Volatility of Volatility: Insights from VVIX

The volatility of volatility index, VVIX, is a measure of the expected volatility of the VIX index itself. In this post, we will discuss its dynamics, compare it with the VIX index, and explore how it can be used to characterize market regimes.

Dynamics of the Volatility of Volatility Index, VVIX

The VVIX, also known as the Volatility of Volatility Index, is a measure that tracks the expected volatility of the CBOE Volatility Index (VIX). As the VIX reflects market participants’ expectations for future volatility in the S&P 500 index, the VVIX provides insights into the market’s perception of volatility uncertainty in the VIX itself.

Reference [1] studied the dynamics of VVIX and compared it to the VIX.

Findings

-The VVIX tracks the expected volatility of the VIX, providing a direct measure of uncertainty around future changes in market volatility itself.

-It shows strong mean-reverting behavior, indicating that large deviations from its average level tend to reverse over time.

-The VVIX responds asymmetrically to S&P 500 movements, typically increasing more sharply during market downturns than it decreases during upswings.

-It experiences sudden jumps in both directions, reflecting its sensitivity to abrupt changes in market sentiment and conditions.

-A persistent upward trend in the VVIX began well before 2020, driven by factors such as rising VIX volatility and an increasing volatility-of-volatility risk premium (VVRP).

-The growth of the VIX options market from 2006 to 2014 improved liquidity, which likely contributed to the VVIX’s upward trend and closer link to the VIX.

-VVIX and VIX innovations are highly correlated, highlighting their structural connection despite often differing in their responses to specific market events.

-VVIX quickly incorporates new market information, with minimal autocorrelation beyond a single day, showing its responsiveness to real-time market changes.

In summary, this paper analyzes the similarities and differences between the VIX and VVIX, offering key insights for traders and hedgers in the VIX options market. Understanding their relationship helps improve risk management, refine hedging strategies, and better assess market sentiment.

Reference

[1]  Stefan Albers, The fear of fear in the US stock market: Changing characteristics of the VVIX, Finance Research Letters, 55

Using Hurst Exponent on the Volatility of Volatility Indices

A market regime refers to a distinct phase or state in financial markets characterized by certain prevailing conditions and dynamics. Two common market regimes are mean-reverting and trending regimes. In a mean-reverting regime, prices tend to fluctuate around a long-term average, with deviations from the mean eventually reverting back to the average. In a trending regime, prices exhibit persistent directional movements, either upwards or downwards, indicating a clear trend.

Reference [2] proposed the use of the Hurst exponent on the volatility of volatility indices in order to characterize the market regime.

Findings

-The study analyzes the volatility of volatility indices using data from five international markets—VIX, VXN, VXD, VHSI, and KSVKOSPI—covering the period from January 2001 to December 2021.

-It employs the Hurst exponent to evaluate long-term memory and persistence in volatility behavior, providing a framework to characterize market regimes over time.

-Different range-based estimators were used to calculate the Hurst exponent on various volatility measures, improving the robustness of the analysis.

-The volatility of volatility indices was estimated through a GARCH(1,1) model, which captures time-varying volatility dynamics effectively.

-The results show that Hurst exponent values derived from volatility of volatility indices reflect market regime shifts more accurately than those from standard volatility indices, supporting the authors’ hypothesis (H1).

-The analysis explores how different trading strategies—momentum, mean-reversion, and random walk—align with the Hurst exponent values, linking theoretical behavior to practical trading outcomes.

-The study highlights the effectiveness of the Hurst exponent as a tool for identifying and interpreting market regimes, which is essential for informed trading and investment decisions.

-Findings are particularly useful for financial analysts and researchers working with volatility indices and market behavior analysis.

-The paper contributes a novel methodological approach by combining Hurst exponent estimation with GARCH modeling and strategy backtesting, offering a comprehensive view of volatility behavior across regimes.

In short, the article highlights the effectiveness of employing the Hurst exponent on the volatility of volatility indices as a suitable method for characterizing the market regime.

Reference

[2] Georgia Zournatzidou and Christos Floros, Hurst Exponent Analysis: Evidence from Volatility Indices and the Volatility of Volatility Indices, J. Risk Financial Manag. 2023, 16(5), 272

Closing Thoughts

In this post, we explored the dynamics of the VVIX index, and how to use the Hurst exponent on it to characterize the market regime, offering a practical lens through which traders can gauge the persistence or randomness in volatility movements. By understanding these dynamics, market participants can better anticipate shifts in sentiment, enhance their hedging strategies, and adapt more effectively to evolving risk conditions in the options market.

Simplicity or Complexity? Rethinking Trading Models in the Age of AI and Machine Learning

When it comes to trading system design, there are two schools of thought: one advocates for simpler rules, while the other favors more complex ones. Which approach is better? This newsletter explores both perspectives through the lens of machine learning.

