Stationarity and Fat Tails

In this post I am going to explore how, starting from the assumption of a stable, Gaussian distribution in a returns process, we evolve to a system that displays all the characteristics of empirical market data, notably time-dependent moments, high levels of kurtosis and fat tails.  As it turns out, the only additional assumption one needs to make is that the market is periodically disturbed by the random arrival of news.

NOTE:  if you are unable to see the Mathematica models below, you can download the free Wolfram CDF player and you may also need this plug-in.

You can also download the complete Mathematica CDF file here.


A stationary process is one that evolves over time, but whose probability distribution does not vary with time. As the word implies, such a process is stable. More formally, the moments of the distribution are independent of time.

Let’ s assume we are dealing with such a process that have constant mean μ and constant volatility (standard deviation) σ.


Here are some examples of Normal probability distributions, with constant mean μ = 0 and standard deviation σ ranging from 0.75 to 2



Chart 1

The moments of Φ are given by:

 Through[{Mean, StandardDeviation, Skewness, Kurtosis}[Φ]]

{μ,  σ,  0,   3}

They, too, are time – independent.

We can simulate some observations from such a process, with, say, mean μ = 0 and standard deviation σ = 1:

ListPlot[sampleData=RandomVariate[Φ /.{μ→0, σ→1},10^4]]


Chart 2


Chart 3

If we assume for the moment that such a process is an adequate description of an asset returns process, we can simulate the evolution of a price process as follows :


Chart 4


An Empirical Distribution

Lets take a look at a real price series, comprising 1 – minute bar data in the June ‘ 14 E – Mini futures contract.

Chart 5

As with our simulated price process, it is clear that the real price process for Emini futures is also non – stationary.

What about the returns process?


Chart 6

Notice the banding effect in returns, which results from having a fixed, minimum price move of $12 .50, rather than a continuous scale.



Chart 7


{-0.00867214,  0.0112353,  2.75501×10-6,   0.,   0.000780895,   0.35467,   26.2376}

The empirical returns distribution doesn’ t appear to be Gaussian – the distribution is much more peaked than a standard Normal distribution with the same mean and standard deviation. And the higher moments don’t fit the Normal model either – the empirical distribution has positive skew and a kurtosis that is almost 9x greater than a Gaussian distribution. The latter signifies what is often referred to as “fat tails”: the distribution has much greater weight in the tails than a standard Normal distribution, indicating a much greater likelihood of an extreme value than a Normal distribution would predict.

Non-Stationarity: Two States

Non – stationarity arises when one or more of the moments of a distribution vary over time. Let’s take a look at how that can arise, and its effects.Suppose we have a Gaussian returns process for which the mean, or drift, or trend, fluctuates over time.

Let’s consider a simple example where the process drift is  μ1 and volatility σ1 for most of the time and then for some proportion of time k, we get addition drift  μ2 and volatility σ2.  In other words we have:



{μ1,   σ1,   0,   3}



{μ2,   σ2,   0,   3}

This simple model fits a scenario in which we suppose that the returns process spends most of its time in State 1, in which is Normally distributed with  drift is  μ1 and volatility σ1, and suffers from the occasional “shock” which propels the systems into a second State 2, in which its distribution is a combination of its original distribution and a new Gaussian distribution with different mean and volatility.

Let’ s suppose that we sample the combined process y =  Φ1 + k  Φ2.   What distribution would it have?  We can represent this is follows :

 y=TransformedDistribution[(x1+k x2),{x11,x22}]




 Plot[PDF[y,x]/.{μ10,μ20,σ1 1,σ2 2, k0.5},{x,-6,6},FillingAxis]

Chart 8

The result is just another Normal distribution. Depending on the incidence k, y will follow a Gaussian distribution whose mean and variance depend on the mean and variance of the two Normal distributions being mixed. The resulting distribution in State 2 may have higher or lower drift and volatility, but it is still Gaussian, with constant kurtosis of 3.

In other words, the system y will be non-stationary, because the first and second moments change over time, depending on what state it is in. But the form of the distribution is unchanged – it is still Gaussian. There are no fat-tails.

Non – Stationarity : Random States

In the above example the system moved between states in a known, predictable way. The “shocks” to the system were not really shocks, but transitions. But that’s not how financial markets behave: markets move from one state to another in an unpredictable way, with the arrival of news.

We can simulate this situation as follows. Using the former model as a starting point, lets now relax the assumption that the incidence of the second state, k, is a constant. Instead, let’ s assume that k is itself a random variable. In other words we are going to now assume that our system changes state in a random way. How does this alter the distribution?

An appropriate model for λ might be a Poisson process, which is often used as a model for unpredictable, discrete events, ranging from bus arrivals to earthquakes.  PDFs of Poisson distributions with means  λ=5, 10 and 20 are shown in the chart below.  These represent probability distributions for processes that have mean  arrivals of 5, 10 or 20 events.


Chart 9

Our new model now looks like this :


The first two moments of the distribution are as follows :



As before, the mean and standard deviation of the distribution are going to vary, depending on the state of the system, and the mean arrival rate of shocks, . But what about kurtosis? Is it still constant?



Emphatically not!  The fourth moment of the distribution is now dependent on the drift in the second state, the volatilities of both states and the mean arrival rate of shocks, λ.

Let’ s look at a specific example.  Assume that in State 1 the process has volatility of 7.5 %, with zero drift, and that the shock distribution also has zero drift with volatility of 65 %. If the mean incidence rate of shocks λ = 10 %, the distribution kurtosis is close to that seen in the empirical distribution for the E-Mini.

 Kurtosis[y] /.{σ10.075,μ20,σ20.65,λ→0.1}


More generally :

 ListLinePlot[Flatten[Kurtosis[y]/.Table[{σ10.075,μ20,σ20.65,λ→i/20},{i,1,20}]],PlotLabelStyle["Kurtosis vs Mean Shock Arrival Rate", FontSize18],AxesLabel->{“Incidence Rate (%)”, “Kurtosis”},FillingAxis, ImageSizeLarge]


Chart 10

Thus we can see how, even if the underlying returns distribution is Gaussian in form, the random arrival of news “shocks” to the system can induce non – stationarity in overall drift and volatility. It can also result in fat tails. More specifically, if the arrival of news is stochastic in nature, rather than deterministic, the process may exhibit far higher levels of kurtosis than in its original Gaussian state, in which the fourth moment was a constant level of 3.

