statistical arbitrage – Relative Value Arbitrage

PCA in Action: From Commodity Derivatives to Dispersion Trading

Principal Component Analysis (PCA) is a dimensionality reduction technique used to simplify complex datasets. It transforms the original variables into a smaller set of uncorrelated variables called principal components, ranked in order of their contribution to the dataset’s total variance. In this post, we’ll discuss various applications of PCA.

Pricing Commodity Derivatives Using Principal Component Analysis

Due to the seasonal nature of commodities, pricing models should be able to take into account seasonality and other deterministic factors.

Reference [1] proposed a new, multi-factor pricing method based on Principal Component Analysis (PCA). It introduces a multi-factor model designed to price commodity derivatives, with a particular focus on commodity swaptions.

Findings

– The model calibration process consists of two key steps: offline and online.

– The offline step, conducted infrequently, determines mean reversion rates, the ratio of long and short factor volatilities, and the correlation between the factors using historical data.

– The online step occurs every time the model is used to price an option or simulate price paths.

– Empirical analysis demonstrates that the model is highly accurate in its predictions and applications.

– Swaptions, which are relatively illiquid commodities, present a challenge due to their one-sided natural flow in the market.

– Model calibration strategies are divided into seasonal and non-seasonal categories, considering the asset’s characteristics. For seasonal assets like power or gas, local volatilities are calibrated separately for each contract, while a boot-strapping strategy is employed for non-seasonal assets like oil.

– Currently, the multi-factor model lacks a term structure for volatility ratios and mean reversions. However, it can be easily extended to incorporate a time dependency, which would facilitate fitting market prices of swaptions across various tenors.

Reference

[1] Tim Xiao, Pricing Commodity Derivatives Based on A Factor Model, Philarchive

Dispersion Trading Using Principal Component Analysis

Dispersion trading involves taking positions on the difference in volatility between an index and its constituent stocks.

Reference [2] examined dispersion trading strategies based on a statistical index subsetting procedure and applied it to the S&P 500 constituents

Findings

– This paper introduces a dispersion trading strategy using a statistical index subsetting approach applied to S&P 500 constituents from January 2000 to December 2017.

– The selection process employs principal component analysis (PCA) to determine each stock’s explanatory power within the index and assigns appropriate subset weights.

– In the out-of-sample trading phase, both hedged and unhedged strategies are implemented using the most suitable stocks.

– The strategy delivers significant annualized returns of 14.52% (hedged) and 26.51% (unhedged) after transaction costs, with Sharpe ratios of 0.40 and 0.34, respectively.

– Performance remains robust across different market conditions and outperforms naive subsetting schemes and a buy-and-hold approach in terms of risk-return characteristics.

– A deeper analysis highlights a correlation between the chosen number of principal components and the behavior of the S&P 500 index.

– An index subsetting procedure was developed, considering the explanatory power of individual stocks, allowing a replicating option basket with as few as five securities.

– An analysis of sector exposure, principal components, and robustness checks demonstrated that the trading systems have superior risk-return characteristics compared to other dispersion strategies.

Reference

[2] L. Schneider, and J. Stübinger, Dispersion Trading Based on the Explanatory Power of S&P 500 Stock Returns, Mathematics 2020, 8, 1627

Closing Thoughts

PCA is a powerful tool in quantitative finance. In this issue, we have demonstrated its effectiveness in pricing commodity derivatives and developing dispersion trading strategies. Its versatility extends beyond these applications, making it a valuable technique for tackling a wide range of problems in quantitative finance.

CAPM, WACC, and Beyond: Beta’s Application in Arbitrage

Beta is a measure of an asset’s sensitivity to market movements, indicating how much its price is expected to change in relation to the overall market. Beta is often used in CAPM and the calculation of WACC. However, it can also be applied in trading, specifically in arbitrage. In this post, I’ll discuss beta arbitrage.

