Using the Black-Scholes Model

Using the Black-Scholes Model

The Black-Scholes Model is an option pricing model that takes six inputs and calculates what the call and put prices should be. The six factors are:

·       Stock price

·       Exercise price (strike price)

·       Risk-free rate of interest

·       Time to expiration

·       Dividends

·       Volatility

Let’s run through an example of how to use it. Assume that the underlying stock is $50, the strike price is $50, there are 60 days to expiration, volatility is 30%, the risk-free interest rate is 2%, and no dividends are paid during the life of the option. If we put these values into the calculator (as shown in Figure 6, following page) and hit the “calculate” bar, we find that the call should be priced at $2.50 and the put at about $2.35 as shown by the circles in Figure 6:

Those circled values are not just some ambiguous values. They are the fair values of the option “bets” based on the six inputs given on the left hand side. In this example, if you pay more than $2.50 for the call option (or $2.35 for the put), you are expected to just break even in the long run. If you pay less, you’ll end up on the winning side and if you pay more you’ll end up on the losing side. This assumes that you are able to make this exact bet hundreds of times, which is an unrealistic assumption. In the real world, you are probably only going to get one shot at any given option scenario. Still, understanding the fair value of such a bet over hundreds of trials gives us a benchmark for what we should be willing to pay for only one shot.

Black-Scholes Input Explanations

Style

The “style” pull down menu asks you to select either ‘American” or “Euro­pean” style options. An American style option is one that can be exercised at any time prior to expiration while the European counterpart may only be exercised at expiration. All equity (stock) options are American style while most index options are European (the OEX is about the only major excep­tion - it is an American style option). If you are valuing stock options, make sure you select the “American” style.

Interest Rate

The interest rate is always the annualized risk-free rate over the time period of the option. In this example, we need the risk-free rate for a sixty-day investment (expressed as an annual rate). However, the CBOEs calculator automatically looks this up for you based on the “days to expiration” field you provide. So the risk-free rate is not an input that you need to worry about finding if you are using this calculator.

Volatility

This is the most important number in determining an option’s value. We used 30% for this example but, in the real world of trading, that number could be anything from a low of about 10% to a high of 300% or more. The volatility number is simply the standard deviation of stock price returns, which we will talk about shortly. The volatility number gives us an idea of how large we expect the future price swings to be.

Dividends

The “dividends” field requires us to type in the amount of any dividend that will be paid during the life of the option.

Of the six factors, volatility is the most critical since it is not directly observable in the market. In other words, if you asked 100 traders to fill in the six fields in the Black-Scholes Model for a particular option, they would all use the same current stock price, the same exercise price, the same risk-free interest rate (or at least very close), the same dividend amount, and the same time to expira­tion. Those five factors are fixed at each moment in time. However, it is pos­sible dial no two traders would use the same volatility since there is no way to say for sure what the correct answer is. This is why volatility is so important to understand.