Use of Machine Learning in Pairs Trading

Machine learning has become an essential tool in modern finance, transforming the way financial institutions and investors approach data analysis and decision-making.

Reference [1] explored the use of machine learning in pairs trading. Specifically, the authors developed an algorithm to trade the classic Pepsi/Cola pair using three predictive methods: (i) fitting a linear model to real datasets of Pepsi and Coca-Cola stocks, (ii) employing a neural network approach to fit non-linear models, and (iii) utilizing an error correction model (ECM).

Findings

-The study investigates the relationship between two correlated stocks, Pepsi and Coca-Cola, using regression modeling and machine learning algorithms.

-The data is split into a training set (75%) and a testing set (25%) to evaluate model performance.

-A simple linear relationship between Pepsi prices (Y) and Coca-Cola prices (X) is modeled using both ordinary least squares (OLS) and a neural network (NN).

– A non-linear model between Y and X was fitted using the neural network (NN) method, and predictions were made for the X series.

-Two co-integrated stationary processes are used to analyze trading performance: the spread (Y − 𝑌^) and the ratio (𝑌^/X).

-The performance of each strategy is evaluated to determine the most effective approach for trading based on the co-movement of Pepsi and Coca-Cola.

– The total profit was computed and compared: the linear model generated a profit of $1.05102, while the neural network model produced $1.049395.

– The NN model’s performance was similar to that of the linear method.

– The NN model can outperform other methods if the optimal number of neurons is used in the hidden layers.

In short, the neural network performs similarly to the linear model method but can be improved by optimizing the number of neurons.

Reference

[1] R. Sivasamy, Dinesh K. Sharma, Sediakgotla, and B. Mokgweetsi, Machine Learning Algorithmic Model for Pairs Trading, in Machine Learning for Real World Applications, Springer 2024.

Can a Complex Trading System Be Profitable

The previous article shows that a more complex system does not lead to higher returns. Reference [2], however, demonstrates that such a complex system can provide better risk-adjusted performance. The authors achieved that by using Machine Learning techniques.

Findings

-Traditional financial literature often relies on simple models with few parameters to predict market returns.

-This study theoretically proves that such simple models significantly understate the potential for return predictability.

-The article provides new theoretical insights into the out-of-sample performance of machine learning portfolios.

-It demonstrates that high-complexity models in machine learning can improve investment strategies, contradicting conventional wisdom.

-Market timing strategies based on ridgeless least squares can generate positive Sharpe ratio improvements, even for highly complex models.

-The study shows that machine learning models can perform better with greater model parameterization, despite having fewer training observations and minimal regularization.

-The findings are supported by random matrix theory and explained through intuitive statistical mechanisms.

-The article argues that out-of-sample R² is a poor measure of a model’s economic value, as models with large negative R² can still generate large economic profits.

-It recommends that the finance profession shift focus from forecast accuracy to evaluating models based on economic metrics, such as Sharpe ratios.

Reference

[2] Kelly, Bryan T., and Malamud, Semyon and Zhou, Kangying, The Virtue of Complexity in Machine Learning Portfolios (2023). Swiss Finance Institute Research Paper No. 21-90

Closing Thoughts

So, should a trading system be simple and intuitive or complex and data-rich? In this edition, we featured research supporting both schools of thought. Perhaps both approaches have merit, depending on the context and objectives. What ultimately matters is not the simplicity or complexity of the model, but whether it has been thoroughly tested, proven robust across different market conditions, and shown to deliver consistent profitability before risking real capital.

Low-Volatility Stocks: Reducing Risk Without Sacrificing Returns

The recent market turbulence highlights the need for improved risk management and strategies to reduce portfolio volatility. In this post, I’ll explore how to enhance portfolio diversification using low-volatility stocks.

Gold and Low-Volatility Stocks as Diversifiers

Gold has long been regarded as a valuable diversification tool in investment portfolios due to its unique characteristics. As an asset class, gold has historically exhibited a low correlation with traditional financial assets such as stocks and bonds.

Reference [1] revisited the role of gold as a diversifier in a traditional stock-bond portfolio. It also proposed adding low-volatility stocks to the portfolio in order to reduce the risks without sacrificing the returns.

Findings

-The primary goal of investing is to avoid capital losses.

-Conservative investors often include gold in their portfolios to reduce downside risk. Although gold is volatile, it serves as a partial safe haven during bear markets.

-The study confirms that modest allocations to gold lower a portfolio’s loss probability, expected loss, and downside volatility.

-However, the downside protection offered by gold comes at the cost of reduced returns.

– In contrast, adding low-volatility stocks enhances a portfolio’s defensiveness without sacrificing returns.

