Cross Section Pricing
Intrinsic Value
Options
Option Price
Stock Price
Cross Section Pricing
Intrinsic Value
Options
Option Price
Stock Price
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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 = Ps[N(d1)] - S[N(d2)]e-rt
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
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
(d2) = - .5056
N(d2) = 1 - .6935 = .3065
(d2) = d1 - v t
Example
What is the price of a call option given the following?.
P = 36 r = 10% v = .40
S = 40 t = 90 days / 365
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
OC = Ps[N(d1)] - S[N(d2)]e-rt
OC = 36[.3794] - 40[.3065]e - (.10)(.2466)
OC = $ 1.70
Call Option
$ 1.70
36 40 41.70
Example
What is the price of a call option given the following?.
P = 36 r = 10% v = .40
S = 40 t = 90 / 365 days
Call OptionExample (same option)
What is the price of a call option given the following?.
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)
Call OptionExample (same option)
What is the price of a call option given the following?.
P = 41 r = 10% v = .42
S = 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 (same option)
P = 41 r = 10% v = .42
S = 40 t = 30 days/ 365
Call Option
$ 1.70
40 41 41.70
Example (same option)
P = 41 r = 10% v = .42
S = 40 t = 30 days/ 365
Call OptionExample (same option)
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
Call OptionExample (same option)
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
Call OptionExample (same option)
P = 41 r = 10% v = .42
S = 40 t = 30 days/ 365
$ 1.70
40 41
$ 2.67
$ 0.97
Call OptionExample (same option)
P = 41 r = 10% v = .42
S = 40 t = 30 days/ 365
Call OptionExample (same option)
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
Example (same option)
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
Example (same option)
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
Binomial Pricing50.78 = price
10.78 = intrinsic value
40.37
.37
32.10
0
25.52
0
45.28
5.60
36
28.62
40.37
32.1036
trdduu ePUPO
Binomial Pricing50.78 = price
10.78 = intrinsic value
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
Binomial Pricing50.78 = price
10.78 = intrinsic value
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
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