Beta Arbitrage Around Macroeconomic Announcements

The macroeconomic announcement premium refers to the phenomenon where financial markets experience higher-than-usual returns on days when significant macroeconomic announcements are made.

Reference [1] studies the dynamics of high-beta stock returns around macroeconomic announcements.

Findings

– Stocks in the top beta-decile show distinct return patterns: negative returns before announcements (-0.075%), positive on announcement days (0.164%), and negative again after (-0.093%).

– The beta premium experiences significant fluctuations around macroeconomic announcements, with a swing driven by high-beta stock returns.

– A long-short strategy involving betting against beta (BAB) before and after announcements, and betting on beta (BOB) on announcement days, can yield an annualized return of 25.28%.

– Liquidity effects explain pre-announcement high-beta returns, while risk has a weak but consistent pattern around announcements.

– Investor risk aversion shifts significantly explain the variation in beta returns around announcements.

– Liquidity, risk, and investor risk appetite only partially account for variations in high-beta stock returns.

Reference

[1] Jingjing Chen, George J. Jiang, High-beta stock valuation around macroeconomic announcements, Financial Review. 2024;1–26.

Beta Arbitrage: Betting on Stock Comovements

This trading strategy is based on the assumption that stock betas tend to mean regress towards one in the long run, leading to exploitable comovement patterns in stock prices.

Reference [2] discusses a model for S&P 500 index changes, involving two beta-based styles: index trackers and beta arbitrageurs. The comovement effect has two components, influenced by low and high beta stocks in pre-event scenarios.

The paper presents a stylized model for S&P 500 index changes, highlighting the distinct components of comovement effects and the exploitable nature of beta arbitrage.

Findings

-Beta arbitrage is a trading strategy that capitalizes on the belief that betas tend to mean regress towards one over time.

– This paper develops a model for S&P 500 index changes, focusing on two beta-based trading styles: index trackers and beta arbitrageurs.

– Index trackers follow the index, while beta arbitrageurs trade both high and low beta event stocks to exploit mean reversion toward one.

– Arbitrageurs employ common or contrarian trading patterns depending on whether a stock’s historical beta is below or above one.

– The overall comovement effect of index changes has two components:

1- Pre-event low beta stocks experience beta increases due to common demand from both indexers and arbitrageurs.

2- Arbitrageurs short high beta additions, reducing or reversing beta increases caused by indexers.

– Similar patterns are observed for stocks deleted from the index.

Reference

[2] Yixin Liao, Jerry Coakley, Neil Kellard, Index tracking and beta arbitrage effects in comovement, International Review of Financial Analysis, Volume 83, October 2022, 102330

Closing Thoughts

Beta is more than a measure of an asset’s sensitivity to market movements or a key component in financial models like CAPM and WACC. Its application extends to trading strategies, particularly beta arbitrage, where investors exploit discrepancies in beta values to identify profitable opportunities.

From Gold to Bitcoin: Exploring the Oldest and Newest Asset Classes

Gold, one of the oldest and most enduring asset classes, had an exceptional run in 2024, capturing attention across financial markets. Its role in investment portfolios continues to spark interest, acting as a hedge against uncertainty. On the other end of the spectrum, cryptocurrencies represent the newest frontier in finance. While opinions remain divided, some are enthusiastic supporters, while others remain skeptical, one thing is undeniable: Bitcoin has just crossed the remarkable $100,000 USD milestone. In this article, I’ll discuss gold’s role in an investment portfolio and pairs trading within the crypto market.

Is Gold a Hedge or a Safe Haven Asset?

Historically, gold has exhibited a low correlation with other asset classes such as stocks and bonds, making it an effective hedge against market volatility and economic uncertainty.

Reference [1] delves deeper into examining the role of gold as a hedge or safe haven asset. It defines a weak, strong hedge, or safe haven asset as follows,

-A weak hedge is an asset that has a negative conditional correlation with another asset or portfolio on average. A strong hedge is an asset that has both a negative conditional correlation and positive conditional coskewness with another asset or portfolio on average.

