Financial EngineeringLecture 3

Option Valuation Methods Genentech call options have an exercise

price of $80 and expire in one year.

Case 1

Stock price falls to $60

Option value = $0

Case 2

Stock price rises to $106.67

Option value = $26.67

Option Valuation Methods If we are risk neutral, the expected return on Genentech

call options is 2.5%. Accordingly, we can determine the price of the option as follows, given equal probabilities of each outcome.

01.13$

)050(.)67.2650(.e

0rise ofy probabilit167.26rise ofy probabilit valueOption

0.1025.

rt

e

Binomial ModelThe price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Genentech example illustrates this fact.

Binomial Pricing

)()( upy Probabilit

dudap

p1downy Probabilit

yearof % as interval time

theu

ed

ea

h

h

rh

The prior example can be generalized as the binomial model and shown as follows.

ExamplePrice = 36 = .40 t = 90/365 t = 30/365 Strike = 40 r = 10%

a = 1.0083u = 1.1215d = .8917Pu = .5075Pd = .4925

Binomial Pricing

40.37

32.1036

37.401215.13610

UPUP

Binomial Pricing

40.37

32.1036

37.401215.13610

UPUP

10.328917.3610

DPDP

Binomial Pricing

50.78 = price

40.37

32.10

25.52

45.28

36

28.62

40.37

32.1036

1 tt PUP

Binomial Pricing

50.78 = price10.78 = intrinsic value

40.37.37

32.100

25.520

45.28

36

28.62

36

40.37

32.10

Binomial Pricing

50.78 = price10.78 = intrinsic value

40.37.37

32.100

25.520

45.285.60

36

28.62

40.37

32.1036

trdduu ePUPO

The greater of

Binomial Pricing

50.78 = price10.78 = intrinsic value

40.37.37

32.100

25.520

45.285.60

36.19

28.620

40.372.91

32.10.10

36

1.51

trdduu ePUPO

Binomial Pricing

Price Comparisons

Black Scholes price= 1.70

Binomial price = 1.51

Volatility Only non-observable variable Historical volatility Predictive models

◦ ARCH (Robert Engel)◦ GARCH

Weighted Average Historical Volatility Implied Volatility VIX – Exchange traded volatility option

◦ 1993◦ S&P 500 Implied Volatility

Implied Volatility is highest where time premium is highest…usually at the money

Time Decay

Option Price

Stock Price

Days to Expiration906030

Volatility Surface Term Structure of Volatilities

Time to Expiration 80% 90% 100% 110% 120%

15 0.150 0.134 0.130 0.134 0.150 30 0.148 0.139 0.135 0.139 0.148 45 0.145 0.144 0.140 0.144 0.145 60 0.151 0.149 0.145 0.149 0.151 75 0.164 0.156 0.152 0.156 0.164 90 0.170 1.604 1.600 1.604 0.170

Stock Price as % of Strike Price

Volatility Smile

Strike PriceAsset Price

Implied Volatility

Volatility Smirk

Strike PriceAsset Price

Implied Volatility

Volatility Smirk

Strike PriceAsset Price

Implied Volatility

Volatility Calculate the Annualized variance of the

daily relative price change Square root to arrive at standard deviation Standard deviation is the volatility

Volatility Develop Spreadsheet Download data from internet

http://finance.yahoo.com

Implied Volatility All variables in the option price can be

observed, other than volatility. Even the price of the option can be

observed in the secondary markets. Volatility cannot be observed, it can only be

calculated. Given the market price of the option, the

volatility can be “reverse engineered.”

Implied VolatilityUse Numa to calculate implied volatility.Example (same option)P = 41 r = 10% PRICE = 2.67EX = 40 t = 30 days / 365 v = ????

Implied volatility = 42.16%

Implied Volatility CBOE Example

Use Actual option ◦ Calculate historical volatility◦ Calculate implied volatility

http://www.math.columbia.edu/~smirnov/options13.htmlhttp://www.cboe.comhttp://www.numa.com

Expected Returns Given a normal or lognormal distribution of

returns, it is possible to calculate the probability of having an stock price above or below a target price.

Wouldn’t it be nice to know the probability of making a profit or the probability of being “in the money?”

Expected ReturnSteps for Infinite Distribution of Outcomes

tVVt

y volatilitadjusted timeConvert to :1 Step

t

Pq

VNq ln pricebelow ty Probabili:2 Step

qbelow prob -1 price abovety Probabili:3 Step

q

Expected Return

Example

Example (same option)P = 41 r = 10% v = .42EX = 40 t = 30 days / 365

1204.42. :1 Step 36530 tVVt

4187.2051.1204.ln40)( Prob:2 Step 41

40

NN

6299.3316.1204.

ln)67.24( Prob:2 Step 4167.42

NN

Expected ReturnExample (same option)P = 41 r = 10% v = .42EX = 40 t = 30 days / 365

$2.67

40 42.67

37%58%

63%

Option Pricing Project

• See handout for specs

• Walk through sample project