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Securities prices follow a log-normal distribution - a winning system?

  • #1
  • First Post: Jun 16, 2011 7:09pm Jun 16, 2011 7:09pm
http://en.wikipedia.org/wiki/Black%E2%80%93Scholes

Black-Scholes model assumes securities prices follow a log-normal distribution, so the logarithm of prices follow a normal distribution.

Let's say Y = log(Price). Y follows a normal distribution.
So if I take the arithmetic mean of Y as an estimate for the mean, and it's for example 1.2 and the current Y is 1. So I know that for example P([1.1, +∞[) is higher than P(]0, 0.9]). So in this example, if I bought that security at the price whose logarithm is 1 (price = e^1) and I took profit if it reached the price whose logarithm is 1.1 (e^1.1) and closed with a loss if it reached e^0.9, I'd win in the long term.

I'm just a begginner in statistics, just started studying statistics this year. But I see many problems why this wouldn't work:
- First of all I don't know if prices really follow a log-normal distribution, I haven't found much information about this. I only know this is assumed in the black-scholes model, and that model is used to price options.
- The distribution function would change in time. As time passes, the mean would vary. Because of this I don't know if I can use this distribution that doesn't consider time.

If anyone knows statistics and probability theory here, what do you think?
  • #2
  • Jun 19, 2011 8:14am Jun 19, 2011 8:14am
They do not follow a normal distribution.

A normal distribution severely underestimates the probability of a large loss. Thats one of the reasons LTCM crashed (they used the black scholes model to trade options)

Stock returns tend to be 'fat tailed', meaning they have a higher frequency of returns that are several standard deviations away from mean. This means you'll get bigger good days but also bigger bad days (stock market crashes etc).

I think the statistic is if stock returns were normally distributed, you'd have a stock market crash every 500 years or so. Of course we know these are much more frequent.

The 'levy curve' is a model that seems to resemble returns with more accuracy. You may like to look into that.

If you are going to develop trading systems based on a normal distribution, please be advised that the model will underestimate volatility in all markets.
  • #3
  • Jun 26, 2011 10:32am Jun 26, 2011 10:32am
Quoting scottyfromau
They do not follow a normal distribution.

A normal distribution severely underestimates the probability of a large loss. Thats one of the reasons LTCM crashed (they used the black scholes model to trade options)

Stock returns tend to be 'fat tailed', meaning they have a higher frequency of returns that are several standard deviations away from mean. This means you'll get bigger good days but also bigger bad days (stock market crashes etc).

I think the statistic is if stock returns were normally distributed, you'd have a stock market crash...
Yes Black-Scholes model doesn't explain big price movements, but that's not a big problem because the possibility of sharp moves isn't so great. And it says prices follow a log-normal distribution, not a normal distribution, and a log-normal distribution can have very large tails: http://en.wikipedia.org/wiki/Log-normal_distribution

I've heard of the 'levy curve' before, I'll definitely look into it. But do you think a statistics approach could make a winning system?
  • #4
  • Jun 26, 2011 10:48am Jun 26, 2011 10:48am
yes, its an area called statistical arbitrage which you might like to look into. Apologies about missing that you wrote 'log normal', was very tired when I wrote that reply.

But I still strongly advise being careful using distributions like that in the market, models very often have assumptions that don't tend to work too well in the real world market.
  • #5
  • Jun 26, 2011 11:01am Jun 26, 2011 11:01am
Quoting scottyfromau
But I still strongly advise being careful using distributions like that in the market, models very often have assumptions that don't tend to work too well in the real world market.
I have that in mind, but Black-Scholes is used widely to price options, so shouldn't it work? If it didn't work it wouldn't be used anymore right? I'm not sure, but if it didn't work the issuers of these options would lose in the long term and they wouldn't keep using that model.
  • #6
  • Jun 26, 2011 11:03am Jun 26, 2011 11:03am
some people still use it yes but I think you'll find more sophsticated instiutions will use improvements and further derivations to find more accurate option prices.

Black scholes is good but there's better tools out now
  • #7
  • Jun 26, 2011 11:22am Jun 26, 2011 11:22am
I've tried all manner of trading and the only way I've been able to really make money is trading index and stock options.
You are in a maze of twisty little passages, all alike.
  • #8
  • Jun 26, 2011 11:23am Jun 26, 2011 11:23am
Quoting scottyfromau
some people still use it yes but I think you'll find more sophsticated instiutions will use improvements and further derivations to find more accurate option prices.

Black scholes is good but there's better tools out now
The model for me is not important, I'll only be using one assumption of the Black-Scholes model - prices follow a log-normal distribution. Maybe a best thing to do is to verify if this really happens, and if it doesn't, try other distributions. Do you know any good statistics program to do this?
  • #9
  • Jun 26, 2011 11:25am Jun 26, 2011 11:25am
ask the people at the willmott forums, they can probably guide you to some whitepapers or books on the subject.

