Jonathan, Author at QUANTITATIVE RESEARCH AND TRADING

May 23, 2021

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 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 http://www.quantifiedstrategies.com/the-greatest-gold-stock-system-you-should-trade/):

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%.

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.

May 12, 2021May 13, 2021

Strategy Backtesting in Mathematica

This is a snippet from a strategy backtesting system that I am currently building in Mathematica.

One of the challenges when building systems in WL is to avoid looping wherever possible. This can usually be accomplished with some thought, and the efficiency gains can be significant. But it can be challenging to get one’s head around the appropriate construct using functions like FoldList, etc, especially as there are often edge cases to be taken into consideration.

A case in point is the issue of calculating the profit and loss from individual trades in a trading strategy. The starting point is to come up with a FoldList compatible function that does the necessary calculations:

CalculateRealizedTradePL[{totalQty_, totalValue_, avgPrice_, PL_, totalPL_}, {qprice_, qty_}] := Module[{newTotalPL = totalPL, price = QuantityMagnitude[qprice], newTotalQty, tradeValue, newavgPrice, newTotalValue, newPL}, newTotalQty = totalQty + qty; tradeValue = If[Sign[qty] == Sign[totalQty] || avgPrice == 0, priceqty, If[Sign[totalQty + qty] == Sign[totalQty], avgPriceqty, price(totalQty + qty)]]; newTotalValue = If[Sign[totalQty] == Sign[newTotalQty], totalValue + tradeValue, newTotalQtyprice]; newavgPrice = If[Sign[totalQty + qty] == Sign[totalQty], (totalQtyavgPrice + tradeValue)/newTotalQty, price]; newPL = If[(Sign[qty] == Sign[totalQty] ) || totalQty == 0, 0, qty(avgPrice - price)]; newTotalPL = newTotalPL + newPL; {newTotalQty, newTotalValue, newavgPrice, newPL, newTotalPL}]

Trade P&L is calculated on an average cost basis, as opposed to FIFO or LIFO.

Note that the functions handle both regular long-only trading strategies and short-sale strategies, in which (in the case of equities), we have to borrow the underlying stock to sell it short. Also, the pointValue argument enables us to apply the functions to trades in instruments such as futures for which, unlike stocks, the value of a 1 point move is typically larger than 1(e.g.50 for the ES S&P 500 mini futures contract).

We then apply the function in two flavors, to accommodate both standard numerical arrays and timeseries (associations would be another good alternative):

CalculateRealizedPLFromTrades[tradeList_?ArrayQ, pointValue_ : 1] := Module[{tradePL = Rest@FoldList[CalculateRealizedTradePL, {0, 0, 0, 0, 0}, tradeList]}, tradePL[[All, 4 ;; 5]] = tradePL[[All, 4 ;; 5]]pointValue; tradePL] CalculateRealizedPLFromTrades[tsTradeList_, pointValue_ : 1] := Module[{tsTradePL = Rest@FoldList[CalculateRealizedTradePL, {0, 0, 0, 0, 0}, QuantityMagnitude@tsTradeList["Values"]]}, tsTradePL[[All, 4 ;; 5]] = tsTradePL[[All, 4 ;; 5]]pointValue; tsTradePL[[All, 2 ;;]] = Quantity[tsTradePL[[All, 2 ;;]], "US Dollars"]; tsTradePL = TimeSeries[ Transpose@ Join[Transpose@tsTradeList["Values"], Transpose@tsTradePL], tsTradeList["DateList"]]]

These functions run around 10x faster that the equivalent functions that use Do loops (without parallelization or compilation, admittedly).

Let’s see how they work with an example:

Trade Simulation

Next, we’ll generate a series of random trades using the AAPL time series, as follows (we also take the opportunity to convert the list of trades into a time series, tsTrades):

trades = Transpose@ Join[Transpose[ tsAAPL["DatePath"][[ Sort@RandomSample[Range[tsAAPL["PathLength"]], 20]]]], {RandomChoice[{-100, 100}, 20]}]; trades // TableForm

Trade P&L Calculation

We are now ready to apply our Trade P&L calculation function, first to the list of trades in array form:

TableForm[
Flatten[#] & /@ 
Partition[
Riffle[trades, 
CalculateRealizedPLFromTrades[trades[[All, 2 ;; 3]]]], 2], 
TableHeadings -> {{}, {"Date", "Price", "Quantity", "Total Qty", 
"Position Value", "Average Price", "P&L", "Total PL"}}]

The timeseries version of the function provides the output as a timeseries object in Quantity[“US Dollars”] format and, of course, can be plotted immediately with DateListPlot (it is also convenient for other reasons, as the complete backtest system is built around timeseries objects):

tsTradePL = CalculateRealizedPLFromTrades[tsTrades]

March 9, 2021

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.

March 2, 2021

Option Prices in the Variance Gamma Model

February 23, 2021

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

References

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.

February 16, 2021

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 parameters, 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 techniques and attempt to illustrate and compare their effectiveness.

The Mathematica notebook is also available here.

Generalized Regression

February 9, 2021

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.

Role
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.

Responsibilities
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 systematic-strategies.com.

No recruiters please.

February 2, 2021February 2, 2021

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.2

Fig. 3

Fig. 4

Fig. 5

PRINCIPAL COMPONENTS ANALYSIS OF ALPHA POWER SPECTRUM
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.

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

January 26, 2021

Market Microstructure Models for High Frequency Trading Strategies

This note summarizes some of the key research in the field of market microstructure and considers some of the models proposed by the researchers. Many of the ideas presented here have become widely adopted by high frequency trading firms and incorporated into their trading systems.

Market MicroStructure Models

January 19, 2021January 19, 2021

Forecasting Financial Markets – Part 1: Time Series Analysis

The presentation in this post covers a number of important topics in forecasting, including:

Stationary processes and random walks
Unit roots and autocorrelation
ARMA models
Seasonality
Model testing
Forecasting
Dickey-Fuller and Phillips-Perron tests for unit roots

Also included are a number of detailed worked examples, including:

Forecasting 2011 - Time Series