__Option Premiums__

Depending on the sophistication of the market maker, option pricing is typically determined either by straightforward use of the original pricing formula developed by Fischer Black and Myron Scholes in 1973 or by the use of later extensions of this formula by Robert Merton, Giovanni Barone-Adesi, and Robert Whaley. In a nutshell, the Black-Scholes model determines option prices for European-style options, ignoring the effects of dividends. The Merton model adjusts the Black-Scholes formula to take dividends into account; the Barone-Adesi and Whaley model adjusts the Black-Scholes-Merton formula for the early exercise potential present in American-style options.

**The Black-Scholes Formula for Puts**

The Black-Scholes formula for the put premium for European-style options with a zero dividend rate is given by:

where:

P is the price of the underlying issue

S is the strike (exercise price) of the option

r is the risk-free interest rate in decimal form

t is the time in years till expiration

v is the annualized volatility in decimal form

and h is given by the formula:

When t approaches zero, h takes on one of two values:

1. if the stock price P equals or is greater than the strike price S

2. if P is less than S, the result of which shows that the value of the put goes to zero if P equals or is greater than S, or it approaches S - P if P is less than S.

The integrals are the standard normal distribution, and the same curve-fitting formula that can be used for their evaluation. Note that the right-hand integral is the probability that the put will wind up in the money.

**The Black-Scholes Formula for Calls**

For completeness, the corresponding Black-Scholes formula for the call premium is given by:

where P, S, r, t, and h are as defined earlier for the put formula. If the put price has already been calculated, one can use the shortcut formula (called the put-call parity formula) for the call, as follows:

**The Effect of Dividends on Option Premiums**

To keep things simple and conservative, all put premiums used in conducting the ten-year economic simulations were calculated using a zero dividend rate. The effect of dividends is to lower the price of calls and to increase the price of puts. If I were to assume that dividends were paid continuously through the year (rather than in the usual quarterly distribution), the Merton variation of the Black-Scholes pricing formula can be used. This formula takes the following form for puts:

where δ is the annual dividend rate, and h is modified as follows:

The right-hand integral is again the probability that the put will wind up in the money and is the one adopted for that purpose.

**Numerical Example**

The effect that dividends have on option premiums, consider the case where the strike price and stock price are both $100, the volatility is 0.35, the risk-free interest rate is 6 percent, there are 24 months till expiration, and the annual dividend rate is 2 percent. Applying the basic Black-Scholes formula yields a put premium for the zero-dividend case of $13.314. For the Merton variation, do the following:

Since S = P = $100, the parameter h is calculated as

Therefore, -h is -0.409112, and is -0.409112 + 0.4949748, or 0.0858628. From tables of the normal distribution or through the use of the curve-fitting formulas, we have N(-0.409112) = 1 - N(0.409112) = 0.341238 and N(0.0858628) = 0.534223. When the time to expiration is 2 years, e-0.02t is 0.960790 and e-0.06t is 0.886920. So from Equation the European put with its 2 percent annual dividend rate is calculated as

Thus, the annual 2 percent dividend has increased the price of the put by about $1.28 over the two-year period involved, thus showing that the effect of dividends on option pricing can indeed be significant.The pricing of American options is so complex that market makers often restrict their pricing formulas to that of Black-Scholes or one of its simpler variations. The pricing of American-style options is examined.

**Implied Volatility**

Now that you've seen how to calculate the LEAP put premium for European-style options using the Black-Scholes-Merton formulas, the question arises as to whether one can work the process backwards and determine volatility if the premium is supplied. The answer to this question is both yes and no. To begin, I first note that it is not possible to solve for the volatility directly. However, the volatility can be determined on a trial-and-error basis using a Newton-Raphson iterative search technique. A relatively short BASIC program for doing this follows. It is then illustrated by two numerical examples.

**Numerical Examples**

As examples of the results obtainable using the computer program, consider the two cases investigated earlier:

Stock Price: $100

Strike Price: $100

Risk-Free Interest Rate (e.g., .06): .06

Annual Dividend Rate (e.g., .02): 0

Time to Expiration in Months: 24

Put Premium: 13.314

No. VOLD VNEW F

1 0.50000 0.38384 7.4605

2 0.38384 0.35707 1.6934

3 0.35707 0.35144 0.3544

4 0.35144 0.35029 0.0723

5 0.35029 0.35005 0.0147

6 0.35005 0.35001 0.0030

Implied Volatility: 0.35003

Stock Price: $100

Strike Price: $100

Risk-Free Interest Rate (e.g., .06): .06

Annual Dividend Rate (e.g., .02): .02

Time to Expiration in Months: 24

Put Premium: 14.596

No .VOLD VNEW F

1 0.50000 0.38456 7.3947

2 0.38456 0.35755 1.7191

3 0.35755 0.35163 0.3758

4 0.35163 0.35036 0.0808

5 0.35036 0.35008 0.0173

6 0.35008 0.35003 0.0037

Implied Volatility: 0.35005

At first blush, this numerical method for determining volatilities from premiums looks like a great shortcut to the laborious process of calculating them from historical price information. All one has to do is to obtain the premium quoted, say, for a LEAP put, and then use the computer program to determine the volatility. The trouble with this is that it is not ordinarily possible to determine what pricing model the market maker used in the first place to arrive at the option premium quoted. The market maker may have used Black-Scholes, or the Merton variation, or a multiperiod binomial model, or even a more sophisticated model for the pricing of American-style options.