black scholes option pricing model
TRANSCRIPT
Basics
Inputs
$69.81 69
71.65% 7
Riskfree Rate- Annual (r) 4.95% 5
Exercise Price (X) $70.00 70
Time To Maturity - Years (T) 0.5777 6
Outputs
BLACK SCHOLES OPTION PRICING
Stock Price Now (S0)
Standard Dev - Annual (s)
d1
d2
N(d1)
N(d2)
Call Price (C0)
-d1
-d2
N(-d1)
N(-d2)
Put Price (P0)
1d Ts
20ln / / 2 /S X r T Ts s
0 0 1 2rTC S N d Xe N d
0 0 1 2rTP S N d Xe N d
Continuous Dividend
Inputs
Option Type FALSE
$69.81 69
71.65% 7
Riskfree Rate- Annual (r) 4.95% 5
Exercise Price (X) $70.00 70
Time To Maturity - Years (T) 0.5777 6
Dividend Yield (d) 1.00% 1
Outputs
BLACK SCHOLES OPTION PRICING
Stock Price Now (S0)
Standard Dev - Annual (s)
d1
d2
N(d1)
N(d2)
Call Price (C0)
-d1
-d2
N(-d1)
$0 $20 $40 $60 $80 $100 $120 $140$0
$10
$20
$30
$40
$50
$60
Dynamic Chart of Black Scholes Option Pricing
Stock Price Now
Op
tio
n P
ric
e
Call Put
20ln / / 2 /S X r d T Ts s
0 0 1 2dT rTC S e N d Xe N d
Option Type
Data Table: Sensitivity of Option Price to Stock Price NowInput Values for Stock Price Now (P)
Output Formula:Option PriceIntrinsic Value
Greeks
Inputs Call Greeks
$69.81 69
71.65% 7
Riskfree Rate- Annual (r) 4.95% 5
N(-d2)
Put Price (P0)
BLACK SCHOLES OPTION PRICING
Stock Price Now (S0)
Standard Dev - Annual (s)
$0 $20 $40 $60 $80 $100 $120 $1400.00
2.00
4.00
6.00
8.00
10.00
12.00
Greeks by Stock Price Now
Stock Price Now
Gre
ek
Va
lue
0 0 1 2dT rTP S e N d Xe N d
Delta
Theta
Gamma
Vega
Rho
Greeks
Exercise Price (X) $70.00 70 Put Greeks
Time To Maturity - Years (T) 0.5777 6
Dividend Yield (d) 1.00% 1
Outputs FALSE
d1
d2
$0 $20 $40 $60 $80 $100 $120 $1400.00
2.00
4.00
6.00
8.00
10.00
12.00
Greeks by Stock Price Now
Stock Price Now
Gre
ek
Va
lue
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00
2.00
4.00
6.00
8.00
10.00
12.00
Greeks by Time To Maturity
Time To Maturity
Gre
ek
Va
lue
Delta
Theta
Gamma
Vega
Rho
Greeks
1dTDelta e N d
1
0
1 1where = normal evaluated at
dTN d eGamma
S T
N d density d
s
0 10 1 2
1 1
2
where = normal evaluated at
dTdT rTS N d e
Theta dS N d e rXe N dT
N d density d
s
Call Greeks
Put Greeks
Delta (¶Call / ¶Stock Price)Theta (-¶Call / ¶Time To Mat)
Gamma (¶2Call / ¶Stock Price2)Vega (¶Call / ¶Std Dev)Rho (¶Call / ¶Riskfree Rate)
Delta (¶Put / ¶Stock Price)Theta (-¶Put / ¶Time To Mat)
Gamma (¶2Put / ¶Stock Price2)Vega (¶Put / ¶Std Dev)Rho (¶Put / ¶Riskfree Rate)
2rTRho XTe N d
1dTDelta e N d
1
0
1 1where = normal evaluated at
dTN d eGamma
S T
N d density d
s
0 1
1 1where = normal evaluated at
dTVega S T N d e
N d density d
Call Gamma
Call Vega
2rTRho XTe N d
0 10 1 2
1 1
2
where = normal evaluated at
dTdT rTS N d e
Theta dS N d e rXe N dT
N d density d
s
Data Table: Sensitivity of the Selected Greek to Stock Price NowInput Values for Stock Price Now
Output Formula:Selected Greek
Time To MaturityData Table: Sensitivity of the Selected Greek to Time To Maturity Index
Input Values for Time To Maturity IndexOutput Formula:Selected Greek
Implied Volatility
InputsOption Type: 1=Call, 2=Put 1 1 1 1 1 2 2 2 2 2
