1. A different kind of Tree Using trees to help price options. Using trees to help price options. Some of the ideas behind Black-Scholes. Some of the ideas behind Black-Scholes. Myron ScholesFischer Black http://hilltop.bradley.edu/~arr/bsm/model.html

2. Option Basics A stock option is a derivative security, because the value of the option is “derived” from the value of the underlying common stock. A stock option is a derivative security, because the value of the option is “derived” from the value of the underlying common stock. There are two basic option types. There are two basic option types. Call options are options to buy the underlying asset. Call options are options to buy the underlying asset. Put options are options to sell an underlying asset Put options are options to sell an underlying asset

3. Example 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 whether or not you would exercise your option: Consider the following scenarios and whether or not you would exercise your 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 your option? How do you know what a good price is for your option?

4. $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.

5. $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.

6. $30 $31 $1 $29 $0 $32 $2 $30 $0 $28 $0 $31 $1 $29 $0 $33 $3 $27 $0

7. The Black and Scholes Model http://hilltop.bradley.edu/~arr/bsm/model.html