Risk = ??

[Bylo Selhi]

Correct, but that's what was used as the "risk free" baseline in the data.

NormR must have got up this morning with a black swancloud hovering over his head

[CROCKD]

Stock market fraud — Why Canada doesn’t scare fraudsters

Douglas Cumming and Sofia Johan, seasoned academic researchers at York University and Holland’s University of Tilburg, respectively, undertook the first effort to compare “fraud risk” among the leading exchanges in Canada, the U.S. and Britain.

[CROCKD]

BEHAVIOURAL FINANCE
Balancing act: Why we misjudge risks

[ghariton]

Here's my attempt at defining risk, inspired by the discussion around the entry for risk on finiki.

Consider an asset, or a portfolio of assets, and a time horizon (one month, one year, one decade, whatever). The risk associated with the asset is wholly captured by the probability distribution of the value of the asset at the end of the time horizon.

Now probability comes in two broad flavours, objective (or frequentist) and subjective (or Bayesian). Objective probability can be calculated by repeated trials or experiments, e.g. flipping a coin repeatedly and observing the proportion of heads. The more flips, the more confident were are that we know the probability. (In technical terms, we can place a confidence interval around the estimated probability. As the number of flips grows, the interval becomes narrower.)

The difficult question then is: What constitutes a relevant trial, and where can we carry them out (or observe them)? Unfortunately in financial matters we seldom have identical repetitions of an event of interest. Many substitute whatever experience they can find, e.g. five years' worth of price movements. But then one has to make the heroic assumption that the next period will be the same as the previous period, at least as regards the variable of interest. Many of us go ahead nevertheless -- at our peril, as we have discovered.

And in many situations, we have so little relevant historical information that the only honest answer is: We don't know.

Subjective probability is usually interpreted as degrees of belief, i.e. how strongly do I believe that a particular outcome will happen. Formation of such beliefs starts with a prior distribution, i.e. beliefs based on instinct, experiance, and gut feel. This prior distribution is then adjusted in light of whatever evidence I have to hand (using Bayes' theorm, hence the label of Bayesian probabilities). The result, called a posterior distribution, is then the probability distribution that defines the risk.

Note a subjective distribution always exists. In the worst case, when I really have no idea, the practice is to attach equal probabilities to all outcomes (if I know that is wrong, then I have some idea after all). My ignorance is reflected by the very wide spread of my posterior distribution. Note also that the distinction between "risk" and "uncertainty", made by some, disappears: Uncertainty is merely a particularly spread out posterior distribution.

Whichever path I choose, I end up with a probability distribution for the value of my asset at the time in question. That tells me the risk of the asset. Note that it is multidimensional and hence often cannot be used to say that one asset is more risky than another. Rather, we can say that one asset has a certain risk profile, while another asset has a different risk profile. Thus in some ways, the first asset is more risky, and in other ways the second asset is more risky.

But that kind of answer is too nuanced for many. They often want one number that summarizes the risk of an asset, And of course, many candidates have been proposed. There is the variance, usually interpeted as volatility. There is the skewness, which can be made to reflect the fact that most people find that the pain of a loss is bigger, dollar for dollar, than the pleasure of a gain, and that this effect grows as the sums involved get bigger. There is the kurtosis, which warns that the "once-in-a-hundred-years" events can pop up every decade. There are the deciles, which tell us the probability of losing more than X dollars, or of losing more than Y dollars (often subsumed under "probability of loss" models, and kissing cousins of Value at Risk measures).

These single-number measures of risk can be very useful. But they are all partial measures, and address different aspects. The only comprehensive measure is the entire probability distribution.

That's where I am now. But I'm still evolving.

George

[deaddog]

Here's my attempt at defining risk, inspired by the discussion around the entry for risk on finiki.

>……………………….<

That's where I am now. But I'm still evolving.

George

Wow that’s an awful lot of words (big ones at that) to say that you don’t know what the future holds and that you can only make an educated guess and that it might be wrong.

[newguy]

There is the kurtosis, which warns that the "once-in-a-hundred-years" events can pop up every decade.

I'm more worried about how much to spend every year. I'm trying to model or predict all income streams and have a 95% or so chance of not running out of money. One thing about Canada and pensions is you can elect to have higher guaranteed payments by delaying them. This means I need a way to predict my various returns and inflation because of the way they work (ie. pension is 1/2 inf adjusted, cpp is 100% adj, stocks are adjusted based on a distribution, bonds are adjusted based on math). I do this all monte carlo style.

