Valuing an Option

Valuing an Option

According to the Black-Scholes Model, the key ingredient in valuing an option is the future volatility - the volatility of the stock’s prices that will occur over the life of the option. We do not need to know anything about the direction of the stock (notice that the model does not ask you for a prediction on the direc­tion of the stock). The reason for this is that stock price changes follow a bell­shaped curve. Once we have the volatility (the standard deviation) we can determine many probabilities since we know the following properties must hold for bell curves:

1.     68% of the data fall within one standard deviation

2.    95% of the data fall within two standard deviations

3.    Essentially all of the data fall within three standard deviations

As an example, if we have a $100 stock with 30% volatility, we know that in one year:

1.     There is a 68% chance tire stock’s price will fall between $70 and $130 (one standard deviation, or 30% on either side of the current $100 price).

2.    There is a 95% chance the stock’s price will fall between $40 and $160 (two standard deviations, or 2* 30% on either side of $100)

3.    There is virtually a 100% chance that the stock’s price would lay between $10 and $190 after one year (three standard deviations, or 3 * 30% on either side of $100).

Once we have a standard deviation (volatility), the Black-Scholes Model can then find the probabilities and payoffs of the option and that means we can find the fair value of the bet. If we are faced with low volatility, then there are only a few possible stock prices that can occur and the option has only a few possible small payoffs. Consequently, the option has a low price. But if there is a lot of volatility then there are a lot of possible stock prices underneath that bell curve and the bet become more valuable (the option has a much higher price). For option trading it is important to remember the following: High vola­tility equals high-priced options (true for both calls and puts). Low volatility equals low-priced options.

Of course, we will never know what the true volatility number is until after the fact but we can get an idea based on past volatilities. There is one more concept that we need to grasp before putting all of this information to­gether. That is, volatility tends to move sideways over time. Let’s take a closer look at why this statement is true.