-Low-volatility stocks are more effective than gold in mitigating losses while maintaining performance.

-Portfolios combining stocks, bonds, gold, and low-volatility stocks can be more resilient and allow for a higher equity allocation relative to bonds.

-The effectiveness of defensive multi-asset portfolios increases with a longer investment horizon.

In short, a stock-bond-gold allocation benefits significantly from incorporating low-volatility stocks, and the effectiveness of this defensive multi-asset portfolio grows with the investment horizon.

Reference

[1] van Vliet, Pim and Lohre, Harald, The Golden Rule of Investing, 2023, SSRN 4404688

Blending Low-Volatility with Momentum Anomalies

The low volatility anomaly in the stock market refers to the phenomenon where stocks with lower volatility tend to provide higher risk-adjusted returns compared to their higher volatility counterparts, contrary to traditional financial theories.

The momentum anomaly in the stock market refers to the tendency of assets that have performed well in the past to continue performing well in the future, and those that have performed poorly to continue performing poorly.

Reference [2] combined the low volatility anomaly with the momentum anomaly and examined whether the low volatility anomaly can enhance risk-adjusted returns in momentum-sorted portfolios.

Findings

-This paper analyzes the profitability of combining low-volatility and momentum strategies in the Nordic stock markets between January 1999 and September 2022.

-Both volatility and momentum strategies are found to remain effective as standalone (pure-play) approaches

-The authors evaluate three combination methods: 50/50 allocation, double screening, and ranking strategies.

-Among long-only portfolios, the momentum-first double screening strategy delivers the highest Sharpe ratio, slightly outperforming the ranking method.

-All long-only combination portfolios outperform the market in terms of risk-adjusted returns.

-Long-short combination strategies provide significantly better risk-adjusted returns compared to pure-play strategies.

-However, after adjusting returns using the Fama and French five-factor model, none of the combination long-short strategies outperform the pure momentum strategy.

In summary, the paper shows that incorporating both momentum and low volatility anomalies yields positive exposure to factors like value and profitability. Returns from these strategies are consistent over time and are more pronounced in later subsamples, with higher robust Sharpe Ratios. For long-only investors, the DS (double-sorted) strategy, which sorts stocks by momentum first and then by low volatility, seems superior to other strategies.

Reference

[2] Klaus Grobys, Veda Fatmy and Topias Rajalin, Combining low-volatility and momentum: recent evidence from the Nordic equities, Applied Economics, 2024

Closing Thoughts

In this post, we have seen how incorporating low-volatility stocks into a stock-gold portfolio can enhance risk-adjusted returns. We also discussed how to select stocks based on momentum and low-volatility criteria, highlighting the effectiveness of combining these factors through methods like double screening or ranking. While momentum tends to drive performance, especially in long-short strategies, low volatility adds defensiveness to the portfolio.

The Calendar Effects in Volatility Risk Premium

I recently covered calendar anomalies in the stock markets. Interestingly, patterns over time also appear in the volatility space. In this post, I’ll discuss the seasonality of volatility risk premium (VRP) in more detail.

Breaking Down the Volatility Risk Premium: Overnight vs. Intraday Returns

The decomposition of the volatility risk premium (VRP) into overnight and intraday components is an active area of research. Most studies indicate that the VRP serves as compensation for investors bearing overnight risks.

Reference [1] continues this line of research, with its main contribution being the decomposition of the variance risk premium into overnight and intraday components using a variance swap approach. The study also tests the predictive ability of these components and examines the seasonality (day-of-week effects) of the VRP.

An interesting finding of the paper is the day-of-week seasonality. For instance, going long volatility at the open and closing the position at the close tends to be profitable on most days, except Fridays.

Findings

-The analysis is conducted on implied variance stock indices across the US, Europe, and Asia.

-Results show that the VRP switches signs between overnight and intraday periods—negative overnight and positive intraday.

-The findings suggest that the negative VRP observed in previous studies is primarily driven by the overnight component.

-The study evaluates the predictive power of both intraday and overnight VRP in forecasting future equity returns.

-The intraday VRP component captures short-term risk and demonstrates predictive ability over 1–3-month horizons.

-The overnight VRP component reflects longer-term risk and shows predictive power over 6–12-month horizons.

Reference

[1] Papagelis, Lucas and Dotsis, George, The Variance Risk Premium Over Trading and Non-Trading Periods (2024), SSRN 4954623

Volatility Risk Premium Seasonality Across Calendar Months

Reference [2] examines the VRP in terms of months of the year. It concluded that the VRP is greatest in December and smallest in October.

An explanation for the large VRP in December is that during the holiday season, firms might refrain from releasing material information, leading to low trading volumes. The combination of low trading volume and the absence of important news releases would result in lower realized volatility.

