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Historical Volatility, Distributions and Odds Calculations.

In this article we will cover historical volatility, distributions and odds calculations.

In future articles we will look at future volatility (implied volatility) the VIX and also ATR (average true range)


There are so many potential assets to trade in the world the prospect of simply deciding what you want to trade and why can be pretty daunting, especially to new retail traders with no prior experience.

I have seen dozens if not hundreds of strategies ranging from highly complicated and convoluted technical analysis ideas, using loads of different technical tools. To very simple technical analysis ideas such as “when this line crosses that one BUY and when this one crosses that one SELL”

Then also strategies which focus mainly on fundamentals some using a bottom up corporate/company analysis of financial performance and some focusing on a top down (global economy) approach.

And of course there are those retail traders which either pick at random or choose fancy names or maybe trade an asset “cus they saw it on the tele” or I don't know just throw a dart at a wall maybe.

No one strategy is right, just as no one strategy is wrong (well apart from maybe the dart at the wall one) but most traders who survive not becoming a part of the 90/90/90 statistic of failure most likely develop a strategy using multiple fundamental and technical elements together and one that fits both their broader outlook and personal psychology.

However before we get into the huge subject of trading strategy (a subject for a future article), I want to focus in this article on the one thing I believe should underline all of it (everything we do as traders) and that is understanding and defining via the processing of data, our odds of making or losing a certain amount of money on any particular asset over any given time horizon.

We are after all in this game to make money and unless you want to go into a trade blind it is our responsibility to understand our odds and the potential outcomes of the asset we are trading (understanding the nature of the beast)

We do this by establishing the historical volatility of the asset we wish to trade as the trading history of the asset up until this point is all we have to go on regarding raw data, everything thing else regardless of how valid it may be is an unknown and at worst pure speculation.

So we are going to start by looking at Distributions.

It is highly likely you already know what a normal distribution is. But for a quick reminder normal distributions are most commonly applicable to the natural world. A real world example of a normal distribution could be explained as the probability of being a certain body weight or height or the probability of being a certain IQ level.

It presents itself as a bell shaped curve, that is symmetrical around the mean of the data sample that is being analysed. The larger the data set being used the more accurate the distribution will be and the greater the tendency towards normality.

Here is an example of what a normal distribution would look like.

In the example of a standard distribution for the attribute of Human IQ level we would plot the IQ number along the X axis and the number of tests results along the Y axis.

In the graph below we can see that the X axis has the numbers 0,1,2 and 3 followed by the greek letter sigma this is what we call the Standard deviation.


We can express the distribution of results along the X axis in the form of a Standard deviation.

34.1% of the total distribution will fall within 1 standard deviation of either side of the mean.

13.6% of the total distribution will fall between 1 and 2 standard deviations of either side of the mean.

2.1% of the total distribution will fall between 2 and 3 standard deviations of either side of the mean.

0.1% of the total distribution will fall beyond 3 standard deviations of either side of the mean.

So put another way.

68.2% of a normal distribution will fall within 1 standard deviation of the mean.

95.4% of a normal distribution will fall within 2 standard deviations of the mean.

98.8% of a normal distribution will fall within 3 standard deviations of the mean.

Now I hear what your saying “This is all well and good Tom but I learnt all this in high school and I thought it was boring then and it is still boring now”.

Fair enough, but what if I told you that this is one thing you learnt that can actually be usefully applied to making you money from the trading.

So instead of using a distribution to present a bunch of useless information regarding statistics like human IQ or height or even the monthly rainfall in the amazon basin or some pointless crap like that, how about we collect data on 20,30,50 years or more of an assets daily returns for any tradable asset we can find data for, it could be a single equity or even the entire stock market itself. With us plotting daily return range on the X axis and the number of occurrences on the Y axis.

Below is my own distribution I have made for the S&P 500 Index covering 50 years of stock market returns, from 1968 to 2018.

This Distribution graph takes 50 years of daily returns that is a total of 12,586 trading sessions for the S&P 500 (an excellent and detailed data set by anyone’s standard) and condenses it into an easy to read distribution from which we can start to draw some very interesting insights regarding potential returns when trading this asset long or short.

We can see that the distribution for the S&P 500 does to some degree resemble a normal distribution.

However there are some differences worth noting mainly that it has a greater peak and fatter tails that a normal distribution.

So let’s have a closer look at the data.

This chart displays both the standard deviation we see from a normal distribution (in red) and the actual standard deviation of the S&P 500 distribution (in green)

From this chart we can see that roughly 10% more of all results fall within 1 standard deviation of the Mean than would normally occur within a normal distribution.