– A weak safe haven is an asset that has a negative conditional correlation with another asset or portfolio in times of market stress or turmoil. A strong safe haven is an asset that has both a negative conditional correlation and positive conditional coskewness with another asset or portfolio in times of market stress or turmoil.

Findings

– The study empirically analyzes the performance of gold across 24 countries over a 40-year period.

– Results show that gold acts as a strong hedge in Brazil, India, Indonesia, Italy, Mexico, Russia, South Korea, Thailand, and Turkey, and as a safe haven in Brazil, France, India, Indonesia, Italy, Mexico, Russia, South Korea, and Turkey.

– The study investigates whether gold can enhance overall portfolio performance as a hedge or safe-haven asset.

– The conditional comoment-based dynamic (CCD) strategy adjusts portfolio allocation to gold based on its properties and adds gold to the stock portfolio during the holding period only if it serves as a hedge or safe haven.

– Findings indicate that the CCD trading strategy outperforms the buy-and-hold strategy, generating higher returns, Sharpe ratio, and skewness when gold is utilized as a hedge or safe-haven asset.

Reference

[1] Lei Ming, Ping Yang, Qianqiu Liu, Is gold a hedge or a safe haven against stock markets? Evidence from conditional comoments, Journal of Empirical Finance, Volume 74, December 2023, 101439

Pairs Trading in the Cryptocurrency Market

Pairs trading is a popular strategy in equity and commodity markets. While successful in equities, limited research exists on pair trading in the cryptocurrency market. Reference [2] examines the application of pairs trading within the cryptocurrency market.

Findings

-The study applied the Distance Method and Cointegration Method to cryptocurrency pairs using both daily and hourly data for formation and trading periods.

-Results showed that the frequency of the selection period (daily or hourly) did not significantly affect the pairs chosen.

-Pairs selected using the Cointegration Method generally outperformed those chosen with the Distance Method.

-Intraday trading proved more profitable than longer-term trading but lost its advantage when a stop-loss was implemented.

-The Cointegration Method performed better than the Distance Method, as the latter incurred higher trading costs due to an increased number of trades.

– Pairs trading outperformed the buy-and-hold long/short strategy in the cryptocurrency market. But it underperformed the traditional Buy and Hold.

Reference

[2] Lesa, Chiara and Hochreiter, Ronald, Cryptocurrency Pair Trading, SSRN, 2023

Closing thoughts

As we navigate an ever-evolving financial landscape, understanding the roles of these two asset classes can help build diversified, forward-looking investment portfolios.

When Correlations Break or Hold: Strategies for Effective Hedging and Trading

It’s well known that there is a negative relationship between an equity’s price and its volatility. This can be explained by leverage or, alternatively, by volatility feedback effects. In this post, I’ll discuss practical applications to exploit this negative correlation between equity prices and their volatility.

A Trading Strategy Based on the Correlation Between the VIX and S&P500 Indices

This paper [1] examines the strong correlation in the S&P 500 and identifies trading opportunities when this correlation weakens or breaks down.

Findings

-The study covers the period from January 1995 to October 2020, utilizing 6,488 daily observations of the VIX and S&P500 indexes.

– In scenarios where the options market indicates increased drawdown risk with higher implied volatility but negative returns have not yet occurred, consider shorting the market.

– The signal to short the market occurs when the negative correlation between the S&P 500 and VIX is broken, and they start exhibiting a positive correlation.

– The test setup involves identifying one or two consecutive days with positive co-movement between the VIX and S&P 500, then setting the transaction date for the day after or at the close of the chosen date.

– Empirical results show that the strategy outperforms the S&P500 index over the 25-year period, achieving higher returns, lower systematic risk, and reduced volatility.

-The findings provide evidence that excess returns can be generated by timing the market using historical data, even after accounting for trading costs.