So the problem...

Like ghariton says about Bayes and given that todays bond rates* are x% and future bond rates have an average and distribution I can then get a more accurate picture of future stock returns and inflation. I can model this using Box-Muller to come up with probable random changes in bond returns (including serial correlation if I want, ie. a rise in rates in more likely followed by another rise in rates). Then I can use that number to predict(guess) the stock return distribution and inflation either with another box-muller run or bayes' theorem. I can't make it too fine grained but I think it offers more realistic results than any retirement calculator. They all assume that the numbers aren't correlated.

None of this will give me a '87 style collapse and very rarely a '08 style crisis. So how do I add the right amount of kurtosis to a normal distribution generated with a box-muller algo?

newguy

*I'm still wondering if change in rates is a better metric than absolute rates

[LadyGeek]

As evidenced by this 14 page thread, and a 6 page thread in the Bogleheads forum here: Redefining risk, there are indeed many ways to define risk. There are also many ways to measure risk. To keep Risk and return - finiki on track, I used the risk definition from Wikipedia because it's used by authoritative experts.

This follows the same methodology used to develop the articles in the Bogleheads wiki. Use "textbook" definitions by accepted authoritative experts to provide a consistent, time-proven approach. It also removes personal points of view.

I'm a Bogleheads forum member and wiki editor. There was a recent effort to rewrite the wiki's section on risk. You can see some of the Bogleheads wiki influence in finiki, as I helped with this development. To be very clear, this effort was driven by Bogleheads' member Kevin M, who wrote the majority of the content in Risk and return: an introduction - Bogleheads.

A follow-on tutorial at more basic level: Risk and return: application - Bogleheads

My personal preference on risk definition comes from this publication: Mismeasurement of risk in financial planning, Richard K. Fullmer, CFA, October 2009.

Additional definitions are in this Bogleheads forum post: Economic textbooks definition of risk

[Bylo Selhi]

Food for thought from one of my favourite restaurateurs: The Intelligent Investor: Polishing the Dimon Principle - WSJ.com

Forget the "Dimon principle." Investors should follow the Feynman principle. When J.P. Morgan Chase's chief executive, James Dimon, disclosed a $2 billion trading loss during a hastily organized conference call on Thursday, he said: "This trading may not violate the Volcker rule, but it violates the Dimon principle." Mr. Dimon didn't say what the Dimon principle is, and a spokesman for the nation's largest bank by assets didn't respond to requests for comment. The Feynman principle, however, is simple: "You must not fool yourself—and you are the easiest person to fool," as the Nobel Prize-winning physicist Richard Feynman put it...

The moguls of J.P. Morgan, in letting a complex risk run wild and denying any potential for error until it was too late, are a reminder that one of the biggest dangers in finance is self-deception. For investors, the bigger the commitment, the more certain they become that they must have been right to make it—and the harder it becomes to let go...So how can investors avoid deceiving themselves?

First of all, remember that "the riskiest moment is when you are right," as the economist Peter Bernstein was fond of saying...

Look at the results of other people and organizations that have tried something similar to the investment actions you are considering. Unless other people have succeeded at it, there isn't any objective reason to believe that you will...

Monitor yourself for vehemence. If you find yourself tempted to ridicule anyone who tells you are wrong, you probably are wrong. The philosopher Bertrand Russell wisely warned that the less evidence someone has that his ideas are right, "the more vehemently he asserts that there is no doubt whatsoever that he is exactly right."

Finally, try the technique that psychologist Gary Klein calls a "pre-mortem." Gather a group of people whose views you respect. Ask them all to imagine looking back, a year from now, at the investment you just made—and that it has turned out to be a disaster. Have them list all the possible causes of the failure. That may well help you see how it might have been avoided.

Above all, remember that the smarter you are, the more easily you can fool yourself.

[ghariton]

Food for thought from one of my favourite restaurateurs

We were told the Millikan story when I was in third-year physics -- 48 years ago. I learned the wrong lesson from it.

On how experts are seldom better at prediction than chimps, in many, many fields, see Dan Gardner, Risk, and also his book Future Babble, books that I recommend very highly. He is a journalist, but builds on the work of political scientists and psychologists, especially Philip Tetlock and Paul Slovic. The main take-away is: the more confident the expert, the more likely he or she will be wrong. (I will leave climate change to another thread.)


