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Stuart_Armstrong comments on When the uncertainty about the model is higher than the uncertainty in the model - LessWrong
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Comments (78)
But some ad hoc adjustments are better than others. Black-Scholes has been clearly wrong for a long time, but there's no better option around. So use it, and add ad hoc adjustments that are more defensible.
First of all, Black-Scholes is not a model of price variations of stocks. It is a model for putting values on options contracts, contracts that give their owner the right to buy or sell a stock at a particular price at a particular time in the future.
You are saying Black-Scholes is a model of stock prices probably because you recall correctly that Black-Scholes includes a model for stock price variations. To find the Black-Scholes options prices, one assumes (or models) stock price variations so that the logarithm of the variations is a normally distributed random variable. The variance, or width, of the normal distribution is simply determined by the volatility of the stock, which is determined from plotting the histogram of price variations of the stock.
The histogram of the logarithm of the variations fits a normal distribution quite well out to one or two sigma. But beyond 2 sigma or so, there are many more high variation events than the normal distribution predicts.
So if you just multiplied the variance by 7 or whatever to account for the prevalence of 20-sigma events, yes, you would now correctly predict the prevalence of 20-sigma events, but you would severely underestimate the prevalence of 1, 2, 3, 4... 19 sigma events. So this is not at all a good fix. You are attempting to fit something that is NOT bell-curved shape by covering it with a wider bell-curve.
Meanwhile, how good is black-scholes, and is there something better for options? For short dated options (less than a year to expiration), its pretty good. The calculation is more influenced by the -2sigma to 2sigma part of the curve which is accurately predicted as a log-normal, than it is by the outliers. But for longer dated options, the outliers become more and more important.
Are there better models than black-scholes for predicting options values? Yes, if you don't mind dealing with GIGO. There are models that effectively let you specify the probability distribution function curve completely, and then calculate various options prices from that. The GIGO, garbage in garbage out, arises because the "correct" probability distribution, in detail, is not known a priori. If it turns out the stock is going up, there will be more excess probability on the positive variations side then the negative variations side. If it turns out the stock is going down, it will be the other way. If you knew ahead of time whether the stock was going up or going down, you wouldn't need a calculation as complex as black-scholes to know how to get rich from that knowledge.
Yes, I should be more careful when using terms. As you said, I used B-S as informal term for the log-normal variation assumptions.
So back when I worked for a major bank, no one was really using Black-Scholes for pricing anymore. For specific uses cases, there very much are (proprietary) better options, and people use them instead of just making ad-hoc adjustments to a broken model.
Why is there nothing better? Given the importance to the financial world of the problem that B-S claims to be a solution to, surely people must have been trying to improve on it?
I really don't know. I did try and investigate why they didn't, for instance, use other stable distributions than the normal one (I've been told that these resulted in non-continuous prices, but I haven't found the proof of that). It might be conservatism - this is the model that works, that everyone uses, so why deviate?
Also, the model tends to be patched (see volatility smiles) rather than replaced.
If you plot a histogram of price variations, you see it is quite well fit by a log-normal distribution for about 99% of the daily price variations, and it is something like 1% of the daily variations that are much larger than the prediction says they should be. Since log-normal fits quite well for 99% of the variations, this pretty much means that anything other than a log-normal will fit way less of the data than does a log-normal. That's why they don't use a different distribution.
The 1% of price variations that are too large are essentially what are called "black swans." The point of Taleb's talking about black swans is to point out that this is where all the action is, this is where the information and the uncertainty are. On 99 out of 100 days you can treat a stock price as if it is log-normally distributed, and be totally safe. You can come up with strategies for harvesting small gains from this knowledge and walk along picking small coins up from trading imperfections and do well. (The small coin usually cited is the American $0.05 coin called a nickel.)
But Taleb pointed out that the math makes picking up these nickels look like a good idea because it neglects the presence of these high variation outliers. You can walk along for 100 days picking up nickels and have maybe $5.00 made, and then on the 101st day the price varies way up or way down and you lose more than $5.00 in a single day! Taleb describes that as walking along in front of steam rollers picking up nickels. Not nearly as good a business as picking up nickels in a safe environment.