Reference

[1] Tuomas Lehtinen, Statistical arbitrage strategy based on VIX-to-market based signal, Hanken School of Economics

Optimal Hedging for Options Using Minimum-Variance Delta

Contrary to the first paper, Reference [2] focuses on the strong correlation between the S&P 500 and its volatility, designing an efficient scheme for hedging an options book.

The authors developed a so-called minimum variance (MV) delta. Essentially, the MV delta is the Black-Scholes delta with an additional adjustment term.

Findings

-Due to the negative relationship between price and volatility for equities, the minimum variance delta is consistently less than the practitioner Black-Scholes delta.

-Traders should under-hedge equity call options and over-hedge equity put options compared to the practitioner Black-Scholes delta.

-The study demonstrates that the minimum variance delta can be accurately estimated using the practitioner Black-Scholes delta and the historical relationship between implied volatilities and asset prices.

-The expected movement in implied volatility for stock index options can be approximated as a quadratic function of the practitioner Black-Scholes delta divided by the square root of time.

-A formula for converting the practitioner Black-Scholes delta to the minimum variance delta is provided, yielding good out-of-sample results for both European and American call options on stock indices.

-For S&P 500 options, the model outperforms stochastic volatility models and models based on the slope of the volatility smile.

-The model works less well for certain ETFs

Reference:

[2] John Hull and Alan White, Optimal Delta Hedging for Options, Journal of Banking and Finance, Vol. 82, Sept 2017: 180-190

Closing Thoughts

These two papers take opposing approaches: one exploits correlation breakdown, while the other capitalizes on the correlation remaining strong. However, they are not mutually exclusive. Combining insights from both can lead to a more efficient trading or hedging strategy.

Educational Video

This seminar by Prof. J. Hull delves into the second paper discussed above.

Abstract

The “practitioner Black-Scholes delta” for hedging equity options is a delta calculated from the Black-Scholes-Merton model with the volatility parameter set equal to the implied volatility. As has been pointed out by a number of researchers, this delta does not minimize the variance of a trader’s position. This is because there is a negative correlation between equity price movements and implied volatility movements. The minimum variance delta takes account of both the impact of price changes and the impact of the expected change in implied volatility conditional on a price change. In this paper, we use ten years of data on options on stock indices and individual stocks to investigate the relationship between the Black-Scholes delta and the minimum variance delta. Our approach is different from earlier research in that it is empirically-based. It does not require a stochastic volatility model to be specified. Joint work with Allan White.

Hurst Exponent Applications: From Regime Analysis to 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.

Examining Contango and Backwardation in VIX Futures

In this post, I will continue exploring various aspects of the volatility index and the associated volatility futures.

Data

To conduct this study, data is essential. Below are the data sources:

Spot VIX: Yahoo Finance provides data but no longer allows direct downloads. With some programming, a workaround can be found, but the most convenient option is to use Barchart.

https://www.barchart.com/stocks/quotes/$VIX/price-history/historical

VX Futures: CBOE offers historical data in CSV format.

https://www.cboe.com/us/futures/market_statistics/historical_data/

Short-Term Futures Index:

While not directly utilized in this issue, I use this data to validate other ideas. For completeness, here is the download link:

https://www.spglobal.com/spdji/en/indices/indicators/sp-500-vix-short-term-index-mcap/#overview

Statistics for spot VIX and VX Futures

The table below provides statistics for the S&P 500 (tracked by SPY), spot VIX, and VX futures. It shows the percentage of days the S&P 500 index is up and the percentage of days VX futures are in contango, where the front-month futures price is lower than the next-month futures price.

From January 2013 to November 2024, the S&P 500 index was up 54.9% of the time, while VX futures were in contango most of the time (85.2%).

The next table presents the number of days VX futures were in backwardation while the spot VIX was in contango. Spot VIX in contango is defined as the 1M spot VIX being less than the 3M spot VIX.

From January 2013 to July 2024, this situation was very rare, occurring only 7% of the time. However, from August 1 to November 4, 2024, this divergence occurred with much higher frequency, 53% of the time.

