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Understanding the Black-Scholes model is fundamental to understanding both the theory of option pricing and the strategies of profitable trading. The mathematical concepts that underlie the model are advanced and complex, but the application of the model is relatively simple, especially with calculators or computers. For the first course in investments, rather than try to derive the option pricing model, it's beneficial simply to explain the concept and show how to use the model to price options.
The foundation of the model rests on the construction of a hypothetical risk-free portfolio, consisting of long call options and short positions in the underlying stock. With proper selection of the number of call options held and the number of stocks sold, the investor can lock in a certain amount of profit. Since this profit is certain, or risk free, the investor earns the risk-free rate on the portfolio. The model uses four directly observable variables (the market price of the stock, the exercise price on the call, the time remaining until expiration on the call, and the risk-free interest rate) and one variable that is fairly easy to estimate (the standard deviation of the stock's returns). With the five variables, the basic Black-Scholes option pricing model calculates the price of a call option as follows:
C = (S)[N(d_{1})] - (X)(exp^{-rt})[N(d_{2})]
where
C = | the price of the call, or the call premium |
S = | the price of the underlying asset, such as a stock price |
X = | the exercise price, or the strike price of the call |
r = | the continuously compounded annual risk-free interest rate |
t = | the time (in fractions of 1 year) to the option's maturity |
exp = | approximately 2.718292; it is included as a function key (sometimes labeled e^{x} on most financial calculators |
N(d_{1}) and N(d_{2}) = | probabilities from the cumulative standard normal distribution (see Table 1) |
Let's look at this equation before we try to use it. It begins by stating that in calculating the price of a call, the difference between the stock price and the exercise price is very important. In fact, at expiration, either the stock price of a call will be zero, if it is out-of-the-money (i.e., if the stock price is less than the strike price), or it will be equal to the difference between the stock price and the strike price, if it is in-the-money. Because we are trying to price the call at a point in time before expiration, we explicitly capture the time value of money by finding the present value of the exercise price. This continuous discounting process is performed by exp^{-rt}.
Also, since we do not know what the stock price will be by the expiration date, we must use a probability distribution to adjust the call premium for the uncertainty involved. The variables N(d_{1}) and N(d_{2}) are probabilities of the stock price being used at a certain price relative to where it is now. The values for d_{1} and d_{2} are used to calculate the probabilities that the stock price at expiration will be a certain number of standard deviations above or below the standardized mean (i.e., 0). Their formulas are:
Notice that the calculation of d_{1}, ln refers to the natural logarithm of the ratio of the stock price and exercise price. This function too is included on most financial calculators. Also note that the equation uses both the standard deviation , and the variance , of the stock's returns. The variable t is included as a fraction of a 365-day year, and r is the risk-free interest rate in decimal form.
Although you will most often want to use the Black-Scholes option pricing formula on a preprogrammed calculator or computer, it is useful to work through an application to see how the model works. Assume that you want to earn income by writing a call option on Gogol MegaCorp stock, which you think will decrease in value. The current stock price is $32 5/8, and you want to write a call with a strike price of $30, to expire in 85 days. The risk-free rate is currently 7%, and you estimate that the volatility of the stock's returns has a standard deviation of .32.
To value this call, we must first compute d_{1} and d_{2}; that is:
Now for d_{2}:
Next, we take these numbers to the cumulative standard normal table to find the associated probabilities. These values are given in Table 1. To use it, we need to first round d_{1} and d_{2} to two decimal places. This gives d_{1} = .73 and d_{2} = .57. Looking these values up in the table, we find the respective probabilities of N (d_{1}) = .7673 and N (d_{2}) = .7157.
We can now complete the OPM by calculating the call price:
C | = (S)N(d_{1}) - (X)(exp^{-rt})[N(d_{2})] |
= 32.625(.7673) - 30 exp^{(-0.07)(.2329)}(.7157) | |
= 25.03 - 30(.9838)(.7157) | |
= 25.03 - 21.12 | |
= $3.91 |
Computer solutions may give slightly different answers, due to rounding and the accuracy involved in using the standard normal table. Also, the solution provided by the Black-Scholes model will not always equal the price at which the option is trading. This basic model ignores the fact that the stock could pay a dividend and that the option is American, not European. Even so, the option price provided by the model (especially by the more complicated extensions of the model) is usually very close to the actual price; if it is not, then a trading opportunity may exist. A number of empirical studies have attempted to test the accuracy of various versions of the option pricing model. Most find that the model gives fairly accurate results.
A change in any of the five variables of the option pricing model will result in a change in the call premium--that is, in the price of the option. A summary of the direction of change is given in Table 2. Most of these changes can be observed by looking at a quotation of options prices and observing the option premiums over a period of time.
Table 2: Determinants of the Call Premium
An Increase in... | Will Cause the Call Premium to... |
Stock price | Increase |
Exercise price | Decrease |
Time to expiration | Increase |
Volatility of returns | Increase |
Risk-free rate | Increase |
A simple extension of the Black-Scholes model can be made to incorporate dividend information. The annual dividend yield of the stock is the expected annual dividend divided by the current stock price. Instead of using the stock price directly in the formulas, the following substitution should be made:
S' = S(exp ^{-Dt})
where D is the annual dividend yield. For example, suppose Gogal MegaCorp is expected to pay $2 per share in dividends in the coming year. This means that the dividend yield is about 6.13% (2 ÷ 32.625). If we use S' = 32.16 instead of S' = 32.625, we get an option price of about $3.50. This result makes intuitive sense: the lower the stock price (S), the lower will be the price of the call (C).
The option pricing model also gives another useful number besides the option price. The variable N(d_{1}) is called the hedge ratio, or the delta, for the call option. The hedge ratio tells how much the option price will change when the underlying stock price changes by some small amount. For example, our option for Gogal MegaCorp had a hedge ratio of about .77. This means that if the price of Gogal increased (or decreased) by one dollar per share, the price of the call option would increase (or decrease) by about $0.77. This information is very useful for option traders who are trying to combine stocks and options into portfolios that will have offsetting movements. The number of call contracts or stocks held can be adjusted using the hedge ratio to produce protected portfolios. Remember, though, that the hedge ratio will change whenever the stock price changes and also as the time to maturity decreases. Thus, the information provided by the hedge ratio is good for only small stock price changes, and for only a short period of time.