Black-Scholes model assumes a normal price distribution

Posted by isg123 on May 30, 2011 (05:39AM)

Hello Everyone and Thanks for all your help!
I recently read the Big Short. In it, a market participant claims that the Black-Scholes model assumes a normal price distribution for underlying stocks and therefore would not be accurate for special situation stocks or very unpopular trades such as buying an SPY Dec 13 900 Put.

If this is true how can I get a feel for potential profit,assuming my thesis is correct and SPY drops to 500 before Dec 13?  (TK's P&L Calculator is based Black_Scholes-based.)

Posted by sublimevotum on May 30, 2011 (01:15PM)

I love option pricing models!  I know, I'm a nerd.  Anyways...BK does assume that option prices are normally distributed.  However, stock return distributions have tended to have "fat" tails.  This means that the probability of "extreme" stock price movements (including either huge price increases or declines) are higher than they would be if they were normally distributed. 

What does this mean for you?  As an example, let's assume that you are right in saying that the S&P goes to 500 by the end of the year.  You will make a ton of money by buying put options.  However, you will make slightly less money than you would make if the market assumed that stock returns were normally distributed.  This becomes more and more relevant the further out of the money you go.

There are option pricing models that take into account the "fat" tails that actual stock return distributions seem to exhibit. 

Does this answer your question?

Posted by OldFart on May 30, 2011 (03:01PM)

sublime - we are looking for geeks and ners to chime in on the Alpha thread just below -

isq - the reality is that the BS assumes lognormal distribution of prices, not normal. The main difference is that price can not drop < 0. The TK P&L calculator is good enough if I understand correctly what u are trying to do. Keep in mind the main shortcoming of any option calculator - you will have to guess what the volatility will be at that time. Today VIX is around 17, if u r right and SPX drops to 500 - 50, 80, 100, .... nobody knows

Posted by spshapiro on May 30, 2011 (04:50PM)

With the S&P @ 1335, we are talking about a very low probability event.  I am not saying impossible, but one of such unlikelihood as to be not meaningful when doing normal calculations of likely scenarios.  I would sooner be looking to another string of Japanese ‘style’ events, than the S&P falling 63% in 2 1/2 years. 

I, for one, would focus on realistic alternatives, and adjust my view when the S & P reached, say 1000. In the other thread you ask under the scenario...could the option expire worthless; it would be worth your while to ask, under normal circumstances, would it be likely that the option would expire worthless.

Posted by Minion on May 30, 2011 (08:14PM)

Much of the time, the market makers compensate for those fat tails by huge bid / ask spread during large moves, or shifting the equilibrium point based on the actual demand.  Stocks that have a history of going up typically have much higher call premiums than ones which have been flat, even though the textbook might give a similar price.

However, there is still an edge most people don't know about - the wave nature of price movements, which is based on waves of group sentiment.  Since most participants are enclosed by the boundary of human nature, they get suckered into believing what everyone else does, bwahahahaha!

That's when the algobots, system traders, and trend followers devour the opportunities they create.

Posted by sublimevotum on May 31, 2011 (12:33AM)

sphapiro makes some good points.  This specific event seems highly unlikely.   I'm curious, why do you think the S&P will go to 500?

Posted by optionsguy on May 31, 2011 (11:57AM)

The below post under my comment is correct here. The Profit and Loss calculator assumes a log normal distribution, accounting for the fact that a stock's price could go up to infinity (in theory) on the upside, but can only go to zero on the downside. Also, it is important to note that the Black/Scholes that is used is a modified model - meaning it will account for possible Early Exercise of an option.

FYI - there are binomial models available if you click the "tools" graphic inside the graph. The issue here is that the binomial models calculate much slower than the B/S model does, but you might want to look at both models in this situation.

Lastly, if you are trading just one option you can attempt to forecast a price for your put in the Option Pricing Calculator tool. This is what I would use if I was doing a large move in the SPX, in a limited time frame, and wanted to adjust the volatility to a specific level.

