A different kind of TreeA different kind of Tree

Using trees to help price options. Using trees to help price options. Some of the ideas behind Black-Some of the ideas behind Black-

Scholes.Scholes.Myron Scholes Fischer Black

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Option BasicsOption Basics

A stock option is a A stock option is a derivative securityderivative security,, because the value of the option is “derived” because the value of the option is “derived” from the value of the underlying common from the value of the underlying common stock.stock.

There are two basic option types. There are two basic option types. Call optionsCall options are options to are options to buybuy the underlying the underlying

asset.asset. Put optionsPut options are options to are options to sellsell an underlying asset an underlying asset

ExampleExample

Suppose Aetna is selling for $30 a share.Suppose Aetna is selling for $30 a share. Suppose the option price is $1.30. Suppose the option price is $1.30. Consider the following scenarios and Consider the following scenarios and

whether or not you would exercise your whether or not you would exercise your option:option: At the expiration date the stock is $28 per share.At the expiration date the stock is $28 per share. At the expiration date the stock is $30 per share. At the expiration date the stock is $30 per share. At the expiration date the stock is $32 per share.At the expiration date the stock is $32 per share.

How do you know what a good price is for How do you know what a good price is for your option?your option?

$30

Price of stock = $32

Value of option = $2

Price of stock = $28

Value of option = $0

Suppose you buy x shares and sell one option.

• If the price goes up to $32 your portfolio is worth 32x-2.

•If the price goes down to $28 your portfolio is worth 28x-0.

$30

Price of stock = $32

Value of option = $2

Price of stock = $28

Value of option = $0

Suppose you buy x shares and sell one option.

•A risk free portfolio will have the same value regardless of what happens:

32x-2 = 28x-0. Solve for x to get x = ½. If you have ½ a share, then regardless of how the price of the stock changes (up to $32 or down to $28) the portfolio is worth 32(1/2)-2=14.

•So a risk free portfolio that contains x shares and sells one option has a value of 1/2x- 1 option = 14. This means an option is worth (1/2)(30) – 1 option = 14. Solve for one option to get 1 option = $1.

$30

$31

$1

$29

$0

$32

$2

$30

$0

$28

$0

$31

$1

$29

$0

$33

$3

$27

$0

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