Lecture 2. Option - Gives the holder the right to buy or sell a security at a specified price during a specified period of time. Call Option - The

Financial EngineeringLecture 2

Option Review

assetbuy toObligationasset sell Right tooptionPut

asset sell toObligationassetbuy Right tooption Call

ShortSeller / ngBuyer / Lo

Options Review Option - Gives the holder the right to buy or sell a security at a

specified price during a specified period of time. Call Option - The right to buy a security at a specified price within a

specified time. Put Option - The right to sell a security at a specified price within a

specified time. Option Premium - The price paid for the option, above the price of

the underlying security. Intrinsic Value - Diff between the strike price and the stock price Time Premium - Value of option above the intrinsic value Exercise Price - (Striking Price) The price at which you buy or sell

the security. Expiration Date - The last date on which the option can be exercised.

Option ReviewOption ends by…1. Expiration2. Exercise3. Sales

American option European option Intrinsic Value = P – E Time Premium = O + E – P Moneyness

◦ In the money◦ Out of the money◦ At the money

Asset Price

Profit

Loss

Option Review

Option Concepts Market Makers Round Trip Lot size is 100 shares Naked positions Covered positions

CBOE Quotes (web) Open interest Volume Bid-ask Prices

Option Value

Price

0 30 60 90 (expiration) Time (days)

Time Decay

Example – Given an exercise price of $55, what are the likely call option premiums, given stock prices of 50, 56, and 60 dollars?

Stock Price

*TP = .30 IV = 0 *TP = .50 IV = 0 *TP = .80 IV = 0

TP = .40 IV = 1.00 TP = .75 IV = 1.00 TP = 1.10 IV = 1.00

TP = .30 IV = 5.00 TP = .50 IV = 5.00 TP = .80 IV = 5.00

* Using (O+E-P), the TP would be 5.30, 5.50, and 5.80, respectively.

30 Days 60 Days 90 Days

60$5.30 $5.50 $5.80

$0.80

56$1.40 $1.75 $2.10

$0.3050

$0.50

Days to Expiration

Intrinsic Value & Time Premium graphed

Time Decay

Days to Expiration

90

60

30Option Price

Stock Price

Exotic Options Swaptions Index options Futures options Currency options Convertible bond Warrant

Barrier Options Knock out options

◦ Down and out◦ Up and out

Knock in options◦ Down and in◦ Up and in

Current Events Executive Stock Options

◦ “To Expense or Not to Expense”

Option Value

Components of the Option Price1 - Underlying stock price2 - Striking or Exercise price3 - Volatility of the stock returns (standard deviation of

annual returns)4 - Time to option expiration5 - Time value of money (discount rate)

6 - PV of Dividends = D = (div)e-rt

Option Value

)()()( 21 EXPVdNPdNOC

Black-Scholes Option Pricing Model

OC- Call Option Price

P - Stock Price

N(d1) - Cumulative normal density function of (d1)

PV(EX) - Present Value of Strike or Exercise price

N(d2) - Cumulative normal density function of (d2)

r - discount rate (90 day comm paper rate or risk free rate)

t - time to maturity of option (as % of year)

v - volatility - annualized standard deviation of daily returns





rteEXEXPV )(

factordiscount gcompoundin continuous1

rt

rt

ee

N(d1)=

tv

trd

vEXP )()ln( 2

1

2


Cumulative Normal Density Function

tv

trd

vEXP )()ln( 2

1

2

tvdd 12

Cumulative Normal Density Function

Call Option

2292.1 d

tv

trd

vEXP )()ln( 2

1

2

5906.)( 1 dN

Example - Genentech

What is the price of a call option given the following?

P = 80 r = 5% v = .4068

EX = 80 t = 180 days / 365

Call Option

4775.5225.1)(

0565.

2

2

12

dN

d

tvdd

Example - Genentech


P = 80 r = 5% v = .4068

EX = 80 t = 180 days / 365

Call Option

98.9$

)80(4775.805906.

)()()(

)365180)(05(.

21

C

C

rtC

O

eO

eEXdNPdNO

Example - Genentech


P = 80 r = 5% v = .4068

EX = 80 t = 180 days / 365

Call Option

3070.1 d

tv

trd

vEXP )()ln( 2

1

2

3794.6206.1)( 1 dN

Example


P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365

.3070 = .3

= .00

= .007

Call Option

3066.6934.1)(

5056.

2

2

12

dN

d

tvdd

Example


P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365

Call Option

70.1$

)40(3066.363794.

)()()()2466)(.10(.

21

C

C

rtC

O

eO

eEXdNPdNO

Example


P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365

Call Option

$ 1.70

36 40 41.70

Example


P = 36 r = 10% v = .40

EX = 40 t = 90 days / 365

Call Option

(d1) =

ln + ( .1 + ) 30/36541

40

.422

2

.42 30/365

(d1) = .3335 N(d1) =.6306

Example


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

Call Option

(d1) =

ln + ( .1 + ) 30/36541

40

.422

2

.42 30/365

(d1) = .3335 N(d1) =.6306

Example


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

(d2) = .2131

N(d2) = .5844

(d2) = d1 - v t = .3335 - .42 (.0907)

Call OptionExample


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

Call Option

OC = Ps[N(d1)] - S[N(d2)]e-rt

OC = 41[.6306] - 40[.5844]e - (.10)(.0822)

OC = $ 2.67

Example


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

Call Option

$ 1.70

40 41 41.70

Example


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

Call Option

Intrinsic Value = 41-40 = 1

Time Premium = 2.67 + 40 - 41 = 1.67

Profit to Date = 2.67 - 1.70 = .97

Due to price shifting faster than decay in time premium

Example


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

$ 1.70

40 41

Call Option Q: How do we lock in a profit? A: Sell the Call

$ 1.70

40 41

$ 2.67


$ 1.70

40 41

$ 2.67

$ 0.97


Call Option

$ 1.70

40 41

$ 2.67

$ 0.97

Q: How do we lock in a profit? A: Sell the Call

Put OptionBlack-Scholes

Op = EX[N(-d2)]e-rt - Ps[N(-d1)]

Put-Call Parity (general concept)

Put Price = Oc + EX - P - Carrying Cost + D Carrying cost = r x EX x t

Call + EXe-rt = Put + Ps

Put = Call + EXe-rt - Ps

Put Option

N(-d1) = .3694 N(-d2)= .4156

Black-Scholes

Op = EX[N(-d2)]e-rt - Ps[N(-d1)]

Op = 40[.4156]e-.10(.0822) - 41[.3694]

Op = 1.34

Example


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

Put-Call ParityPut = Call + EXe-rt - Ps

Put = 2.67 + 40e-.10(.0822) - 41

Put = 42.34 - 41 = 1.34

Put OptionExample


P = 41 r = 10% v = .42

EX = 40 t = 30 days / 365

Put OptionPut-Call Parity & American PutsPs - EX < Call - Put < Ps - EXe-rt

Call + EX - Ps > Put > EXe-rt - Ps + call

Example - American Call

2.67 + 40 - 41 > Put > 2.67 + 40e-.10(.0822) - 41

1.67 > Put > 1.34

With Dividends, simply add the PV of dividends

DividendsExamplePrice = 36 Ex-Div in 60 days @ $0.72t = 90/365 r = 10%

PD = 36 - .72e-.10(.1644) = 35.2917

Put-Call ParityAmer

D+ C + S - Ps > Put > Se-rt - Ps + C + D

EuroPut = Se-rt - Ps + C + D + CC

Documents

Lecture 2. Option - Gives the holder the right to buy or sell a security at a specified price during a specified period of time. Call Option - The