Category: statistical arbitrage

One of my favourite ways to characterize the market regime is by using the Hurst exponent. However, its applications are not limited to identifying market regimes. There are innovative ways to utilize it. In this post, I will discuss two approaches to applying the Hurst exponent.

Using the Hurst Exponent to Time the Market

The Hurst exponent can be employed to directly time the market.  Reference [1] calculated the moving Hurst exponents for rolling windows of 100 and 150 days. The timing signals are subsequently generated by using these calculations.

Findings

-The study suggests that the Moving Hurst (MH) indicator is effective for forecasting and managing volatility in Indian equity markets.

-MH is more effective at capturing profitable trading opportunities than Moving Averages (MA).

-MH is a less lagging indicator than MA, making it more responsive to market changes.

-MH incorporates principles from chaos theory and fractal analysis, offering a unique perspective for market analysis.

-The research was conducted in the Indian stock market. However, it can be readily applied to any stock market.

Reference

[1] Shah, Param, Ankush Raje, and Jigarkumar Shah, Patterns in the Chaos: The Moving Hurst Indicator and Its Role in Indian Market Volatility. Journal of Risk and Financial Management 17: 390, 2024

Using the Hurst Exponent for Pairs Trading

The Hurst method isn’t restricted to single underlying assets; it can also be applied to a pair of stocks to identify pairs trading (statistical arbitrage) opportunities.  Reference [2] proposed a new approach to measure the co-movement of two price series through the Hurst exponent of the product.

Findings

– The Hurst exponent of the product series, referred to as HP, can measure the existence of a relationship between two series.

– The HP method is a new way to measure the dependence between two series, detecting various types of relationships, including correlation, cointegration, and non-linear relationships, even when the relationship is weak or given by a copula.

– This method is particularly useful for studying financial series as it gives more weight to high increments than low increments, unlike other correlation measures.

– The efficiency of the HP method was tested through a statistical arbitrage technique for pairs selection and compared with the classical correlation method.  Results indicate that the HP method performs better in most cases.

Reference

[2] José Pedro Ramos-Requena, Juan Evangelista Trinidad-Segovia, and Miguel Ángel Sánchez-Granero, An Alternative Approach to Measure Co-Movement between Two Time Series, Mathematics 2020, 8, 261

Educational Video

This seminar by Markis Vogl presents the theory and application of the Hurst exponent.

Abstract

My presentations elaborates on the meaning of Hurst exponents, namely, that instead of long memory, fractal trends are measured instead (contradicting Mandelbrot’s conception). Further, the talk encompasses the generation of rolling window (time varying) Hurst exponent series based upon the cascadic level 12 wavelet filtered (denoised) S&P500 logarithmic return series (2000-2020). The Hurst exponent series are then analysed with a generalizable nonlinear analysis framework, which allows the determination of the underlying empirical data generating process.

Closing thoughts

The Hurst exponent is an effective tool for gaining insights into market dynamics. Whether for timing the market or identifying pairs trading opportunities, it offers traders an edge in strategy development.

Pairs trading is a market-neutral strategy that involves trading two correlated stocks or assets. The idea is to identify pairs that historically move together, and then take a long position in one and a short position in the other when they diverge, with the expectation that they will eventually revert to their mean relationship.

The popularity of pairs trading has risen over the years. Naturally, this raises the question: is pairs trading still profitable, and is it worth investing time, money, and resources to find profitable pairs trading strategies?

Pairs Trading: No Longer Profitable

There is a perspective among some researchers and traders that pairs trading may have lost its profitability over time due to increased competition and the efficiency of modern markets.

Reference [1] argues that pairs trading is no longer profitable, especially when using basic approaches for pairs selection.

Findings

• This paper focuses on the German stock market from 2000 to 2023, a market with relatively few analyses in this area.
• Basic strategies based on spread distance and cointegration barely cover transaction costs and often break even.
• A copula-based method, especially when combined with simpler strategies, shows stronger performance, yielding an average portfolio return of around 170 basis points (bps) per month after transaction costs.
• The strategies are designed to be uncorrelated with systemic market risk, and empirical results confirm this.
• Sensitivity analyses indicate the robustness of the copula-based method and suggest possible refinements for further strategy enhancement.

Reference

[1] Sascha Wilkens, Pairs Trading in the German Stock Market: There’s Life in the Old Dog Yet.

Pairs Trading: Still Profitable

On the other hand, some argue that pairs trading remains profitable. Reference [2] supports this view, showing evidence of profitability even with classical pairs selection methods like the spread distance approach.

Findings

• The paper replicates Gatev et al.’s [3] pairs trading strategy using twenty years of stock price data, affirming robustness despite transaction costs in the current market.
• The top strategy achieves a compounded annual excess return of 6.2%, a notable finding given market dynamics.
• A broader stock pool mitigates outlier effects from events like delistings or stock splits, enhancing strategy performance compared to typical literature.
• The study examines two profit determinants in pairs trading: medium-term momentum and the default spread, correlating with the investor risk premium.
• These findings support Gatev et al.’s [3] hypothesis on arbitrage compensation for restoring market efficiency.

References

[2] Xuanchi Zhu, Examining Pairs Trading Profitability, 2024, Yale University

[3] Gatev, E., Rouwenhorst, K. G., and Goetzmann, W. (2006). Pairs trading: Performance of a relative value arbitrage rule.

Closing Thoughts

In my opinion, pairs trading is still profitable. However, it requires using a pairs selection method that isn’t obvious or widely adopted by others. I was somewhat surprised that, in Reference [2], the author still finds pairs trading profitable using a classical selection method.

