Option Pricing

Junya Namai

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 Model for Stockt0 t1

up

down

$80

$55

P(u) = 0.6

P(d) = 0.4

= = $64.81r = 0.08

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)

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)

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)

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

Call Option - Multi Period

Binomial Distribution (Pascal's triangle)

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 Finite steps 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

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

Quick Quiz 2 If interest rates becomes higher, does the

option price go up or down?

Question

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