Cross Section Pricing Intrinsic Value Options Option Price Stock Price.

Cross Section Pricing

Intrinsic Value

Options

Option Price

Stock Price

Interest Rates

Settlement

Projects

Computer software

Options

Options

Components of the Option Price

1 - Underlying stock price = Ps

2 - Striking or Exercise price = S

3 - Volatility of the stock returns (standard deviation of annual returns) = v

4 - Time to option expiration = t = days/365

5 - Time value of money (discount rate) = r

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

Black-Scholes Option Pricing ModelBlack-Scholes Option Pricing Model

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

Black-Scholes Option Pricing ModelBlack-Scholes Option Pricing Model


OC- Call Option Price

Ps - Stock Price

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

S - 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 (days/365)

v - volatility - annual standard deviation of returns

(d1)=

ln + ( r + ) tPs

S

v2

2

v t

32 34 36 38 40

Cumulative Normal Density FunctionCumulative Normal Density Function

N(d1)=

(d1)=

ln + ( r + ) tPs

S

v2

2

v t

Cumulative Normal Density FunctionCumulative Normal Density Function

(d2) = d1 - v t

Call OptionExample

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

P = 36 r = 10% v = .40

S = 40 t = 90 days / 365

Call Option

(d1) =

ln + ( r + ) tPs

S

v2

2

v t

(d1) = - .3070 N(d1) = 1 - .6206 = .3794

Example


P = 36 r = 10% v = .40

S = 40 t = 90 days / 365

Call Option

(d2) = - .5056

N(d2) = 1 - .6935 = .3065

(d2) = d1 - v t

Example


P = 36 r = 10% v = .40

S = 40 t = 90 days / 365

Call OptionExample


P = 36 r = 10% v = .40

S = 40 t = 90 days / 365


OC = 36[.3794] - 40[.3065]e - (.10)(.2466)

OC = $ 1.70

Call Option

$ 1.70

36 40 41.70

Example


P = 36 r = 10% v = .40

S = 40 t = 90 / 365 days

Call OptionExample (same option)


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

(d1) =

ln + ( .1 + ) 30/36541

40

.422

2

.42 30/365

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

(d2) = .2131

N(d2) = .5844

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



P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

Call Option


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

OC = $ 2.67

Example (same option)

P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

Call Option

$ 1.70

40 41 41.70


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

• Intrinsic Value = 41-40 = 1

• Time Premium = 2.67 + 40 - 41 = 1.67

• Profit to Date = 2.67 - 1.70 = .94

• Due to price shifting faster than decay in time premium


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

Q: How do we lock in a profit?

A: Sell the Call

$ 1.70

40 41

$ 1.70

40 41

$ 2.67


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

$ 1.70

40 41

$ 2.67

$ 0.97


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

$ 1.70

40 41

$ 2.67

$ 0.97

Put Option

Black-Scholes

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

Put-Call Parity (general concept)Put Price = Oc + S - P - Carrying Cost + D

Carrying cost = r x S x t

Call + Se-rt = Put + Ps

Put = Call + Se-rt - Ps

Put OptionExample (same option)

P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

Calculate the Value of The Put

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

Black-Scholes

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

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

Op = 1.34


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365

Calculate the Value of The Put

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

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

Put = 42.34 - 41 = 1.34

Put Option

Put OptionPut-Call Parity & American Puts

Ps - S < Call - Put < Ps - Se-rt

Call + S - Ps > Put > Se-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

Volatility

• Calculate the Annualized variance of the daily relative price change

• Square root to arrive at standard deviation

• Standard deviation is the volatility

Implied Volatility

O Price Volume Implied V

Jan30C 4.50 50 .34

Jan35C 1.50 90 .28

Apr35C 2.50 55 .30

Apr40C 1.50 5 .38

200

• Recalculate the volatility using volume & price deviation

Implied Volatility

Volume Volume Weights

Jan30C 50 50/200 = .25

Jan35C 90 90/200 = .45

Apr35C 55 55/200 = .275

Apr40C 5 5 / 200=.025

200

Implied Volatility

Distance Factor (25% tolerance)

Jan30C [(3/33)-.25]2 / .252 = .41

Jan35C [(2/33)-.25]2 / .252 = .57

Apr35C [(2/33)-.25]2 / .252 = .57

Apr40C [(7/33)-.25]2 / .252 = .02

Weight Adjusted Implied volatility =

=.41x.25x.334 + .57x.45x.28 +... = .298298

.41x.25 + .57x.45 + ...