Findings

-The paper identifies a “December effect” in option returns, where delta-hedged returns on stock and S&P 500 index options are significantly lower in December than in other months.

-This effect is attributed to investors overvaluing options at the start of December due to underestimating the typically low volatility that occurs in the second half of the month.

– The reduced volatility is linked to lighter stock trading during the Christmas holiday season.

– A trading strategy that involves shorting straddles at the beginning of December and closing the position at the end of the month yields a hedged return of 13.09%, with a t-value of 6.70.

-This return is much higher than the unconditional sample mean of 0.88%, highlighting the strength of the effect.

The paper is the first in academic literature to document and analyze this specific December anomaly in option markets. It is another important contribution to the understanding of the VRP.

Reference

[2] Wei, Jason and Choy, Siu Kai and Zhang, Huiping, December Effect in Option Returns (2025). SSRN 5121679

Closing Thoughts

In this post, I have discussed volatility patterns in terms of both days of the week and months of the year. Understanding this seasonality is crucial for traders and portfolio managers, as it can inform better timing of volatility trades and risk management strategies.

Stock-Bond Correlation: What Drives It and How to Predict It

The correlation between stocks and bonds plays a crucial role in portfolio allocation and diversification strategies. In this issue, I discuss stock-bond relationships, the factors that influence their correlation, and techniques for forecasting it.

What Influences Stock-Bond Correlation?

Correlation between stocks and bonds is crucial for portfolio allocation and diversification, but this correlation can vary over time due to factors like inflation and real returns on short-term bonds.

Reference [1] conducts a study on stock-bond correlation spanning an extended timeframe. Their findings indicate that contrary to conventional assumptions, stock-bond correlation generally tends to exhibit a positive or near-zero relationship. Exceptions, where the correlation drops below -0.2, were notably observed during the early 1930s, the late 1950s, and most of the 2000s.

Findings

-The correlation between stock and bond returns is a key component in asset allocation decisions. This correlation is not stable and can vary significantly over time, affecting how portfolios should be constructed.

– The recent market environment has shown that stock-bond correlation can turn positive, potentially impacting diversified portfolios negatively.

– The article suggests that contrary to conventional assumptions, stock-bond correlation generally tends to be positive or near-zero.

– Exceptions to positive correlation occurred during the early 1930s, late 1950s, and most of the 2000s.

– Factors such as inflation, real returns on short-term bonds, and uncertainty surrounding inflation play pivotal roles in determining the direction and strength of stock-bond correlation.

– Time variation in stock and bond volatility can also affect the impact of stock-bond correlation.

– Bond risk premia are positively correlated with estimates of the stock-bond correlation.

– The correlation between stocks and bonds can significantly fluctuate over time and across countries.

In short, the correlation between stocks and bonds can significantly fluctuate over time. Factors such as inflation and real returns on short-term bonds, along with the associated uncertainty regarding inflation, play pivotal roles in determining both the direction and strength of the stock-bond correlation.

Reference

[1] Molenaar, Roderick and Senechal, Edouard and Swinkels, Laurens and Wang, Zhenping, Empirical evidence on the stock-bond correlation (2023), SSRN 4514947

Forecasting Short-Term Stock-Bond Correlation

Reference [2] employs a country’s Correlation Outlook, Prospective Inflation Volatility, the Yield Curve Momentum Regime, and the Trailing 3-month stock-bond correlation to build a predictive model.

Findings

-This paper extends a macroeconomic framework that explains long-term changes in stock-bond correlation.

-Prior research explains around 70% of the variation in 10-year rolling stock-bond correlations using the relative volatility and correlation of growth and inflation.

-The authors shift focus to forecasting short-term (three-month) variations in stock-bond correlation.

-Their method uses indicators based on whether individual forecasters expect stock and bond markets to move in the same or opposite directions.

– This approach improves the ability to forecast stock-bond correlations over tactical, short-term horizons.

This paper complements previous work by focusing on short-term horizons, showing that detailed forecast data can help predict high-frequency changes in stock-bond correlation. It also highlights the value of granular forecast data, especially the correlation between responses, which may be missed in standard survey summaries.

Reference

[2] Flannery, Garth and Bergstresser, Daniel, A Changing Stock-Bond Correlation: Explaining Short-term Fluctuations (2023). SSRN 4672744

Closing Thoughts

As we have seen, stock-bond correlation plays a crucial role in portfolio management and asset allocation. We have discussed how this correlation shifts over time, influenced by macroeconomic factors such as inflation and growth volatility, and how it can be forecasted. Accurately anticipating these shifts enables more informed portfolio construction and risk management.

Profitability of Dispersion Trading in Liquid and Less Liquid Environments

Dispersion trading is an investment strategy used to capitalize on discrepancies in volatilities between an index and its constituents. In this issue, I will feature dispersion trading strategies and discuss their profitability.

