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(1)

Options

Unit V

(2)

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?

(3)

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

(4)

• 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.

(5)

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

(6)

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

(7)

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

(8)

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

(9)

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

(10)

Payoff of a call options

(11)

Payoff of a put options

(12)

Payoff to Call Writer - (ST - X) if ST >X

0 if ST < X Profit to Call Writer

Payoff + Premium

(13)

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:

(14)

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

(15)

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

(16)
(17)

Option valuation

(18)

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)

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)

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

References

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