Hence why the S&P 500 distribution appears peakier.

So what does this mean for anyone wanting to trade the S&P 500. Well we can illustrate this in another chart.

Daily Return Range: Our returns are calculated as the range between the opening price for the session and the closing price for the session and not the high and the low for the session. The high and low range is known as the ATR (average true range) and we are not using it as we are focused on the return for the days whole session and not a specific window within that day.

Let’s focus for a minute on the range highlighted in green.

This covers the probability of the S&P 500 moving 1% either way, up or down on any given day in the last 50 years or 12,586 daily sessions.

So all we need to do is add together the percentages of that green range and we get 77.65%.

Remember that 1 standard deviation from the mean for the S&P 500 is 78.58%.

So basically that green range is essentially our 1 standard deviation.

But you should note that almost 50% of all trading activity in the S&P 500 occurs within 0.5% of the mean from -0.5% to 0% and from 0% to 0.5%.

So in layman's terms roughly 80% of the time the S&P 500 does very little!

77.65% of the time the S&P 500 yields a daily return of 1% up or down.

50% of the time the S&P 500 yields a daily return of 0.5% up or down.

So also in layman's terms roughly 50% of the time the S&P 500 does fuck all!

So what simple conclusions can we draw from this data.

Well I would say that the most obvious one is that Day Trading as a concept when applied to the S&P 500 is a waste of one’s valuable time. And believe me as I have crunched the numbers on multiple other asset classes the same principle applies to pretty much all major indices and the FOREX markets also.

Of course if you are happy to trade x100 leverage then there is profit to be made, but I am pretty sure I have expressed my opinion on borrowing 100 times the capital you commit to a trade from your broker in previous articles. Suffice to say it is a mugs game and you will eventually get it wrong and blow up!

So in hindsight of this information let's look at an example of a stupid trade.

A retail trader deposits $1,000 to their account and assigns 50% of it to a single long trade on the S&P 500 at lets say x10 leverage (mistake number 1).

So $500 at x10 leverage buys him $5,000 of S&P 500 Index. He does this most likely just because he thinks it will go up today, maybe it was written in his morning bowl of porridge or worse he heard something on a financial media channel and took it as sound advice (God forbid).

So we have our 1% (100 basis points) target and set a 0.3% (or 33 bps) stop loss on the trade.

Upside = $50

Downside = -$16.50

So a Risk to Reward Ratio of 1 to 3 or 33% Breakeven threshold (sounds pretty good yeah?)

Put simple he would need to be right just 33% of the time when placing this exact trade in order to make his desired profit.

But hang on he didn’t account for the spread. (mistake number 2)

Lets use a nice round figure of 0.2% or 20bps for the spread.

So $5000 of the S&P 500 x 20 bps = $10

So factoring in spread Net upside = $40

Net Downside = - $26.50

This is now a risk to reward ratio of 1.5 to 1 or 66% Breakeven threshold

So now put simple he would have to be right 66% of the time when placing this trade to make his desired profit.

Doesn’t sound so good now does it?

However What is the likely hood of the S&P 500 returning 1% in a day in the first place which is the belief behind his whole trade.

We need to adjust this trade for historical probability. So let’s take a look at our data chart.

Well we know that the likelihood of a return between 0.5% and 1% is 14.21% and the likelihood of a return between 1% and 1.5% is 6.28% so let's not overcomplicate things and just take an average of these two figures as a guide this works out at roughly 10%.

So now we have to take our spread adjusted target and further adjust it for probability so $40 multiplied by 0.10 = $4

So we are now entering a trade with a downside of -$26.5

and an upside of $4

Not adjusting for probability (mistake number 3)

This is a risk to Reward ratio of 1.1 to 1 or 90% Break even threshold.

Or put simple he would need to be right 90% of the time when placing this exact trade in order to make his desired profit.

I call this Mathematical confirmation that this trader is an Idiot.

I have been this idiot, if we are honest with ourselves most of us have when we first started out.

I made trades without taking into account anything other than a simple conviction in my desired outcome.

Although accounting for probability greatly increases our odds of being right it is by no means fool proof and only the result when we eventually close the trade will confirm the accuracy of both our conviction and probability combined.

So is trading the S&P 500 a waste of time?

The answer is a resounding NO!

Perhaps as the numbers show trying to make 1% on any given day is close too impossible but that is only if we have brought into the concept of day trading.

However if we are willing to adjust our time horizon and extend it significantly then things start to drastically turn in out favour regarding going long in the S&P 500.

Below is a another table that shows the total number of sessions over this 50 year period when broken into positive and negative returns plus the average daily return.

