The big question, of course, is: How can we rely on a formula with a series of variables that are provably inaccurate and based on a flawed assumptions, exponentially inaccurate variables, and outdated models about the nature of options?
The pricing model under the Black-Scholes (B-S) formula is premised on several assumptions. Today, in spite of advances and changes in the options market, this model continues to be used by many as the standard for theoretical options pricing. However, the original formula and its assumptions, as well as subsequent modifications, contain many flaws.
The initial assumptions of the formula were:
1. Exercise. Options will be exercised in the European model, meaning no early exercise is possible. In fact, U.S. listed stocks are exercised in the American model, meaning exercise may occur at any time prior to expiration. This makes the original calculation inaccurate, since exercise is one of the key attributes of valuation.
2. Dividends. The underlying security does not pay a dividend. Today, many stocks pay dividends and, in fact, dividend yield is one of the major components of stock popularity and selection, and a feature affecting option pricing as well. (This flaw in the original model was corrected by Black and Scholes after the initial publication once they realized that most stocks do pay dividends.)
3. Calls but not puts. Modeling was based on analysis of call options values only. At the time of publication, no public trading in puts was available. Once puts began to trade, the formula was again modified. However, if traders continue relying on the original BSM, even for put valuation, they may be missing a fundamental inaccuracy in the price attributes.
4. Taxes. Tax consequences of trading options are ignored or non-existent. In fact, option profits are taxed at both federal and state levels and this affects net outcome directly. In some instances, holding the underlying over a one-year period may lead to short-term capital gains taxation due to the nature of options activity, for example. The exclusion of tax rules makes the model applicable as a pre-tax pricing model, but that is not realistic. In fairness to the model, everyone pays different tax rates combining federal and state, that any model has to assume pre-tax outcomes.
5. Transaction costs. No transaction costs apply to options trades. This is another feature affecting net value, since it's impossible to escape the brokerage fees for both entry and exit into any trade. This is a variable, of course; fee levels are all over the place and, making it even more complex, the actual options fee is reduces as the number of contracts traded rises. The model just ignored the entire question, but every trades knows that commissions can turn a marginally profitable trade into a net loss.
6. Interest does not change. A single interest rate may be applied to all transactions and borrowing; interest rates are unchanging and constant over the life span of the option. The interest component of B-S is troubling for both of these assumptions. Single interest rates do not apply to everyone, and the effective corresponding rates, risk-free or not, are changing continually. What might have been applicable, at least in theory, in 1973, is clearly not true today. Even adjusted pricing models since the original BSM tend to overlook this fact in how value is determined.
7. Volatility is a constant. Volatility remains constant over the life span of an option. Volatility is also a factor independent of the price of the underlying security. This is among the most troubling of the BSM assumptions. Volatility changes daily, and often significantly, during the option life span. It is not independent of the underlying and, in fact, implied volatility is related directly to historical volatility as a major component of its change. Furthermore, as expiration approaches, volatility collapse makes the broad assumption even more inaccurate.
8. Trading is continuous. Trading in the underlying security is continuous and contains no price gaps. Every trader recognizes that price gaps are a fact of life and occur frequently between sessions. It would be difficult to find a price chart that did not contain many common gaps. It is understandable that in order to make the pricing model work, this assumption was necessary as a starting point. But the unrealistic assumption further points out the flaws in the model.
9. Price movement is normally distributed. Price changes in the short term in the underlying security are normally distributed. This statistical assumption is based on averages and the behavior of price; but studies demonstrate that the assumption is wrong. It is one version of the random walk theory, stating that all price movement is random. But influences like earnings surprises, merger rumors, and sector, economic and political news, all affect price in a very non-random manner. Sheldon Natenberg (Option Volatility and Pricing, 1994) concluded that price changes are not normally distributed (p. 400-401).
One big question often overlooked in all of the debate over pricing models: Do traders even need the model itself? Or are traders much more concerned with levels of volatility and the momentum of change in volatility? If this is the case, then focusing on delta and gamma makes much more sense than trying to identify the theoretical price of the option. Even volatility is useful only until the last week of the option's life. In this final period, volatility collapse makes even delta and gamma unreliable. This is especially true on the Thursday and Friday, when volatility tracking becomes quite unreliable, as pointed out by Jeff Augen in his book, Trading Options at Expiration: Strategies and Models for Winning the Endgame (FT Press, 2009). If you want to track delta and gamma, one of the best ways is to use the free calculator provided by the Chicago Board Options Exchange (NASDAQ:CBOE). Go to CBOE and link to "Tools" and from there to "Options calculator."
This discussion of pricing models and alternatives is very interest, and I invite everyone to visit ThomsettOptions.com and continue this dialogue. Please go to ThomsettOptions.com to read more about this and topics of interest of options traders. I hope to see you there.