Profitability of a Dispersion Trading Strategy

Reference [1] provided an empirical analysis of a dispersion trading strategy to verify its profitability. The return of the dispersion trading strategy was 23.51% per year compared to the 9.71% return of the S&P 100 index during the same period. The Sharpe ratio of the dispersion trading strategy was 2.47, and the portfolio PnL had a low correlation (0.0372) with the S&P 100 index.

Findings

-The article reviews the theoretical foundation of dispersion trading and frames it as an arbitrage strategy based on the mispricing of index options due to overestimated implied correlations among the index’s constituents.

-The overpricing phenomenon is attributed to the correlation risk premium hypothesis and the market inefficiency hypothesis.

-Empirical evidence shows that a basic dispersion trading strategy—using at-the-money straddles on the S&P 100 and a representative subset of its stocks—has significantly outperformed the broader stock market.

-The performance of the dispersion strategy demonstrated a very low correlation to the S&P 100 index, highlighting its diversification potential.

-This study reinforces the idea that sophisticated options strategies can uncover persistent market inefficiencies.

This article proved the viability of the dispersion trading strategy. However, there exist two issues related to execution,

-The analysis assumes no transaction costs, which is a key limitation; in practice, only market makers might replicate the back-tested performance due to the absence of slippage.

-Another limitation is the simplified delta hedging method used, which was based on daily rebalancing.

-A more optimized hedging approach could potentially yield higher returns and partially offset transaction costs.

Reference

[1] P. Ferrari, G. Poy, and G. Abate, Dispersion trading: an empirical analysis on the S&P 100 options, Investment Management and Financial Innovations, Volume 16, Issue 1, 2019

Dispersion Trading in a Less Liquid Market

The previous paper highlights some limitations of the dispersion strategy. Reference [2] further explores issues regarding liquidity. It investigates the profitability of dispersion trading in the Swedish market.

Findings

-Dispersion trading offers a precise and potentially profitable approach to hedging vega risk, which relates to volatility exposure.

-The strategy tested involves shorting OMXS30 index volatility and taking a long volatility position in a tracking portfolio to maintain a net vega of zero.

-The backtesting results show that vega risk can be accurately hedged using dispersion trading.

– Without transaction costs, the strategy yields positive results.

-However, after accounting for the bid-ask spread, the strategy did not prove to be profitable over the simulated period.

– High returns are offset by substantial transaction costs due to daily recalibration of tracking portfolio weights.

– Less frequent rebalancing reduces transaction costs but may result in a worse hedge and lower correlation to the index.

In short, the study concluded that if we use the mid-price, then dispersion trading is profitable. However, when considering transaction costs and the B/A spreads, the strategy becomes less profitable.

I agree with the author that the strategy can be improved by hedging less frequently. However, this will lead to an increase in PnL variance. But we note that this does not necessarily result in a smaller expected return.

Reference

[2] Albin Irell Fridlund and Johanna Heberlei, Dispersion Trading: A Way to Hedge Vega Risk in Index Options, 2023, KTH Royal Institute of Technology

Closing Thoughts

I have discussed the profitability of dispersion strategies in both liquid and illiquid markets. There exist “inefficiencies” that can be exploited, but doing so requires a more developed hedging approach and solid infrastructure. The “edge” is apparent, but consistently extracting it demands a high level of skill, discipline, and operational capability. In reality, it is this latter part, i.e. the ability to build and maintain the necessary infrastructure, that represents the true edge.

Machine Learning in Financial Markets: When It Works and When It Doesn’t

Machine learning (ML) has made a lot of progress in recent years. However, there are still skeptics, especially when it comes to its application in finance. In this post, I will feature articles that discuss the pros and cons of ML. In future editions, I’ll explore specific techniques.

How Accurate is Machine Learning Prediction in Finance?

Machine Learning has many applications in finance, such as predicting stock prices, detecting fraudulent activities, and automating investment decisions. However, the accuracy of ML prediction can vary widely depending on the type of data used and the model chosen.

Reference [1] discusses the problems that Machine Learning is facing in finance.

Findings

-Recent research suggests that machine learning (ML) is valuable in asset pricing due to its ability to capture nonlinearities and interaction effects that traditional models often miss.

-Machine learning is highly effective for applications with large datasets and high signal-to-noise ratios, but financial market data often lacks these characteristics.

-Financial markets evolve over time, meaning anomalies detected by ML can be arbitraged away, rendering past data less relevant for future predictions.

-An analogy highlights the challenge: once an ML algorithm learns to recognize “cats” in an image, all “cats” could morph into “dogs,” requiring the algorithm to relearn from scratch.

