lect options

8/8/2019 Lect Options

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Lecture 21: Options Markets


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Options

With options, one pays money to have a

choice in the future

Essence of options is not that I buy the

ability to vacillate, or to exercise free will.

The choice one makes actually depends

only on the underlying asset price Options are truncated claims on assets


3/31

Options Exchanges

Options are as old as civilization. Option to

buy a piece of land in the city

Chicago Board Options Exchange, a spinoff

from the Chicago Board of Trade 1973,

traded first standardized options

American Stock Exchange 1974, NYSE1982


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Terms of Options Contract

Exercise date

Exercise price

Definition of underlying and number of

shares


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Two Basic Kinds of Options

Calls, a right to buy

Puts, a right to sell


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7/31

Two Basic Kinds of Options

American options can be exercised any

time until exercise date

European options can be exercised only

on exercise date


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Buyers and Writers

For every option there is both a buyer and a

writer

The buyer pays the writer for the ability to

choose when to exercise, the writer must

abide by buyers choice

Buyer puts up no margin, naked writermust post margin


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In and Out of the Money

In-the-money options would be worth

something if exercised now

Out-of-the-money options would be

worthless if exercised now


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Exercise Price = 20

-5

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45

Stock Price

IntrinsicValueCall


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Exercise Price = 20

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40 45

Stock Price

IntrisnicValuePut


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Put-Call Parity Relation

Put option price call option price =

present value of strike price + present value

of dividends price of stock For European options, this formula must

hold (up to small deviations due to

transactions costs), otherwise there wouldbe arbitrage profit opportunities


13/31

Put Call Parity Relation Derivation

-5

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35 40 45

Stock Price

Stock Price

Intrinsic Value Put

Intrinsic Value Call

Exercise Price


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Limits on Option Prices

Call should be worth more than intrinsic

value when out of the money

Call should be worth more than intrinsic

value when in the money

Call should never be worth more than the

stock price


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Exercise Price = 20, r=5%, T=1,sigma=.3

-5

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45

Stock Price

CallPrice

Intrinsic Value of Call

Call Price (Black Scholes)


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Binomial Option Pricing

Simple up-down case illustrates

fundamental issues in option pricing

Two periods, two possible outcomes only

Shows how option price can be derived

from no-arbitrage-profits condition


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Binomial Option Pricing, Cont.

S= current stock price

u = 1+fraction of change in stock price if

price goes up

d= 1+fraction of change in stock price if

price goes down

r = risk-free interest rate


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Binomial Option Pricing, Cont.

C= current price of call option

Cu= value of call next period if price is up

Cd= value of call next period if price is

down

E= strike price of option H= hedge ratio, number of shares

purchased per call sold


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Hedging by writing calls

Investor writes one call and buysHshares

of underlying stock

If price goes up, will be worth uHS-Cu

If price goes down, worth dHS-Cd

For whatHare these two the same?

Sdu

CCH

du

)(

=


20/31

Binomial Option Pricing Formula

One investedHS-Cto achieve riskless

return, hence the return must equal (1+r)

(HS-C)

(1+r)(HS-C)=uHS-Cu=dHS-C

d

Subst forH, then solve forC

)1

)(1

()1

)(1(

r

C

du

ru

r

C

du

drC

du

+

+

+

+=


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22/31


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Black-Scholes Formula

T

TrTE

S

d

T

TrTE

S

d

dEdSC

2/)ln(

2/)ln(

where

)(N)(N

2

2

2

1

21

+

=

++

=

=


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VIX Implied Volatility

Weekly, 1992-2004

0

50

100

150

200

250

300

350

400

5 /7 /1 99 0 9 /1 9/ 19 91 1 /3 1/ 19 93 6 /1 5/ 19 94 1 0/ 28 /1 99 5 3 /1 1/ 19 97 7 /2 4/ 19 98 1 2/ 6/ 19 99 4 /1 9/ 20 01 9 /1 /2 00 2 1 /1 4/ 20 04 5 /2 8/ 20 05


26/31

Implied and Actual Volatility

Monthly Jan 1992-Jan 2004Implied Volatility & Actual Vo latility, Monthly, Jan 1992-Jan 2004

0

50

100

150

200

250

300

350

400

1990 1992 1994 1996 1998 2000 2002 2004 2006

Year

0

1

2

3

4

5

6

7

Implied

Actual


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Actual S&P500 Volatility

Monthly1871-2004Six-Month Moving Standard Deviation of S&P 500 Price Change, 1871-2004

0

5

10

15

20

25

30

1860 1880 1900 1920 1940 1960 1980 2000 2020

Year


28/31

Using Options to Hedge

To put a floor on ones holding of stock,one can buy a put on same number of shares

Alternatively, one can just decide to sellwhenever the price reaches the floor

Doing the former means I must pay theoption price. Doing the latter costs nothing

Why, then, should anyone use options tohedge?


29/31

Behavioral Aspects of Options

Demand Thalers mental categories theory

Writing an out-of-the-money call on a stock one

holds, appears to be a win-win situation (Shefrin) Buying an option is a way of attaining a moreleveraged, risky position

Lottery principle in psychology, people

inordinately attracted to small probabilities ofwinning big

Margin requirements are circumvented by options


30/31

Option Delta

Option delta is derivative of option price with respect tostock price

For calls, if stock price is way below exercise price, delta

is nearly zero For calls, if option is at the money, delta is roughly a

half, but price of option may be way below half the priceof the stock.

For calls, if stock price is way above the exercise price,delta is nearly one and one pays approximately stock

price minus pdv of exercise price, like buying stock withcredit pdv(E)


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Volatility of Call Return / Volatility of Stock Return, Exercise Price = 20

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45

Stock Price

dln(callprice)/dln(sto

ckprice)

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