Jump Diffusion

Nobel – prize winning economist Robert Merton extended this basic concept to the realm of stochastic calculus.

In Merton’s jump diffusion model, the stock price follows the random process

∂St / St =μdt + σdWt+(J-1)dNt

The first two terms are familiar from the Black–Scholes model : drift rate μ, volatility σ, and random walk Wt (Wiener process).The last term represents the jumps :J is the jump size as a multiple of stock price, while Nt is the number of jump events that have occurred up to time assumed to follow the Poisson process.


where λ is the average frequency with which jumps occur.

The jump size J follows a log – normal distribution

 PDF[LogNormalDistribution[m, ν], s]

where m is the average jump size and v is the volatility of the jump size.

In the jump diffusion model, the stock price St follows the random process dSt/St=μ dt+σ dWt+(J-1) dN(t), which comprises, in order, drift, diffusive, and jump components. The jumps occur according to a Poisson distribution and their size follows a log-normal distribution. The model is characterized by the diffusive volatility σ, the average jump size J (expressed as a fraction of St), the frequency of jumps λ, and the volatility of jump size ν.

The Volatility Smile

The “implied volatility” corresponding to an option price is the value of the volatility parameter for which the Black-Scholes model gives the same price. A well-known phenomenon in market option prices is the “volatility smile”, in which the implied volatility increases for strike values away from the spot price. The jump diffusion model is a generalization of Black–Scholes in which the stock price has randomly occurring jumps in addition to the random walk behavior. One of the interesting properties of this model is that it displays the volatility smile effect. In this Demonstration, we explore the Black–Scholes implied volatility of option prices (equal for both put and call options) in the jump diffusion model. The implied volatility is modeled as a function of the ratio of option strike price to spot price.


Posted in Fat Tails, Mathematica, Regime Shifts, Regime Switching, Uncategorized, Volatility Modeling | Tagged , , , , , , | Comments Off

The Mathematics of Scalping

NOTE:  if you are unable to see the Mathematica models below, you can download the free Wolfram CDF player and you may also need this plug-in.

You can also download the complete Mathematica CDF file here.

In this post I want to explore aspects of scalping, a type of strategy widely utilized by high frequency trading firms.

I will define a scalping strategy as one in which we seek to take small profits by posting limit orders on alternate side of the book. Scalping, as I define it, is a strategy rather like market making, except that we “lean” on one side of the book. So, at any given time, we may have a long bias and so look to enter with a limit buy order. If this is filled, we will then look to exit with a subsequent limit sell order, taking a profit of a few ticks. Conversely, we may enter with a limit sell order and look to exit with a limit buy order.
The strategy relies on two critical factors:

(i) the alpha signal which tells us from moment to moment whether we should prefer to be long or short
(ii) the execution strategy, or “trade expression”

In this article I want to focus on the latter, making the assumption that we have some kind of alpha generation model already in place (more about this in later posts).

There are several means that a trader can use to enter a position. The simplest approach, the one we will be considering here, is simply to place a single limit order at or just outside the inside bid/ask prices – so in other words we will be looking to buy on the bid and sell on the ask (and hoping to earn the bid-ask spread, at least).

One of the problems with this approach is that it is highly latency sensitive. Limit orders join the limit order book at the back of the queue and slowly works their way towards the front, as earlier orders get filled. Buy the time the market gets around to your limit buy order, there may be no more sellers at that price. In that case the market trades away, a higher bid comes in and supersedes your order, and you don’t get filled. Conversely, yours may be one of the last orders to get filled, after which the market trades down to a lower bid and your position is immediately under water.

This simplistic model explains why latency is such a concern – you want to get as near to the front of the queue as you can, as quickly as possible. You do this by minimizing the time it takes to issue and order and get it into the limit order book. That entails both hardware (co-located servers, fiber-optic connections) and software optimization and typically also involves the use of Immediate or Cancel (IOC) orders. The use of IOC orders by HFT firms to gain order priority is highly controversial and is seen as gaming the system by traditional investors, who may end up paying higher prices as a result.

Another approach is to layer limit orders at price points up and down the order book, establishing priority long before the market trades there. Order layering is a highly complex execution strategy that brings addition complications.

Let’s confine ourselves to considering the single limit order, the type of order available to any trader using a standard retail platform.

As I have explained, we are assuming here that, at any point in time, you know whether you prefer to be long or short, and therefore whether you want to place a bid or an offer. The issue is, at what price do you place your order, and what do you do about limiting your risk? In other words, we are discussing profit targets and stop losses, which, of course, are all about risk and return.

Risk and Return in Scalping

Lets start by considering risk. The biggest risk to a scalper is that, once filled, the market goes against his position until he is obliged to trigger his stop loss. If he sets his stop loss too tight, he may be forced to exit positions that are initially unprofitable, but which would have recovered and shown a profit if he had not exited. Conversely,  if he sets the stop loss too loose, the risk reward ratio is very low – a single loss-making trade could eradicate the profit from a large number of smaller, profitable trades.

Now lets think about reward. If the trader is too ambitious in setting his profit target he may never get to realize the gains his position is showing – the market could reverse, leaving him with a loss on a position that was, initially, profitable. Conversely, if he sets the target too tight, the trader may give up too much potential in a winning trade to overcome the effects of the occasional, large loss.

It’s clear that these are critical concerns for a scalper: indeed the trade exit rules are just as important, or even more important, than the entry rules. So how should he proceed?

Theoretical Framework for Scalping

Let’s make the rather heroic assumption that market returns are Normally distributed (in fact, we know from empirical research that they are not – but this is a starting point, at least). And let’s assume for the moment that our trader has been filled on a limit buy order and is looking to decide where to place his profit target and stop loss limit orders. Given a current price of the underlying security of X, the scalper is seeking to determine the profit target of p ticks and the stop loss level of q ticks that will determine the prices at which he should post his limit orders to exit the trade. We can translate these into returns, as follows:

to the upside: Ru = Ln[X+p] – Ln[X]

and to the downside: Rd = Ln[X-q] – Ln[X]

This situation is illustrated in the chart below.

Normal Distn Shaded

The profitable area is the shaded region on the RHS of the distribution. If the market trades at this price or higher, we will make money: p ticks, less trading fees and commissions, to be precise. Conversely we lose q ticks (plus commissions) if the market trades in the region shaded on the LHS of the distribution.

