Black-Scholes:

Volatility:

Statistics:

Black-Scholes option valuation: an overview in plain language.

The following article is expanded from a post I made to the CompuServe Investor's Forum in 1986. You can now find incredibly detailed (and likely more accurate) technical explanations of the equation elsewhere on the web, but if you're new to it, this overview might be helpful because it uses plain language and avoids the mathematics.

Disclaimer: I do NOT consider option trading to be a viable route to profits in the financial markets, whether one uses the Black-Scholes equation as a valuation guide or not. The reasons are given in a previous article.

Introduction

The Black-Scholes option valuation formula was developed as an attempt to determine the fair economic value of an option for both buyer and seller, that is, to determine at what price (exclusive of commissions) both a buyer and seller would break even on the option (and thus lose only their commissions). It uses probability theory, discounted cash flows, and expected value calculations.

The model assumes:

• That stock price fluctuations are random (not predictable), that they are a random walk.
• That the random fluctuations are described by a normal (bell-shaped) curve that is slightly biased toward uptrend.
• That the volatility of a particular stock, the amount by which its price fluctuates over the course of a given time interval, is a stable measurement for that stock. That is, one can expect that stock to exhibit the same volatility in the future that it has had in the past.

A summary of the Black-Scholes method might read like this:

Our stock has a characteristic volatility that we can expect it to exhibit during the life of our option. Although we cannot predict the future price of our stock, we can (from the volatility) calculate the statistical probability that on the option's expiration date the stock will have reached or exceeded each possible target price. At each target price, the option will have a knowable value which is its intrinsic value, the amount by which it is in the money.

Because we know each possible outcome of this situation (the various target prices), and because we know the statistical likelihood of each outcome, and because we know what our return will be from each of the possible outcomes (the option's value), we have enough information to calculate the expected value of this option by using a payoff table. The actual return we get this time might not be this average value, but if we were to repeat this situation many times, this would be the average return of all those times.

In very simple terms, it is similar to this: if we are about to flip a coin, we have no way to predict whether it will come up heads or tails, but we do know that it will come up heads 50% of the time and tails 50% of the time. That is useful information even if it does not allow us to predict the result of this one coin toss. For example, if we will earn $1 for each head but lose $10 for each tail, the 50/50 statistic tells us that we should not play this game because we will lose badly at it. On average, we will lose $4.50 with every toss. How to calculate this is shown below.

Why the stock's volatility is important

Let's use a concrete example, a $50 stock that on average fluctuates 2% per day.

Based on its historical volatility, its price tomorrow will be, with a 67% probability, between $49 and $51, and it is slightly more likely to be above $50 than below it.

1. The best single guess about tomorrow's price is that it will be the same as today's, midway within the predicted range.
2. It might make its full 2% ($1.00) move in either direction, but that is slightly less likely.
3. It might even make a $10 move in either direction, but that is even less likely.
4. And so on. The larger the move, the less likely it is.

A big move on one day might be offset by a big move in the other direction the next day, or by small moves on subsequent days, but on average, this stock fluctuates about 2% per day. The 2% historical volatility figure makes it possible for us to estimate the range of prices within which this stock will tend to stay over a given day, week, month, or year.

A low volatility stock will stay within a narrow range. A high volatility stock will tend to wander farther, in either direction, from its current price, making its put and call options theoretically more valuable because the stock price is more likely to reach extremes.

Payoff Table

A payoff table or decision matrix is a way of estimating the expected value of a situation that has multiple possible outcomes. Each possible outcome has a probability of occurring and an associated payoff that will result if it does occur. The payoff table calculates the average expected return from the situation. The situation won't return that exact amount each time, but if you encounter that situation many times, that will be the average return you get from it.

Examples

If something will return $20 and the probability of it happening is 100%, then the expected value is $20. The calculation is easy. It is:

100% x $20 = $20.

If another situation returns $20, but there is only a 50% chance of it happening, and there is a 50% chance of it returning nothing, then the expected value is cut in half, as follows, to $10:

As applied to a stock

If the stock is now at 50 and its historical volatility is 2% per day, then there is a statistical method (incorporated into the Black-Scholes equation) for estimating the probability that the stock will reach (for our simple example) each of the prices 40, 45, 50, 55, 60 at a given date (our expiration date), and we know what the value of the option will be on that date because at expiration an option has exactly its intrinsic value (the amount by which it is in the money).

We can therefore create a payoff table, using a $50 call option, like this. Note that the probabilities reflect that the most likely price is the current price, and extreme prices in either direction are increasingly unlikely, as described above:

In this simplified table, it appears that the fair value of the call to a buyer should be $2.00, but that will be its value in the future, and we are buying it now, so we discount it (using the T-Bill interest rate, as an example) to present value. So let's say we come up with $1.95 (just guessing).

