耶鲁-Robert Shiller-金融市场-Lect21-Options Markets

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
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
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”
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
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
Exercise Price = 20
25
20
15
Intrinsic Value Call
10
5
0 0 5 10 15 20 25 30 35 40 45
-5 Stock Price
Exercise Price = 20
20
15
Intrisnic Value Put
10
5
0 0 5 10 15 20 25 30 35 40 45
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
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
Two Basic Kinds of Options
• American options – can be exercised any time until exercise date • European options – can be exercised only on exercise date
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
79/2001
9/1/2002
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
Exercise Price = 20, r=5%, T=1,sigma=.3
25
20
15
Call Price
10
Intrinsic Value of Call Call Price (Black Scholes)
5
0 0 5 10 15 20 25 30 35 40 45
-5 Stock Price
300 5 250 4 200 3 150 2 100 Implied Actual
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
-5 Stock Price
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
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
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
Put Call Parity Relation Derivation
45
40
35
30
25 Stock Price 20 Intrinsic Value Put Intrinsic Value Call Exercise Price 15
10
5
0 0 -5 Stock Price 5 10 15 20 25 30 35 40 45
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.)