Under our assumptions, the probability of ending up in the RHS shaded region is:

probWin = 1 – NormalCDF(Ru, mu, sigma),

where mu and sigma are the mean and standard deviation of the distribution.

The probability of losing money, i.e. the shaded area in the LHS of the distribution, is given by:

probLoss = NormalCDF(Rd, mu, sigma),

where NormalCDF is the cumulative distribution function of the Gaussian distribution.

The expected profit from the trade is therefore:

Expected profit = p * probWin – q * probLoss

And the expected win rate, the proportion of profitable trades, is given by:

WinRate = probWin / (probWin + probLoss)

If we set a stretch profit target, then p will be large, and probWin, the shaded region on the RHS of the distribution, will be small, so our winRate will be low. Under this scenario we would have a low probability of a large gain. Conversely, if we set p to, say, 1 tick, and our stop loss q to, say, 20 ticks, the shaded region on the RHS will represent close to half of the probability density, while the shaded LHS will encompass only around 5%. Our win rate in that case would be of the order of 91%:

WinRate = 50% / (50% + 5%) = 91%

Under this scenario, we make frequent, small profits  and suffer the occasional large loss.

So the critical question is: how do we pick p and q, our profit target and stop loss?  Does it matter?  What should the decision depend on?

Modeling Scalping Strategies

We can begin to address these questions by noticing, as we have already seen, that there is a trade-off between the size of profit we are hoping to make, and the size of loss we are willing to tolerate, and the probability of that gain or loss arising.  Those probabilities in turn depend on the underlying probability distribution, assumed here to be Gaussian.

Now, the Normal or Gaussian distribution which determines the probabilities of winning or losing at different price levels has two parameters – the mean, mu, or drift of the returns process and sigma, its volatility.

Over short time intervals the effect of volatility outweigh any impact from drift by orders of magnitude.  The reason for this is simple:  volatility scales with the square root of time, while the drift scales linearly.  Over small time intervals, the drift becomes un-noticeably small, compared to the process volatility.  Hence we can assume that mu, the process mean is zero, without concern, and focus exclusively on sigma, the volatility.

What other factors do we need to consider?  Well there is a minimum price move, which might be 1 tick, and the dollar value of that tick, from which we can derive our upside and downside returns, Ru and Rd.  And, finally, we need to factor in commissions and exchange fees into our net trade P&L.

Here’s a simple formulation of the model, in which I am using the E-mini futures contract as an exemplar.

 WinRate[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, tgtReturn, slReturn,tgtDollar, slDollar, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;
probWin=1-CDF[NormalDistribution[0, periodVolatility],tgtReturn];
probLoss=CDF[NormalDistribution[0, periodVolatility],slReturn];
{expProfit, winRate}]

For the ES contract we have a min price move of 0.25 and the tick value is $12.50.  Notice that we scale annual volatility to the size of the period we are trading (15 minute bars, in the following example).

Scenario Analysis

Let’s take a look at how the expected profit and win rate vary with the profit target and stop loss limits we set.  In the following interactive graphics, we can assess the impact of different levels of volatility on the outcome.

Expected Profit by Bar Size and Volatility

Expected Win Rate by Volatility

Notice to begin with that the win rate (and expected profit) are very far from being Normally distributed – not least because they change radically with volatility, which is itself time-varying.

For very low levels of volatility, around 5%, we appear to do best in terms of maximizing our expected P&L by setting a tight profit target of a couple of ticks, and a stop loss of around 10 ticks.  Our win rate is very high at these levels - around 90% or more.  In other words, at low levels of volatility, our aim should be to try to make a large number of small gains.

But as volatility increases to around 15%, it becomes evident that we need to increase our profit target, to around 10 or 11 ticks.  The distribution of the expected P&L suggests we have a couple of different strategy options: either we can set a larger stop loss, of around 30 ticks, or we can head in the other direction, and set a very low stop loss of perhaps just 1-2 ticks.  This later strategy is, in fact, the mirror image of our low-volatility strategy:  at higher levels of volatility, we are aiming to make occasional, large gains and we are willing to pay the price of sustaining repeated small stop-losses.  Our win rate, although still well above 50%, naturally declines.

As volatility rises still further, to 20% or 30%, or more, it becomes apparent that we really have no alternative but to aim for occasional large gains, by increasing our profit target and tightening stop loss limits.   Our win rate under this strategy scenario will be much lower – around 30% or less.

Non – Gaussian Model

Now let’s address the concern that asset returns are not typically distributed Normally. In particular, the empirical distribution of returns tends to have “fat tails”, i.e. the probability of an extreme event is much higher than in an equivalent Normal distribution.

A widely used model for fat-tailed distributions in the Extreme Value Distribution. This has pdf:




EVD pdf




EVD Variance

In order to set the parameters of the EVD, we need to arrange them so that the mean and variance match those of the equivalent Gaussian distribution with mean = 0 and standard deviation . hence:

EVD params

The code for a version of the model using the GED is given as follows

WinRateExtreme[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, alpha, beta,tgtReturn, slReturn,tgtDollar, slDollar, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
beta = Sqrt[6]*periodVolatility / Pi;
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;
probWin=1-CDF[ExtremeValueDistribution[alpha, beta],tgtReturn];
probLoss=CDF[ExtremeValueDistribution[alpha, beta],slReturn];
{expProfit, winRate}]



We can now produce the same plots for the EVD version of the model that we plotted for the Gaussian versions :

Expected Profit by Bar Size and Volatility – Extreme Value Distribution

Expected Win Rate by Volatility – Extreme Value Distribution

Next we compare the Gaussian and EVD versions of the model, to gain an understanding of how the differing assumptions impact the expected Win Rate.

Expected Win Rate by Stop Loss and Profit Target

As you can see, for moderate levels of volatility, up to around 18 % annually, the expected Win Rate is actually higher if we assume an Extreme Value distribution of returns, rather than a Normal distribution.If we use a Normal distribution we will actually underestimate the Win Rate, if the actual return distribution is closer to Extreme Value.In other words, the assumption of a Gaussian distribution for returns is actually conservative.

