Options Prepared by Paul A. Spindt. A Call Option Gives its owner the right (not obligation) underlying to buy an asset (the underlying) exercise price.

OptionsOptions

Prepared by

Paul A. Spindt

A Call OptionA Call Option

Gives its owner the right (not obligation)

to buy an asset (the underlyingunderlying) at a specified price (the exercise exercise

priceprice) on (and perhaps before) a given

date (the expiration dateexpiration date)

A Put OptionA Put Option

Gives its owner the right (not obligation)

to sell an asset (the underlyingunderlying) at a specified price (the exercise exercise

priceprice) on (and perhaps before) a given

date (the expiration dateexpiration date)

Options LingoOptions Lingo

option premium intrinsic value; time value European; American long; short covered; naked

ExampleExample

Here’s a quote from Wednesday’s Wall Street Journal:

-Call- -Put-

SunMic

583/4 60 Jan 540 35/8 1105 37/8

Wednesday’s closing stock

price

Option strike price

Expiration date

Volume

Premium

π =cT − c

Option Payoff: CallsOption Payoff: Calls

0

Payoff

Stock Price (at Expiration)

Strike Price

cT =max0,ST −X( )

π =pT − p

Option Payoff: PutsOption Payoff: Puts

0

Payoff

Stock Price (at Expiration)

Strike Price

pT =maxX −ST ,0( )

Option ValueOption Value

Intrinsic value The intrinsic value of a call is

max(0,S-X)max(0,S-X) The intrinsic value of a put is max(X-max(X-

S,0)S,0) Time value

Time value is the option premium minus intrinsic value

The Value of a CallThe Value of a CallValue

X }When S is greater than X, the intrinsic value of

the call is $(S-X)

(S-X)

SThis call option has an intrinsic value of $3.75 and time value of $3.50

-Call- -Put-

SunMic

583/4 55 Jan 166 71/4 109 2

{

When S is less than X, the intrinsic value of

the call is $0

The Value of a CallThe Value of a CallValue

X

This call option has an intrinsic value of $0 and

time value of $3.625

-Call- -Put-

SunMic

583/4 60 Jan 540 35/8 1105 37/8

The Value of a PutThe Value of a PutValue

{

When S is less than X, the intrinsic value of

the put is $(X-S)

X

This put option has an intrinsic value of $1.25 and

time value of $2.625.

-Call- -Put-

SunMic

583/4 60 Jan 540 35/8 1105 37/8

The Value of a PutThe Value of a PutValue

{

When S is greater than X, the intrinsic value of

the put is $0

X

This put option has an intrinsic value of $0 and

a time value of $2.00

-Call- -Put-

SunMic

583/4 55 Jan 166 71/4 109 2

Put-Call ParityPut-Call Parity

In an efficient market two investments with the same payoff ought to have the same price.

Put-Call ParityPut-Call Parity

This principle implies that the current stock price plus the price of

a put

S + p=c+ Xe−rt

Put-Call ParityPut-Call Parity

should equal the price of a call plus the PV of the

exercise price

S + p=c+ Xe−rt

$50

The Payoff on a StockThe Payoff on a Stock

Payoff

Stock Price at Expiration

A stock is currently selling at $45. A call and a put each with a strike price of $50 and an expiration date 6 months from now are available.

$50

Terminal value of investment in stock

The Payoff on a PutThe Payoff on a Put

Payoff

$50

Stock Price at Expiration

Terminal value of investment in put

Terminal value of investment in stock (minus $50)

$0

The Payoff on a Stock The Payoff on a Stock and a Putand a Put

Payoff

$50

Stock Price at Expiration

Terminal value of investment in both stock and put

$50

The Payoff on a CallThe Payoff on a Call

Payoff

$50

Stock Price at Expiration$0

Terminal value of investment in call

The Payoff on a BondThe Payoff on a Bond

Payoff

$50

Stock Price at Expiration$50

Terminal value of investment in bond

Terminal value of investment in call (plus $50)

The Payoff on a Call and The Payoff on a Call and a Bonda Bond

Payoff

$50

Stock Price at Expiration$50

Terminal value of investment in bond

Terminal value of investment in call and bond

For Example:For Example: Here’s a put and a call on SunMic. Each

has a strike price of $60. The current stock price is $58.75, so the call is out of the money and the put is in the money. Both expire in one month.

-Call- -Put-

SunMic

583/4 60 Jan 540 35/8 1105 37/8

Put-call parity implies that

For Example:For Example:

c−p=S−Xe−rt

=$58.75 − $60 e−.05 /12( ) = −$1.00

-Call- -Put-

SunMic

583/4 60 Jan 540 35/8 1105 37/8

Determinants of Option Determinants of Option ValueValue

The price of the underlying assetprice of the underlying assetThe value of a call rises (the value of a

put falls) as the price of the underlying asset rises, all other things equal.

The amount an option’s premium changes when the price of the underlying asset changes is called the option’s deltadelta.

Determinants of Option Determinants of Option ValueValue

The strike pricestrike priceThe value of a call falls (the value of a

put rises) as the strike price rises, all other things equal.

Determinants of Option Determinants of Option ValueValue

Time to expirationTime to expirationThe value of both puts and calls rises

as the time to expiration increases, all other things equal.

The amount an option’s premium changes when its time to maturity changes is called the option’s thetatheta.

Determinants of Option Determinants of Option ValueValue

VolatilityVolatilityThe volatility of the underlying asset is a

measure of how uncertain we are about future changes in an asset’s value.

The more volatility increases, other things equal, the greater the chance that an option will do very well.

The value of both puts and calls rises as the volatility of the underlying asset increases.

Determinants of Option Determinants of Option ValueValue

The risk-free rate of interestThe risk-free rate of interestThe value of a call rises (the value of a

put falls) when the risk-free interest rate rises.

The Black-Scholes The Black-Scholes FormulaFormula

The Black-Scholes pricing formula for a “plain vanilla” call option when the stock price is S, the strike price is X, the risk-free rate is r per annum and the time to expiration is t years is:C =SN(d1) −Xe−rtN(d2 )

N(*) is the cumulative standard normal distribution evaluated at *, and

The Black-Scholes The Black-Scholes FormulaFormula

d1 and d2 are functions of the stock price, the strike price, the interest rate, time and volatility:

d1 =ln

SX ⎛ ⎝

⎞ ⎠+ r +

σ 2

2 ⎛ ⎝ ⎜ ⎞

⎠t

σ td2 =

lnSX ⎛ ⎝

⎞ ⎠+ r −

σ 2

2 ⎛ ⎝ ⎜ ⎞

⎠t

σ t

compare

Normal DistributionNormal Distribution

ExampleExample

B-S Option Calculator

Stock Price 47Exercise Price 45Years to Maturity 0.08Volatility 0.1Risk-free Rate 5.00%d1 1.665139789N(d1) 0.952057586d2 1.636272276N(d2) 0.949108705Call Value 2.21

TelMex Jul 45 143 CB 23/8 -5/16 47 2,703

AssignmentAssignment

Option Price Calculator Ito’s Dilemma (A) Ito’s Dilemma (B)