Expected Return

Example P = 41 40C=2.67

Possible

Price Prob Profit ProbxProfit

35 .10 -7.67 -.767

38 .20 -4.67 -.934

41 .40 -1.67 -.668

44 .20 1.33 .266

47 .10 4.33 .433

-1.67

Expected Profit = - 1.67

Expected Return

Steps for Infinite Distribution of Outcomes

1 - convert annual V to time adjusted V

Vt = V (t.5)

2 - Prob(below a price q ) = N [ln(q/p) /Vt]

3 - Prob (above price q ) = 1 - Prob (below)

Expected Return

Example

Vt = .42 (30/365).5 = .1204

Prob (<40) = N[ln(40/41) /.1204] = N[-.2051] = .4187

Prob (<42.67) = N[ln(42.67/41) /.1204] = N[.3316] = .6299


P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365 Call = 2.67

Expected ReturnExample (same option)

P = 41 r = 10% v = .42

S = 40 t = 30 days/ 365 Call = 2.67

$2.67

40 42.67

37%58%

63%

DividendsExample

Price = 36 Ex-Div in 60 days @ $0.72

t = 90/365 r = 10%

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

Put-Call Parity

Amer

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

Euro

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

Binomial Pricing Model

Binomial Pricing

Outcome Trees

Example - one month option

Price = $20 Possible outcomes = 22 or 18

21call = ? Monthly risk free rate = 1%

Intrinsic Value @ 22 = 1

Intrinsic Value @ 18 = 0

T0 T1 Value X Shares

Pa=22 22x -1

P=20

Pb=18 18x

22x - 1 = 18x

x= .25

at .25 shares A=B

Binomial Pricing

at .25 shares A=B

T1 Value = 22(.25) - 1 = 4.5

T0 Value = 20 (.25) - Call = 5 - Call

(T0 Value) (1+ r) = 4.5

(5-call) (1.01) = 4.5

call = .5446

Binomial Pricing

Probability Up = p = (a - d) Prob Down = 1 - p

(u - d)

a = ert d =e-[t].5 u =e[t].5

t = time intervals as % of year

Binomial Pricing

Example

Price = 36= .40 t = 90/365 t = 30/365

Strike = 40 r = 10%

a = 1.0083

u = 1.1215

d = .8917

Pu = .5075

Pd = .4925

Binomial Pricing

Binomial Pricing

40.37

32.10

36

37.401215.13610

UPUP

Binomial Pricing

40.37

32.10

36

37.401215.13610

UPUP

10.328917.3610

DPDP

50.78 = price

40.37

32.10

25.52

Binomial Pricing45.28

36

28.62

40.37

32.10

36

1 tt PUP

Binomial Pricing50.78 = price

10.78 = intrinsic value

40.37

.37

32.10

0

25.52

0

45.28

36

28.62

36

40.37

32.10



40.37

.37

32.10

0

25.52

0

45.28

5.60

36

28.62

40.37

32.1036

trdduu ePUPO



40.37

.37

32.10

0

25.52

0

45.28

5.60

36

.19

28.62

0

40.37

2.91

32.10

.10

36

1.51

trdduu ePUPO

Project• Select a Call option (w/ high vol & expires

next month)

• Use spreadsheet to calc BS value for this Friday

• Calc volatility (include div if necessary)

• Calc Expected Return Probability Intervals

• Use spreadsheet to calc Binomial value. Use weekly intervals.

• Chart Black Scholes position

• Create a cross section price chart (showing time value decay) - Calculate option price at various stock prices for 0, 30, 60, 90 days.

• Include 1 page executive summary

Cross Section Pricing Intrinsic Value Options Option Price Stock Price.

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