Distribution of Probability

“Tomorrow never happens. It’s all the same fucking day, man. ”  Janis Joplin

Probability distribution

Analyzing history and hoping it will somehow repeat itself  is the big hope of all quantitative traders. This article is about the distribution of market returns, but not about normal distribution, Gauss and standard deviation. This article is about the visualization of market returns and what can be learned from it.

Probability distribution diagrams show the probability of a specific outcome. How likely is it that the market will be at a specific price sometimes in the future?  How does a specific bullish or bearish indicator signal affect the future market behavior on a statistical basis?

Distribution of returns

The chart below shows the distribution of returns of the American Express stock.

The probability of a specif future return is coded in shades of grey. The darker the distribution diagram, the higher the probability that the market will be there in the future. To generate this kind of chart all 1 to 50 day returns over the last 10 years have been analysed. A 50 day projection of returns is shown.

When changing the contrast settings of the distribution indicator, one gets a better impression on how likely specific events will be. As there are only 255 types of grey from white to black, a multiplication of the density by a contrast setting will overexpose the likely events. Everything that is not black, is very unlikely to happen.

As you can see, the shape of he probability distribution is something different than a normal distribution; and  the shape of the distribution will differ largely from market to market. This is the representation if different, inherent attributes of each market. The distribution of a Nasdaq will look different than that of EURUSD. To see these differences in the distribution helps when designing custom trading strategies, price targets and option prices.

The next chart is a returns projection for the VXX volatility ETF. We already had an article regarding the volatility of VIX ;this indicator offers a more intuitive representation of future returns than the classic distribution diagrams used back then. Data since 2011 has been used to generate this projection for the next 100 days.

 

Signal response and returns distribution

That different markets behave differently is nothing new to the experienced trader, this indicator just helps to visualize this fact.

An interesting usage of this distribution indicator is to analyze the behavior of the market after a specific signal has occurred. The chart below shows the answer of the yearly German power contract to a simple trend following signal. The loop shows the distribution of German power trading returns after a bullish-neutral-bearish signal. Significant differences between the expected returns after these 3 signals can be observed. These differences will finally lead to a trading strategy tailored to the specific signal-market response.

Also have a look at Demystifying the 200 day average for further ideas on how to analyze the quality of signals.

A picture says more than a thousand words..

Sorry Janis, it seems it`s not always the same f****** day…

 

Tradesignal Equilla indicator code

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Bet on Bollinger

Ever since John Bollinger introduced his Bollinger Bands in the early 1980s they have been a favourite indicator to all technical trades. This article is about the probabilities of Bollinger bands.

How good are the chances to be outside or inside of the bands in the future? How do these probabilities relate to the current position of the market relative to the Bollinger bands? What impact has overall volatility on these statistics? These questions will be answered.

Bollinger Forward Prediction

By definition of the indicator most of of the times the market will trade inside the Bollinger band. But what happens in the future? Where will the market be in some days from now. These are the questions which interest me from a truing point of view.

So I did some test on the forward prediction qualities of the Bollinger band indicator. For all tests I used the 20 day, 2 standard deviations setting, which is the standard setting for most charting packages. I then looked at the positioning of the market 20 days after relative to its current Bollinger band.

 

The first test was done on the S&P500 index, using data since 1983. The indicator shown on the screenshot below gives the probability of the market being outside today’s Bollinger band in 20 days from now.

The red line is the probability of being outside todays band if we are already out of the band today.  The green line shows the probability of being outside of today’s band in 20 days from now if we are trading inside of the band today.

The outside starters being outside the band in the future more likely than the inside starters, is most probably due to the trend lines of the market. Once a break is done there is a high probability it either vigorously reverses or that it carries on. Both events lead to an outside of today’s Bollinger band event.

100%-outside probability is the probability to be inside todays band in 20 days from now. It is less than 50%, regardless of today’s market relative to the band.

Stability of results

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Scanning for Support and Resistance Probabilities

I have been in search for a signal I could use for a short vertical spread or naked short option strategy. So my main concern has been to find a level, which will most probably not be penetrated over the next few bars.

This is what I came up with.

Support and Resistance

We are all familiar with oscillators like the RSI indicator. It gives an idea if the market is oversold or overbought.

The chart gives a basic idea of the signal I am looking for. Once the indicator is leaving the overbought / oversold area, there should be a good chance that the market actually stays above or below it’s previous high or low. If this probability is high enough, it would be a great signal for a short vertical spread or to sell a naked put / call option. (be aware of the unlimited risk in the naked short trade!) Both strategies win, if the selected level is not penetrated at expiry.

What is manually drawn on the chart above can also be done automatically. The following chart shows how it looks like if you use the code given at the end of the article.

Every time the RSI leaves the extreme zones the indicator will draw the previous high or low for a given prognosis interval. To enhance the chances and not to get too many signals in a trending market I also made use use of the ADX indicator. So to see a signal on the chart, RSI has to leave the extreme level while ADX signals a sideway market. This should give the best signal quality.

The three signals shown would have resulted in a winning trade as the market did not cross the shown support / Resistance levels. But how does it work out in the long term?

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Weekend Reading Recommendation

The markets will go up and down, and usually it’s not my business why they do it, I am just interested in making  my luck with a position on the right side of the trade.

But of course markets don’t move just because of  fear and greed, but because of demand and supply. And these two factors are deeply founded in the “real” world.

Michael Roberts, a London based economist with lots of markets experience, is doing an fantastic blog which explains the foundation of the markets with a lot of nicely prepared data and based on a sound economic theory – Marxism.  Don’t let us start a political discussion over here, but have a look at his blog, see the data, read his arguments and get a broader view of the market than you would get by just watching the charts and reading the daily news.