-There is a risk of positive publication bias, overfitting, and reliance on the assumption that past relationships will persist in the future.

-Human expertise remains crucial due to the low signal-to-noise ratio in financial data and the limitations of ML models.

In summary, the paper concludes that while ML does show promise, its superior performance is often overstated. When practical challenges are taken into account, the performance gap between ML and traditional methods narrows. However, investors who follow a rigorous and disciplined research process can still benefit meaningfully from ML-based strategies.

Reference

[1] Blitz, David and Hoogteijling, Tobias and Lohre, Harald and Messow, Philip, How Can Machine Learning Advance Quantitative Asset Management? (2023), SSRN 4321398

Machine Learning: Is More Data Always Better?

Reference [2] discusses the question whether more data is always beneficial in machine learning.

Findings

-The paper delves into the nuanced aspect of data quantity, questioning the assumption that more data necessarily leads to better machine learning outcomes.

-It argues that older data may lose relevance over time and including it can actually reduce model accuracy.

-Increasing the flow of data, or collecting data at a higher rate, tends to improve model accuracy but requires more frequent model retraining.

– Quality vs. Quantity: It discusses the trade-off between the quality and quantity of data, suggesting that the relevance and quality of the data are crucial factors in the effectiveness of machine learning models.

-The business value of machine learning models does not necessarily scale with the amount of stored data, especially if the data becomes outdated.

-Firms should adopt a growth policy that balances the retention of historical data with the acquisition of fresh data.

-Real-world Applications: Examples from various industries, such as healthcare, are presented to illustrate scenarios where the volume of data may not be the sole determinant of success in machine learning applications.

What implication does this paper have for trading and portfolio management? Should we use more data?

The short answer is probably no. In fact, using more data can actually lead to sub-optimal results. The reason is that, in the financial world, data is often noisy and contains a lot of irrelevant information. If you use too much data, your machine learning models will end up picking up on this noise, which can lead to sub-optimal results.

Reference

[2] Valavi, Ehsan, Joel Hestness, Newsha Ardalani, and Marco Iansiti. Time and the Value of Data. Harvard Business School Working Paper, No. 21-016

Closing Thoughts

In this post, I discussed the advantages and disadvantages of machine learning techniques as applied in finance. However, as the field is progressing rapidly, many of the current limitations, such as overfitting, interpretability, and data relevance, are being actively addressed by researchers and practitioners. With a disciplined research process and model design, investors can harness the strengths of machine learning to enhance forecasting, risk management, and strategy development.

Do Calendar Anomalies Still Work? Evidence and Strategies

Calendar anomalies in the stock market refer to recurring patterns or anomalies that occur at specific times of the year, month, or week, which cannot be explained by traditional financial theories. These anomalies often defy the efficient market hypothesis and provide opportunities for investors to exploit market inefficiencies. In this post, I will feature some calendar anomalies and discuss whether they work in the current market or not.

Do Calendar Anomalies Still Exist?

Calendar anomalies were discovered long ago. Reference [1] examines whether they still persist in the present-day stock market. Specifically, the author investigates the turn-of-the-month (TOM), turn-of-the-quarter (TOQ), and turn-of-the-year (TOY) effects in the US stock market.

Findings

– The paper identifies the presence of the turn-of-the-month (TOM), turn-of-the-quarter (TOQ), and turn-of-the-year (TOY) effects in the US stock market, with the TOY effect being the most prominent.

-The analysis uses panel regression models on four-day return windows for individual stocks listed on the NYSE, AMEX, and NASDAQ from 1986 to 2021.

– The TOM, TOQ, and TOY effects are found to be present, and their strength varies based on firm characteristics.

– The TOY effect primarily affects small stocks with volatile prices, indicating that individual investors may sell their losses for tax purposes before the year-end.

– Stocks with low momentum are more susceptible to the TOY effect, suggesting that institutional investors may engage in performance hedging by selling underperforming stocks.

– The calendar effects have evolved over time, with the TOM and TOY effects resurfacing in recent decades, while the TOQ effect has diminished, potentially due to increased disclosure regulations.

– Companies with low Google search volumes are significantly more impacted by all three effects, indicating a relationship between information accessibility and the magnitude of calendar anomalies.

-A trading strategy is developed to identify stocks with the highest expected returns over TOM and TOY windows. The return exceeds realistic trading costs, indicating that calendar effects can be used to construct profitable trading strategies.

In summary, calendar anomalies continue to exist in the US stock market. Furthermore, they can be exploited to gain abnormal returns. For instance, every four-day TOY window yields an average profit of 1.66% when holding all stocks exclusively over the TOY windows. Similarly, an average profit of 0.55% is generated every four-day TOM window by exclusively holding all stocks over the TOM windows.

