Black-Scholes: The maths formula linked to the financial crash | Hacker News Reader

Black-Scholes: The maths formula linked to the financial crash

bbc.co.uk

41 pointswr147227 comments

This article is deeply flawed, primarily because it parrots the common misconception leveled against the formula, and options market by association (by none other than Taleb himself - who obviously knows better but prefers to act like a self-important bafoon).

Here is the crux of this misconception - that traders assume a Gaussian (ie Normal) distribution of events, while the world is far from "normal". THIS IS ABSOLUTELY WRONG. Traders and investors know that the world is full of "Fat Tails" and price the options accordingly. This is what's known as the Volatility Smile [1]

[1] http://en.wikipedia.org/wiki/Volatility_smile

Moreover - the formula is not used to predict anything. It simply converts a trader's expectations of future events (ie volatility) into dollar figures - price of the option. So the sentence "Black-Scholes changed the culture of Wall Street, from a place where people traded based on common sense, experience and intuition, to a place where the computer said yes or no" - is absolute nonsense. People still trade based on common sense and intuition - BS is merely a way to convert views in numerical figures.

The Normal distribution is part of the parametric derivation of the Black-Scholes pricing model [1]. It really does seem to be part of the model's key assumptions.

Can you expand on how the Volatility Smile demonstrates that traders model fat tails in their options pricing strategies. It wasn't clear to me from the wiki page you linked.

[1] http://en.wikipedia.org/wiki/Black-Scholes#Notation

If you calculate the implied volatility (volatility is not readily observable) from the option prices at different strikes, you will get different values. Ergo, the traders are not simply plugging a single volatility value into the formula for pricing.

BS does assume a normal distribution; the point is that people do not strictly take the output from BS and say "tada, here's the price."

Sure, I'll try. Here is an example:

Assume that you want to buy an option on Apple stock - 3 months in the future. Assume that price today is $500. If you want to buy an option with strike=500, you will pay 50 vol (BS as a formula will just convert the price in vol terms into a dollar cash price). But - if you want to buy an option struck at 400 - you might pay 75 vol. And if you want a 300 strike - it will be 100 vol.

As you see, volatility will RISE the further you move away from spot price (ie where it is today). So while BS calculation assumes constant volatility to convert vol to dollars, traders always ask for higher vol, as you move out to less likely scenarios. That is how they deal with the world being non-normal. It really is options 101 - so any writer who talks about BS assuming Gaussian probabilities is either lying or doesn't know anything about the subject.

The most common derivation of Black-Scholes absolutely does assume a standard normal distribution. There are other models that don't (i.e., Mandelbrot tried to get people to use Levy distributions, and a lot of newer models are built around martingale theory), and traders have adopted rules-of-thumb that pull prices out of line with what BS says they should be, but it's certainly not the case that anyone who speaks of BS assuming Gaussians is lying or speaking from ignorance.

Uggh, no. Let me try and explain. BS is not used as a model that explains how the world works.

It's just a formula to convert volatility to a dollar price of an option. But the actual vol surface that people use is anything but Gaussian. So the job of the trader is to construct a vol surface. However in order for 2 people to trade an option, they have to agree to a cash price - which is when BS is used. But the entire trading universe revolves around vol surfaces.

Think of it as making a painting. BS is just a single paint pigment - whereas the painter will take 20 pigments and mix them up to produce millions of colours. So to accuse those who use BS formula of assuming Gaussian probability distribution is akin to saying that painters use one color to paint.

If you ignore the absurdly stupid title, this is actually worth a quick read. It's excerpts from interviews with Myron Scholes and Ian Stewart woven into an oversimplified but not completely awful overview of Black-Scholes and its impact.