 

 

 

 

Backtesting Market Volatility

If you want to predict volatility, you can place a bet on the option market. Just buy an at the money put and call, and at expiry day you will either win or lose, depending on the actual market move since you bought the straddle and the price you paid for the straddle. To put it simple, if the market moves more than you paid for the two options you will win, otherwise you will lose.

The fair price for volatility

When I look at the S&P500 I could buy or short a straddle with 16 business days until expiry right now for around 70$. That’s the implied volatility.

When I look at the standard deviation of 16 day returns, using the last 30 days to calculate it, it shows me a volatility of around 30$. That’s historical volatility.

When I use my own fair bet KVOL Volatility, it gives me a volatility of about 50$

Now I got three measures for volatility, but which one is the best prediction for future market volatility? And how big will the error (=wins and losses) be if we place this bet over and over again?

Backtesting volatility

Placing an perpetual bet on future volatility using the payback profile of a short straddle will give me an idea on how good historical volatility and Kahler’s volatility was able to predict future volatility. In a perfect world this virtual test strategy should be zero sum game; if not, future volatility is either over or underestimated by these 2 indicators.

If I know which indicator gives me the best volatility assumption, I can use this information to find out, if the current implied volatility of the market is too high or too low. Sell high, buy low…

The chart above gives you an idea on how I did the backtest. I place a weekly bet on volatility, based on a short straddle trade. So if I close outside the of the projected volatility, I have a loss. If the market closes inside the projection, I win. The maximum win will be the price of the volatility indicator at the beginning of the bet, the max loss is the point move within the week minus the price I got for volatility at the beginning of the bet.

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Demystifying the 200 day average

The 200 day average is usually considered as a key indicator to tell if markets are bullish or bearish. But can this be proved statistically, or is it just an urban legend handed down from one generation to the next?

Let’s do some studies and find out.

The 200 day moving average

Looking at the chart of the S&P500 index and it’s 200 day average let’s me think that the 200 day average is actually a useful indicator to separate the bull and bear phases of the market. But the eye tends to see what the brain is looking for, so you might have focused on the crash 2008 and the bull market afterwards, but have ignored all these little breaches below the 200 day moving average which happened in between. As a trader I can`t make any money with knowing that there has been a long period under the 200 day average, i need to make my decision as soon as the market drops below the 200 day average.

Distribution of returns above and below the 200 day moving average

The chart above (on the right side) shows the returns distribution of 10 day returns. The green distribution represents the 10 day returns if S&P500 is trading above it`s 200 day moving average, the red line represents the 10 day returns returns when the market is trading below its 200 day moving average. Data from 1980 up to now has been used.

What are the curves telling us?

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Google EOD csv stock price data download

Sometimes my data provider has not got the data I am looking for. Searching for downloadable csv data I recently came across google spreadsheets. It provides an easy way to get historical stock price data. Save it as csv and use it with your Tradesignal.

The only thing you will have to do is to open a google spreadsheet in your browser, add a formula as shown in one cell and the data will be pulled. Copy&Paste the data to another spreadsheet and save it as csv.

Money for nothing

We already had a post regarding the mean reverting tendency of Volatility, now it`s time to make some money using this information.

Trading Volatility

The VIX volatility index on the chart above looks like an easy to trade instrument, just buy when it is around 10 and sell when it has doubled, tripled, quadrupled…

But unfortunately life is not that easy, VIX is just an index and you will not be able to buy or sell it. You might try to trade volatility using options, but there is a better plan to make money on this wonderful asset class, the VXX, BRCL BK IPTH S&P 500 VIX SH FTRS ETN.

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Seasonal trouble ahead

If a bitchy prime minister and a crazy president weren’t enough, for the upcoming months the seasonal chart is also indicating further price setbacks.

Seasonality of DAX

Analyzing the average monthly performance of the German DAX index a distinct pattern of seasonality can be observed. On average June has been down 0.6%, but the big trouble is yet to come.

 

The chart shows the average monthly performance of the DAX index, using data since 1999.

Each bar of the histogram represents a specific months percent performance. Starting with the dark blue bar in January, the green bar right now represents the average June performance.

Seasonal  and Volatility Prognosis:

As you can see on the above chart June always has had a bearish prognosis over the last years. July might bring some relief (the positive magenta bar behind June), but therefore August and September surely got a strong bearish setup. Although the markets have been bullish over the last 20 years, the average combined performance of August and September is -4%.

The average performance of a month is not a good indication for the actual magnitude of the upcoming market move, it just is an indication for its direction.

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KVOL Volatility part 2

How to calculate volatility based on the expected return of a straddle strategy has been shown in part 1 of fair bet volatility KVOL.

Using and Displaying Volatility:

KVOL uses the given amount of historic returns to calculate an expected value of an at the money put and call option. The sum of these prices are the historic fair value for implied volatility. It can be used to compare current market implied volatility to historic fair values.

Beside calculating KVOL for a specific return period it can also be used to show it as a projection indicator on the chart.

The example on the chart gives such an expectation channel for the s&P500 at the beginning of each month. The 250 days before are used to calculate KVOL. The line underneath the chart is running KVOL for 13 trading days.

Simplified trading:

to win, with higher volatility expected: you would have bought a straddle at the beginning of the month, expiring at the end of the month. You should not have paid more than a KVOL for 25 bars (working days to expiry) would have suggested. You win if the chart is outside of the projection at the end of the month.

The shown example uses the 250 daily bars before  the beginning of the month to calculate the returns and the price of KVOL. The projected lines represent the winning boundaries of the straddle at expiry.

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