Options Pricing

Spencer Lin

Sebastian Ruf

5/1/2012

Outline

� Background

� Model of Stock Evolution

� Black-Scholes Equation

� Binomial Method

Options

� Contract/Agreement between two parties

� Defining parameters:

� What underlying security (what stock)

� Quantity of the underlying (number of shares)

� Option Type (Put or Call)

� Strike Price

� Option expiration date

� Style (European, American, etc.)

� Other legal terms (not important to us)

Option Options

� Put:� Seller pays premium (to buyer) for the right to sell the underlying at strike price

� Buyer is obligated to purchase the underlying from Seller

� Call:� Buyer pays premium (to seller) for right to purchase the underlying at strike price

� Seller is obligated to sell the underlying to Buyer

� American:� Option may be exercised at on any trading day before expiration date

� European:� Option may only be exercised on day of expiration

Black Scholes Model Assumptions

� No transaction costs or taxes

� Trading happens in a continuous manner

� No dividends or splits

� Risk-free interest rate is constant

� Options are European style

� The underlying stock behavior follows a geometric brownian motion, with constant drift and volatility

Wiener Process

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Generalized Wiener Process

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�: random, normally distributed value on [0,1]

Figure from Hull J.C., “Options Futures and Other Derivatives”, p221

Stock Behavior

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�: Stock value

: expected rate of return

�: volatility

Ito’s Lemma

� If x follows a Generalized Wiener Process

� A function G(x,t) follows

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Black Scholes Model

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Solution to Black Scholes

� Process:

� Variable substitution

� Heat equation

� Self-similar solution

Solution to Black Scholes cont.

Simulations – raw data

0 50 100 150 200 250 300460

480

500

520

540

560

580

600

620

640

660Google Stock Price in 2011

trading day of 2011

opening stock price [$]

Simulations – changing strike price

300 400 500 600 700 800 900 1000 1100 1200 13000

100

200

300

400

500

600

Strike Price vs Option ValueGoogle 2011 Data

Stock Price=$642.0, Volatility=29.2

Strike Price [$]

Option Value [$]

Put

Call

Simulations – Volatility and Expiration (Put)

Nominal Strike: $642 stock value: $642 risk free rate: 18% per annum

Put Option

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.80

20

40

60

80

expiration time [yrs]

Changing Volatility and Expiration Time

volatility

option value [$]

Simulations – Volatility and Expiration (Call)

Nominal Strike: $642 stock value: $642 risk free rate: 18% per annum

Call Option

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.80

50

100

150

200

expiration time [yrs]

Changing Volatility and Expiration Time

volatility

option value [$]

Simulations – Volatility and Strike (Put)

Expiration: 0.5 years stock value: $642 risk free rate: 18% per annum

Put Option

600620

640660

680700

0

0.2

0.4

0.6

0.80

20

40

60

80

100

strike price [$]

Changing Volatility and Strike Price

volatility

option value [$]

Simulations – Volatility and Strike (Call)

Expiration: 0.5 years stock value: $642 risk free rate: 18% per annum

Call Option

600

650

700

00.1

0.20.3

0.40.50

20

40

60

80

100

120

140

strike price [$]

Changing Volatility and Strike Price

volatility

option value [$]

Simulations – Expiration and Strike (Put)

Put Option

600620

640660

680700

0

0.5

10

10

20

30

40

50

60

strike price [$]

Changing Expiration Time and Strike Price

expiration time [yrs]

option value [$]

Volatility: 29% per annum stock value: $642 risk free rate: 18% per annum

Simulations – Expiration and Strike (Call)

Call Option

Volatility: 29% per annum stock value: $642 risk free rate: 18% per annum

600620

640660

680700

0

0.5

10

50

100

150

200

strike price [$]

Changing Expiration Time and Strike Price

expiration time [yrs]

option value [$]

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Binomial Method

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Figure modified from Hull J.C., “Options Futures and Other Derivatives”, p207

0 10 20 30 40 50 6020

22

24

26

28

30

32

34

36Value of euro style put options, binomial method vs Black Scholes

Number of time steps

Value of option now [$]

Binomial Method Simulation (Euro)

Black Scholes

0 10 20 30 40 50 6027

28

29

30

31

32

33

34

35

36Value of amer style put options, binomial method vs Black Scholes

Number of time steps

Value of option now [$]

Binomial Method Simulation (American)

Black Scholes

Further Work

� Exploration into Modification of Black-Scholes:

� Other Option Styles

� Dividends

� Varying Volatility