Now, on the other hand, it is also the case that at higher levels of volatility the assumption of Normality will tend to over – estimate the expected Win Rate, if returns actually follow an extreme value distribution. But, as indicated before, for high levels of volatility we need to consider amending the scalping strategy very substantially. Either we need to reverse it, setting larger Profit Targets and tighter Stops, or we need to stop trading altogether, until volatility declines to normal levels.Many scalpers would prefer the second option, as the first alternative doesn’t strike them as being close enough to scalping to justify the name.If you take that approach, i.e.stop trying to scalp in periods when volatility is elevated, then the differences in estimated Win Rate resulting from alternative assumptions of return distribution are irrelevant.

If you only try to scalp when volatility is under, say, 20 % and you use a Gaussian distribution in your scalping model, you will only ever typically under – estimate your actual expected Win Rate.In other words, the assumption of Normality helps, not hurts, your strategy, by being conservative in its estimate of the expected Win Rate.

If, in the alternative, you want to trade the strategy regardless of the level of volatility, then by all means use something like an Extreme Value distribution in your model, as I have done here.That changes the estimates of expected Win Rate that the model produces, but it in no way changes the structure of the model, or invalidates it.It’ s just a different, arguably more realistic set of assumptions pertaining to situations of elevated volatility.

Monte-Carlo Simulation Analysis

Let’ s move on to do some simulation analysis so we can get an understanding of the distribution of the expected Win Rate and Avg Trade PL for our two alternative models. We begin by coding a generator that produces a sample of 1,000 trades and calculates the Avg Trade PL and Win Rate.

Gaussian Model

GenWinRate[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, randObs, tgtReturn, slReturn,tgtDollar, slDollar, nWins,nLosses, perTradePL, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;



Now we can generate a random sample of 10, 000 simulation runs and plot a histogram of the Win Rates, using, for example, ES on 5-min bars, with a PT of 2 ticks and SL of – 20 ticks, assuming annual volatility of 15 %.

Histogram[Table[GenWinRate[1900,0.15,5,2,-20,0.25,12.50,3][[2]],{i,10000}],10,AxesLabel{“Exp. Win Rate (%)”}]


Histogram[Table[GenWinRate[1900,0.15,5,2,-20,0.25,12.50,3][[1]],{i,10000}],10,AxesLabel{“Exp. PL/Trade ($)”}]


Extreme Value Distribution Model

Next we can do the same for the Extreme Value Distribution version of the model.

GenWinRateExtreme[currentPrice_,annualVolatility_,BarSizeMins_, nTicksPT_, nTicksSL_,minMove_, tickValue_, costContract_]:=Module[{ nMinsPerDay, periodVolatility, randObs, tgtReturn, slReturn,tgtDollar, slDollar, alpha, beta,nWins,nLosses, perTradePL, probWin, probLoss, winRate, expWinDollar, expLossDollar, expProfit},
nMinsPerDay = 250*6.5*60;
periodVolatility = annualVolatility / Sqrt[nMinsPerDay/BarSizeMins];
beta = Sqrt[6]*periodVolatility / Pi;
tgtReturn=nTicksPT*minMove/currentPrice;tgtDollar = nTicksPT * tickValue;
slReturn = nTicksSL*minMove/currentPrice;
randObs=RandomVariate[ExtremeValueDistribution[alpha, beta],10^3];

Histogram[Table[GenWinRateExtreme[1900,0.15,5,2,-10,0.25,12.50,3][[2]],{i,10000}],10,AxesLabel{“Exp. Win Rate (%)”}]


Histogram[Table[GenWinRateExtreme[1900,0.15,5,2,-10,0.25,12.50,3][[1]],{i,10000}],10,AxesLabel{“Exp. PL/Trade ($)”}]





The key conclusions from this analysis are:

  1. Scalping is essentially a volatility trade
  2. The setting of optimal profit targets are stop loss limits depend critically on the volatility of the underlying, and needs to be handled dynamically, depending on current levels of market volatility
  3. At low levels of volatility we should set tight profit targets and wide stop loss limits, looking to make a high percentage of small gains, of perhaps 2-3 ticks.
  4. As volatility rises, we need to reverse that position, setting more ambitious profit targets and tight stops, aiming for the occasional big win.


Posted in Futures, Mathematca, Mathematica, Scalping, Trading, Volatility Modeling | Tagged , , , , , , , , , , , , , , , , | Comments Off

How to Spot a Fake

One of the issues that comes up regularly is how, as an investor or other interested party, one can protect oneself from unscrupulous scam artists posing as professional traders or money managers. This is a particular problem on web sites featuring trader forums, where individuals with unverified track records claiming stellar trading histories use their purported trading “prowess” to try to impress and intimidate other participants, usually impressionable newbies. The purpose of this post is to provide some guidance to help investors, traders and other fellow travelers sort the wheat from the chaff. We’ll be doing some forensic analysis on the track record for a strategy in NG futures that one such character recently posted in one of these forums, as a classic example of the kind of fakery I am describing.

One thing you should understand about scam artists operating on forums, is that they don’t work alone: usually they have a bunch of groupies who will shill for them at every opportunity and who will try to shout down any investigative questioning. Don’t be deterred. These know-it-alls are usually just ignorant dupes, who understand no more about trading than the scam artist. They may just as easily be fellow-scam artists themselves.

Anyone claiming to be a CTA or professional money manager (or whose shills claim he is one) has to have a track record that is freely available in the public domain. So how does a scam artist overcome a challenge to produce it? He will claim that he “can’t advertise”, or make some other, similar excuse. Don’t accept that at face value. Ask him to PM it to you. If he won’t, there’s already a high probability he’s a con artist.

Let’s say our suspect meets the challenge and produces a track record. Ideally this will be an audited P&L statement, but let’s assume for the purposes of this discussion that he produces something along the lines of the Performance Reports produced by a product like Tradestation or MultiCharts, i.e. we are dealing with a simulated back-test.

If your suspect produces a back-test, you can be pretty sure it’s going to look good – otherwise he wouldn’t produce it. The task now is to dig into those reports to spot the red flags that give clues as to whether it might be fake.
Now of course any trading system is going to make assumptions – about fill rates, slippage, commissions, capacity etc. All that is fine, as long as the assumptions are clearly stated. You might want to challenge any or all of the assumptions, and the trader may disagree with you about some or all of them. That’s perfectly ok – it’s an honest, open discussion about a set of investment assumptions that have been revealed at the outset.

