Home > Blogs > Options Trading - the B-S Pricing Model

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of variables that are provably inaccurate and based on a flawed assumptions, exponentially inaccurate variables, and outdated models about the nature of options?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. In fact, 15 years after the original Black-Scholes paper was published, one of its authors, Fischer Black, wrote about the model and its flaws. To read his analysis of the model, visit http://www.scribd.com/doc/74806381/Holes-in-Black-Scholes and read his own words.

The original formula and its assumptions 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 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 (CBOE). Go to cboe.com and link to "Tools" and from there to "Options calculator." The Black-Scholes pricing model did serve a valuable purpose: It laid the groundwork for analysis of option pricing in academia. The debate began in 1973 and has continued to this day, with numerous rewrites to the original. At the time of its publication, however, the options market had just begun and the Chicago Board Options Exchange (CBOE) opened at about the same time, offering public trading for the first time (in calls, but not in puts), but only on 16 companies. So no one could foresee the dozens of strategies and applications of options trading at the level we enjoy today. Even with this launch of the debate concerning pricing models, a distinction has to be made. On the one hand, price models like Black-Scholes serve a purpose in academia. But for traders trying to determine when to enter and exit positions, timing based on volatility and underlying price, volume and momentum is more realistic.

Is it heresy to say that traders don't care about the
option's price? Like many heresies, it is true. Traders care about the
level of volatility, the direction it is moving, and the momentum of
that change. I doubt that any serious trader relies on Black-Scholes to
time actual trades, using their own money, and believing that the
assumptions underlying the formula are reliable or accurate in making
those trade decisions.

*Michael C. Thomsett has written several books on the topic of options trading, published by FT Press, including The Options Trading Body of Knowledge and Options Trading for the Conservative Investor.* He writes fulltime and also teaches several options classes at the New York Institute of Finance (NYIF).

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