Brian Overby (OG)

isq - the reality is that the BS assumes lognormal distribution of prices, not normal. The main difference is that price can not drop < 0. The TK P&L calculator is good enough if I understand correctly what u are trying to do. Keep in mind the main shortcoming of any option calculator - you will have to guess what the volatility will be at that time. Today VIX is around 17, if u r right and SPX drops to 500 - 50, 80, 100, .... nobody knows


Posted by sublimevotum on May 31, 2011 (02:52PM)

I’m looking at page 113 through 114 of The Big Short.  Here, the author is discussing how Capital One had problems with regulators, and how its stock would “either collapse to zero or jump to $60.”  The author goes on to say that the market was pricing long-term options on Capital One using the Black-Scholes model.  The assumptions of this model were wrong because “When Capital One stock moves, as it surely would, it was more likely to move by a lot than by a little.”  In this example, the market is assuming that the stock’s return distribution is normal, or it could more correctly be assumed to be lognormal (as others have pointed out).  A certain investor “couldn’t believe people would sell us these long-term options so cheaply.”  The reason these options were so cheap was because the market, at this time, was pricing these options assuming a lognormal distribution.  This distribution was not appropriate, according to the author, because relatively extreme price movements were more likely than this distribution implies.  Again, the tails of actual stock price returns are “fatter” than are the tails of the lognormal distribution.


Is this the part of the book that you are referring to?  Also, perhaps I could examine how "fat" the tails are for the expected stock return distribution going forward.  Again, if the tails aren't very "fat" and the market is expecting a lognormal distribution going forward, then your trade would be more profitable than it would be if the market is currently pricing in more extreme price movements than is implied by a lognormal distribution.

Posted by isg123 on May 31, 2011 (04:38PM)


This is the part of the book I was referring to.

We are in a secular bear. Research the bottom P/Es and dividend yields of the bear markets of 1964-81 and1929-49. We have not even come near them yet.

The fact that sphapiro and everyone else calls it highly unlikely makes it even more likely. Actually I hope that more people respond like he does. Then I will be more comfortable  knowing that the clever virus (the stock market) will mutate to hurt the maximum # of investors, as it always has and always will and that I am indeed of a tiny minority. Does this situation sound familiar? Have you read the various book like Lowenstein's, etc.

We will have high interest rates and more moves up later in the decade.

 The dollar has probably ended its decline, the Long Bond has started to move up. Maybe the market has already peaked, I can't be sure. Insiders are BIG sellers, commodity cos. are down. GDX hasn't followed its 2.5 X the price of gold and is telling us to expect deflation for the next~2yrs.

Could you please examine how "fat" the tails are for the expected stock return distribution going forward.

Thanx to all and remember it's not different this time.

Posted by optionsguy on June 02, 2011 (10:57AM)

This is the concept also discussed in,

The Black Swan: The Impact of the Highly Improbable
by Nassim Nicholas Taleb

Many say the actual price movement of a stock is fatter is the middle and in the tails than a log normal distribution implies. ie It either goes nowhere or makes a big move. Which makes sense if you think about most stocks (not the ones that are talked about each day on TV), the stock sits until there is a news item and then the market "overreacts" to the news and it causes a big swing.

Brian (Og)

Posted by sublimevotum on June 02, 2011 (07:38PM)

Haven't had a chance to read The Black Swan yet; I'm hoping I'll get a chance sometime soon.

Posted by sublimevotum on June 03, 2011 (11:32PM)

BTW The “fatness” of a distribution’s tail is technically defined as kurtosis. 

You can gain an intuitive idea of the kurtosis implied by current option prices by simply viewing the implied volatility for options with different strike prices.  Part of the reason that OTM options have higher IV than ATM options is because the market has priced in kurtosis.  Like oldfart and Brian suggested, playing around with the Options Pricing or P&L calculators may also give you some insight.  For example, consider what happens to the price of an OTM option if you calculate its price using the IV for an ATM option. 

I knew it wouldn’t be easy to calculate the kurtosis implied by current option prices.  But as I have thought about it I’ve realized it would require much more work than I had anticipated.  I’m sorry-I doubt I’ll have time to do it anytime soon.

If I do somehow find the time, my plan is to carry out the following ad-hoc analysis:  1. Make a reasonable assumption on the “true” volatility of the S&P 500 going forward.  This would likely be based on past values of volatility.  2. Calculate the Black-Scholes (or another similar model) value for an option(s) and compare that to its actual/market value.  3. Assume that the difference in these values is caused entirely by kurtosis (ignoring other factors such as skew makes this analysis easier but less reliable). 4. Calculate the option(s) kurtosis using the value found in step three and certain equations developed in one of several papers I have considered using.  5. Compare the kurtosis to the kurtosis seen in past actual returns to determine if the market is expecting relatively high or low values for kurtosis going forward.  Again, this analysis would be somewhat ad-hoc; there is no “standard” approach to carrying out such an analysis as far as I know.  Does anyone have any thoughts or suggestions?

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