15.1

Options on Stock Indices and Currencies

Chapter 15

15.2

Index Options

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)

Contracts are on 100 times index; they are settled in cash; OEX is American; the XEO and all other options are European.

15.3

Index Option Example

Consider a call option on an index with a strike price of 560

Suppose 1 contract is exercised when the index level is 580

What is the payoff?

(580–560)100=2000

Portfolio Insurance with Index Puts

Type of Protective Put (but applied to portfolios) Problem: Portfolio not the same as index Must determine #puts and K # of puts = betaX(Vport/(100xVindex)) Strike price, K: Derived from Vprotect and SML

(security market line) Rindex & Rport each split into 2 components:

capital gains & dividends

15.5

Using Index Options for Portfolio Insurance

Suppose the value of the index is S0 and the strike price is K

If 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 held

If the is not 1.0, the portfolio manager buys put options for each 100S0 dollars held

In both cases, K is chosen to give the appropriate insurance level

15.6

Example 1

Portfolio has a beta of 1.0 It is currently worth $500,000 The index currently stands at 1000 Portfolio & index: same dividend yield of 4% q.c. What trade is necessary to provide insurance

against the portfolio value falling below $450,000 by end of quarter?

Risk-free rate = 12% with quarterly compound. (compounding period matches time horizon)

Buys 5 puts with K = 900

15.7

Example 2 (same as Example 1 except for higher beta)

Portfolio has a beta of 2.0 It is currently worth $500,000 and index

stands at 1000 Interest rates quoted: quarterly compounding The risk-free rate is 12% per annum The dividend yield on both the portfolio and

the index is 4% Portfolio insurance: #puts = 10; K = 960

Effect of beta increase on portfolio insurance

#puts increases. Why? K, strike price, also increases. Why? Increase in beta means greater probability

mass below the threshold (Vprotect). Thus, require higher indemnity (payoff from the insurance policy). Ergo, more insurance contracts, i.e. puts.

15.9

If index rises to 1040, it provides a 40/1000 or 4% return in 3 months

Total return (incl. dividends)=5% Excess return of index over risk-free

rate= 2% = (5-3)% Excess return for portfolio=4%= 2(2%) Increase in Portfolio Value=4–1+3=6% Portfolio value= $530,000 =

$500,000(1.06)

Calculating Relation Between Index Level and Portfolio Value in 3 months

15.10

Determining the Strike Price:nexus between Vprotect & K is SML

Value of Index in 3 months (K)

Expected Portfolio Value in 3 months (Vprotect $)

1,080 570,000 1,040 530,000 1,000 490,000 960 450,000 920 410,000 880 370,000

An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value from $500,000.

15.11

Currency Options Currency options trade on the Philadelphia

Exchange (PHLX) There also exists an active over-the-counter

(OTC) market Currency options are used by corporations

to buy insurance when they have an FX exposure

15.12

European Options on StocksPaying Dividend Yields

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 dividend yield = q

2. The stock starts at price S0e–q T

and provides no income

15.13

European Options on StocksPaying Dividend Yieldcontinued

We can value European options by reducing the stock price to S0e–q T and then behaving as though there is no dividend

15.14

Extension of Chapter 10 Results

rTqT KeeSc 0

Lower Bound for calls:

Lower Bound for puts

qTrT eSKep 0

Put Call Parity

qTrT eSpKec 0

15.15

Extension of Chapter 13 Results

T

TqrKSd

T

TqrKSd

dNeSdNKep

dNKedNeScqTrT

rTqT

)2/2()/ln(

)2/2()/ln(

)()(

)()(

02

01

102

210

where

15.16

The Binomial Model (p is the sole parameter that changes)

S0u ƒu

S0d ƒd

S0

ƒ

p

(1– p )

f = e-rT[pfu+(1– p)fd ]

15.17

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 q

The probability, p, of an up movement must therefore satisfy

pS0u+(1 – p)S0d=S0e (r-q)T

so that

pe d

u d

r q T

( )

15.18

Valuing European Index Options

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

Set S0 = current index level

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

15.19

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 dollars

The return from investing at the foreign rate is rf S0 dollars

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

15.20

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

15.21

Formulas for European Currency Options

T

Tf

rrKSd

T

Tf

rrKSd

dNeSdNKep

dNKedNeScTrrT

rTTr

f

f

)2/2()/ln(

)2/2()/ln(

)()(

)()(

0

2

0

1

102

210

where

15.22

Alternative Formulas(combine cost-of-carry model with previous model)

F S e r r Tf

0 0 ( )Using

Tdd

T

TKFd

dNFdKNep

dKNdNFecrT

rT

12

20

1

102

210

2/)/ln(

)]()([

)]()([