I recently read a book on essays edited by Francis Diebold, entitled The Known, the Unknown, and the Unknowable in Financial Risk Management: Measurement and Theory Advancing Practice. It has a lot of big-name authors, but I found the result very disappointing. If anyone wants my copy, please PM me with a mailing address.

Since risk is so difficult to define, I figured that the next best thing is to see what "risk managers" do. Presumably their actions are intended to reduce risk, so, whatever risk is, it must be what they are trying to reduce. The first book I read on the subject was by Michel Crouhy et al, The Essentials of Risk Management. I gather the book is a classic. Anyhow, it's an easy read, with very little mathematics and lots of narrative. It lays out the usual: market risk, credit risk, liquidity risk, operational risk. Use futures contracts and options to defease (reduce) your risks. Match durations. Delta hedging and all that.

Another book was John C. Hull, Risk Management. I found it a bit simplistic. If anyone wants my copy, send me a PM with a mailing address.

Finally there was Philippe Joriot, Financial Risk Manager Handbook. This one is into the nitty-gritty of how you actually measure risk. I found it better for my purposes than the other two books (although the reviews on Amazon are mixed -- perhaps because of the price). I was interested to find that this is actually the basic textbook for a program in Financial Risk Management, with certification available, upon passing two exams, by something called GARP (Global Association of Risk Professionals). GARP positions its certificationf as a more specialized version of the CFA (specialties available in management of financial risk and energy risk). New to me. Has anyone else ever heard of GARP and whether its certification is actually worth anything?

Bottom line: Risk is multifaceted and cannot be summarized by a single number or simple phrase. If you are going to take risk seriously, you have to work at understanding it in contest.

George

[LadyGeek]

As an engineer, I have a very hard time understanding the many definitions of investing risk. Risk is simple- just take the probability of failure for each component and do some statistics (root sum square, standard deviation, etc.) to come up with the final system risk. However in engineering, past performance must predict future performance or you did something wrong. This may be why the investing model gets a lot more complicated.

The investing risk publication I referred to in my previous post (Mismeasurement of risk in financial planning) just makes sense to me - I don't see how it can be any different.

[ghariton]

As an engineer, I have a very hard time understanding the many definitions of investing risk. Risk is simple- just take the probability of failure for each component and do some statistics (root sum square, standard deviation, etc.) to come up with the final system risk

Suppose that the underlying probability distribution is Cauchy distribution of the kind

f(x) = 1/[pi (1 + x) ^ 2]

How would you interpret root sum square or standard deviation in such a case?

(Tails don't have to be very fat before the population mean becomes meaningless and the population standard deviation explodes.)

However in engineering, past performance must predict future performance or you did something wrong.

Yes. Unfortunately, often the way that you discover you did something wrong is when your bridge collapses or your telephone system fails.

This may be why the investing model gets a lot more complicated.

Maybe. Or it could be that investing involves humans as elements of the model, and humans are notoriously unpredictable.

I don't see how it can be any different.

I take that as a challenge I'll try to come up with some cogent arguments when I have more time.

George

[George$]

As an engineer, I have a very hard time understanding the many definitions of investing risk. Risk is simple- just take the probability of failure for each component and do some statistics (root sum square, standard deviation, etc.) to come up with the final system risk. However in engineering, past performance must predict future performance or you did something wrong. This may be why the investing model gets a lot more complicated.

The investing risk publication I referred to in my previous post (Mismeasurement of risk in financial planning) just makes sense to me - I don't see how it can be any different.

I also cannot find any definition of investing risk as complete or fully solid.

Several references come to mind.

(1) James Montier -
Pseudoscience and finance: the tyranny of numbers and the fallacy of safety

In the world of modern finance, a love of numbers has replaced a desire for critical thinking. As long as something has a number attached to it, then it is taken as gospel truth. Research shows that people are often fooled by the use of pseudoscience. Simply making things sound complex makes people believe them more! Risk managers, analysts and consultants are all guilty of using pseudoscience to promote an illusion of safety. We all need to be on our guard against the artificial deployment of meaningless numbers. Critical thinking and scepticism are the most unrated (and scarce) tools in our world.

or anothere reference from Montier -- Mind Matters --Clear and present danger: the trinity of risk - and a few words

Despite risk appearing to be one of finance’s favourite four letter words, it remains finance’s most misunderstood concept. Risk isn’t a number, it is a concept or a notion. From my perspective, risk equates to what Ben Graham called a “permanent loss of capital”. Three primary (although interrelated) sources of such danger can be identified: valuation risk, business/earnings risk, and balance sheet/financial risk. Rather than running around obsessing on the pseudoscience of risk management, investors should concentrate on understanding the nature of this trinity of risks.