What’s your experience with pairs trading? Let me know in the comments section.

Pairs selection is a critical step in developing a winning trading system. In a future issue, I’ll cover different pairs selection methods that could enhance profitability.

Would you rather choose a trading system that wins small amounts most of the time but when it loses, the loss is big? Or would you rather choose a trading system that loses small amounts most of the time but when it wins, the gain is big? In this blog post, we will examine such systems from the risk of ruin perspective.

The risk of ruin is the probability of an investor’s eventual bankruptcy due to a series of losses that exceed his/her capital. It is essential for any trader to understand their risk of ruin, as it will heavily influence the trading system they ultimately develop.

We will use Monte Carlo simulations to perform our analysis. We examine the following 3 trading systems

For each system, we generate 1000 trades randomly. If a trade is a win, then we’ll make the amount in the column “Win”, and if it’s a loss, then we’ll lose the amount in the column “Loss”. For example, for system A, if we win, we make $90 and if we lose, we lose $10.

To simplify the analysis, we assume that all 3 systems have zero expectancies. The total winning/losing amounts of each system equal $9000. We start with an initial capital of $1000. The figure below shows the first 100 simulated paths for system A. It’s clear that the system has zero expectancy as the terminal wealth equals the starting capital.

To calculate the probability of ruin, we first count the number of paths that go below $0 at any time in their evolutions. We then divide the number of such paths by the number of iterations which is 10000.

The table below summarizes the results.

We can see from the results that system B has the lowest probability of ruin. This confirmed that traders should avoid systems that have high win percentages but suffer occasional large losses. Short volatility trading systems are in this category. Traders should also avoid systems that have low percentage win rates.

In a previous post, we discussed how the dynamics of assets are priced in the options prices. We recently came across a newly published article [1] that explored the same topic but from a different perspective that does not involve options.

The conclusion of the new article [1] is consistent with the previous one [2]; that is, the volatilities of mean-reverting assets are smaller than those of assets that follow the GBM process. The reverse applies to trending assets.

In this post, we are going to investigate whether the mean-reverting/trending property of an asset has any impact on a trading strategy’s PnL volatility.

To do this, we first generate asset prices using Monte Carlo simulations. We evolve the asset prices in both mean-reverting and trending regimes for 500 days. We then apply a simple trading system to the simulated asset prices. The trading system is as follows,

Go LONG when the Relative Strength Index <40, SHORT when the Relative Strength Index >70

The picture below shows the Autocorrelation Functions (ACF) of the asset returns. Panels (a) and (b) present ACFs of the trending and mean-reverting assets respectively. It’s clear that the assets are trending and mean-reverting at lag 3, respectively.

The picture below shows the simulated equity curves of the trading strategy applied to the trending (a) and mean-reverting (b) assets. The starting capital is $100 in both cases.

Visually, we do not observe any difference in terms of PnL dispersion. Indeed, the standard deviation of the terminal wealth at day 500 is $18.9 in the case of the trending asset (a), and $17.3 in the case of the mean-reverting asset (b). Is the difference statistically significant? We don’t think so.

This numerical experiment shows that the PnL volatility of a trading strategy has little to do with the underlying asset’ mean-reverting/trending property. Maybe it depends more on the strategy itself? (Note that in this example, we utilize a mean-reverting strategy). What would happen at the portfolio level?

References

[1] L. Middleton, J. Dodd, S. Rijavec, Trading styles and long-run variance of asset prices, 2021, arXiv:2109.08242

[2] Liao, S.L. and Chen, C.C. (2006), The valuation of European options when asset returns are autocorrelated, Journal of Futures Markets, 26, 85-102.

This post is a follow-up to the previous one on a simple system for hedging long exposure during a market downturn. It was inspired by H. Krishnan’s book The Second Leg Down, in which he referred to an interesting research paper [1] on the power-law behaviour of the equity indices.  The paper states,

We find that the distributions for ∆t ≤4 days (1560 mins) are consistent with a power-law asymptotic behavior, characterized by an exponent α≈ 3, well outside the stable Levy regime 0 < α <2. .. For time scales longer than ∆t ≈4 days, our results are consistent with slow convergence to Gaussian behavior.

Basically, the paper says that the equity indices exhibit fatter tails in shorter time frames, from 1 to 4 days. We apply this idea to our breakout system.  We’d like to see whether the 4-day rule manifests itself in this simple strategy. To do so, we use the same entry rule as before, but with a different exit rule.   The entry and exit rules are as follows,

Short at the close when Close of today < lowest Close of the last 10 days

Cover at the close T days after entry (T=1,2,… 10)

The system was backtested on SPY from 1993 to the present. Graph below shows the average trade PnL as a function of number of days in the trade,

We observe that if we exit this trade within 4 days of entry, the average loss (i.e. the cost of hedging) is in the range of -0.2% to -0.4%, i.e. an average of -0.29% per trade. From day 5, the loss becomes much larger (more than double), in the range of -0.6% to -0.85%. The smaller average loss incurred during the first 4 days might be a result of the fat-tail behaviour.

This test shows that there is some evidence that the scaling behaviour demonstrated in Ref [1] still holds true today, and it manifested itself in this system.  More rigorous research should be conducted to confirm this.

 References

[1] Gopikrishnan P, Plerou V, Nunes Amaral  LA, Meyer M, Stanley HE, Scaling of the distribution of fluctuations of financial market indices, Phys Rev E, 60, 5305 (1999).