If the question interests you, I recommend An Engine, Not a Camera by Donald MacKenzie. It goes into great detail about the history of Black-Scholes, its application in options trading, and how it (or, rather, financial modeling in general) tied into the 1987 crash and the failure of LTCM. I have a small writeup about it at http://jasonfager.com/1080-an-engine-not-a-camera/

Another thumbs up for "An Engine, Not A Camera." A great book on the philosophy of financial modeling and how models influence the markets (the "engine"), rather than providing static snapshots of information (the "camera").

Here is an interesting take on the formula from The Black Swan by Nassim Taleb:

> Things got a lot worse in 1997. The Swedish academy gave another round of Gaussian-based Nobel Prizes to Myron Scholes and Robert C. Merton, who had improved on an old mathematical formula and made it compatible with the existing grand Gaussian general financial equilibrium theories—hence acceptable to the economics establishment. The formula was now “useable.” It had a list of long forgotten “precursors,” among whom was the mathematician and gambler Ed Thorp, who had authored the bestselling Beat the Dealer, about how to get ahead in blackjack, but somehow people believe that Scholes and Merton invented it, when in fact they just made it acceptable. The formula was my bread and butter. Traders, bottom-up people, know its wrinkles better than academics by dint of spending their nights worrying about their risks, except that few of them could express their ideas in technical terms, so I felt I was representing them. Scholes and Merton made the formula dependent on the Gaussian, but their “precursors” subjected it to no such restriction. /.../

> And the option formula went on bearing the name Black-Scholes-Merton, instead of reverting to its true owners, Louis Bachelier, Ed Thorp, and others.

Did Ed Thorp ever publish his formula? I'm flipping through his book Beat the Market now, and so far I've only seen two rules of thumb for pricing an option, which would only be useful for identifying the most absurdly mispriced options.

An option should be worth at least its intrinsic value, I agree. But how much more? You don't get Nobel prizes for stating something that's common sense.

   You don't get Nobel prizes for stating something that's common sense.

True. You get Nobel prize for assuming that returns follow Normal distribution, which goes against common sense.

At least now, title does not go as far as claiming that this is formula that _caused_ the crash, unlike this sensationalist piece[1]. The other popular formula to pick on is Li's Gaussian Copula [2] that did not address tail risks and dynamics of assets' co-dependence.

The important thing to remember is that finance is not physics and any model of reality is just that - model. As long as model's assumptions are close to reality, model works fine. It is failure to understand limitations or when assumptions do not hold any more that led to crash.

Actually the over-reliance of quants on models like Black-Scholes or Gaussian copula is a testament to how well these models work (or should I say worked?) for specific market conditions.

[1] http://www.guardian.co.uk/science/2012/feb/12/black-scholes-...

[2] http://www.wired.com/techbiz/it/magazine/17-03/wp_quant?curr...

> The important thing to remember is that finance is not physics and any model of reality is just that - model.

Side remark: this is true of physics too, only the models are better tested and we are better at understanding their scope of application.

Indeed. The map is not the territory -- even if some people get confused by terms such as "physical law".

But physics doesn't pay attention to our models and intentionally change its behavior to stymie us. The financial market does, since financial modelling is the recursive introspective study of its own modelling process.

1. only use numbers based on models you fully understand

2. always remember: all models are wrong, (but some of them are usefull.)

p.s.: does soembody know a really good book about system modelling? i`m searching for one.

Not what you asked but you might find it very interesting: https://class.coursera.org/pgm/auth/welcome

thx, sadly i don't do video (there are readers and there are listeners in this world, i'm definitely a reader), but i'm giving here book a chance http://www.amazon.com/Probabilistic-Graphical-Models-Computa...

I'm more of a reader too, but these videos are very different. Give it a try to a non-introductory video. They have quizzes and you need to follow with the material and even do some coding.

This is a fairly sensationalist article. Misaligned incentives and poor risk management cause crashes, not formulas and models.

Every option trader I've ever met is 100% aware of the limitations of Black-Scholes and are not using it blindly to price options/complex derivatives.

