Cathy O’Neil: Black Scholes and the normal distribution

By Cathy O’Neil, a data scientist. Cross posted from mathbabe

There have been lots of comments and confusion, especially in this post, over what people in finance do or do not assume about how the markets work. I wanted to dispel some myths (at the risk of creating more).

First, there’s a big difference between quantitative trading and quantitative risk. And there may be a bunch of other categories that also exist, but I’ve only worked in those two arenas.

Markets are not efficient

In quantitative trading, nobody really thinks that “markets are efficient.” That’s kind of ridiculous, since then what would be the point of trying to make money through trading? We essentially make money because they aren’t. But of course that’s not to say they are entirely inefficient. Some approaches to removing inefficiency, and some markets, are easier than others. There can be entire markets that are so old and well-combed-over that the inefficiencies (that people have thought of) have been more or less removed and so, to make money, you have to be more thoughtful. A better way to say this is that the inefficiencies that are left are smaller than the transaction costs that would be required to remove them.

It’s not clear where “removing inefficiency” ends and where a different kind of trading begins, by the way. In some sense all algorithmic trades that work for any amount of time can be thought of as removing inefficiency, but then it becomes a useless concept.

Also, you can see from the above that traders have a vested interest to introduce new kinds of markets to the system, because new markets have new inefficiencies that can be picked off.

This kind of trading is very specific to a certain kind of time horizon as well. Traders and their algorithms typically want to make money in the average year. If there’s an inefficiency with a time horizon of 30 years it may still exist but few people are patient enough for it (I should add that we also probably don’t have good enough evidence that they’d work, considering how quickly the markets change). Indeed the average quant shop is going in the opposite direction, of high speed trading, for that very reason, to find the time horizon at which there are still obvious inefficiencies.


A long long time ago, before Black Monday in 1987, people didn’t know how to price options. Then Black-Scholes came out and traders started using the Black-Scholes (BS) formula and it worked pretty well, until Black Monday came along and people suddenly realized the assumptions in BS were ridiculous. Ever since then people have adjusted the BS formula. Everyone.

There are lots of ways to think about how to adjust the formula, but a very common one is through the volatility smile. This allows us to remove the BS assumption of constant volatility (of the underlying stock) and replace it with whatever inferred volatility is actually traded on in the market for that strike price and that maturity. As this commenter mentioned, the BS formula is still used here as a convenient reference to do this calculation. If you extend your consideration to any maturity and any strike price (for the same underlying stock or thingy) then you get a volatility surface by the same reasoning.

Two things to mention. First, you can think of the volatility smile/ surface as adjusting the assumption of constant volatility, but you can also ascribe to it an adjustment of the assumption of a normal distribution of the underlying stock. There’s really no way to extricate those two assumptions, but you can convince yourself of this by a thought experiment: if the volatility stays fixed but the presumed shape of the distribution of the stocks gets fatter-tailed, for example, then option prices (for options that are far from the current price) will change, which will in turn change the implied volatility according to the market (i.e. the smile will deepen). In other words, the smile adjusts for more than one assumption.

The other thing to mention: although we’ve done a relatively good job adjusting to market reality when pricing an option, when we apply our current risk measures like Value-at-Risk (VaR) to options, we still assume a normal distribution of risk factors (one of the risk factors, if we were pricing options, would be the implied volatility). So in other words, we might have a pretty good view of current prices, but it’s not at all clear we know how to make reasonable scenarios of future pricing shifts.

Ultimately, this assumption of normal distributions of risk factors in calculating VaR is actually pretty important in terms of our view of systemic risks. We do it out of computational convenience, by the way. That and because when we use fatter-tailed assumptions, people don’t like the answer.

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About David Dayen

David is a contributing writer to He has been writing about politics since 2004. He spent three years writing for the FireDogLake News Desk; he’s also written for The New Republic, The American Prospect, The Guardian (UK), The Huffington Post, The Washington Monthly, Alternet, Democracy Journal and Pacific Standard, as well as multiple well-trafficked progressive blogs and websites. His has been a guest on MSNBC, CNN, Aljazeera, Russia Today, NPR, Pacifica Radio and Air America Radio. He has contributed to two anthology books, one about the Wisconsin labor uprising and another on the fight against the Stop Online Piracy Act in Congress. Prior to writing about politics he worked for two decades as a television producer and editor. You can follow him on Twitter at @ddayen.


