Lecture 21: Options Markets
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
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, NYSE
1982
Terms of Options Contract
• Exercise date
• Exercise price
• Definition of underlying and number of
shares
Two Basic Kinds of Options
• Calls, a right to buy
• Puts, a right to sell
Two Basic Kinds of Options
• American options – can be exercised any
time until exercise date
• European options – can be exercised only
on exercise date
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 buyer’s choice
• Buyer puts up no margin, naked writer
must post margin
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
Exercise Price = 20
25
20
Intrinsic Value Call
15
10
5
0
0
5
10
15
20
25
-5
Stock Price
30
35
40
45
Exercise Price = 20
20
Intrisnic Value Put
15
10
5
0
0
5
10
15
20
25
-5
Stock Price
30
35
40
45
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 would
be arbitrage profit opportunities
Put Call Parity Relation Derivation
45
40
35
30
25
Stock Price
Intrinsic Value Put
20
Intrinsic Value Call
Exercise Price
15
10
5
0
0
5
10
15
20
25
-5
Stock Price
30
35
40
45
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
Exercise Price = 20, r=5%, T=1,sigma=.3
25
20
Call Price
15
Intrinsic Value of Call
10
Call Price (Black Scholes)
5
0
0
5
10
15
20
25
-5
Stock Price
30
35
40
45
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
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
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
Hedging by writing calls
• Investor writes one call and buys H shares
of underlying stock
• If price goes up, will be worth uHS-Cu
• If price goes down, worth dHS-Cd
• For what H are these two the same?
Cu  C d
H
(u  d ) S
Binomial Option Pricing Formula
• One invested HS-C to achieve riskless
return, hence the return must equal
(1+r)(HS-C)
• (1+r)(HS-C)=uHS-Cu=dHS-Cd
• Subst for H, then solve for C
1  r  d Cu
u  1  r Cd
C (
)(
)(
)(
)
u  d 1 r
u  d 1 r
Formula does not use probability
• Option pricing formula was derived without
regard to the probability that the option is
ever in the money!
• In effect, the price S of the stock already
incorporates this probability
• For illiquid assets, such as housing, this
formula may be subject to large errors
Black-Scholes Option Pricing
• Fischer Black and Myron Scholes derived
continuous time analogue of binomial formula,
continuous trading, for European options only
• Black-Scholes continuous arbitrage is not really
possible, transactions costs, a theoretical exercise
• Call T the time to exercise, σ2 the variance of oneperiod price change (as fraction) and N(x) the
standard cumulative normal distribution function
(sigmoid curve, integral of normal bell-shaped
curve) =normdist(x,0,1,1) Excel (x,
mean,standard_dev, 0 for density, 1 for cum.)
Black-Scholes Formula
C  SN (d1 )  EN (d 2 )
where
S
ln( )  rT   2T / 2
E
d1 
 T
S
ln( )  rT   2T / 2
E
d2 
 T
Implied Volatility
• Turning around the Black-Scholes formula,
one can find out what σ would generate
current stock price.
• σ depends on strike price, “options smile”
• Since 1987 crash, σ tends to be higher for
puts or calls with low strike price, “options
leer” or “options smirk”
VIX Implied Volatility
Weekly, 1992-2004
400
350
300
250
200
150
100
50
0
5/7/1990
9/19/1991
1/31/1993
6/15/1994
10/28/1995
3/11/1997
7/24/1998
12/6/1999
4/19/2001
9/1/2002
1/14/2004
5/28/2005
Implied and Actual Volatility
Monthly Jan 1992-Jan 2004
Im plied Volatility & Actual Volatility, Monthly, Jan 1992-Jan 2004
400
7
350
6
300
5
250
4
Implied
200
Actual
3
150
2
100
1
50
0
1990
1992
1994
1996
1998
Y ear
2000
2002
2004
0
2006
Actual S&P500 Volatility
Monthly1871-2004
Using Options to Hedge
• To put a floor on one’s holding of stock, one
can buy a put on same number of shares
• Alternatively, one can just decide to sell
whenever the price reaches the floor
• Doing the former means I must pay the
option price. Doing the latter costs nothing
• Why, then, should anyone use options to
hedge?
Behavioral Aspects of Options
Demand
• Thaler’s 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 more
leveraged, risky position
• Lottery principle in psychology, people
inordinately attracted to small probabilities of
winning big
• Margin requirements are circumvented by options
Option Delta
• Option delta is derivative of option price with
respect to stock 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 price of 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 with credit pdv(E)
Volatility of Call Return / Volatility of Stock Return, Exercise Price = 20
25
dln(call price)/dln(stock price)
20
15
10
5
0
0
5
10
15
20
25
Stock Price
30
35
40
45