Circumventing the Limitations of Black-Scholes

Mathematical or quantitative model-based trading continues to gain momentum, despite major failures like the financial crisis of 2008–09, which was attributed to the flawed use of trading models. Complex trading instruments such as derivatives continue to gain popularity, as do the underlying mathematical models of valuation. While no model is perfect, being aware of limitations can help in making informed trading decisions, rejecting outlier cases and avoiding costly mistakes that may result in huge losses. (See also: Build a Profitable Trading Model In 7 Easy Steps).

We'll discuss the limitations of the Black-Scholes model, which is one of the most popular models for options pricing. Some of the standard limitations of the Black-Scholes model are:

• Assumes constant values for risk free rate of return and volatility over the option duration—none of those may remain constant in the real world
• Assumes continuous and costless trading—ignoring liquidity risk and brokerage charges
• Assumes stock prices to follow lognormal pattern, i.e., a random walk (or geometric Brownian motion pattern)—ignoring large price swings that are observed more frequently in the real world
• Assumes no dividend payout—ignoring its impact on the change in valuations
• Assumes no early exercise (i.e., fits only European options)—the model is unsuitable for American options
• Other assumptions, which are operational issues, include assuming no penalty or margin requirements for short sales, no arbitrage opportunities and no taxes—in reality, all these do not hold true; either additional capital is needed or realistic profit potential is decreased  

This section describes how the above-mentioned limitations impact the day-to-day trading and whether any prevention or remedial actions can be taken. Among other problems, the biggest limitation of the Black-Scholes model is that while it provides a calculated price of an option, it remains dependent on the underlying factors that are

• assumed to be known
• assumed to remain constant during the life of the option

Unfortunately, none of the above is true in the real world. Underlying stock price, volatility, risk-free rate and dividend are unknown, and may change in short duration with high variance. This leads to high fluctuations in option prices. It does provide significant profit opportunities to experienced option traders (or ones with luck on their side). But it comes at the cost to the counterparts—especially newbies or ignorant speculators or punters—who are often unaware of the limitations and are at the receiving end.

It doesn't only have to be high-magnitude changes; the frequency of such changes can also lead to problems. Large price changes are more frequently observed in the real world, than those expected and implied by the Black-Scholes model. This higher volatility in the underlying stock price results in substantial swings in option valuations. It often leads to disastrous results, especially for short option sellers who may end up being forced to close positions at huge losses for the want of margin money, or being assigned the American options if exercised by the buyer. To prevent any high losses, option traders should keep a constant watch on changing volatility and remain prepared with pre-determined stop-loss levels. Model-based valuation should be complemented by realistic and pre-determined stop-loss levels. Intermittent remedial alternatives also include being prepared for averaging techniques (dollar-cost and value), as per the situation and strategies. (See also: The Black-Scholes Option Valuation Model).

Stock prices never show lognormal returns, as assumed by Black-Scholes. Real world distributions are skewed. This discrepancy leads to the Black-Scholes model substantially underpricing or overpricing an option. Traders unfamiliar with such implications may end up buying overpriced or shorting underpriced options, thereby exposing themselves to loss if they blindly follow the Black-Scholes model. As a preventive measure, traders should keep an eye on volatility changes and market developments—attempt to buy when volatility is in lower range (for instance, as observed over the past duration of the intended option holding period) and sell when it is in the high range to get maximum option premium.  

An additional implication of geometric Brownian motion is that volatility should remain constant during option duration. (See also: Monte Carlo Simulation With GBM). It also implies that moneyness of option should not impact implied volatility, i.e. ITM, ATM and OTM options should display similar volatility behavior. But in reality, the volatility skew curve is observed (instead of the volatility smile curve) where higher implied volatility is perceived for lower strike prices. Black-Scholes overprices ATM options, and underprices deep ITM and deep OTM options. That is why most trading (and hence highest open interest) is observed for ATM options, rather than for ITM and OTM. Short sellers get maximum time decay value for ATM options (leading to the highest option premium), compared with that for ITM and OTM options, which they attempt to capitalize on. Traders should be cautious and avoid buying OTM and ITM options with high time decay values (part of option premium = intrinsic value + time decay value).

In a nutshell, price movements are assumed with absolute applicability and there is no relation or dependency from other market developments or segments. For example, the impact of the 2008–09 market crash, attributed to the housing bubble bust leading to an overall market collapse, cannot be accounted for in the Black-Scholes model (and possibly cannot be accounted for in any mathematical model). But it did lead to low-probability extreme events of high declines in stock prices, causing massive losses for option traders. The forex and interest rate markets did follow the expected price patterns during that crisis period but could not remain shielded from the impact all across.

The Black-Scholes model does not account for changes due to dividends paid on stocks. Assuming all other factors remaining the same, a stock with a price of $100 and a dividend of $5 will come down to $95 on dividend ex-date. Option sellers utilize such opportunities to short call options/long put options just prior to the ex-date and square-off the positions on the ex-date, resulting in profits. Traders following Black-Scholes pricing should be aware of such implications and use alternative models such as Binomial pricing that can account for changes in payoff due to dividend payment. Otherwise, the Black-Scholes model should only be used for trading European non-dividend-paying stocks.  

The Black-Scholes model does not account for the early exercise of American options. In reality, few options (such as long put positions) do qualify for early exercises, based on market conditions. Traders should avoid using Black-Scholes for American options or look at alternatives such as the Binomial pricing model. (See also: How to Build Valuation Models Like Black-Scholes?).

• It fits very well for very popular delta hedging strategy on European options for non-dividend-paying stocks.
• It is simple and provides a readymade value.
• Overall, when the entire (or a majority of the) market is following it, prices tend to get calibrated to the ones computed from Black-Scholes.

Blindly following any mathematical or quantitative trading model leads to uncontrolled risk exposure. Financial failures of 2008–09 are attributed to the flawed use of trading models. Despite the challenges, model usage is here to stay thanks to the constantly evolving markets, with a variety of instruments and the entry of new participants. Models will continue to be the primary basis for trading, especially for complex instruments such as derivatives. A cautious approach with clear insights about the limitations of a model, their repercussions, available alternatives and remedial actions can lead to safe and profitable trading.