    1. jake chase

      I have been trading options for thirty-three years and have never paid the slightest attention to BS. In 1981 I attended a one day seminar sponsored by the CBOE (I think) which was trying to encourage traders to become what it called “market makers” by teaching them to count up BS deltas (buy 100 of this, sell 300 of that, etc.) and crib a small profit despite eggregious commissions then being charged by any broker that offered options accounts. The guy who had taught me everything I knew about options had made himself $4 million in two years and lost all of it in one month, trading options on a single stock, ASA. I had made a whole lot less and still had enough of it left to think it might be possible to make a business out of this, at a time when I had begun to believe lawyering simply didn’t pay.

      Perhaps these market makers were destined not to pay commissions, I didn’t stay for the coffee and donuts. After twenty minutes, the only thing I had written in my notebook was, ‘this is utter bullshit.’ The very attractive woman next to me was writing down equations as fast as she could. I hope she is still solvent. BS seemed to me then an excellent way to go broke.

      The thing about volatility is, it changes minute to minute and day to day. Yesterday’s volatility is no more useful than last month’s weather forecast.

    2. H. Alexander Ivey

      BS is the best description of the finance industry. What is being discussed by Cathy, inefficiencies, prices, stock, thingys, are not “real” manufactured goods or non-FIRE services. They are pure money – fiat currency or credit / debt – that no longer are used by or available to the real economy. In a real sense the models are for gambling. There is no other end purpose than to gamble – to get a high from “winning” or from beating the other guy, not to do some socially useful activity. So BS, which means to lie, not for the primary purpose of deceiving for financial gain but for the primary purpose to make the BSer appear more important or knowledgeable, is perfect for Black-Scholes.

  1. Keiran

    The better shops don’t use normal distributions in their VaR calculation. It’s not hard to fit a curve to the observed distribution of returns for an instrument then sample from that to generate possible future market scenarios. Furthermore, if you’re repricing your instruments using a modern pricing engine (eg stochastic volatility for options) then you get the fat tailed prices at your fat tailed scenarios, which is a reasonable VaR model.

    The bigger issue is the lookback period for VaR calculations. In 2008, nobody was including historical data which had volatility anything like what was observed, so the VaR was understated. Smart practitioners were aware of this limitation of the modelling, less smart commentators sounded, well less smart.

    1. Ben Johannson

      Of course it it isn’t hard to project future scenarios from current data. Scenarios are products of imagination; anyone can do that. What’s hard is knowing which scenario will be most/least accurate before we get to that point in time.

    2. Ruben

      Some years ago when he was young idealist and capitalist a friend of mine worked on modeling monthly returns of the S&P 500 index from the end of the 19th century to the present. Underlying the normal distribution model of the returns is the assumption that the time series of returns is a random walk.

      My friend found that instead of a normal distribution the returns (both capital and total) were composed of a mixture of normal distributions. Thus returns walked a random path interspersed by regime shifts to another random path.

      He went on to study regime shift. This part of the modeling was less complete though, but he published a paper in a peer reviewed journal.

      Nowadays my friend is still thinking about it and is coming around the idea that the tipping point for regime shift can be predicted by calculating the characteristic return time of the path to the current equilibrium after small disturbances: the slower the return from the disturbance to the current equilibrium, the closer to the tipping point, to the regime shift to a new equilibrium.

      My friend still engages me in conversation now and then and tells me that apart from being dishonest and carry out insiders jobs, the crux to being successful in finance is not to master the contortions of volatility but to stay with the crowd in the middle in normal times and being able to anticipate regime shift but some method such as the one outlined above.

  2. vlade

    The main problem with BS, VaR etc. is that it gives nice and simple answer in one number.

    A good practicioner understand that there’s lots of information compressed into that one number (hidden in assumptions) and compensates for it (usally based on heuristcs learned the hard way).

    The average manager in my experience doesn’t. Just trying to explain to someone that 99% VaR is NOT the number you expect to lose, it’s the +/- MINIMUM amount amount you expect to lose (but can lose much more, unquantifiably so) tends to leave them flabbergasted.