(2) Howard Marks - who I value, his common insights - and practical description of his perceptions.


(3) On the limitations of the sometimes limitations of engineering understanding - I recall hearing that aironautical engineering, in studying the parameters of a bumble bee, - size, weight, wing span, etc in the context of their engineering theory - concluded that on the basis of aeronautical engineering theory it was impossible for bumble bees to fly. ( Years ago I did hear this years ago - but never confirmed it fully.) - The legitimate point being that I think realities may not always fully cover everything one wishes or understands in mathematics or engineering or science.

[Shakespeare]

(root sum square, standard deviation, etc.)

Root sum square assumes zero correlation. But in finance, almost everything correlates on the downside.

Standard deviation assumes a Gaussian distribution. Gaussians are used because they have nice mathematics, but in finance are usually not realistic and underestimate the risk severely.

[Shakespeare]

cornrisk.gif (28.01 KiB) Viewed 885 times

(Read the slide.)

[FinEcon]

(root sum square, standard deviation, etc.)

Root sum square assumes zero correlation. But in finance, almost everything correlates on the downside.

Standard deviation assumes a Gaussian distribution. Gaussians are used because they have nice mathematics, but in finance are usually not realistic and underestimate the risk severely.

root sum square, covariance, standard deviation, Gaussian, Cauchy, serial autocorellation, heteroskedasticity, total derivative, Box Jenkins, etc, etc.

When anything like or even remotely related to anything like the above, applied to investing, amounts to man with a hammer syndrome. For investors, anything beyond basic (discrete) financial math is not necessary and will probably make said investor worse off as he/she focuses time and effort on useless confidence building excercises dressed up an elegant mathematics rather than focusing thought on simple, relevant, yet difficult do pin down variables. Example: rather than performing statistical calcs on financial data for REITs with heavy condo focus in TO, one should think about how dramatic increases in supplied sqft will affect residential rents going forward.

[ghariton]

For investors, anything beyond basic (discrete) financial math is not necessary and will probably make said investor worse off as he/she focuses time and effort on useless confidence building excercises dressed up an elegant mathematics rather than focusing thought on simple, relevant, yet difficult do pin down variables.

To each, his own.

I can think of three reasons why I want to go beyond the basics:

(1) I like to have a toolkit that is larger than just the tools I anticipate using. While almost all the work I do around the house requires just screwdriver, hammer and drill, I like to have a saw too, just in case.

(2) The big boys who are moving the market are using some pretty sophisticated tools. While I don't expect to take them on nose-to-nose, I think that it helps to have at least some idea of what they are up to.

(3) Financial institutions keep inventing new products. While 99% of them are not useful to me, from time to time they come up with one that is useful. Two examples are RRBs and ETFs. But before I invest in something new, I need to understand as much as I can about it.

Example: rather than performing statistical calcs on financial data for REITs with heavy condo focus in TO, one should think about how dramatic increases in supplied sqft will affect residential rents going forward.

Increases in the housing stock is certainly one should look at. But it is not the only factor. See some of patriot1's posts for some of these other factors.

George

[FinEcon]

I appreciate the debate of the subject at hand but in my view you are referencing methodology and a framework for a different set of problems particularly when risk is viewed from the point of view of a single, non institutional, investor.

(1) I like to have a toolkit that is larger than just the tools I anticipate using. While almost all the work I do around the house requires just screwdriver, hammer and drill, I like to have a saw too, just in case.

For work around the house a saw is an appropriate tool, the Mars Rover however, not so much. Complicated, sophisticated, incredible technology but has nothing to do with the problem under consideration.

(2) The big boys who are moving the market are using some pretty sophisticated tools. While I don't expect to take them on nose-to-nose, I think that it helps to have at least some idea of what they are up to.

This has been done to death by every competent investor who has written on the subject (Buffet, Whitman, Marks, etc). You can't focus on what the other guy (in aggregate, Mr. Market) is doing. The investor must examine only deals available to him or her. Engaging in nth level thinking of what the other guys is doing or what the other guy is thinking of doing, etc is a waste of time. The results of some regression analysis or GDP numbers are not going to play a part in the analysis of a successful multi family building or car wash investment.