This situation presented a high reward/risk trade opportunity. For instance, one could structure a trade to capitalize on the high likelihood of VX futures returning to contango and a decline in the overall volatility level. One potential trade is buying a put option in VXX. We’ll discuss this strategy in an upcoming webinar.

Seasonality of Volatility

With the holiday season approaching, the equity world often discusses the “Santa Rally.” This raises the question: Is there any seasonality in the volatility market?

The graph below shows the average and median monthly implied and historical volatilities. A clear seasonal pattern is observed, with low volatility between April and July and high volatility in October. However, for December, there is no discernable pattern—volatility can be either high or low during this period.

Educational Video

This webinar by Prof Andrew Papanicolaou covers fundamental concepts of VX futures, such as contango, backwardation, and roll yield. It also presents an approach to modeling the VX futures term structure.

Abstract:

We study VSTOXX, VSTOXX futures and VSTOXX exchange-traded notes (ETNs) econometrically. We find that different rates of mean reversion capture fluctuations in the short and long maturities, respectively. We fit an exponential Ornstein-Uhlenbeck (OU) model to the data and find it to capable of simulating ETN time series that have similar properties to the historical observed ETN time series. We compare these results to a similar study performed on ETNs and futures for VIX. We also look at the joint behavior of VIX and VSTOXX futures, and explore portfolio allocation strategies among ETNs for both markets.

Closing Thoughts

The volatility market offers unique insights and opportunities for investors. By understanding concepts like contango, backwardation, and seasonality, we can structure strategies with favorable risk-reward profiles.

Let me know your thoughts in the comments below.

Rethinking Pairs Trading: Can Traditional Methods Still Deliver Returns?

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.

The Weekend Effect in The Market Indices

The weekend (or Monday) effect in the stock market refers to the phenomenon where stock returns exhibit different patterns on Mondays compared to the rest of the week. Historically, there has been a tendency for stock prices to be lower on Mondays. Various theories attempt to explain the weekend effect, including investor behaviour, news over the weekend, and the impact of events occurring during the weekend on market sentiment.

In this post, we’ll investigate the weekend effect in the market indices using data from Yahoo Finance spanning January 2001 to December 2023. Specifically we choose SPY, which tracks the SP500, and the volatility index, VIX.

Our strategy involves taking a long position in SPY at Friday’s close and exiting the position at Monday’s close, or the next business day’s close if Monday is a holiday. The figure below depicts the cumulative, non-compounded return of the strategy.

Cumulative return of holding SPY over the weekend

From the figure, we observe that holding SPY over the weekend resulted in negative returns during the GFC, Covid pandemic, and the recent 2022 bear market. The overall return is flat-ish, indicating a low reward/risk ratio for holding the SPY over the weekend.

Next, we analyze the change in the VIX index during the weekend. We compute the change in the VIX index from Friday’s close to the close of the next business day and plot the cumulative difference in the figure below. A noticeable upward trend is observed in the cumulative difference. This result indicates that maintaining a long vega/gamma position over the weekend would offer a favourable reward-to-risk trade.

Cumulative difference of the VIX index over the weekend

It’s important to note, however, that investing directly in the spot VIX is not possible. To confirm and capitalize on the weekend effect in the volatility index, one would have to:

Trade a volatility ETN, or
Trade a delta-hedged option position

Each of these approaches introduces additional risk factors, specifically 1- the roll yield and contango, and 2- PnL originating from gamma and theta. These issues will be addressed in the next installment.

Which System Has The Lowest Risk of Ruin?

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

System	Percent win	Win	Loss	Expectancy
A	10%	$90	$10	0
B	50%	$18	$18	0
C	90%	$10	$90	0

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.

System	Risk of Ruin
A	10.22%
B	0.18%
C	9.95%

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.

Asset Price Dynamics and Trading Strategy’s PnL Volatility

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.

Autocorrelation Functions of asset returns

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.