    As the worlds is becoming more complex, we seek more simple answers hiding the complexity – without understaning what it is we lose.

  3. larry

    Taleb considers VaR to be an instance of charlatinism.

    To fudge a false formula is ok if you realize that that is what you are doing, but not a good idea if you think that the only problem is that the formula needs to be altered somewhat but that the distribution on which it is based is grounded in reality. You can then easily fool yourself about what your fudge is accomplishing.

  4. Philip Pilkington

    The EMH is a tautology. It says that the “average investor” canot beat the market consistently. But what is “the market”? Well, its the outcome of the interactions of the investors therein. That means that it’s just the result of the “average” of investor decisions.

    So, what the EMH says is that the average investor cannot beat the market. But since “the market” is basically just the outcome of the average of investment decisions what the “theory” really says is that the market cannot beat the market. Or, that the average investor cannot beat the average investor.

    It’s a tautology and its meaningless. I gave a presentation on this the other day. People find itvery amusing when you work through this and point it out. The language of the EMH is also meaningless when you see it for what it is.

    1. craazyman

      be carefulL that means they can’t lose consistently either, since probability is a two-sided mirror. but lots of them do! something weird is going on here that doesn’t fit on a Cartesian graph. that’s one reason I think Descartes was just making sh*t up in his head.

      1. Bill Smith

        Gol Dangit. Did a google search on “naked” lookin fer some porn sites and Naked Capitalism popped up. Said to myself, “Whoa, Berlusconi gotta website. Betcha its even free!”.

        Then I got here and the pics weren’t quite what I expected, and I look around and see it’s some sort of econ site. Like I don’t already know econ. The Art of Boiling Frogs – Slowly. That’s ’bout all you need to know.

        But then I thought, maybe they do finance here too? I wouldn’t mind getting rich. And the quicker the better. I figure, Hell’s Bells, we gotta new Pope. What could go wrong? So bring it on, I’m all geared up for some “Big Money” talk. Some red wine and crackers, 2 beers and maybe 3,4, or 6 bong hits should do it and I’ll be ready to pull the trigger and short volitiliy all the way down to zero!

    2. skippy

      See Criticism

      VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 1997 debate between Nassim Taleb and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR:[27]

      Ignored 2,500 years of experience in favor of untested models built by non-traders

      Was charlatanism because it claimed to estimate the risks of rare events, which is impossible

      Gave false confidence

      Would be exploited by traders

      Skippy… in moderation hell…

    3. skippu

      See Criticism

      VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 1997 debate between Nassim Taleb and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR:[27]

      Ignored 2,500 years of experience in favor of untested models built by non-traders

      Was charlatanism because it claimed to estimate the risks of rare events, which is impossible

      Gave false confidence

      Would be exploited by traders

      Skippy… in moderation hell…

      PS. VaR and derivitives = delusion

        1. Lambert Strether

          Actually, no. It’s algorithmic. Since people aren’t brutalizing the commons with ad hominem attacks, there is no need for me to become involved to enforce Yves’s policy here.

    4. Doug Terpstra

      The market is an animal farm where the average investor is sheared for fleece and slaughtered for muppet mutton.

  5. craazyman

    well Cathy you sound like you know what yer talking about. so when’s best time to buy UVXY for a two-bagger in 5 days. I can’t wait forever to get rich quick and I think something big is coming up maybe this year or next, or the one after. soon, for sure. i don’t like black sholes cause that’s where ships wreck in storms at night. I can’t afford to lose even $1. If somebody can make $5 million in a year in a chair doing math on a computer. I’d say that’s efficient. Why theorize when reality is right in front of you?

    1. skippy

      See Criticism

      VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 1997 debate between Nassim Taleb and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR:[27]

      Ignored 2,500 years of experience in favor of untested models built by non-traders

      Was charlatanism because it claimed to estimate the risks of rare events, which is impossible

      Gave false confidence

      Would be exploited by traders

      Skippy… in moderation hell…

      PS VaR and derivitives = delusion

      1. craazyman

        It’s an insanely efficient way of getting people to hand over their money.