But here is what is NOT ok: any opacity about which data was used to build the trading model and which data was used to test it. The former, the in-sample (IS) data set, used to construct the model, must be entirely separate and distinct from the out-of-sample (OOS) data set. It is trivially easy using a tool like Tradestation to produce a trading system that shows stellar results in-sample, but which will immediately crash and burn when it is used in live trading. This is known as curve-fitting. And it’s by far the most common method by which scam artists try to dupe investors.

In order to demonstrate the robustness of the system prior to risking real money, a genuine trader will test his system OOS and show you the results. What you are looking for ideally is congruity between the IS and OOS results. Now by congruity, I don’t mean that they should be identical. Far from it – markets evolve and strategy performance will vary over time. But what you are hoping is that the key performance metrics in the OOS and IS periods, such as annual returns, Sharpe ration, PNL per contract, profit ratio and win rate, will be comparable. At the very least, you would like to be able to identify some portion of the IS data set for which the strategy performance characteristics are similar to those in the OOS period.

Any – I mean ANY – ambiguity or lack of clarity about which data was used to build the model and which was used for OOS testing is a HUGE red flag. Chances are, your scam artist is already trying to fudge the issue that he curve-fitted the system.
This was the case in the recent forum post we are using as a test case. The trader made no attempt whatsoever to clarify which data was used for model development and which for testing. Immediately, I was suspicious and began looking for other evidence of curve fitting. It didn’t take me long to find it.

The first item I turned to in the performance reports was the equity curve and I immediately spotted two rather large clues that I was dealing with a fake.

The first clue was the large sign on the chart labelled “live start date”. What does this mean? This is a back-test, so all of the results are theoretical, including those after the supposed “live start date” sometime in 2013. What the faker is trying to do is imply the part of the equity curve shown after that date indicate actual performance results. He doesn’t actually claim this, so he has plausible deniability if you call him on it (“I said it was just a back test”). But he hopes that you won’t, and that, by default, you’ll accept these results are real. But they aren’t.

The second clue of fakery is much more important: the equity curve itself. When someone shows you and equity curve like the one reported by this trader, rising in a straight line from the lower left to upper right quadrants, you can be 99% confident that you are dealing with a fake.
You see, in finance there are almost never any straight lines. They are as rare as unicorns. Especially when it comes to strategy performance. They only time you will EVER see an equity curve like this is when you are looking at the equity curve of (i) a high frequency market making trading system or (ii) a fake, produced by curve fitting a strategy to the ENTIRE data set.
And this strategy was not high frequency – as we shall see, it operated on 15 minute bars, holding positions overnight.

EC Chart

I said that straight line equity curve were extremely rare. In fact, even God’s equity curve isn’t often a straight line. What does that mean?

Suppose you had a strategy that could predict with 100% accuracy whether the market would go up or down over the next bar (whether you are using daily bars, or 15 minute bars, as in our example). The system would buy (or hold) when the market was forecast to rise, and sell when the market was predicted to fall. What would the performance of such a perfect system look like? Pretty stellar, obviously. And most people would guess that the system’s equity curve would be a straight line, or maybe even exponential in shape. In fact that’s typically not the case. God’s equity curve will be sloped and kinked, just like any other equity curve. And if your suspect’s equity curve is real, it should show some commonality with God’s equity curve, by which I mean it should show changes in slope and level that reflect those seen in the perfect equity curve.

What does God’s Equity Curve look like in NG futures?

Gods EC

As you can see it’s not straight. In fact it’s concave. So a REAL equity curve should have similar characteristics, like this one, for example:


As you can see, the equity curve of the real trading system track’s God’s Equity Curve, albeit at a much lower level. It’s concave, with an upswing during the final few months of trading, just like God’s. That’s a good sign that the strategy back-test is very likely genuine (which it is – I produced it).

Why is Gods’ Equity Curve the shape it is? The answer will vary from market to market. In the case of NG, the suggestion is that the market is becoming more efficient: simple trading strategies based on technical indicators work less well than they did five years ago. We have seen something very similar in F/X markets. During the 1970’s and 1980’s when Soros was active in the field, simple strategies like moving average crossovers made great returns, but these entirely dissipated in the 1990’s, with the advent of widely available computing power.

When I posted my analysis, which clearly indicated fakery by this well known forum participant, I was immediately flamed by one of his supporters who shouted something to the effect that (i) everyone knows that the downward slope of God’s Equity Curve was caused by volatility and (ii) the star trader, unlike God, or me, knows about position sizing.

This attempt at misdirection in the face of awkward facts is a classic sign of fakery. What distinguishes the shill post is:

(i) Immediacy – clearly no attempt has been made to evaluate the argument or analysis. The shill simply attempts to drown out the critic with a lot of noise, as quickly as possible.

(ii) Plausibility – shills will throw around terms that lend plausibility to their objection, but which after a moment’s reflection are entirely irrelevant or, as in this case, detrimental to their own cause.

(iii) Invective – the more intemperate the post, the more likely the shill is simply trying to provide cover for the faker.

So let’s take a moment to dispose of the plausible sounding objections posted by the shill in this example.
I am going to take it as read that everyone understands that trading profitability is positively correlated with volatility. There is a huge amount of empirical research supporting that finding, but to keep it simple we can appeal to one of the cornerstones of modern finance: risk and return. The higher the volatility, i.e. the greater the risk, the greater the return traders and investors in the markets will require on their capital. This is a principle of modern financial theory that even a graduate of the Scranton college of fine art should be expected to appreciate.

So what’s the story with NG volatility? You can see the time series of NG volatility in the chart below. One feature stands out above all others: the upward slope of the curve. NG volatility has RISEN over the sample period from 2008 to 2014. Consequently, returns from trading NG futures should also have RISEN rather than fallen. One thing we can say for sure, whatever caused the concave shape in God’s Equity Curve in NG futures, it was NOT volatility!

NG Volatility

Turning to the shill’s next, plausible sounding, but dubious “explanation”, position sizing: this really is completely irrelevant. Because, as we shall see from an examination of the performance report, the track record was created by trading a constant one-lot! So this was just an attempt to sound “sophisticated” by someone trying to misdirect the reader away from the increasingly obvious evidence of fakery.

One of the highly unusual features of our faker’s equity curve is it’s exceptional smoothness. Low volatility in the equity curve is, in and of itself, an indicator the track record results from curve fitting. But we can get even more insight by digging into the performance report, shown below.

