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OPT-2
Options Review
• Mechanics of Option Markets
• Properties of Stock Options• Valuing Stock Options:
–The Black-Scholes Model
OPT-4
Option Basics
• Option = derivative security– Value “derived” from the value of the
underlying asset
• Stock Option Contracts
– Exchange-traded
– Standardized
• Facilitates trading and price reporting.
– Contract = 100 shares of stock
OPT-5
Put and Call Options• Call option
– Gives holder the right but not the obligation to buy the underlying asset at a specified price at a specified time
• Put option– Gives the holder the right but not the
obligation to sell the underlying asset at a specified price at a specified time
OPT-6
Options on Common Stock
1. Identity of the underlying stock
2. Strike or Exercise price
3. Contract size
4. Expiration date or maturity
5. Exercise cycle• American or European
6. Delivery or settlement procedure
OPT-7
Option Exercise
• American-style– Exercisable at any time up to and
including the option expiration date– Stock options are typically American
• European-style– Exercisable only at the option expiration
date
OPT-8
Option Positions• Call positions:
– Long call = call “holder”• Hopes/expects asset price will increase
– Short call = call “writer”• Hopes asset price will stay or decline
• Put Positions:– Long put = put “holder”
• Expects asset price to decline– Short put = put “writer”
• Hopes asset price will stay or increase
OPT-9
Option Writing
• The act of selling an option
• Option writer = seller of an option contract – Call option writer obligated to sell the
underlying asset to the call option holder– Put option writer obligated to buy the
underlying asset from the put option holder– Option writer receives the option premium
when contract entered
OPT-10
Option Payoffs & Profits
Notation:• S0 = current stock price per share• ST = stock price at expiration • X = option exercise or strike price• C = American call option premium per share
• c = European call option premium
• P = American put option premium per share• p = European put option premium
• r = risk free rate• T = time to maturity in years
OPT-11
Payoff to Call Holder
(S - X) if S >X
0if S < X
Profit to Call Holder
Payoff - Option Premium
Profit =Max (S-X, 0) - C
Option Payoffs & ProfitsCall Holder
= Max (S-X,0)
OPT-12
Payoff to Call Writer
- (S - X) if S > X = -Max (S-X, 0)
0 if S < X = Min (X-S, 0)
Profit to Call Writer
Payoff + Option Premium
Profit = Min (X-S, 0) + C
Option Payoffs & ProfitsCall Writer
OPT-13
Payoff & Profit Profiles for Calls
Call Payoff and ProfitX = $20 c = $5.00
StockPrice Payoff Profit Payoff Profit
0 $0 -$5 $0 $5$10 $0 -$5 $0 $5$20 $0 -$5 $0 $5$30 $10 $5 -$10 -$5$40 $20 $15 -$20 -$15
Call Holder Call Writer
Payoff: Max(S-X,0) -Max(S-X,0) Profit: Max (S-X,0) – c -[Max (S-X, 0)-p]
OPT-14
Payoff & Profit Profiles for Calls
Profit
Stock Price
0
Call Writer Profit
Call Holder Profit
Payoff
OPT-15
Payoffs to Put Holder
0 if S > X(X - S) if S < X
Profit to Put Holder Payoff - Option PremiumProfit = Max (X-S, 0) - P
Option Payoffs and Profits Put Holder
= Max (X-S, 0)
OPT-16
Payoffs to Put Writer
0 if S > X = -Max (X-S, 0)-(X - S) if S < X = Min (S-X, 0)
Profits to Put Writer
Payoff + Option PremiumProfit = Min (S-X, 0) + P
Option Payoffs and Profits Put Writer
OPT-17
Put Payoff and ProfitX = $20 p = $5.00
StockPrice Payoff Profit Payoff Profit
0 $20 $15 -$20 -$15$10 $10 $5 -$10 -$5$20 $0 -$5 $0 $5$30 $0 -$5 $0 $5$40 $0 -$5 $0 $5
Put Holder Put Writer
Payoff & Profit Profiles for Puts
Payoff: Max(X-S,0) -Max(X-S,0) Profit: Max (X-S,0) – p -[Max (X-S, 0)-p]
OPT-19
CALL PUT
Holder: Payoff Max (S-X,0) Max (X-S,0)(Long) Profit Max (S-X,0) - C Max (X-S,0) - P
“Bullish” “Bearish”
Writer: Payoff Min (X-S,0) Min (S-X,0) (Short) Profit Min (X-S,0) + C Min (S-X,0) + P
“Bearish” “Bullish”
Option Payoffs and Profits
S = P = Value of firm at expiration X = Face Value of Debt
OPT-20
Long Call
Call option premium (C) = $5, Strike price (X) = $100.