(3) Financial institutions keep inventing new products. While 99% of them are not useful to me, from time to time they come up with one that is useful. Two examples are RRBs and ETFs. But before I invest in something new, I need to understand as much as I can about it.

I don't believe a person needs knowledge of statistics to evaluate assets, simple arithmetic and a dash of financial math will do. Fancy technique makes sense when you are looking to get a job at a bank and get paid to crank out impressive looking reports which are really just business news articles with a dash of quant makes sense but other than than, the techniques belong in a domain where they are more useful.

[Thorn]

In most investing topics, it is best not to get too caught up in the extensive math and stats literature. The models fall far short of the real-world complexities in the money, fixed-income and equity markets.

For me, "risk" is the probability that I won't achieve the promised or expected returns on one investment or on my entire portfolio. In each case I try to consider how I might manage the risk.

I tend to consider 5 items:
(1) Currency - is there a non-CDN currency component to this investment? If I am unwilling to take chances on fluctuations in the global currency market which could undermine my returns, then I will stay with CDN or CDN-hedged securities.
(2) Inflation - invisible but insidious over investing time horizons - I only consider securities which return a minimum of inflation + taxes
(3) Interest - fixed income securities of longer durations are at greater risk of losing value if future rates rise. Of course, in the current environment it is unlikely this will happen for years as my corporate bonds return 5%+.
(4) Specific - refers to the possiblity that I make a mistake in selecting a security, through my own fault or that of others (a fund manader drops the ball). This suggests that second opinions from financial advisors or knowledgeable friends is very helpful over time, but still imperfect.
(5) Market - refers to malaise in the domestic or global markets, which we have seen for several years now. As a poor defensive retail investor, I have little defence except to hold, expecting a rally sometime in the future, buy bargains as they appear and back out into cash if I become too apprehensive.

Remember that modern portfolio theory states that NO individual investment or portfolio is risk-free. We all used to think that government bonds were risk-free until sovereign debt levels and defaults (e.g. Greece) arose. This also applies to US treasuries, from a country with a debt equal their annual GDP, an entrenched inability to run a budget surplus and no plan in place to address either problem.

I do calculate standard deviation of my total portfolio $ weekly (cash + fixed income + equity) and retain it for consideration on an historical basis, as it reflects the extent of swings in my total $ and their implied impact on income and capital returns.

[Peculiar_Investor]

Tip of the hat to a discussion on the Bogleheads forum, The Five Dimensions of Risk - Rick Ferri. Reading the article has given me a different viewpoint on risk to consider.

As I read back through this topic it is clear that there are many viewpoints on risk, which IMHO provides an opportunity for each investor to conduct a thought experience regarding their own viewpoint of risk and the many views that have been expressed here.

[ghariton]

Eric Falkenstein has an interesting -- and infuriating -- blog, and now he has a fascinating -- and infuriating -- book (self-published as evidenced by the lack of any editing).

For the last eighteen years, ever since his Ph.D. thesis, he has argued that returns are not positively related to risk. If anything, riskier assets experience slightly lower returns. The risk premium is essentially zero, or in the jargon, "risk is not priced". About a third of the present book is devoted to empirical support for this assertion, and most of the rest gives a theorietic model and intuitive analogies. (The theoretic model postulates that people are interested in relative income, not absolute income, i.e. they don't care how much income they have in actual dollars, but rather how well they are doing compared to others. Markets are driven by envy rather than greed.)

Importantly, Falkenstein equates risk to volatility, measured by either beta or by standard deviation.

The most important practical conclusion is that one should invest in low-volatility assets -- a point that NormR has made several times on this forum.

I'm still not convinced. For example, when Falkenstein measures average returns, he uses geometric means. In turn, those are completely determined by the two end-points of his time series. If one of those end points is unusual, the whole result is thrown off.

For example, comparing SPY (an ETF representing the S & P 500) with SPLV (an ETF representing the 100 least volatile of the 500), year by year, I don't see any great advantage to SPLV.

I do agree that volatility, whether measured by beta or be standatd deviations, by itself doesn't explain much of the return. But then I don't much like the Fama-French three-factor model, or its four-factor extension. (Neither does Falkenstein.)