Replication

Vahid Asghari and his team at Academic Quant Lab have replicated the strategy presented in this paper. The results and codes can be found here.

Reference

[1] Idunn Myrvang Hatlemark and Maria Grohshennig, Calendar Effects in the US Stock Market: Are they still present?, 2022, Norwegian University of Science and Technology

How End-of-Month Returns Predict the Next Month’s Performance

Reference [2] introduced a novel calendar anomaly known as the end-of-month reversal effect. The study showed that end-of-month returns, i.e. returns from the fourth Friday to the last trading day of the month, are negatively correlated with returns in the following month.

Findings

-This paper identifies a novel 1-month aggregate market reversal pattern, which is driven by the previous end-of-month market return.

– It demonstrates that end-of-the-month returns of the S&P 500 are negatively correlated with returns one month later.

-The reversal effect is statistically significant both In-Sample and Out-of-Sample, confirming its robustness.

-Unlike traditional cross-sectional reversals, this pattern is stronger in high-priced and liquid stocks and follows an economic cycle.

-A simple rule-based trading strategy and more sophisticated models leveraging this pattern generate significant economic gains. The strategy is cyclical in nature and does not rely on short-selling.

-The reversal effect strengthens over the following month, aligning with pension fund inflows and reinforcing the payment cycle explanation.

In short, a simple trading strategy based on this effect, that is buying if the end-of-month return is negative and selling if it is positive, outperforms the buy-and-hold strategy over a 45-year period.

The author also provides an explanation for this anomaly, attributing it to pension funds’ liquidity trading, as they adjust their portfolios to meet pension payment obligations.

Reference

[2] Graziani, Giuliano, Time Series Reversal: An End-of-the-Month Perspective, 2024, SSRN

Closing Thoughts

In this post, I discussed several calendar anomalies. Some of these patterns were discovered long ago and have proven to be persistent in today’s market. One of them represents a newly identified anomaly with promising characteristics. In all cases, profitable trading strategies were developed to take advantage of these recurring effects, highlighting the continued relevance of calendar-based insights in quantitative investing.

Catastrophe Bonds: Modeling Rare Events and Pricing Risk

A catastrophe (CAT) bond is a debt instrument designed to transfer extreme event risks from insurers to capital market investors. They’re important for financial institutions, especially insurers and reinsurers, because they offer a way to manage large, low-probability. In this post, I feature research on CAT bonds, how they’re priced, and why they matter more than ever in a world of rising tail risks.

A Pricing Model for Earthquake Bonds

An earthquake bond is a type of catastrophe bond, in which an insurer, reinsurer, or government, transfers a portion or all of the earthquake risk to investors in return for higher yields. Earthquake bonds are crucial in countries prone to earthquakes. However, pricing them presents challenges.

Reference [1] developed a pricing model for pricing earthquake bonds. The authors modeled the risk-free interest rate using the Cox–Ingersoll–Ross model. They accommodated the variable intensity of events with an inhomogeneous Poisson process, while extreme value theory (EVT) was used to model the maximum strength.

Findings

– Earthquake bonds (EBs) connect insurance mechanisms to capital markets, offering a more sustainable funding solution, though pricing them remains a challenge.

– The paper proposes zero-coupon and coupon-paying EB pricing models that incorporate varying earthquake event intensity and maximum strength under a risk-neutral framework.

– The models focus on extreme earthquakes, which simplifies data processing and modeling compared to accounting for continuous earthquake occurrences.

– The earthquake event intensity is modeled using an inhomogeneous Poisson process, while the maximum strength is handled through extreme value theory (EVT).

– The models are tested using earthquake data from Indonesia’s National Disaster Management Authority covering 2008 to 2021.

– Sensitivity analyses show that using variable intensity instead of constant intensity significantly affects EB pricing.

– The proposed pricing model can help EB issuers set appropriate bond prices based on earthquake risk characteristics.

– Investors can use the sensitivity findings to select EBs that align with their individual risk tolerance.

In summary, the authors modeled the risk-free interest rate using the Cox–Ingersoll–Ross model. They accommodated the variable intensity of events with an inhomogeneous Poisson process, while extreme value theory (EVT) was used to model the maximum strength.

Reference

[1] Riza Andrian Ibrahim, Sukono, Herlina Napitupulu and Rose Irnawaty Ibrahim, Earthquake Bond Pricing Model Involving the Inconstant Event Intensity and Maximum Strength, Mathematics 2024, 12, 786

No-arbitrage Model for Pricing CAT Bonds

Pricing models for catastrophic risk-linked securities have primarily followed two methodologies: the theory of equilibrium pricing and the no-arbitrage valuation framework.

Reference [2] proposed a pricing approach based on the no-arbitrage framework. It utilizes the CIR stochastic process model for interest rates and the jump-diffusion stochastic process model for losses.