Perf 1
Perf 2

As you can see from the second page of the report, the strategy holds positions for an average of 57 15-minute bars, equivalent to slightly over 14 hours. So this is a low frequency strategy that takes overnight risk. Now, as any trader will know, overnight gap risk in a product like NG can be very significant and likely to be produce much larger drawdowns over a 5 year period than the $8,470 reported here.

The only other possible explanation is that the strategy is traded continuously through both day and night sessions. But this is not only itself improbable, it gives rise to another implausibility: liquidity in the overnight session is so poor that the strategy is unlikely to be able to trade more than 1-2 contracts, at most. This would be of little value to a CTA, or its customers, whatever the star trader’s protestations that his “clients are happy”.

There is no plausible way to resolve the disconnection between the low drawdown, overnight gap risk and market illiquidity. The most plausible explanation: the back-test is a curve fitting exercise.

As any experienced strategy developer knows, you can get some of the things you want, but you can never achieve all of them. Amongst the desirable features to be maximized are
• Profit factor
• Average PNL per contract
• Percentage win rate

There is a trade-off between the features. A high PNL per contract typically means you are trading less frequently, with longer hold periods, and consequently the percentage win rate tends to be lower. Alternatively, you can increase the win rate, at the cost of lowering the average PNL per contract and/or the profit factor. And so on.

This strategy purports to have it all: a high average PNL per contract resulting from low frequency trading, coupled with good percentage win rate of over 50% and profit factor. A win rate of much over 40% is highly unusual for a momentum strategy entering and exiting with market or stop orders – and its almost inconceivable for a strategy with a PNL per contract and profit factor as large as suggested here.

This back-test fails the sniff test on so many levels, I would rate the chance of it being real as less than 1 in 1000.
The final, conclusive proof of fakery is that the “star trader” responsible for producing the report was unable and/or unwilling to attempt to answer even a single one of the criticisms.

So, be warned. If you see forum members banding about track records like this one, you can be sure that they and their strategies are likely to be fake, and not to be trusted.

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A Study in Gold

I want to take a look at a trading strategy in the GDX Gold ETF that has attracted quite a lot of attention, stemming from Jay Kaeppel’s article: The Greatest Gold Stock System You’ll Probably Never Use (

The essence of the approach is that GDX has reliably tended to trade off during the day session, after making gains in the overnight session. One possible explanation for the phenomenon is offer by Adrian Douglas in his article Gold Market is not “Fixed”, it’s Rigged (see in which he takes issue with the London Fixing mechanism used to set the daily price of gold

In any event there has been a long-term opportunity to exploit what appears to be a market inefficiency using an extremely simple trading rule, as described by Oddmund Grotte (see

1) If GDX rises from the open to the close more than 0.1%, buy on the close and exit on the opening next day.
2) If GDX rises from the open to the close more than 0.1% the day before, sell short on the opening and exit on the close (you have to both sell your position from number 1 but also short some more).

Unfortunately this simple strategy has recently begun to fail, producing (substantial) negative returns since June 2013. So I have been experimenting with a number of closely-related strategies and have created a simple Excel workbook to evaluate them. First, a little notation:
RCO = Return from prior close to market open
ROC = return from open to close
RCC = Return from close to close

I also use the suffix “m1″ to denote the prior period’s return. So, for example, ROCm1 is yesterday’s return, measured from open to close. And I use the slash symbol “/” to denote dependency. So, for instance, RCO/ROCm1 means the return from today’s open to close, given the return from open to close yesterday.

In the accompanying workbook I look at several possible, closely related trading rules, and evaluate their performance over time. The basic daily data spans the period from May 2006 to July 2013 and in shown in columns A-H of the workbook. Columns I-K show the daily RCO, ROC and RCC returns. The returns for three different strategies are shown in columns L, N and P, and the cumulative returns for each are shown in columns M, N and Q, respectively.

The trading rules for each of the strategies are as follows:

COL L: RCO/ROCm1>0.5%. Which means: buy GDX at the close if the intra-day return from open to close exceeds 0.5%, and hold overnight until the following morning.

COL N: ROC/ROCm1>1%. Which means: sell GDX at today’s open if the intra-day return from open to close on the preceding day exceeds 1% and buy at today’s close.

COL P: ROC/RCCm1>1%. Which means: sell GDX at today’s open if the return from close to close on the preceding day exceeds 1% and buy at today’s close.

COL R shows returns from a blended strategy which combines the returns from the strategies in columns N and P on Mondays, Tuesdays and Thursdays only (i.e. assuming no trading on Wednesdays or Fridays). The cumulative returns from this hybrid strategy are shown in column S.

We can now present the results from the four strategies, over the 7 year period from May 2006 to July 2013, as shown in the chart and table below. On their face, the results are impressive: all four strategies have Sharpe ratios in excess of 3, with the blended strategy having a Sharpe of 4.57, while the daily win rates average around 60%.

Cumulative Returns

Performance Stats

How well do these strategies hold up over time? You can monitor their performance as you move through time by clicking on the scrollbar control in COL U of the workbook. As you do so, the start date of the strategies is rolled forward, and the table of performance results is updated to include results from the new start date, ignoring any prior data.

As you can see, all of the strategies continue to perform well into the latter part of 2010. At that point, the performance of the first strategy begins to decline precipitously, although the remaining three strategies continue to do well. By mid-2012, the first strategy is showing negative performance, while the Sharpe ratios of the remaining strategies begin to decline. As we reach the end of Q1, 2013, only the Sharpe ratio of the ROC/RCCm1 strategy remains above 2 for the period Apr 2013 to July 2013.

The conclusion appears to be that there is evidence for the possibility of generating abnormal returns in GDX lasting well into the current decade. However these have declined considerably in recent years, to a point where the effects are likely no longer important.

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Implied Volatility in Merton’s Jump Diffusion Model

The “implied volatility” corresponding to an option price is the value of the volatility parameter for which the Black-Scholes model gives the same price. A well-known phenomenon in market option prices is the “volatility smile”, in which the implied volatility increases for strike values away from the spot price. The jump diffusion model is a generalization of Black\[Dash]Scholes in which the stock price has randomly occurring jumps in addition to the random walk behavior. One of the interesting properties of this model is that it displays the volatility smile effect. In this Demonstration, we explore the Black-Scholes implied volatility of option prices (equal for both put and call options) in the jump diffusion model. The implied volatility is modeled as a function of the ratio of option strike price to spot price.

