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Black-Scholes option valuation: an overview in plain language
The following article is expanded from a long post I made to the CompuServe Investor's Forum in 1986. You can now find incredibly detailed and more accurate technical explanations of the equation elsewhere on the web, but if you're new to it, this overview might be helpful to start with because it uses plain language and avoids almost all of the mathematics.
The Black-Scholes option valuation formula attempts to calculate the fair economic value of an option for both buyer and seller, the price at which (exclusive of commissions) both a buyer and seller would, on average, break even if they repeated this trade many times. Thus, the calculated price does not favor either the buyer or seller, so it is a price they can agree on. In real world trading, it means that, on average, both the buyer and seller will lose money on the transaction, by the amount of the commissions they paid. The valuation model uses probability theory, discounted cash flows, and expected value calculations.
The model assumes:
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.
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.
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.
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 of $50, 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 to present value using, for example, the interest rate on a T-Bill of the same time period as the expected holding period of our option. So let's say we come up with $1.95 (just guessing).
The basic Black-Scholes model assumes that the stock pays no dividends. However, some stocks do pay dividends, and dividends do affect the prices of options.
The morning a stock goes ex-dividend, its price is dropped by the amount of the dividend. The lowering of the price now also lowers the price projections into the future. Because the stock's projected price is lower than it would have been, a dividend decreases the value of call options (because the stock is less likely to be above the call's strike price) and increases the value of put options (because the stock is more likely to be below the put's strike price).
Various refinements of Black-Scholes attempt to take these dividends into account. One approach is to subtract the amount of the dividend from the current stock price before doing the calculation (giving a lower price to start from). Since the dividend will be paid in the future, it is not the actual amount of the dividend that is used, but the present value of it.
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 can be less expensive than buying a put option, the prices of real puts 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 overvalues puts and makes all traded puts appear to be undervalued.
Trading strategy implied by Black-Scholes
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 longer term to raise the option's market value to equal the "fair value" that you calculated before you bought it. The goal would be to take advantage of a temporary market inefficiency to purchase the stock's volatility at a bargain rate.
This is no formula for riches. Since the computed fair value reflects only a breakeven proposition, your expected profit is limited to the amount that it was undervalued when you bought it minus your in and out commissions.
The Black-Scholes formula requires as one of its inputs the stock's volatility, the "annualized standard deviation of the underlying stock price's daily logarithmic returns".
I've created a volatility calculator. In addition to calculating historical volatility from a price history you provide, it describes the calculation, has links to articles with greater detail, and shows how it compares against other easier methods that were once common.
There was a time when one could only get the volatility estimate (or the historical price data necessary for computing it) from expensive subscription services, so various alternative formulas were in use.
Even today, volatility computation is a field of study in its own right. With the discovery (hardly a surprise) that volatility is not as stable as the Black-Scholes model assumes, there are alternative calculations that attempt to predict more precisely what the volatility is likely to be during the specific holding period. For example, a weighted volatility might be used, with values from recent weeks weighted more heavily than those from longer ago.
The different calculation methods can produce quite different estimates. If the volatility calculation is critical, and if no method is definitive and universally accepted, which do you trust?
This brings us to a real life truth: A thing is worth what anyone is willing to pay for it at a particular time, regardless of what any model says it is supposed to be worth. If you cling steadfastly to the Black-Scholes valuation model, or to one particular volatility calculation method, but other people are willing to pay more than the "fair price" that your model calculates, then the other traders, using their valuation methods, will drive prices to the valuations calculated by their methods (or lack of method).
Nonetheless, if you want to actually find economically undervalued or overvalued options using the Black-Scholes formula, you must use the true volatility calculation that it requires or one of the newer alternatives that adheres completely to its intent but attempts to improve the quality of the estimate.
If you want to get the estimated "normal" price of an option, there are other valuation methods besides Black-Scholes that take different approaches.
In 1987, it seemed that option traders tended to accept under- or over-valuations as conditions of a particular option series that they expected to remain fairly constant. If they had to buy at an overvaluation, they'd expect to sell at one, too, when that time came.
That approach treats options as independent instruments (which they are) that have their own supply/demand characteristics independent of those of the underlying stock (which also is often the case), and whose prices can therefore stray significantly from valuations calculated by Black-Scholes or other models (which they often do).
Despite its attempt at mathematical precision for option valuation, for Black-Scholes the devil is in the details of the volatility calculation and in the market realities of the trading floor. It is a precise mathematical formula which requires an input value that is impossible to determine precisely.
Nonetheless, because its principle seems sound, many traders do use Black-Scholes valuation as a guide to pricing.
Related page: Black-Scholes option calculator
Questions and comments are welcome in the discussion forum.
Copyright ©2012 Steven Whitney. Last modified Sun 07/29/2012 10:52:10 -0700.