The Black Scholes (Merton) Derivative Pricing Model was revolutionary! . . In 1970. In 1997 a Nobel prize in economic science was awarded for Black & Scholes work in derivative pricing. It may be the most widespread used formula in finance, ignoring CAPM (trust me, ignore CAPM), perhaps millions of times a day computure programs and human traders alike price an option with Black-Scholes; why?

Let's just take a peak at the formula, I'll focus on call options as the logic is the same for puts, without diving too deep:

Okay...

C(S, t) = This represents the price of the call option, given the current stock price, S, at the current time, t. This formula gives us the options current price given these underlying observable facts.

S = The current stock price.

K = The strike price, the exercise price, the amount for which you are entitled to buy the security for.

t = The current time, normally represented as a year, or a distance from the expiration date of the option, T.

T = The expiry time of the option, represented as a fraction of years. A one year option would have a big T of 1 or 365 current days until the expiry, divided by 365 days.

r = The risk free rate for the relative time period. Normally the yield/rate on a risk free US government bond or treasury bill.

Sigma/standard deviation = The volatility of the stock. Normally calculated as LN(Day(n) price/Day(n-1) price)*Square root of 252, the number of trading days in a year, to brinf us to an annualized volatility number.

d1, d2, N(d1), N(d2) = The normalized probabilities, under a standard normal distribution, N is just the notation to say that we are calculating the probability under normal distribution.

d2 = d2 is the probability that the option will expire in the money i.e. spot above strike for a call. N(D2) gives the expected value (i.e. probability adjusted value) of having to pay out the strike price for a call.

d1 = d1 is a conditional probability. A gain for the call buyer occurs on two factors occurring at maturity. One, the spot has to be above strike price. (Direction). Two, the difference between spot and strike prices at maturity (Quantum). Imagine, a call at strike price $100. If the spot price of the stock is $101 or $150, the first condition is satisfied. The second condition is about whether the gain is $1 or $50. The term D1 combines these two into a conditional probability that if the spot at maturity is above, what will be its expected value in relation to current spot price.

If you can get past all that mumbo jumbo here's a few problems with Black-Scholes:

(i) assuming constant volatility (it does not reflect the volatility smile prevalent in options since the 1987 crash)

a) throughout strike prices, and;

b) throughout the life of the option

(ii) it assumes asset prices follow a random walk;

This random walk is something like a Geometric Brownian Motion. If that sounds like nonsense to you it's because it is. Forget I even mentioned it and tell anyone that thinks GBM applies to the real world to go study for their actuary exams cause that's the only place they'll encounter it.

Now, if I hit refresh a few times I can get the prices to match pretty well. Sure, I can wait for the 1/1,000,000.00 times that the random walk of the market equals history. Or, we can agree that stock prices are not random. They exhibit a positive skew (stocks go up, in the long run) and are very much reliant on momentum, positive momentum, when they go down its not for long and when they go up its for years at a time.

(iii) it assumes that stock moves are normally distributed;

What should be glaringly obvious from this, what should really jump out and smack you in the face, is that the actual stock market returns, the red lines, are far, far, FAR, greater than what a standard normal random variable would predict. The below graph shows that the probability of seeing the actual daily returns of the stock market is less than 50%. It exhibits a fat left tail. Extreme losses are far more probable that extreme gains.

So, investors realize that a black swan event, a COVID-19, a financial crisis, a 9/11, an X, Y, Z disaster is far more likely than the president hopping on live TV and declaring that it is illegal for stocks to go down, capital gains taxes are 0%, corporate tax is now eliminated and wall street bets are now running the SEC - anything that would represent a right tail event.

Clearly this probability of extreme losses can be exploited, right?

Here's a probability density histogram of the stock market vs a standard normal simulation of returns.

Okay, let's just agree that the stock market isn't random and let me get to the point, after I break down the remaining bogus assumptions in the model.

(iv) it assumes that interest rates (risk free interest rate) are constant;

Interest rates change - who would have thought. Apparently not two nobel prize winners.

(v) the asset (i.e. stock) does not pay any dividends;

Neat.

(vi) it assumes no commissions and transactions costs; and

Seems a bit on the high end but who am I to argue with the all knowing google.

(vii) it assumes markets are perfectly liquid.

Here's the market for a January 20th 2023 Put option I bought on a 3x S&P 500 fund. Ah yes, a 16.67%-20.00% spread between the bid and ask.

Much liquid. Very efficient. Fully perfect.

mArKeTs ArE pErFeCtLy LiQuId.

Let's tie this off with the main point:

https://www.thestreet.com/investing/options/vxx-options-are-different-11211872

"As anyone who still remembers their introductory statistics course knows that the standard deviation is calculated by summing the squares of the price then taking the square root. But the problem is that summing the squares of a random variable gives you a Chi Squared distribution and not a normal one. So the fundamental assumption in the Black Scholes model is destroyed. Essentially the distribution is the square root of a Chi Squared distribution."

So can we build a better option pricing model?

I'll put up a better option pricing model in my next post.

Sources:

Some stuff I copied from money show https://www.moneyshow.com/articles/optionsidea-42955/

Both articles refer to the original formulaâ€™s assumptions which contain many flaws. The initial assumptions of the formula were:

Exercise. Options will be exercised in the European model, meaning no early exercise is possible. In fact, US 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.

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 many stocks do pay dividends.)

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.

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.

Transaction costs. No transaction costs apply to options trades. This is another feature affecting net value, since itâ€™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 reduced as the number of contracts traded rises. The model just ignored the entire question, but every trader knows that commissions can turn a marginally profitable trade into a net loss.

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 ratesâ€”risk-free or notâ€”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.

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.

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.

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 optionâ€™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).

Is it heresy to say that traders donâ€™t care about the optionâ€™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.

By Michael Thomsett of ThomsettOptions.com

Some other implications, that I copied and pasted from Investopedia https://www.investopedia.com/articles/active-trading/041015/how-circumvent-limitations-blackscholes-model.asp that don't hold up:

Assumes no early exercise (e.g., fits only European options). That makes the model 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.

Quick price changes blow up Black-Scholes. 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 that are expected and implied by the Black-Scholes model.

Additionally:

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 developmentsâ€”attempt to buy when volatility is in the 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. It also implies that moneyness of option should not impact implied volatility, for example, that 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). Similarly, educated traders sell ATM options to get higher premiums when volatility is high, buyer should look for purchasing options when volatility is low, leading to low premiums to be paid.

Extreme Events

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 2008â€“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.

Regarding Dividends

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.

Published September 15, 2021

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