Posted in Options, Stochastic Differential Equations, Volatility Modeling | Tagged , , , | Comments Off

Option Prices in the Variance Gamma Model

Posted in Mathematica, Options, Volatility Modeling | Tagged , , , | Comments Off

Measuring Toxic Flow for Trading & Risk Management

A common theme of microstructure modeling is that trade flow is often predictive of market direction.  One concept in particular that has gained traction is flow toxicity, i.e. flow where resting orders tend to be filled more quickly than expected, while aggressive orders rarely get filled at all, due to the participation of informed traders trading against uninformed traders.  The fundamental insight from microstructure research is that the order arrival process is informative of subsequent price moves in general and toxic flow in particular.  This is turn has led researchers to try to measure the probability of informed trading  (PIN).  One recent attempt to model flow toxicity, the Volume-Synchronized Probability of Informed Trading (VPIN)metric, seeks to estimate PIN based on volume imbalance and trade intensity.  A major advantage of this approach is that it does not require the estimation of unobservable parameters and, additionally, updating VPIN in trade time rather than clock time improves its predictive power.  VPIN has potential applications both in high frequency trading strategies, but also in risk management, since highly toxic flow is likely to lead to the withdrawal of liquidity providers, setting up the conditions for a flash-crash” type of market breakdown.

The procedure for estimating VPIN is as follows.  We begin by grouping sequential trades into equal volume buckets of size V.  If the last trade needed to complete a bucket was for a size greater than needed, the excess size is given to the next bucket.  Then we classify trades within each bucket into two volume groups:  Buys (V(t)B) and Sells (V(t)S), with V = V(t)B + V(t)S
The Volume-Synchronized Probability of Informed Trading is then derived as:

Typically one might choose to estimate VPIN using a moving average over n buckets, with n being in the range of 50 to 100.

Another related statistic of interest is the single-period signed VPIN. This will take a value of between -1 and =1, depending on the proportion of buying to selling during a single period t.

Fig 1. Single-Period Signed VPIN for the ES Futures Contract

It turns out that quote revisions condition strongly on the signed VPIN. For example, in tests of the ES futures contract, we found that the change in the midprice from one volume bucket the next  was highly correlated to the prior bucket’s signed VPIN, with a coefficient of 0.5.  In other words, market participants offering liquidity will adjust their quotes in a way that directly reflects the direction and intensity of toxic flow, which is perhaps hardly surprising.

Of greater interest is the finding that there is a small but statistically significant dependency of price changes, as measured by first buy (sell) trade price to last sell (buy) trade price, on the prior period’s signed VPIN.  The correlation is positive, meaning that strongly toxic flow in one direction has a tendency  to push prices in the same direction during the subsequent period. Moreover, the single period signed VPIN turns out to be somewhat predictable, since its autocorrelations are statistically significant at two or more lags.  A simple linear auto-regression ARMMA(2,1) model produces an R-square of around 7%, which is small, but statistically significant.

A more useful model, however , can be constructed by introducing the idea of Markov states and allowing the regression model to assume different parameter values (and error variances) in each state.  In the Markov-state framework, the system transitions from one state to another with conditional probabilities that are estimated in the model.

An example of such a model  for the signed VPIN in ES is shown below. Note that the model R-square is over 27%, around 4x larger than for a standard linear ARMA model.

We can describe the regime-switching model in the following terms.  In the regime 1 state  the model has two significant autoregressive terms and one significant moving average term (ARMA(2,1)).  The AR1 term is large and positive, suggesting that trends in VPIN tend to be reinforced from one period to the next. In other words, this is a momentum state. In the regime 2 state the AR2 term is not significant and the AR1 term is large and negative, suggesting that changes in VPIN in one period tend to be reversed in the following period, i.e. this is a mean-reversion state.

The state transition probabilities indicate that the system is in mean-reversion mode for the majority of the time, approximately around 2 periods out of 3.  During these periods, excessive flow in one direction during one period tends to be corrected in the
ensuring period.  But in the less frequently occurring state 1, excess flow in one direction tends to produce even more flow in the same direction in the following period.  This first state, then, may be regarded as the regime characterized by toxic flow.

Markov State Regime-Switching Model

Markov Transition Probabilities

P(.|1)       P(.|2)

P(1|.)        0.54916      0.27782

P(2|.)       0.45084      0.7221

Regime 1:

AR1           1.35502    0.02657   50.998        0

AR2         -0.33687    0.02354   -14.311        0

MA1          0.83662    0.01679   49.828        0

Error Variance^(1/2)           0.36294     0.0058

Regime 2:

AR1      -0.68268    0.08479    -8.051        0

AR2       0.00548    0.01854    0.296    0.767

MA1     -0.70513    0.08436    -8.359        0

Error Variance^(1/2)           0.42281     0.0016

Log Likelihood = -33390.6

Schwarz Criterion = -33445.7

Hannan-Quinn Criterion = -33414.6

Akaike Criterion = -33400.6

Sum of Squares = 8955.38

R-Squared =  0.2753

R-Bar-Squared =  0.2752

Residual SD =  0.3847

Residual Skewness = -0.0194

Residual Kurtosis =  2.5332

Jarque-Bera Test = 553.472     {0}

Box-Pierce (residuals):         Q(9) = 13.9395 {0.124}

Box-Pierce (squared residuals): Q(12) = 743.161     {0}


A Simple Trading Strategy

One way to try to monetize the predictability of the VPIN model is to use the forecasts to take directional positions in the ES
contract.  In this simple simulation we assume that we enter a long (short) position at the first buy (sell) price if the forecast VPIN exceeds some threshold value 0.1  (-0.1).  The simulation assumes that we exit the position at the end of the current volume bucket, at the last sell (buy) trade price in the bucket.

This simple strategy made 1024 trades over a 5-day period from 8/8 to 8/14, 90% of which were profitable, for a total of $7,675 – i.e. around ½ tick per trade.

The simulation is, of course, unrealistically simplistic, but it does give an indication of the prospects for  more realistic version of the strategy in which, for example, we might rest an order on one side of the book, depending on our VPIN forecast.