I come back to positive skewness (i.e. asymmetry towards the right) as a key factor. Others have suggested skewness and kurtosis in addition to volatility as key factors explaining relative returns, but I think that skewness has been under-explored and could explain many of the anomalies presented by Falkenstein. Certainly, if people are willing to pay a premium for positive skewness, that might explain lottrery tickets and property insurance, as well as the popularity in 2000 of Internet stocks and IPOs more generally. Yet Falkenstein never mentions skewness as a possible factor.

George

[DenisD]

The above brings to mind one of my favorite investing books: What Works on Wall Street by O'Shaughnessy.

For some of the strategies, he compares the results of 50, 25 and 10 company portfolios. Usually, as the number of companies go down, the returns and the volatility both go up.

Unfortunately, my copy is currently at the library. So I can't provide any numbers.

[NormR]

The most important practical conclusion is that one should invest in low-volatility assets -- a point that NormR has made several times on this forum.

I think avoiding the highly volatile bit is the more useful point. I'm more interested in the "what not to do" story

Too bad Eric's newest book needs an edit. I've not read it yet but his previous one contained a slew of typos.

The envy argument is an interesting one. But just like trying to boil everything down to greed, it seems like a mistake to do the same with envy. (Or rationality, etc)

I also wonder how related the volatility anomaly is to the momentum anomaly. That is, are they both just picking off the falling knife effect? (I dimly remember reading something about it but I can remember the details. Sigh.)

[NormR]

For example, comparing SPY (an ETF representing the S & P 500) with SPLV (an ETF representing the 100 least volatile of the 500), year by year, I don't see any great advantage to SPLV.

Isn't it too early to tell? Yahoo seems to indicate that SPLV started in 2011. Importantly, I'd want to keep an eye on the tax efficiency of such funds. Turnovers might be a bit high.

[ghariton]

For example, comparing SPY (an ETF representing the S & P 500) with SPLV (an ETF representing the 100 least volatile of the 500), year by year, I don't see any great advantage to SPLV.

Isn't it too early to tell? Yahoo seems to indicate that SPLV started in 2011. Importantly, I'd want to keep an eye on the tax efficiency of such funds. Turnovers might be a bit high.

Yes, SPLV started in May 2011, so it really is early. On the other hand, the two ETFs are run by the same institutiojn, so presumably are comparable in management approach, etc. Importantly, they are equally easy to invest in.

The usual tests of low-volatility portfolios may go back farther over time. But I'm not sure how easy it is for the individual investor to have actually replicated any of those, or what transaction costs would have been incurred. I find comparing two ETFs more clear-cut.

George

[ghariton]

Jack Schwager announces that risk is more than volatility:

Volatility is often viewed as being synonymous with risk—a confusion that lies at the heart of the mismeasurement of risk. Volatility is only part of the risk picture—the part that can be easily quantified, which is no doubt why it is commonly used as a proxy for risk. A comprehensive risk assessment, however, must also consider and weigh hidden (or event) risks, especially since these risks may often be far more important.

The confusion between volatility and risk often leads investors to equate low-risk funds with low-volatility funds. The irony is that many low-volatility funds may actually be far riskier than high-volatility funds. The same strategies that are most exposed to event risk (e.g., short volatility, long credit) also tend to be profitable a large majority of the time. As long as an adverse event does not occur, these strategies can roll along with steadily rising NAVs and limited downside moves. They will exhibit low volatility (relative to return) and look like they are low risk. But the fact that an adverse event has not occurred during the track record does not imply that the risk is not there.

I think his argument comes down to the assertion that traditional volatility measures do not adequately capture adverse events that have low probability but very large impact when they do occur.

To which I would reply:

(1) Risk measures should include kurtosis (the fourth moment) as well as volatility (the second moment)

(2) It is very dangerous to draw inferences from time series analyses, especially when the observed time period is short

For those who think that kurtosis is too abstruse a subject for practical investors to worry about: John Hull tells the story of the early days of the Black-Scholes formula, in the late 1970s and early 1980s. The formula prices options using the simplifying assumption that the underlying probability distribution is Gaussian (actually, that the underlying random process is a Weiner process). In fact, it is more fat-tailed than that (high kurtosis). Investors who realized this early on made a lot of money betting on out-of-the money options which were underpriced by too great a belief in Black-Scholes and its assumptions.

(Eventually the knowledge became too widespread -- among professional onvestors at least -- to make any money at this game.)

George