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.
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.
Stock price volatility from data. The random variable is log(P2/P1), the log of the later price to the earlier price. In this plot, P2 occurs 0.1 years after P1. A histogram is plotted with a logarithmic axis for the histogram count. A gaussian (bell curve) is fitted to the histogram, with a logarithmic y-axis, a gaussian is just a parabola. You can see the fit is great, except for some outliers on the positive side. Some of these outliers are quite a lot higher than the fitted gaussian, these are events that occur MANY TIMES more often that a log-normal distribution would suggest.
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:
If the prices gap, you cannot maintain the arbitrage and the Black-Scholes based strategy which was making you steady money is all the sudden faced with a large loss that more than wipes out your gains.*
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.
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”.
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.
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:
If the prices gap, you cannot maintain the arbitrage and the Black-Scholes based strategy which was making you steady money is all the sudden faced with a large loss that more than wipes out your gains.*
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.
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”.
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.