Interesting, and somewhat in line with my impressions - but do you have a short reference for this?
Sorry can't give you a reference. I wrote code a few years ago to look at this effect. I found that code and here is one figure I plotted. This is based on real stock price data for QCOM stock price 1999 through 2005. In this figure, I am looking at stock prices about 36 days apart.
Thanks, that's very useful!
Stuart, since you asked I spent a little bit of time to write up what I had found and include a bunch more figures. If you are interested, they can be found here: http://kazart.blogspot.com/2014/12/stock-price-volatility-log-normal-or.html
Cheers!
Interested financial outsider - what would it mean for prices to be non-continuous?
A stock closed at $100/share and opened at $80/share -- e.g. the company released bad earnings after the market closed.
There were no prices between $100 and $80, the stock gapped. Why is this relevant? For example, imagine that you had a position in this stock and had a standing order to sell it if the price drops to $90 (a stop-loss order). You thought that your downside is limited to selling at $90 which is true in the world of continuous prices. However the price gapped -- there was no $90 price, so you sold at $80. Your losses turned out to be worse than you expected (note that sometimes a financial asset gaps all the way to zero).
In the Black-Scholes context, the Black-Scholes option price works by arbitrage, but only in a world with continuous prices and costless transactions. If the prices gap, you cannot maintain the arbitrage and the Black-Scholes price does not hold.
Right, good explanation. Just to make it clearer in an alternate way, I would reword the last sentence:
Well, that's not quite what I mean.
There are many ways to derive the Black-Scholes option price. One of them is to show that that in the Black-Scholes world, the BS price is the arbitrage-free price (see e.g. here). The price being arbitrage-free depends on the ability to constantly be updating a hedge and that ability depends on prices being continuous.
If you change the Black-Scholes world by dropping the requirement for continuous asset prices, the whole construction falls apart. Essentially, the Black-Scholes formula is a solution to a particular stochastic differential equation and if the underlying process is not continuous, the math breaks down.
The real world, however, is not the Black-Scholes world and there ain't no such thing as a "Black-Scholes based strategy which was making you steady money".
If your function isn't continous you can't use Calculus and therefore you lose your standard tools. That means a lot of what's proven in econophysics simply can't be used.
I don't know. I assume it means that stock prices would be subject to jumps at all scales. I just know this was a reason given for using normal distributions.
Primarily because real world options pricing is influenced by a near infinite number of variables, many of which are non financial and BS is a model with a few variables all of which are financial in nature. I don't think there could be one model that featured all potnecialy relevant non financial variables. If there was, it wouldn't be computationaly tractable.
BS tends to be off on tail risk where specifc non financial events can have a big impact on a company or a specifc option. So the best aproach is to model the core financial risk with BS and use ad hoc adjustments to increase tail risk based on relevent non financial factors for a specific company.
BS fails even on purely financial issues - its tails are just too thin.
So why don't you become rich by exploiting this failure? If Black-Scholes fails in an obvious (to you) manner, options in the market must be mispriced and you can make a lot of money from this mispricing.
The market can stay irrational longer than you can stay solvent.
In this case you don't have to wait for the market to become rational. If the options are mispriced, you will be able to realize your (statistically expected) gains at the expiration.
Financial instruments that expire (like options or, say, most bonds) allow you to take advantage of the market mispricing even if the market continues to misprice the securities.
True, but if most of your statistically expected gains comes from rare events, you can still go broke before you get a winning lottery ticket, even if the lottery is positive expected value. I have no idea if there are any real-world financial instruments that work like this, though.
True -- that's why risk management is a useful thing :-)
And yes, options are real-world financial instruments that work like that.
Ok, here's an obvious failure: volatility smiles. Except that that's known and you can't exploit it. And people tend to stop using BS for predicting large market swings. Most of the opportunities for exploiting the flaws of BS are already covered by people who use BS+patches. There might be some potential for long term investments, though, where investors are provably less likely to exploit weaknesses.
Even if there's a known failure, though, you still might be unable to exploit it. In some situations, irrational noise traders can make MORE expected income that more rational traders; this happens because they take on more risk than they realise. So go broke more often, but, still, an increasing fraction of the market's money ends up in noise traders' hands.