$1,513 $1,513 $1,513 $1,513 $1,513 $1,513 $1,513 $1,513 $1,513 $1,51321.66% 17.31% 14.77% 16.01% 13.81% 15.66% 14.73% 14.02% 11.25% 6.31%
Riskfree Rate- Annual (r) 4.62% 4.62% 4.62% 4.62% 4.62% 4.62% 4.62% 4.62% 4.62% 4.62%Exercise Price (X) $1,450 $1,500 $1,520 $1,540 $1,600 $1,450 $1,500 $1,520 $1,540 $1,600Time To Maturity - Years (T) 0.2416 0.2416 0.2416 0.2416 0.2416 0.2416 0.2416 0.2416 0.2416 0.2416Dividend Yield (d) 1.62% 1.62% 1.62% 1.62% 1.62% 1.62% 1.62% 1.62% 1.62% 1.62%Observed Option Price $105.60 $63.40 $45.50 $40.00 $14.50 $17.70 $32.40 $39.50 $42.00 $76.50
Outputs
SolverDifference (observed - model)
BLACK SCHOLES OPTION PRICING
Stock Price Now (S0)Standard Dev - Annual (s)
d1
d2
N(d1)
N(d2)
Model Call Price (C0)
-d1
-d2
N(-d1)
N(-d2)
Model Put Price (P0)
GraphImplied Volatility from CallsImplied Volatility from Puts
Exotic Options
Continuous Dividend Black Scholes Inputs Additional Exotic Option Inputs
$69.81 69 $70.00 70
71.65% 7 70.1% 7
Riskfree Rate- Annual (r) 4.95% 5 0.20 12
Exercise Price (X) $70.00 70 2.0% 2
Time To Maturity - Years (T) 0.5777 6 Time to Chooser Decision (tc) 0.20 2
BLACK SCHOLES OPTION PRICING
Stock Price Now (S0) Asset 2 Stock Price Now (S2)
Standard Dev - Annual (s) Asset 2 Standard Dev (s2)
Corr(Asset 1, Asset 2) (r)
Asset 2 Dividend Yield (d2)
$1,425 $1,450 $1,475 $1,500 $1,525 $1,550 $1,575 $1,600 $1,6253%
5%
7%
9%
11%
13%
15%
17%
19%
21%
23%
"Scowl" Pattern of Implied Volatilities
CallsPuts
Exercise Price
Imp
lied
Vo
lati
lity
Dividend Yield (d) 1.00% 1 Cash Payoff (Z) $60.00 60
Gap Amount (g) $4.00 4
$70.00 70
Continuous Dividend Black Scholes Outputs $75.00 75
11
Supershare Lower Bound (XL)
Supershare Upper Bound (XH)d1
d2
N(d1)N(d2)Call Price (C0)
-d1
-d2
N(-d1)N(-d2)Put Price (P0)
$0 $20 $40 $60 $80 $100 $120 $1400.00
2.00
4.00
6.00
8.00
10.00
12.00
Exotic Options by Stock Price Now
Stock Price NowE
xo
tic
Op
tio
n V
alu
e
Exchange Option
Min of Two Assets
Max of Two Assets
Chooser
Cash-Or-Nothing Call
Cash-Or-Nothing Put
Asset-Or-Nothing Call
Asset-Or-Nothing Put
Exotic Options
Gap Call
Gap Put
Supershare
2 22 22s s rss
22 0 2
1
ln 0.5S S d d Tw
T
Exotic Options
Exchange Option
Minimum of Two AssetsMaximum of Two Assets
Chooser
Binary Options(Digital Options): Cash-or-Nothing Call Cash-or-Nothing Put Asset-or-Nothing Call Asset-or-Nothing Put Gap Call Gap Put
w1
w2
w3
wL
wH
2 1w w T
22 1 0 2
d T dTExchange S e N w S e N w
0 Min of two assets S Exchange
0 Max of two assets S Exchange
2- - rTCash Or Nothing Call Ze N d
2- - rTCash Or Nothing Put Ze N d
0 2- - dTAsset Or Nothing Call e S N d
0 2- - dTAsset Or Nothing Put e S N d
20
3
ln 0.5S X r d T tcw
tc
ss
3 0 3rT dTChooser Call Xe N w tc S e N ws
Supershare
Data Table: Sensitivity of the Selected Exotic Option to Stock Price Now
20ln 0.5L
L
S X r d Tw
T
s
s
20ln 0.5H
H
S X r d Tw
T
s
s
2
0d T
L HL
S eSupershare N w N w
X
0 1 2 dT rTGap Call e S N d X g e N d
2 0 1 rT dTGap Put X g e N d e S N d
0 2- - dTAsset Or Nothing Put e S N d
Input Values for Stock Price NowOutput Formula:Selected Exotic Option