I'm sure there'd be a tonne of papers with people trying to proove/disproove this.
  • #10
  • Jun 26, 2011 11:33am Jun 26, 2011 11:33am
Where are all the Quants?
  • #11
  • Jun 26, 2011 11:39am Jun 26, 2011 11:39am
Mathematical modeling probably works best on options because how else can you come up with a value for a derivative? I'm sure some equation exists that can be used to determine the price of an option based on a number of variables including the price of the underlying asset. Maybe you can do the the same thing when it comes to FX futures............I dunno.
  • #12
  • Jun 26, 2011 12:10pm Jun 26, 2011 12:10pm
Quoting Jasus
I'll only be using one assumption of the Black-Scholes model - prices follow a log-normal distribution.
Let me tell you a joke:
Quote
A physicist, a chemist and an economist are stranded on an island, with nothing to eat. A can of soup washes ashore. The physicist says, "Let's smash the can open with a rock." The chemist says, "Let's build a fire and heat the can first." The economist says, "Let's assume that we have a can-opener...
Black-Scholes makes a lot of invalid assumptions: log-normal price distribution, no transaction costs, continuous prices, no liquidity gaps, ...
  • #13
  • Jun 26, 2011 2:22pm Jun 26, 2011 2:22pm
Quoting Jasus
Let's say Y = log(Price). Y follows a normal distribution.
So if I take the arithmetic mean of Y as an estimate for the mean, and it's for example 1.2 and the current Y is 1. So I know that for example P([1.1, +∞[) is higher than P(]0, 0.9]). So in this example, if I bought that security at the price whose logarithm is 1 (price = e^1) and I took profit if it reached the price whose logarithm is 1.1 (e^1.1) and closed with a loss if it reached e^0.9, I'd win in the long term.

I'm just a begginner in statistics, just started studying statistics...
Return or price change of all financial instruments do not follow normal distribution. You may compare the Q-Q plot for real GBPUSD data and excel random generator here.
  • #14
  • Jun 26, 2011 5:30pm Jun 26, 2011 5:30pm
Quoting scottyfromau
ask the people at the willmott forums, they can probably guide you to some whitepapers or books on the subject.

I'm sure there'd be a tonne of papers with people trying to proove/disproove this.
Thanks, just joined the forums, waiting for approval

Quote
Return or price change of all financial instruments do not follow normal distribution. You may compare the Q-Q plot for real GBPUSD data and excel random generator here.

But BS model says it follows a log-normal distribution, not a normal distribution. But that's interesting, that's what I'd like to do.
  • #15
  • Jun 26, 2011 11:04pm Jun 26, 2011 11:04pm
Quoting Troikaone1
Mathematical modeling probably works best on options because how else can you come up with a value for a derivative? I'm sure some equation exists that can be used to determine the price of an option based on a number of variables including the price of the underlying asset. Maybe you can do the the same thing when it comes to FX futures............I dunno.

I don't think this guy read the (very short) thread b4 posting
  • #16
  • Edited 1:04am Jun 27, 2011 12:39am | Edited 1:04am
Yes, Prices do follow a log normal distribution over the long run.
This is one reason buy and hold tends to work well.

One obvious challenge with using this distribution as a system launching point (which you pointed out), is that the characteristics of such a distribution change over time (mean, etc)-- they are not fixed, nor static.

Get to know and work with R or excel if you can; you can study this phenomena and test out your ideas for free.
  • #17
  • Edited 5:58am Jun 27, 2011 5:45am | Edited 5:58am
Quoting jamjamjam
Yes, Prices do follow a log normal distribution over the long run.
This is one reason buy and hold tends to work well.

One obvious challenge with using this distribution as a system launching point (which you pointed out), is that the characteristics of such a distribution change over time (mean, etc)-- they are not fixed, nor static.

Get to know and work with R or excel if you can; you can study this phenomena and test out your ideas for free.
I already downloaded historical data and have been working on excel. I only tested on 1 forex pair on the daily TF, but the distribution of the logarithm of prices really approximated to a normal distribution

But how could I test trading strategies on excel?

And about the trading system, I couldn't do what normally one does on a problem like this. On a problem like this, we'd take the biggest number possible of prices, calculate the mean estimate and work with that. But in trading obviously I can't take the mean of the last 2000 bars and work with that, I'd have to take an average of a few bars. It's a poor man's solution to the problem of the population's distribution changing over time, and nothing tells me it would work.
  • #18
  • Last Post: Jun 27, 2011 6:14am Jun 27, 2011 6:14am
Quoting Jasus
But BS model says it follows a log-normal distribution, not a normal distribution. But that's interesting, that's what I'd like to do.
it says log-return follows log-normal distribution, so that means return follows normal distribution.

Quoting Jasus
I already downloaded historical data and have been working on excel. I only tested on 1 forex pair on the daily TF, but the distribution of the logarithm of prices really approximated to a normal distribution
Dont trust your naked eye, it looks like a normal distribution to your naked eye but it is actually far from normal distribution. Check its kurtosis and Quartile-to Quartile-Plot Chart.

Quote

But how could I test trading strategies on excel?

And about the trading system, I couldn't do what normally one does on a problem like this. On a problem like this, we'd take the biggest number possible of prices, calculate the mean estimate and work with that. But in trading obviously I can't take the mean of the last 2000 bars and work with that, I'd have to take an average of a few bars. It's a poor man's solution to the problem of the population's distribution changing over time, and nothing tells me it would work.

try GARCH, but I doubt that it is useful in trading strategy. It is just too troublesome for me to calculate it.
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