Figure 2 – Cumulative Trade PL


Easley, D., Lopez de Prado, M., O’Hara, M., Flow Toxicity and Volatility in a High frequency World, Johnson School Research paper Series # 09-2011, 2011

Easley, D. and M. O‟Hara (1987), “Price, Trade Size, and Information in Securities Markets”, Journal of Financial Economics, 19.

Easley, D. and M. O‟Hara (1992a), “Adverse Selection and Large Trade Volume: The Implications for Market Efficiency”,
Journal of Financial and Quantitative Analysis, 27(2), June, 185-208.

Easley, D. and M. O‟Hara (1992b), “Time and the process of security price adjustment”, Journal of Finance, 47, 576-605.


Posted in Algorithmic Trading, ARMA, Direction Prediction, Econometrics, Econophysics, Forecasting, High Frequency Finance, Market Microstructure, Order Flow, Risk Management, Time Series Modeling, Toxic Flow | Tagged , , , | Comments Off

Generalized Regression

Linear regression is one of the most useful applications in the financial engineer’s tool-kit, but it suffers from a rather restrictive set of assumptions that limit its applicability in areas of research that are characterized by their focus on highly non-linear or correlated variables.  The latter problem, referred to as colinearity (or multicolinearity) arises very frequently in financial research, because asset processes are often somewhat (or even highly) correlated.  In a colinear system, one can test for the overall significant of the regression relationship, but one is unable to distinguish which of the explanatory variables is individually significant.  Furthermore, the estimates of the model paramaters, the weights applied to each explanatory variable, tend to be biased.

Over time, many attempts have been made to address this issue, one well-known example being ridge regression.  More recent attempts include lasso, elastic net and what I term generalized regression, which appear to offer significant advantages vs traditional regression techniques in situations where the variables are correlated.

In this note, I examine a variety of these technqiues and attempt to illustrate and compare their effectiveness.

You can downlaod a pdf here.

A Mathematica notebook is also available here.


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Hiring High Frequency Quant/Traders

I am hiring in Chicago for exceptional HF Quant/Traders in Equities, F/X, Futures & Fixed Income.  Remuneration for these roles, which will be dependent on qualifications and experience, will be in line with the highest market levels.

Working closely with team members including developers, traders and quantitative researchers, the central focus of the role will be to research and develop high frequency trading strategies in equities, fixed income, foreign exchange and related commodities markets.

The analyst will have responsibility of taking an idea from initial conception through research, testing and implementation.  The work will entail:

  • Formulation of mathematical and econometric models for market microstructure
  • Data collation, normalization and analysis
  • Model prototyping and programming
  • Strategy development, simulation, back-testing and implementation
  • Execution strategy & algorithms

Qualifications & Experience

  • Minimum 5 years in quantitative research with a leading proprietary trading firm, hedge fund, or investment bank
  • In-depth knowledge of Equities, F/X and/or futures markets, products and operational infrastructure
  • High frequency data management & data mining techniques
  • Microstructure modeling
  • High frequency econometrics (cointegration, VAR,error correction models, GARCH, panel data models, etc.)
  • Machine learning, signal processing, state space modeling and pattern recognition
  • Trade execution and algorithmic trading
  • PhD in Physics/Math/Engineering, Finance/Economics/Statistics
  • Expert programming skills in Java, Matlab/Mathematica essential
  • Must be US Citizen or Permanent Resident

Send your resume to: jkinlay at

No recruiters please.

Posted in High Frequency Finance, High Frequency Trading, Jobs, Quant/Traders, Recruitment | Tagged , , , , | Comments Off

Alpha Spectral Analysis

One of the questions of interest is the optimal sampling frequency to use for extracting the alpha signal from an alpha generation function.  We can use Fourier transforms to help identify the cyclical behavior of the strategy alpha and hence determine the best time-frames for sampling and trading.  Typically, these spectral analysis techniques will highlight several different cycle lengths where the alpha signal is strongest.

The spectral density of the combined alpha signals across twelve pairs of stocks is shown in Fig. 1 below.  It is clear that the strongest signals occur in the shorter frequencies with cycles of up to several hundred seconds. Focusing on the density within
this time frame, we can identify in Fig. 2 several frequency cycles where the alpha signal appears strongest. These are around 50, 80, 160, 190, and 230 seconds.  The cycle with the strongest signal appears to be around 228 secs, as illustrated in Fig. 3.  The signals at cycles of 54 & 80 (Fig. 4), and 158 & 185/195 (Fig. 5) secs appear to be of approximately equal strength.
There is some variation in the individual pattern for of the power spectra for each pair, but the findings are broadly comparable, and indicate that strategies should be designed for sampling frequencies at around these time intervals.

Fig. 1 Alpha Power Spectrum



Fig. 3

Fig. 4

Fig. 5

If we look at the correlation surface of the power spectra of the twelve pairs some clear patterns emerge (see Fig 6):

Fig. 6

Focusing on the off-diagonal elements, it is clear that the power spectrum of each pair is perfectly correlated with the power spectrum of its conjugate.   So, for instance the power spectrum of the Stock1-Stock3 pair is exactly correlated with the spectrum for its converse, Stock3-Stock1.

But it is also clear that there are many other significant correlations between non-conjugate pairs.  For example, the correlation between the power spectra for Stock1-Stock2 vs Stock2-Stock3 is 0.72, while the correlation of the power spectra of Stock1-Stock2 and Stock2-Stock4 is 0.69.

We can further analyze the alpha power spectrum using PCA to expose the underlying factor structure.  As shown in Fig. 7, the first two principal components account for around 87% of the variance in the alpha power spectrum, and the first four components account for over 98% of the total variation.

PCA Analysis of Power Spectra

Fig. 7

Stock3 dominates PC-1 with loadings of 0.52 for Stock3-Stock4, 0.64 for Stock3-Stock2, 0.29 for Stock1-Stock3 and 0.26 for Stock4-Stock3.  Stock3 is also highly influential in PC-2 with loadings of -0.64 for Stock3-Stock4 and 0.67 for Stock3-Stock2 and again in PC-3 with a loading of -0.60 for Stock3-Stock1.  Stock4 plays a major role in the makeup of PC-3, with the highest loading of 0.74 for Stock4-Stock2.

Fig. 8  PCA Analysis of Power Spectra


Posted in Forecasting, Fourier Transforms, High Frequency Finance, Pairs Trading, Principal Components Analysis, Signal Processing, Statistical Arbitrage | Tagged , , , , , | Comments Off