Download - 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
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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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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
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Stock Price
IntrinsicValueCall
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Exercise Price = 20
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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
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Put Call Parity Relation Derivation
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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
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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
)(
=
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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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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
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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
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Implied and Actual Volatility
Monthly Jan 1992-Jan 2004Implied Volatility & Actual Vo latility, Monthly, Jan 1992-Jan 2004
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1990 1992 1994 1996 1998 2000 2002 2004 2006
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Implied
Actual
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Actual S&P500 Volatility
Monthly1871-2004Six-Month Moving Standard Deviation of S&P 500 Price Change, 1871-2004
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1860 1880 1900 1920 1940 1960 1980 2000 2020
Year
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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?
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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
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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
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Stock Price
dln(callprice)/dln(sto
ckprice)