        It’s amazing. Even simple stuff like standard deviation, correlation coefficient, R-squared, tracking error — people walk around and say these like mantras and have no clue what they even mean — mathematically speaking.

        It’s nothing but 8th grade algebra. It’s ludicrously simple. And ludicrously swallowed with blind gullibility. folks will see a pitchbook with charts and graphs and stnd devs and tracking errors and correlation grids and their heads nod like a potato on a spring and out comes the money, haha hahahahahahahahah

        VAR is great, it’s a lot more efficient ’cause it only digs deeper in the pocket with less charts.

        Who said markets aren’t efficient? Today’s markets are the single most efficient wealth transfer process in history.

  6. TomDor

    Where is the real market in all this? – Just looks like a way to skim off money to no good use….gone into someones pocket never to be spent back into the real market.
    Is this just a self-reinforcing loop? some computations introduce inefficiencies so that the same computations can remove the cash?

  7. Kurt Sperry

    QA seems to have a lot of the same underpinnings as engineering. Much of the jargon is similar–curve fitting, deltas, calculus terminology and on and on… Even the methodologies are often similar–the construction of mathematical models to try to make some useful sense of messy reality and then the near inevitable confusion of model outputs with that reality because the inputs aren’t definitive or contain built in assumptions to make the models run. But just as models of even well understood Newtonian processes like fluid dynamics break down when real world complexities like turbulence and vorticities are introduced, how can models with variables and black box inputs dependent on human behavior and psychology have any robustly reliable predictive utility?

    Now I’m sure economic models are useful as thought exercises for looking at fundamentals and probably have some macro descriptive and even predictive value but surely beyond that all this mathematical legerdemain is just obfuscatory cover for looting the system. How does the application of these quantitative models serve society in any useful way? Are there altruistically motivated or socially conscious analyses being done or is it always motivated by simple greed?

    At least engineers can often rightfully claim to be working in the broad public interest.

    1. dutch

      The physical quantities modelled by physics and engineering exist independent of the observer. Those modelled by economists exist only in the mind of the observer. Economics, if a science at all, is a behavioral science, and mathematical models of human behavior are simply worthless.

  8. RR

    “That and because when we use fatter-tailed assumptions, people don’t like the answer.”

    That, right there, is the problem. People really want to believe that it’s possible to consistently extract a profit without taking a significant risk. People really are not that much interested in truth.

    Why would pay some analyst to tell them that they can’t get rich quick without taking the risk of a ruinous outcome?

  9. RR

    Use of any market-implied factors in a widely-used pricing model creates a positive (self-reinforcing) feedback loop that at worst results in a market bubble.

    Market activity that is based on a pricing model that is based on price-implied factors results in a market, where prices are based on prices, i.e. a bubble that is totally detached from reality.

    This was a factor in the CDO calamity. Pricing was mainly based on implied credit risk, but the market prices were in fact made by the CDOs themselves in a neat little feedback loop, not very different from Winnie-the-Pooh following his own tracks.

  10. Hugh

    Look at this way, if government backstops and Helicopter Ben’s endless iterations of ZIRP and QE went away, all the current BS and VaR would mean absolutely nothing. If you just look at how a market behaves and not what underpins it, you’re just telling yourself stories to make yourself feel better about flying blind.

  11. steelhead23

    I readily admit that partial differential equations leave me gasping for air, but there is something missing here – a kind of superset, subset kind of thingy. It is the same something that was missing from a lot of bank’s risk models in 2007 – correlation. This may sound stupid or ignorant, but I posit that the market clearing price of everything is related to the market clearing price of everything else. Hence, if the price of gas rises, the price of homes 40 miles from the city, decline – and default risk increases. Now, some correlations are quite strong and others vanishingly weak, but it is unwise to ignore black swans. Hence, I would suggest that to the extent that BS is mathematically correct, it is correct only for a stable or only mildly variable market and larger economy. Also, even accepting the concept of variable price volatility, it is unlikely that quants would evaluate never before seen levels of volatility (or may merely project the tails of the smile – but would that be an accurate estimate?) and thus I would argue that the BS price, even assuming variable volatility, is likely a tad low – the risks are at least a little larger than your estimate. And no, I cannot imagine mathematically proving what I just wrote. Its all your’n Mathbabe.

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