Dividends

During the course of the option's life, dividends may be paid on the stock. The morning the stock goes ex-dividend, the stock price will be dropped by the amount of the dividend, increasing the likelihood that the stock will be at a lower price at expiration. Dividends paid out will decrease the value of call options and increase the value of put options. The Black-Scholes model takes these dividends into account. However, once again, since the dividend is being paid in the future, we must use not the actual amount of the dividend, but the present value of it.

Expiration date

So far, the only date referred to is the expiration date. The Black-Scholes model only concerns itself with that date. It does not concern itself with the probability that a stock will reach or exceed a given price at some point during the intervening time, but only with the probability that it will be at or above (or below, for puts) that price at the expiration date. It makes the assumption that options are held to expiration. This is how European style options work, but it is not how American style options work.

Valuing put options

To use the Black-Scholes formula for put options you must incorporate a routine to calculate the price of an equivalent "synthetic put", which involves selling the stock short and buying a call option on it. Because this is cheaper than buying a put option, the prices of put options are driven down to the values of their equivalent synthetic puts. Without the synthetic put (also known as "conversion put") calculation, the Black-Scholes formula significantly overvalues puts and thus makes all traded puts appear to be undervalued.

Implied strategy

The only strategy that seems to be implied by the Black-Scholes model is that of buying undervalued options (or selling overvalued ones) and holding them to expiration even in the face of any and all apparent setbacks in the position, trusting that the stock's historical volatility characteristic will be fulfilled over the long term. The goal would be to purchase the stock's volatility at a bargain rate. Even so, since the fair value reflects only a breakeven proposition, your gain is limited to the amount that it was undervalued when you bought it minus your in and out commissions. This paints a bleak picture of option trading.

Volatility calculations are often wrong

The Black-Scholes formula uses as the volatility input the "standard deviation of stock price fluctuations". To get this figure for one year, enter into a statistical calculator the closing price for the stock for each day for one year. Then hit the "standard deviation" key, and you will get a number that is the standard deviation of the stock price for that year. To make it a percentage (fraction) of the average stock price, divide it by the average of the daily closing prices over that one-year period, which you can calculate using the same data you just entered by pressing the "arithmetic mean" (average) key () on the calculator. If you want the standard deviation for a longer period such as 5 years, do it for each day of the period.

I am not absolutely certain that this is the exact input required by the Black-Scholes equation, but it is the statistical standard deviation of the daily stock price. Turning it into the required fraction by expressing it as a percentage of the average price for the year is the part that was a reasoned guess. It does have to be expressed as a fraction, and the average price over the course of the year is the only number that is reasonable to make it a fraction of.

The "volatility" calculation commonly used by traders (or that used to be, in 1986) is not the standard deviation, and uses only 2 historical price figures (for any period) instead of the approximately 250 required for each year! (one for each trading day):

(high - low) / (high + low)

I made a comparison between the two methods, and here's what I found:

The standard deviations of the stock price fluctuations give a smaller volatility estimate than the (hi-lo)/(hi+lo) method, and therefore its resulting option valuations are substantially lower. In fact, they are so much lower that I never found an undervalued option using the standard deviation method. The implication is that most option traders are using the shortcut method and driving option prices to the valuations derived from it. (Or using another valuation model, or trading by the seat of their pants and using no valuation model at all.)

This brings us to a real life truth: A thing is worth what anyone is willing to pay for it. If you cling steadfastly to the Black-Scholes valuation model, but everyone else is willing to pay more than the "fair price" it calculates, then you will wind up sadly out of luck because you'll never find the opportunity to make a trade.

My conclusions are that if you want to get the estimated "normal" price of an option, then use the shortcut, but if you plan to actually find undervalued or overvalued options using the Black-Scholes formula, then you must use the true standard deviation figure, and you will spend a LOT of time looking for undervalued options that you may never find, because no one else is using the same formula as you, and they are willing to bid the prices higher than you are.

This seems to imply that option prices are heavily in favor of sellers [in 1987], and that may be true, but it is beyond the scope of this discussion.

More often than not, it appears that option traders accept under- or over-valuations as conditions of a particular option series that will remain fairly constant. Thus, if we buy at an overvaluation, we expect sell at one, too, when that time comes. This could be realistic, but it renders the Black-Scholes formula less useful unless you incorporate a "fudge factor", which is hardly in the spirit of the original concept.

There is a thread in the discussion forum where you can ask questions.