Black-Scholes Option Valuation

| ## ...what determines the value of a call option? At expiration the holder may deliver an amount equal to the Strike price in return for one share which she can turn around and sell. So at expiration the value is strike price minus stock price with the exception that its value cannot be less than $0.0. Before expiration the value of a call is determined by the liklihood that the stock's price at the expiration date will exceed the strike price and by the time value of money. With respect to calls which will be excercised, the owner effectively owns the stock but hasn't paid for it yet. She is borrowing and should therefore except to pay more if the interest rate is high.

## ...an option pricing formula

Fischer Black and Myron Scholes invented a formula which predicts an option price as a function of time, volatility, stock price, strike price, and the interest rate. Assuming a distant expiration date, the model predicts that a $1.00 change in stock price causes a small change in option price - perhaps $0.50. This makes intuitive sense because the call owner profits only from a high stock price at expiration and a small increase today might easily be canceled out by another small change in the opposite direction tomorrow. If the stock price becomes quite high, then the option value will increase by nearly $1.00 for each $1.00 increase in the stock price. This also makes sense because when the stock is way above the strike price, it's almost certain that the option buyer will excerise the option.

## ...visualizing Black-Scholes

The red surface shows Black-Scholes call values for a variety of stock and strike prices. A Sequencer in the visualization shows the effect over time. You can Pickon a location on the surface which selects of combination of stock and strike and reports the Black-Scholes value. Then you can run the sequencer to watch the price change. Note that at very high stock prices, the slope of the surface approaches 45 degrees. In other words a $1.00 change in stock price translates to a $1.00 change in call price. When strike and stock are close together, a $1.00 change in stock produces about $0.60 change in call value.