Ch16HullOFOD8thEdition

8/12/2019 Ch16HullOFOD8thEdition

1/24

Chapter 16

Options on Stock I ndices

and Currencies

Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 1


2/24

I ndex Options (page 345-347)

The most popular underlying indices in the U.S. are

The S&P 100 Index (OEX and XEO)

The S&P 500 Index (SPX)The Dow Jones Index times 0.01 (DJX)

The Nasdaq 100 Index (NDX)

Exchange-traded contracts are on 100 times index;they are settled in cash; OEX is American; the XEOand all others are European



3/24

I ndex Option ExampleConsider a call option on an index with astrike price of 880

Suppose 1 contract is exercised whenthe index level is 900

What is the payoff?



4/24

Using Index Options for Portfol io

I nsurance

Suppose the value of the index is S0and the strikeprice isK

If a portfolio has a bof 1.0, the portfolio insurance isobtained by buying 1 put option contract on the indexfor each 100S0 dollars held

If the bis not 1.0, the portfolio manager buys bputoptions for each 100S0 dollars held

In both cases,K is chosen to give the appropriateinsurance level



5/24


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Example 2Portfolio has a beta of 2.0

It is currently worth $500,000 and indexstands at 1000

The risk-free rate is 12% per annum

The dividend yield on both the portfolio andthe index is 4%

How many put option contracts should bepurchased for portfolio insurance?



7/24

Calculating Relation Between I ndex Level and

Portfol io Value in 3 months

If index rises to 1040, it provides a40/1000 or 4% return in 3 months

Total return (incl. dividends) = 5%Excess return over risk-free rate = 2%

Excess return for portfolio = 4%

Increase in Portfolio Value = 4+31=6%Portfolio value=$530,000



8/24

Determining the Str ike Price (Table 16.2,page 347)

Value of Index in 3months

Expected Portfolio Valuein 3 months ($)

1,080 570,000

1,040 530,0001,000 490,000

960 450,000920 410,000


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


9/24

European Options on Assets

Providing a Known Yield

We get the same probability distribution

for the asset price at time T in each of thefollowing cases:

1. The asset starts at price S0 and

provides a yield = q2. The asset starts at price S0eqT and

provides no income



10/24

European Options on Assets

Providing KnownYieldcontinued

We can value European options by reducingthe asset price to S0e

qT and then behaving asthough there is no income



11/24

Extension of Chapter 10 Results(Equations 16.1 to 16.3)


rTqT KeeSc 0

Lower Bound for calls:

Lower Bound for puts

qTrT eSKep 0

Put Call Parity

rTrTqTrT eFpKeceSpKec 00


12/24

Extension of Chapter 14 Results(Equations 16.4 and 16.5)


TTqrKSd

T

TqrKSd

dNeSdNKep

dNKedNeSc

qTrT

rTqT

)2/

2

()/ln(

)2/2()/ln(where

)()(

)()(

02

01

102

210


13/24

Alternative Formulas (page 353)

Tqr

rT

rT

eSF

Tdd

T

TKFd

dNFdKNep

dKNdNFec

)(

/)/ln(

)]()([

)]()([

00

12

20

1

102

210

where

2



14/24

Valuing European I ndex OptionsWe can use the formula for an option onan asset paying a dividend yield

Set S0= current index levelSet q= average dividend yield expectedduring the life of the option



15/24

Impl ied Forward Prices and

Dividend YieldsFrom European calls and puts with the same strikeprice and time to maturity

These formulas allow term structures of forward pricesand dividend yields to be estimated

OTC European options are typically valued using theforward prices (Estimates of qare not then required)

American options require the dividend yield termstructure


0

0 ln1)(SKepc

TqepcKF

rTrT


16/24

The Binomial Model


S0u

u

S0d

d

S0

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


17/24

The Binomial Model continued

In a risk-neutral world the asset pricegrows at rqrather than at rwhen there isa dividend yield at rate q

The probability,p, of an up movementmust therefore satisfy

pS0u+(1p)S0d = S0e(rq)T

so that


p e d

u d

r q T

( )


18/24

Currency Options

Currency options trade on NASDAQ OMX

There also exists a very active over-the-counter (OTC) market

Currency options are used bycorporations to buy insurance when theyhave an FX exposure



19/24

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 aput with strikeK1and buying a call with strikeK2(withK2 >K1)

When currency is to be received it involvesbuying a put with strikeK

1and selling a call with

strikeK2 Normally the price of the put equals the price of

the call



20/24

Range Forward Contract continuedF igure 16.1, page 348


Payoff

AssetPrice

K1 K2

Payoff

AssetPrice

K1 K2

ShortPosition

LongPosition


21/24

The Foreign I nterest Rate

We denote the foreign interest rate by rfWhen a U.S. company buys one unit of the

foreign currency it has an investment of S0

dollars

The return from investing at the foreign rateis rf S0dollars

This shows that the foreign currencyprovides a yield at rate rf



22/24

Valuing European Currency

Options

A foreign currency is an asset thatprovides a yield equal to rf

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

S0 = current exchange rate

q = r



23/24

Formulas for European Currency

Options (Equations 16.11 and 16.12, page 355)


T

TfrrKS

d

T

TfrrKS

d

dNeSdNKep

dNKedNeSc

TrrT

rTTr

f

f

)2/2()/ln(

)2/2()/ln(where

)()(

)()(

0

2

0

1

102

210


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Alternative Formulas (Equations 16.13 and 16.14)


F S e r r Tf

0 0( )Using

Tdd

T

TKF

d

dNFdKNep

dKNdNFecrT

rT

12

20

1

102

210

2/)/ln(

)]()([

)]()([

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