Of course, this makes no sense. Volatility changes constantly, ands this points out the problem with any model. It must assume that volatility remains unchanged for the math to work out. Even so, the assumption is so flawed that it makes the model unreliable. Making matter worse, the BSM has many other flaws as well. In math, one flaw is bad enough; but when you face at least 8 flaws (as BSM dies), it means there is zero reliability. [1]
But it gets worse.
Volatility is unpredictable. Even with the high number of flaws that create unaccounted for variables, the volatility problem is more severe. With fixed volatility, the degree of standard deviation is predictable, but this is not how things work in the real world. Price movement is chaotic, meaning that volatility is also chaotic and unpredictable. High and low volatility occur when price behavior is narrow and, on the other extreme, when it is broad. But how can this be predicted? It cannot. Volatility never remains unchanged, and it never changes in a predictable manner, or in a straight line.
Volatility does not anticipate direction or degree of price change. Even though the BSM assumes volatility remains unchanged, another problem must be recognized. Even if the degree of today’s volatility remained unchanged, which direction will price take? Will it rise, fall, or remain unchanged? High daily volatility can occur within a range of price, but from day to day exhibit no significant movement. This is a factor never anticipated in BSM or, for that matter, in any pricing model. The flaws about volatility are more complex than the initial assumption that volatility does not change. Beyond direction of price movement, even fixed volatility does not reveal the degree of price movement in the underlying. A 5% move in a stock selling for $30 per share implies a 1½ point change. But if the price per share is $90, the same 1½ point movement is 4½ points. The assumed degree of volatility is not fixed but varies based on the underlying price range.
The timing of price changes also affects volatility. Does the underlying advance and then decline, or does it move in the opposite direction? Are there extended periods of consolidation? Every underlying behaves different, causing great variability in how volatility reacts. The option contract changes based on underlying price movement, and a correlation between the timing of price trends, and the volatility of the option, cannot be overlooked. The time required and the sequence of movement are further affected by moneyness and time to expiration.
Moneyness also affects volatility. The BSM assumption is normally applied to any option, regardless of its proximity to the strike. This is also unrealistic. ATM options have the highest gamma levels, so there are obvious differences between ATM, OTM and ITM contracts. And the greater the distance to strike, the greater the effect on volatility. It cannot be assumed in any situation that the option is ATM and will remain there. If it would, then no pricing model is needed. Nothing moves. But in practice, as an option moves ITM or OTM, gamma will change as well. Volatility changes as the distance grows.
Time to expiration also affects volatility. A shorter-term option is likely to exhibit higher gamma than longer-term options, and the time span affects volatility directly. As expiration approaches, gamma should increase as well (assuming offsetting movement in moneyness does not change the calculation). In applying a price model, it is unrealistic to base assumptions on an option remaining ATM because, as movement occurs (and as expiration nears), the entire matter of gamma changes drastically. As expiration nears, volatility behavior also changes. Even if a trader could know the volatility level near expiration, a pricing model is likely to undervalue an ATM option as volatility rises, and to overvalue the ATM option as volatility declines.
Type of option trade distorts volatility assumptions. Is the trade a long contract or a short contract? Is it a spread or a straddle? The nature and attributes of trades matter and volatility is going to vary based on the trade itself. A related issue is the historic volatility of the underlying. A highly volatile underlying price will directly affect option premium and its implied volatility. When this point is expanded to different types of trades, the overall problem also becomes clearer. Not all trades are the same, so BSM assumptions about volatility are more complex for some trades and combinations, than for others.
Adjusting for stochastic volatility (SV) creates yet another variable. Under the BSM model, volatility is assumed to remain constant. Applying Stochastic volatility, the assumption is added that volatility varies as time passes. This does not make the calculation more reliable; it only adds one more random variable. This may allow for analysis of a range of possible pricing outcomes, but it remains a guess to matter how many variations of the model are used.
The problem remains: No pricing model can accurately forecast option prices in the future. Those few theorists who swear by BSM must ignore the facts, but any model contains flaws and imperfections. The solution is not to develop better methods for calculating volatility, because it is entirely unpredictable. The solution is to identify strategies and risk limiting methods to survive in an uncertain world.
[1] The 8 most serious flaws of BSM are: (1) Volatility remains constant (2) there is no restriction on buying or selling the underlying; (3) no tax consequences apply to profits; (4) interest rates are fixed and available to all; (5) no transaction costs are in effect; (6) trading is continuous without any gaps in price movement; (7) volatility is independent from underlying price; and (8) price changes are normally distributed.
Michael C. Thomsett is a widely published author with over 80 business and investing books, including the best-selling Getting Started in Options, coming out in its 10th edition later this year. He also wrote the recently released The Mathematics of Options. Thomsett is a frequent speaker at trade shows and blogs on his website at Thomsett Publishing as well as on Seeking Alpha, LinkedIn, Twitter and Facebook.
Original source: https://steadyoptions.com/articles/pricing-models-and-volatility-problems-r598/