Black-Scholes assumes returns are lognormally distributed. The market knows this not to be true and you can back out this fact from option prices. Inputs such as kurtosis and skew are used to determine a more accurate representation of returns.

coming soon -- Compound interest: the maths formula linked to the financial crash

Who blames their equation for the financial crash? Afaik, it's a tool to valuate options, not an algorithm to trade or valuate mortgages. The link is that it "caused investment banks to hire people who had quantitative or mathematical skills". Which is like blaming Marie Curie for the Fukishima disaster.

One thing that is quite easy to forget:

A formula is a formula, it's a tool

It does not crash markets, people who misuse it are the ones who cause trouble.

"Guns don't kill people... people kill people"?? "But I'm pretty sure the gun, helps"

More or less

But I can have an accident with a gun. The gun is a physical, tangible instrument for delivering damage.

You don't see people getting killed directly by formulas.

Also, you are responsible for feeding the parameters in the formula. The formula doesn't interpret or act in the values it provided.

It's not a good article. Apparently the point is that Black-Scholes assigns too low a probability to large price swings.

Even if true, I don't know what it has to do with this particular financial crisis, and the article doesn't speak to that either.

This is so retarded.

Everyone has known for decades that these models (Black Scholes / Gaussian Copula) are wildly inaccurate in the tails. where the real risks live. They are roughly accurate on quiet days. Mandelbrot has been publishing on this since the 1960s! Anyone who pretends to believe these models is running a scam of some kind.

It is true that the GC played a minor role in the recent crisis but Black Scholes did not. See below for details.

The factors in the 2007-2012 crisis:

1. Fraudulent lending practices and falsified loan applications.

2. Excessive borrowings by households fueled by the Fed keeping interest rates too low for too long.

3. Belief that present trends would continue forever and house prices would continue to the moon.

4. Greed blinded people to the risks they were taking.

5. Lax to nonexistent regulation which allowed banks and related organizations to leverage to insane levels. It also allowed companies like AIG to sell insurance that they could not pay off on.

6. Risks were ignored due to perceived government guarantees (Fannie Mae and her ilk).

7. Fraudulently selling subprime toxic garbage as AAA securities. This is where the Gaussian Copula came in. Given known wrong and bogus assumptions (eg that house prices would never fall across the whole USA), it allowed the investment banks to pretend that the top tiers of the subprime securities were AAA ie secure. Internal emails showed they knew they were not really AAA. However the GC was only the vehicle; the underlying problem was fraudulent and criminal intent.

8. Rating agencies were paid large sums of money to rate the toxic waste as AAA. They either knew or did not care that the securities were toxic waste as long as they got the cash.

9. Pension funds and other naive investors believed that the rating agencies and investment banks were not lying when they said the AAA-rated securities were OK.

10. More recently we have seen the crisis in Europe which is the result of the failure to rein in housing bubbles caused by too-loose credit, and by governments which borrowed more than they could afford to pay back, and which made commitments that they could never fulfill (eg excessive pensions).

BS played a role in the near meltdown in 1998 when LTCM went down, and also in the 1987 stock market crash.

In both cases idiots pushed the models outside their sphere of validity. LTCM was leveraged to the hilt and assumed that short term historical correlations would continue to prevail. A cursory examination of history would show this is not the case.

In 1987 a technique called "portfolio insurance" was invented which supposedly allowed the user to simulate a protective "put option" at no cost. Portfolio insurance required selling stocks when the market fell. The BS model assumes infinite liquidity and no price jumps and if these are true PI should work. Again these are not valid assumptions and when the technique was implemented on the overvalued October 1987 market it accelerated and intensified the crash.

Even in physics most models are inaccurate outside a certain range of validity. It requires a degree of honesty and intellectual integrity to refrain from using them when they are not valid. Eg you cannot use Newtonian mechanics at 99.999% of the speed of light.

TL;DR this was not a mathematical mistake - it was fraud.

Reply on news.ycombinator.com