Clearly here we can see that the S&P 500 records more positive daily returns than negative one’s. “Well duh Tom! that’s why the stock markets only ever go up over time!”

The average return also hints at what a more conservative/realistic daily target would be via the average return column, however once we adjust for both spread and probability aiming for a return of 0.36% is not worth sitting in front of a screen for. So once once again I reiterate why would you ever day trade this thing when it is almost impossible to make 1% in a day but guaranteed to go up over a longer time horizon.

So we have learnt the “nature of the beast” regarding the S&P 500 and why a longer term time horizon is essential to making a profit when trading it.

Please always keep in mind that understanding the odds and distribution of an asset and the best way to trade it does not guarantee a profit.

When you enter a trade and exit it is down to you and although the odds of making a profit on for example a 3 month long trade in the S&P 500 are greatly in your favour if you got in at the most recent top or worse before a crash then these historical probabilities are not going to bring you much comfort.

So we have looked at volatility in the S&P 500 and established there isn’t any, 80% of the time it moves no more than 1% either way, only 20% of the time does the volatility increase and the market actually move in any meaningful way on the daily time frame.

But does that mean that we can only ever trade medium/long term time horizons?

Well lets answer that by taking a look at a different asset how about a individual equity from the NASDAQ 100 one of the most popular tech companies being traded on Etoro NVIDIA (NVDA).

Well the first thing you should notice is that the X Axis (daily return range) is not measured from more than -2% to more than 2% like the S&P 500 but actually I have had to use a much wider range of, from less than -5% to more than 5%.

This immediately tells us that the daily historical volatility is massively higher in this individual asset than in the S&P 500.

The next thing is that this distribution for NVDA bears little resemblance to a normal distribution it is nowhere near as peaky and has fatter tails than the female backup dancers in a rap video.

Infact you are more likely to see a daily move of more or less than 5% than you are a 3%-3.5%, 3.5%-4%, 4%-4.5% and 4.5%-5%.

In other words day trading might still not be the best strategy but is a damn sight more viable from an odds point of view than when trading the S&P 500.

But shorter time horizons of days or weeks can potentially yield big returns or of course losses.

The potential for shorter term trading opportunities is obvious when looking at the above chart especially in comparison to the S&P 500 Index.

Once again the daily returns are by no means mind blowing considering non leveraged day trading but significantly better than the S&P 500.

But a non leveraged trade over a longer time horizon has the potential to be very rewarding once again with the caveat of choosing a good entry point whether long or short.

The stand out point of notice is the potential for the stock to move more than 5% up or down in a single day’s session this kind of movement accounts for 5.31% of all of NVDA recorded trading history.

And believe when I tell you that there are far more volatile assets available for trade than this particular one.

We can also see from this frequency table that the daily return range is much closer to our day traders target of 1% from the stupid S&P 500 trade example earlier.

However the skewness to the positive on a daily return basis is less than the S&P 500, but bear in mind we only have 10 years of data to analyse rather than 50.

Also keep in mind that the trend of a single stock can reverse and remain in that new trend for many months or even years where as the broader market recovers much sooner, and of course in some situations eventually lead to bankruptcy which would see the figures skew heaverly to the negative before final collapse.

Well there you have it folks I hope you have made it through this article in one piece and now have a greater appreciation for the value of understanding historical volatility and what it means for our odds of profiting from trading specific assets over any given time horizon.

I also hope that looking back over the long term distribution of returns from any given asset helps you appreciate what kind of reasonable expectations you should have of a particular trade you have made or perhaps even currently have open.

For me analysing the historical volatility and distribution of an asset can really help snap me out of short term tunnel vision. Sometimes we can find ourselves holding out for more profit from a long or short trade that if we looked at the historical volatility of that asset we would realise we have little right or chance to expect given the nature of that particular beast.

Believe it or not but there is still a lot to cover regarding volatility.

We will cover implied volatility in a future article and also learn about the VIX index and what it means for our trading strategy and profit expectations.

We will also briefly cover the ATR (average true range) of assets as a means of measuring the rate of volatility over the history of that asset so whether it is becoming more or less volatile over time.

I intend (EVENTUALLY) to include on my website a page devoted to containing spreadsheets with all relevant data for odds and distributions calculations and Average true ranges for each asset in the the portfolio plus a back catalog of previous holdings. Please bear in mind this is a very labour intensive process and will grow over time so give me a few months.

All raw data downloaded from

All credit for the method of calculation used for spreadsheet/statistic and graph production goes to the ITPM (institute of trading and portfolio management)

Happy trading.

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