Findings

– This paper explores the concept of CAT bonds and explains how they are modeled using financial mathematics.

– Through a semi-discretization approach, a PIDE and a first-order differential equation were derived.

– A key component, the market price of risk of damage, was unavailable, so a quadratic term was constructed using market ask and bid prices to estimate this variable.

– By utilizing the Euler-Lagrange equation, a Poisson PDE was derived.

– The paper concludes by presenting an approach and numerical results for determining the market price of risk.

We find the stochastic model, equation (1), to be particularly insightful and effective in describing catastrophic losses.

Last year has witnessed numerous hurricanes across Asia, Europe, and America, leading to significant claims for insurers. This paper represents a contribution to advancing risk-sharing practices in the insurance industry.

Reference

[2] S. Pourmohammad Azizi & Abdolsadeh Neisy, Inverse Problems to Estimate Market Price of Risk in Catastrophe Bonds, Mathematical Methods of Statistics, Vol. 33 No. 3 2024

Closing Thoughts

In this post, I discussed catastrophe bonds and why they matter for investors navigating extreme event risks. The first paper focused on earthquake bonds, which present a challenge to model due to their rare and severe nature. Interestingly, both pricing models in the paper relied on the Cox–Ingersoll–Ross framework for interest rates, a reminder that even in the world of tail-risk instruments, some core quantitative models remain consistent.

Breaking Down Volatility: Diffusive vs. Jump Components

Implied volatility is an important concept in finance and trading. In this post, I further discuss its breakdown into diffusive volatility and jump risk components.

Decomposing Implied Volatility: Diffusive and Jump Risks

Implied volatility is an estimation of the future volatility of a security’s price. It is calculated using an option-pricing model, such as the Black-Scholes-Merton model.

Reference [1] proposed a method for decomposing implied volatility into two components: a volatility component and a jump component. The volatility component is the price of a portfolio only bearing volatility risks and the jump component is the price of a portfolio only bearing jump risks. The decomposition is made by constructing two option portfolios: a delta- and gamma-neutral but vega-positive portfolio and a delta- and vega-neutral but gamma-positive portfolio. These portfolios bear volatility and jump risks respectively.

Findings

– The study examines the return patterns of straddles and their component portfolios, focusing on jump risk and volatility risk around earnings announcements.

– The findings show that straddle returns closely resemble those of the jump risk portfolio, suggesting that the options market prioritizes earnings jump risk during these events.

– The research highlights the significant role of earnings jump risk in financial markets, as it is substantially priced into straddles and influences both options and stock market behavior.

– A proposed straddle price decomposition method and the S-jump measure could be applied to other market events, such as M & A and natural disasters, to assess risk and pricing dynamics.

This paper discussed an important concept in option pricing theory; that is, the implied volatilities, especially those of short-dated options, comprise not only volatility but also jump risks.

Reference

[1] Chen, Bei and Gan, Quan and Vasquez, Aurelio, Anticipating Jumps: Decomposition of Straddle Price (2022). Journal of Banking and Finance, Volume 149, April 2023, 106755

Measuring Jump Risks in Short-Dated Option Volatility

Unlike long-dated options, short-dated options incorporate not only diffusive volatility but also jump risks. One of the earliest works examining the jump risks is by Carr et al [2].

Reference [3] developed a stochastic jump volatility model that includes jumps in the underlying asset. It then constructed a skew index, a so-called crash index.

Findings

-This paper introduces a novel methodology to measure forward-looking crash risk implied by option prices, using a tractable stochastic volatility jump (SVJ) model.

-The approach isolates the jump size component from the stochastic volatility embedded within uncertainty risk, extending beyond the Black-Scholes-Merton framework.

-The methodology parallels the construction of implied volatility surfaces, enabling the development of an option-implied crash-risk curve (CIX).

-The CIX is strongly correlated with non-parametric option-implied skewness but offers a more refined measure of crash risk by adjusting for stochastic volatility (Vt) and emphasizing tail risk dynamics.

-In contrast, option-implied skewness reflects both crash and stochastic volatility risks, presenting smoother characteristics of the risk-neutral density.

-Empirical analysis reveals a notable upward trend in the CIX after the 2008 financial crisis, aligning with narratives on rare-event risks and emphasizing the value of incorporating such beliefs into asset pricing frameworks.

References

[2] P Carr, L Wu, What type of process underlies options? A simple robust test, The Journal of Finance, 2003

[3] Gao, Junxiong and Pan, Jun, Option-Implied Crash Index, 2024. SSRN

Closing Thoughts

In this issue, I discussed the breakdown of volatility into diffusive and jump components. Understanding this distinction is important for trading, and risk management in theory and practice.