30
20
10
0-5
70 80 90 100
110 120 130
Profit ($)
Terminalstock price (S)
Long Call Profit = Max(S-X,0) - C
OPT-22
Notation
c = European call option price (C = American)
p = European put option price (P = American)
S0 = Stock price today
ST =Stock price at option maturity
X = Strike price
T = Option maturity in years
= Volatility of stock price
r = Risk-free rate for maturity T with continuous compounding
OPT-23
American vs. European Options
An American option is worth at least as much as the corresponding European option
C c
P p
OPT-24
Factors Influencing Option Values
Call Put
Underlying stock price S + -Strike price of option contract X - +Time remaining to expiration T + +Volatility of the underlying stock price σ + +Risk-free interest rate r + -
Input Factor
Effect on Option Value
American
OPT-25
Effect on Option Values Underlying Stock Price (S) & Strike Price (K)
• Payoff to call holder: Max (S-X,0)– As S , Payoff increases; Value increases– As X , Payoff decreases; Value decreases
• Payoff to Put holder: Max (X-S, 0)– As S , Payoff decreases; Value decreases– As X , Payoff increases; Value increases
OPT-26
Option Price QuotesCalls
MSFT (MICROSOFT CORP) 25.98$
July 2008 CALLS
Strike Last Sale Bid Ask Vol Open Int
15.00 10.85 10.95 11.10 10 85
17.50 10.54 8.45 8.55 0 33
20.00 6.00 6.00 6.05 4 729
22.50 3.60 3.55 3.65 195 3891
24.00 2.30 2.24 2.27 422 2464
25.00 1.50 1.45 1.48 3190 10472
26.00 0.83 0.83 0.85 2531 15764
27.50 0.31 0.29 0.31 2554 61529
OPT-27
Option Price QuotesPuts
MSFT (MICROSOFT CORP) 25.98$
July 2008 PUTS
Strike Last Sale Bid Ask Vol Open Int
15.00 0.01 0.00 0.01 0 2751
17.50 0.01 0.00 0.02 0 2751
20.00 0.01 0.01 0.02 0 5013
22.50 0.03 0.03 0.04 13 4788
24.00 0.11 0.11 0.12 50 25041
25.00 0.25 0.24 0.25 399 7354
26.00 0.45 0.45 0.47 10212 51464
27.50 0.80 0.82 0.84 2299 39324
OPT-28
Effect on Option Values Time to Expiration = T
• For an American Call or Put:– The longer the time left to maturity, the
greater the potential for the option to end in the money, the grater the value of the option
• For a European Call or Put:– Not always true due to restriction on exercise
timing
OPT-29
Option Price Quotes
MSFT (MICROSOFT CORP) 25.98STRIKE = $25.00
CALLS Last Sale Bid Ask Vol Open IntJuly 2008 1.42 1.45 1.48 355 10472August 2008 1.80 1.85 1.87 257 927October 2008 2.36 2.43 2.46 41 3309January 2009 3.10 3.15 3.20 454 59244
PUTS Last Sale Bid Ask Vol Open IntJuly 2008 0.47 0.45 0.47 419 51464August 2008 0.81 0.80 0.82 401 1591October 2008 1.43 1.39 1.41 215 25323January 2009 2.09 2.06 2.08 2524 155877
OPT-30
Effect on Option Values Volatility = σ
• Volatility = a measure of uncertainty about future stock price movements
– Increased volatility increased upside potential and downside risk
• Increased volatility is NOT good for the holder of a share of stock
• Increased volatility is good for an option holder– Option holder has no downside risk– Greater potential for higher upside payoff
OPT-31
Effect on Option Values Risk-free Rate = r
• As r : –Investor’s required return increases
–The present value of future cash flows decreases
= Increases value of calls
= Decreases value of puts
OPT-33
BSOPMBlack-Scholes (-Merton) Option Pricing Model
• “BS” = Fischer Black and Myron Scholes– With important contributions by Robert Merton
• BSOPM published in 1973
• Nobel Prize in Economics in 1997
• Values European options on non-dividend paying stock
OPT-34
Concepts Underlying Black-Scholes
• Option price and stock price depend on same underlying source of uncertainty
• A portfolio consisting of the stock and the option can be formed which eliminates this source of uncertainty (riskless).– The portfolio is instantaneously riskless
– Must instantaneously earn the risk-free rate
OPT-35
Assumptions Underlying BSOPM
1. Stock price behavior corresponds to the lognormal model with μ and σ constant
2. No transactions costs or taxes. All securities are perfectly divisible
3. No dividends on stocks during the life of the option
4. No riskless arbitrage opportunities5. Security trading is continuous6. Investors can borrow & lend at the risk-free
rate7. The short-term rate of interest, r, is constant
OPT-36
Notation
• c and p = European option prices (premiums)
• S0 = stock price
• X = strike or exercise price• r = risk-free rate• σ = volatility of the stock price• T = time to maturity in years
OPT-37
Formula Functions
• ln(S/X) = natural log of the "moneyness" term
• N(d) = the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
• N(d1) and N(d2) denote the standard normal probability for the values of d1 and d2.
• Formula makes use of the fact that:
N(-d1) = 1 - N(d1)
OPT-38
The Black-Scholes Formulas
Tdd
T
T)2/2r()X/Sln(d
)d(NS)d(Nep
)d(NeX)d(NSc
12
01
102rT
2rT
10
: where
X
OPT-39
BSOPM Example
Given:
S0 = $42 r = 10% σ = 20%
X = $40 T = 0.5
6278.050.020.07693.0d
Tdd
7693.050.020.0
5.0)220.010.0()4042ln(d
T
T)2/2r()X/Sln(d
2
12
2
1
01
OPT-40
BSOPMCall Price Example
$4.76 c
(0.7349)42e - 40(0.7791) c
.5.10-
)d(NeX)d(NSc 2rT
10
d1 = 0.7693 N(0.7693) = 0.7791
d2 = 0.6278 N(0.6278) = 0.7349