Post on 02-Apr-2018
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Whats an Option?
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Option Positions
Long call
Long put Short call
Short put
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Long Call on eBay
Profit from buying one eBay European call option: option
price = $5, strike price = $100, option life = 2 months
30
20
10
0-5
70 80 90 100
110 120 130
Profit ($)
Terminalstock price ($)
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Short Call on eBay
Profit from writing one eBay European call option: option
price = $5, strike price = $100
-30
-20
-10
05
70 80 90 100
110 120 130
Profit ($)
Terminalstock price ($)
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Long Put on IBM
Profit from buying an IBM European put option: option
price = $7, strike price = $70
30
20
10
0
-770605040 80 90 100
Profit ($)
Terminal
stock price ($)
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Short Put on IBM
Profit from writing an IBM European put option: option
price = $7, strike price = $70
-30
-20
-10
7
070
605040
80 90 100
Profit ($)
Terminalstock price ($)
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Payoffs from OptionsWhat is the Option Position in Each Case?K= Strike price, ST= Price of asset at maturity
Payoff Payoff
ST STK
K
Payoff Payoff
ST STK
K
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Valuing Options
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Topics Covered
Simple Option Valuation Model
Binomial Model
Black-Scholes Model Black Scholes in Action
Option Values at a Glance
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)(
)(upyProbabilit
du
dap
Binomial Pricing
p
1downyProbabilit
yearof%asintervaltime
th
eu
ed
ea
h
h
rh
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ExamplePrice = 36 = .40 t = 90/365 t = 30/365
Strike = 40 r = 10%
Binomial Pricing
a = 1.0083
u = 1.1215
d = .8917
Pu = .5075Pd = .4925
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40.37
32.10
36
37.401215.136
10
U
PUP
Binomial Pricing
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40.37
32.10
36
37.401215.136
10
U
PUP
10.328917.36
10
DPDP
Binomial Pricing
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50.78 = price
40.37
32.10
25.52
45.28
36
28.62
40.37
32.10
36
1 tt PUP
Binomial Pricing
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50.78 = 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 Pricing
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50.78 = 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
The greater of
Binomial Pricing
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50.78 = 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 Pricing
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Option Value
Components of the Opt ion Pr ice
1 - Underlying stock price
2 - Striking or Exercise price
3 - Volatility of the stock returns (standard deviation ofannual returns)
4 - Time to option expiration
5 - Time value of money (discount rate)
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Option Value
B lack -Scho les Opt ion Pric ing Model
)()()( 21 EXPVdNPdNOC
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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
B lack-Scho les Opt ion Pric ing Model
)()()( 21 EXPVdNPdNOC
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B lack-Scho les Opt ion Pric ing Model
)()()( 21 EXPVdNPdNOC
rteEXEXPV )(
factordiscountgcompoundincontinuous1 rt
rt
ee
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32 34 36 38 40
N(d1)=
B lack-Scho les Opt ion Pric ing Model
tv
trdv
EXP )()ln( 2
1
2
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Cumulative Normal Dens i ty Funct ion
tv
trd
vEXP )()ln( 2
1
2
tvdd 12
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Call Option
3070.1 d
tv
trd
vEXP )()ln( 2
1
2
ExampleWhat is the price of a call option given the
following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
3794.6206.1)( 1 dN
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Call Option
3065.6935.1)(5056.
2
2
12
dN
d
tvdd
ExampleWhat is the price of a call option given the
following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
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Call Option
70.1$
)40(3065.363794.
)()()(
)2466)(.10(.
21
C
C
rt
C
OeO
eEXdNPdNO
ExampleWhat is the price of a call option given the
following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365
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Expanding the binomial model to allow more
possible price changes
1 step 2 steps 4 steps
(2 outcomes) (3 outcomes) (5 outcomes)
etc. etc.
Binomial vs. Black Scholes
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Binomial vs. Black Scholes
ExampleWhat is the price of a call option given the
following?
P = 36 r = 10% v = .40
EX = 40 t = 90 days / 365Binomial price = $1.51
Black Scholes price = $1.70
The limited number of binomial outcomes produces thedifference. As the number of binomial outcomes is expanded,
the price will approach, but not necessarily equal, the Black
Scholes price.
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How estimated call price changes asnumber of binomial steps increasesNo. of steps Estimated value
1 48.1
2 41.0
3 42.1
5 41.8
10 41.4
50 40.3
100 40.6
Black-Scholes 40.5
Binomial vs. Black Scholes
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Numericals
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Numericals
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Numericals
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Numericals
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Numericals
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Numericals