Option Pricing
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Transcript of Option Pricing
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Option Pricing
Junya Namai
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Agenda Current Option Price for Netflix Binomial Model for Stock Binomial Options Pricing for Call Option Binomial Options Pricing for Put Option Binomial Options Pricing for Call Option – Multi
period Black-Scholes Model Quiz Questions
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Current Option Price for Netflix http://
finance.yahoo.com/q/op?s=NFLX&m=2013-05
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Binomial Model for Stockt0 t1
up
down
$80
$55
P(u) = 0.6
P(d) = 0.4
= = $64.81r = 0.08
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Binomial options pricing for Call Optiont0 t1
up
down
$80
$55
P(u) = 0.6P(d) = 0.4
= = $5.556
K = $70$10
$0
r = 0.08
Max(0, Price - K)
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Binomial options pricing for Put Optiont0 t1
up
down
$80
$55
P(u) = 0.6P(d) = 0.4
= = $5.556
K = $70$0
$15
r = 0.08
Max(0, K-Price)
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Call Option - Multi Period t0 t1 t2 t3 t4
$90
$80
$70
$60
$50P(u) = 0.6P(d) = 0.4
K = $70r = 0.08
0.6
0.4
0.4
0.40.4
0.40.4
0.4
0.4
0.4
0.40.6
0.60.6
0.6
0.6
0.6
0.6
0.6
0.6
$20
$10
$0
$0
$0
Max(0, Price-K)
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Call Option - Multi Period t4
$90
$80
$70
$60
$50
Path
14
6
4
1
call$20
$10
$0
$0
$0
4ups
3ups + 1down
2ups + 2downs
3downs + 1up
4downs
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Call Option - Multi Period
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Binomial Distribution (Pascal's triangle)
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Black-Scholes Formula (5 parameters) Stock Price Exercise (Strike) Price Time to Expiration Volatility of Stock Risk-Free Rate
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Black-Scholes Formula Value of call option =
cumulative normal probability density function = exercise price of option; PV(EX) is calculated by
discounting at the risk-free interest rate rf t = number of periods to exercise date P = price of stock now = standard deviation per period of (continuously
compounded) rate of return on stock
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Black-Scholes Formula P=430, EX=430, =0.4068, t=0.5(6 months),
rf=.05 = = 0.1956 = 0.195 – 0.4068 = -0.0921
= N(-0.0921) = 1-N(0.0921) = 0.4633 Use Normsdist function in Excel
= 0.5775430 – (0.4633430/1.015) = 52.04 $52.04
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Binomial vs Black-Scholes Binomial
Flexible Finite steps Discrete Values American Values complexities
Black-Scholes Limited Infinite Continuous
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Quick Quiz 1 If volatility of stock price becomes higher,
does the option price go up or down?
Black-Scholes Calculator
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Lognormal Distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Percent price changes
Prob
abili
ty
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Quick Quiz 2 If interest rates becomes higher, does the
option price go up or down?
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Question
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Reference http://stattrek.com/probability-distributions/binomial.aspx http://en.wikipedia.org/wiki/Binomial_distribution http://
www.tradingtoday.com/black-scholes?callorput=c&strike=70&stock=70&time=180&volatility=48&interest=8
http://en.wikipedia.org/wiki/Binomial_options_pricing_model
http://www.optiontradingpedia.com/free_black_scholes_model.htm
http://www.optiontradingpedia.com/free_black_scholes_model.htm
http://easycalculation.com/statistics/binomial-distribution.php
http://www.hoadley.net/options/bs.htm
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Risk-Neutral Valuation (Backup)
Expected return
rf = 1.5%Expected return 1.5 = 33p - 25(1-p)1.5 = 33p + 25p -25 p = 45.6%
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Risk-Neutral Valuation (Backup)
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Risk-Neutral Valuation (Backup)
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Up and Down Changes to STD 1+upside change = u = 1+downside change = d = 1/u
e = 2.718 = standard deviation of stock returns h = interval as fraction of a year
To find the standard deviation given u, we turn the formula around