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RichardKennaway comments on When the uncertainty about the model is higher than the uncertainty in the model - Less Wrong
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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.
Yes. Or, rather, there is a Black-Scholes options pricing model which gives rise to the Black-Scholes equation.
No, it does not predict, it specifies this relationship.
Heh. And how are you going to disambiguate between your volatility estimate being wrong and the model being wrong?
Let me repeat again: Black-Scholes does not price options in the real world in the sense that it does not tell you what the option price should be. Black-Scholes is two things.
First, it's a model (a map in local terms) which describes a particular simple world. In the Black-Scholes world, for example, prices are continuous. As usual, this model resembles the real world in certain aspects and does not match it on other aspects. Within the Black-Scholes world, the Black-Scholes option price holds by arbitrage -- that is, if someone offers a market in options at non-BS prices you would be able to make riskless profits off them. However the real world is not the Black-Scholes world.
Second, it's a converter between price and implied volatility. In the options markets it's common to treat these two terms interchangeably in the recognition that given all other inputs (which are observable and so known) the Black-Scholes formula gives you a specific price for each volatility input and vice versa, gives you a specific implied volatility for each price input.
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.
Yes, to a degree. However in this particular case I can get exposure to both negative shocks AND positive shocks -- and those certainly 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.
As a historical note, the LTCM crisis was caused by Russias default, but LTCM did not bet on Russia or rely on Russian banks. LTCMs big bet was on a narrowing of the price difference between 30 year treasurys and 29 year treasurys. When Russia defaulted people moved out of risky assets into safe assets and lots of people bought 30 years. That temporary huge burst in demand led to a rise in the price of 30s. Given the high leverage of LTCM that was enouph to make them go bust.