binomial Tree Example

A 1-month call option on futures has a strike price of 29. Risk-Free Rate is 6% .

30

Consider the Portfolio: long futuresshort 1 call option

Portfolio is riskless when 3– 4 = -2 or = 0.8

3– 4

-2

Setting Up a Riskless Portfolio

31

Valuing the Portfolio

The riskless portfolio is: long 0.8 futuresshort 1 call option

The value of the portfolio in 1 month is -1.6The value of the portfolio today is

-1.6e – 0.06 = -1.592

32

Valuing the Option

The portfolio that is long 0.8 futuresshort 1 option

is worth -1.592The value of the futures is zeroThe value of the option must therefore be 1.592

33

Generalization of Binomial Tree Example

A derivative lasts for time T & is dependent on a futures

F0u ƒu

F0d ƒd

F0 ƒ

34

Generalization(continued)

Consider the portfolio that is long futures and short 1 derivative

The portfolio is riskless when

ƒu df

F u F d0 0

F0u F0 – ƒu

F0d F0– ƒd

35


Value of the portfolio at time T is F0u –F0 – ƒu

Value of portfolio today is – ƒHence

ƒ = – [F0u –F0– ƒu]e-

rT

36


Substituting for we obtainƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

pd

u d

1

37

Growth Rates For Futures Prices

A futures contract requires no initial investmentIn a risk-neutral world the expected return should be zeroThe expected growth rate of the futures price is thereforezeroThe futures price can therefore be treated like a stock paying a dividend yield of r

38

Futures price vs. expected future spot ptice

In a risk-neutral world, the expected growth rate of the futures price is zero,so

Because FT=ST, so

)(ˆ0 TFEF

)(ˆ0 TSEF

39

Valuing European Futures Options

We can use the formula for an option on a stock paying a continuous dividend yield

Set S0 = current futures price (F0)

Set q = domestic risk-free rate (r )Setting q = r ensures that the expected growth of F in a risk-neutral world is zero

40

Black’s Formula

The formulas for European options on futures are known as Black’s formulas

c e F N d X N d

p e X N d F N d

dF X T

T

dF X T

Td T

rT

rT

0 1 2

2 0 1

10

20

1

2 2

2 2

where

( ) ( )

( ) ( )

ln( / ) /

ln( / ) /

Futures Style Options (page 344-45)

A futures-style option is a futures contract on the option payoff

Some exchanges trade these in preference to regular futures options

A call futures-style option has value

A put futures style option has value

41

)()( 210 dKNdNF

)()( 102 dNFdKN

42

European Futures Option Prices vs Spot Option Prices

If the European futures option matures at the same time as the futures contract, then the two options are in theory equivalent.If the European call future option matures before the futures contract, it is worth more than the corresponding spot option in a normal market, and less in an inverted market.

43

American Futures Option Prices vs Spot Option Prices

There is always some chance that it will be optimal to exercise an American futures option early. So, if futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spotWhen futures prices are lower than spot prices (inverted market) the reverse is true

44

Summary of Key Results

We can treat stock indices, currencies, & futures like a stock paying a continuous dividend yield of q For stock indices, q = average

dividend yield on the index over the option life

For currencies, q = rƒ

For futures, q = r

options on stock indices, currencies, and futures

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