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# Option Pricing - PowerPoint PPT Presentation

Option Pricing. Junya Namai. Agenda. Current Option Price for Netflix Binomial Model for Stock Binomial Options P ricing for Call Option Binomial Options P ricing for Put Option Binomial Options P ricing for Call Option – Multi period Black-Scholes Model Quiz Questions.

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### Option Pricing

JunyaNamai

• 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

• http://finance.yahoo.com/q/op?s=NFLX&m=2013-05

Binomial Model for Stock

t0

t1

\$80

P(u) = 0.6

up

P(d) = 0.4

down

\$55

r = 0.08

= = \$64.81

K = \$70

t0

t1

Max(0, Price - K)

P(u) = 0.6

\$10

\$80

P(d) = 0.4

up

r = 0.08

down

\$55

\$0

= = \$5.556

K = \$70

t0

t1

Max(0, K-Price)

P(u) = 0.6

\$0

\$80

P(d) = 0.4

up

r = 0.08

down

\$55

\$15

= = \$5.556

t0

t1

t2

t3

t4

Max(0, Price-K)

\$20

\$90

0.6

0.6

0.4

\$10

\$80

0.6

0.6

0.4

0.6

0.6

0.4

0.4

\$70

\$0

0.4

0.6

0.4

0.6

0.4

0.6

0.4

\$0

K = \$70

\$60

0.6

P(u) = 0.6

0.4

r = 0.08

P(d) = 0.4

0.4

\$0

\$50

Path

call

t4

1

4ups

\$90

\$20

4

3ups + 1down

\$80

\$10

6

\$0

2ups + 2downs

\$70

4

\$60

\$0

3downs + 1up

1

\$0

\$50

4downs

Call Option - Multi Period

• Stock Price

• Exercise (Strike) Price

• Time to Expiration

• Volatility of Stock

• Risk-Free Rate

• 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

• 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

Binomial vs Black-Scholes

• Binomial

• Flexible

• Finitesteps

• Discrete

• Values American

• Values complexities

• Black-Scholes

• Limited

• Infinite

• Continuous

• If volatility of stock price becomes higher, does the option price go up or down?

• Black-Scholes Calculator

• If interest rates becomes higher, does the option price go up or down?

• http://stattrek.com/probability-distributions/binomial.aspx

• http://en.wikipedia.org/wiki/Binomial_distribution

• http://en.wikipedia.org/wiki/Binomial_options_pricing_model

• http://easycalculation.com/statistics/binomial-distribution.php

Expected return

rf = 1.5%

Expected return

1.5 = 33p - 25(1-p)

1.5 = 33p + 25p -25

p = 45.6%

Risk-Neutral Valuation (Backup)

Risk-Neutral Valuation (Backup)

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