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### Option Pricing

JunyaNamai

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

Current Option Price for Netflix

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

Binomial options pricing for Call Option

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

Binomial options pricing for Put Option

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

Call Option - Multi Period

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

Call Option - Multi Period

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

Black-Scholes Formula (5 parameters)

- Stock Price
- Exercise (Strike) Price
- Time to Expiration
- Volatility of Stock
- Risk-Free Rate

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

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

Binomial vs Black-Scholes

- Binomial
- Flexible
- Finitesteps
- Discrete
- Values American
- Values complexities

- Black-Scholes
- Limited
- Infinite
- Continuous

Quick Quiz 1

- If volatility of stock price becomes higher, does the option price go up or down?
- Black-Scholes Calculator

Quick Quiz 2

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

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

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%

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

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