Why is it a failure and a failure of what, precisely?
Sense make not. Black-Scholes is a formula describing a relationship between several financial variables, most importantly volatility and price, based on certain assumptions. You don't use it to predict anything.
I don't quite understand that sentence as applied to reality. The NBER paper presents a model which it then explores, but it fails to show any connection to real life. As a broad tendency (with lots of exceptions), taking on more risk gives you higher expected return. How is this related to Black-Scholes, market failures, and inability to exploit market mispricings?
Come on now, be serious. Would you ever write this:
"General relativity is a formula describing a relationship between several physical variables, most importantly momentum and energy, based on certain assumptions. You don't use it to predict anything." ?
I am serious. The market tells you the market's forecast for the future via prices. You can use Black-Scholes to translate the prices which the market gives you into implied volatilities. But that's not a "prediction" -- you just looked at the market and translated into different units.
Can you give me an example of how Black-Scholes predicts something?
BS is not just an equation, it is also a model. It predicts the relationships between the volatility and price of the underlying assets, and price of the derivative (and risk free rate, and a few other components). In as much as you can estimate the volatility (the rest is pretty clear) you can see whether the model is correct. And it often is, but not always:
See for instance: http://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model#Black.E2.80.93Scholes_in_practice
Now you might say that the equation shouldn't be used as a model... but it is, and as such, makes predictions.
The assumption here is that the options are being priced with Black-Scholes, which I don't think is true.
Just because you can pick a strategy that should have an expected postivie return doesn't mean that you automatically get rich. People do drown in a river of average depth of 1 meter.
Knowing that there is a mispricing doesn't tell you what the correct price actually is, which is what you need to know in order to make better money than random-walk models.
In this particular case the problem mentioned is too thin tails of the underlying distribution. If you believe the problem is real, you know the sign of the mispricing and that's all you need.
For this particular example, this basically means that you can predict that LTCM will fail spectacularly when rare negative events happen. But could you reliably make money knowing that LTCM will fail eventually? If you buy their options that pay off when terrible things happen, you're trusting that they'll be able to pay the debts you're betting they can't pay. If you short them, you're betting that the failure happens before you run out of money.
Just LTCM, no. But (if we ignore the transaction costs which make this idea not quite practicable) there are enough far-out-of-the-money options being traded for me to construct a well-diversified portfolio that would allow me to reliably make money -- of course, only if these options were Black-Scholes priced on the basis of the same implied volatility as the near-the-money options and in reality they are not.
This assumes the different black swans are uncorrelated.
LTCM should not be your counter-party! Also, using a clearinghouse eliminates much of the risk.
IIRC, LCTM ended up in disaster not only because of a Russian default/devaluation. They had contracts with Russian banks that would have protected them, except that the Russian government also passed a law making it illegal for Russian banks to pay out on those contracts. It's hard to hedge against all the damage a government can do if it wants.
You keep asserting that but provide no arguments and don't explain what do you mean by "better".
Huh? Black-Scholes doesn't tell you what the price of the option is because you don't know one of the inputs (volatility). Black-Scholes is effectively a mapping function between price and volatility.
I don't understand what do you mean. In situations where Black-Scholes does not apply (e.g. you have discontinuities, aka price gaps) people use different models. Volatility smile is not a "patch" on Black-Scholes, it's an empirically observed characteristic of prices in the market and Black-Scholes is perfectly fine with it (again, being a mapping between volatility and price).
The two examples in the post here are not sufficient?
From the Wikipedia article on the subject "This anomaly implies deficiencies in the standard Black-Scholes option pricing model which assumes constant volatility..."
The two examples being the 20-sigma move and the volatility smile?
In the first example, I don't see how applying an ad hoc multiplier to a standard deviation either is "better" or makes any sense at all. In the second example, I don't think the volatility smile is an ad hoc adjustment to Black-Scholes.
The Black-Scholes model, like any other model, has assumptions. As is common, in real life some of these assumptions get broken. That's fine because that happens to all models.
I have the impression that you think Black-Scholes tells you what the price of the option should be. That is not correct. Black-Scholes, as I said, is just a mapping function between price and implied volatility that holds by arbitrage (again, within the assumptions of the Black-Scholes model).