Options
Unit V
An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time.
What is a financial option?
• Call option: An option to buy a specified number of shares of a security within some future period.
• Put option: An option to sell a specified
number of shares of a security within some future period.
• Exercise (or strike) price: The price stated in
the option contract at which the security can be bought or sold.
Option Terminology
• Option price: The market price of the option contract.
• Expiration date: The date the option matures.
• Exercise value: The value of a call option if it were exercised today = Current stock price - Strike price.
Note: The exercise value is zero if the stock price is less than the strike price.
Options terminology
• In the Money - exercise of the option would be profitable
– Call: exercise price< market price – Put: exercise price > market price
• Out of the Money - exercise of the option would not be profitable
– Call: exercise price> market price – Put: exercise price < market price
• At the Money - exercise price and asset price are equal
Options terminology
• American - the option can be exercised at any time before expiration or maturity
• European - the option can only be exercised on the expiration or maturity date
• Intrinsic value - profit that could be made if the option was immediately exercised
– Call: stock price - exercise price – Put: exercise price - stock price
• Time value - the difference between the option price and the intrinsic value
Option Values
• Intrinsic value - profit that could be made if the option was immediately exercised
– Call: stock price - exercise price – Put: exercise price - stock price
• Time value - the difference between the option price and the intrinsic value
Notation
Stock Price = ST Exercise Price = X Payoff to Call Holder
(ST - X) if ST >X
0 if ST < X Profit to Call Holder
Payoff - Purchase Price
Payoffs and Profits at Expiration - Puts
Payoffs to Put Holder
0 if ST > X (X - ST) if ST < X Profit to Put Holder
Payoff - Premium
Payoff of a call options
Payoff of a put options
Payoff to Call Writer - (ST - X) if ST >X
0 if ST < X Profit to Call Writer
Payoff + Premium
Put-Call Parity
Put-call parity establishes an exact
relationship among the current stock price, the call price and the put price at any given moment. It can be written as:
Put Call Parity Theorem
IF C1 IS THE TERMINAL VALUE OF THE CALL OPTION
C1 = MAX [(S1 - E), 0]
P1 = MAX [(E - S1 ), 0]
S1 = TERMINAL VALUE
E = AMOUNT BORROWED C1 = S1 + P1 - E
Put-call parity: Payoffs just before Expiration Date
If S1<EI If S1>EI
1.Buy the equity stock S1 S1
2.Buy a put option E-S1 0
3.Borrow an amount equal to the exercise price
-E -E
1+2+3= Buy a call option 0 S1-E
Option valuation
Determinants of the Values of Call and Put Options
Variable C – Call
Value
P – Put Value
S – stock price ( +) Increase Decrease
X – exercise price (+) Decrease Increase
σ – stock price volatility (+) Increase Increase T – time to expiration ) (+) Increase Increase r – risk-free interest rate (+) Increase Decrease
19
Black- Scholes options Model
T d
d
T
T K R
S d
d N Ke
d SN
C RT
1 2
2
1
2 1
and ln 2 where
) ( )
(
20
The Model (cont’d)
• Variable definitions:
S = current stock price K = option strike price
e = base of natural logarithms R = riskless interest rate
T = time until option expiration
= standard deviation (sigma) of returns on the underlying security
ln = natural logarithm N(d1) and
N(d2) = cumulative standard normal distribution functions