Category: statistical arbitrage

Machine learning has many applications in finance, such as asset pricing, risk management, portfolio optimization, and fraud detection. In this post, I discuss the use of machine learning in forecasting volatility.

Using Machine Learning to Predict Market Volatility

The unpredictability of the markets is a well-known fact. Despite this, many traders and portfolio managers continue to try to predict market volatility and manage their risks accordingly. Usually, econometric models such as GARCH are used to forecast market volatility.

In recent years, machine learning has been shown to be capable of predicting market volatility with accuracy. Reference [1] explored how machine learning can be used in this context.

Findings

-Machine learning models can accurately forecast stock return volatility using a small set of key predictors: realized volatility, idiosyncratic volatility, bid-ask spread, and returns.

-These predictors align with existing empirical findings, reinforcing the traditional risk-return trade-off in finance.

-ML methods effectively capture both the magnitude and direction of predictor impacts, along with their interactions, without requiring pre-specified model assumptions.

-Large current-period volatility values strongly predict higher future volatility; small values have a muted or negative impact.

-LSTM models outperform feedforward neural networks and regression trees by leveraging temporal patterns in historical data.

-An LSTM using only volatility and return history over one year performs comparably to more complex models with additional predictors.

-LSTM models function as distribution-free alternatives to traditional econometric models like GARCH.

-Optimal lag length remains critical in LSTM performance and must be selected through model training.

-The study reports an average predicted realized volatility of 44.1%, closely matching the actual value of 43.8%.

-Out-of-sample R² values achieved are significantly higher than those typically reported in related volatility forecasting literature.

In short, the paper aimed to demonstrate the potential of machine learning for modeling market volatility. In particular, the authors have shown how the LSTM model can be used to predict market volatility and manage risks. The results suggest that this is a promising alternative approach to traditional econometric models like GARCH.

Reference

[1] Filipovic, Damir and Khalilzadeh, Amir, Machine Learning for Predicting Stock Return Volatility (2021). Swiss Finance Institute Research Paper No. 21-95

Machine Learning Models for Predicting Implied Volatility Surfaces

The Implied Volatility Surface (IVS) represents the variation of implied volatility across different strike prices and maturities for options on the same underlying asset. It provides a three-dimensional view where implied volatility is plotted against strike price (moneyness) and time to expiration, capturing market sentiment about expected future volatility.

Reference [2] examines five methods for forecasting the Implied Volatility Surface of short-dated options. These methods are applied to forecast the level, slope, and curvature of the IVS.

Findings

-The study evaluates five methods—OLS, AR(1), Elastic Net, Random Forest, and Neural Network—to forecast the implied volatility surface (IVS) of weekly S&P 500 options.

-Forecasts focus on three IVS characteristics: level, slope, and curvature.

-Random Forest consistently outperforms all other models across these three IVS dimensions.

-Non-learning-based models (OLS, AR(1)) perform comparably to some machine learning methods, highlighting their continued relevance.

-Neural Networks forecast the IVS level reasonably well but perform poorly in predicting slope and curvature.

-Elastic Net, a linear machine learning model, is consistently outperformed by the non-linear models (Random Forest and Neural Network) for the level characteristic.

-The study emphasizes the importance of model selection based on the specific IVS characteristic being forecasted.

-Performance evaluation is supported using the cumulative sum of squared error difference (CSSED) and permutation variable importance (VI) metrics.

-The research highlights the utility of Random Forest in capturing complex, non-linear patterns in IVS dynamics.

-Accurate IVS forecasting is valuable for derivative pricing, hedging, and risk management strategies.

This research highlights the potential of machine learning in forecasting the implied volatility surface, a key element in options pricing and risk management. Among the five methods studied, Random Forest stands out as the most consistent and accurate across multiple IVS features.

Reference

[2] Tim van de Noort, Forecasting the Characteristics of the Implied Volatility Surface for Weekly Options: How do Machine Learning Methods Perform? Erasmus University, 2024

Closing Thoughts

These studies highlight the growing effectiveness of machine learning in financial forecasting, particularly for market volatility and implied volatility surfaces. Models like LSTM and Random Forest demonstrate clear advantages over traditional methods by capturing complex patterns and dependencies. As financial markets evolve, leveraging such tools offers a promising path for enhancing predictive accuracy and risk management.

Being able to anticipate a market correction or an economic recession is important for managing risk and positioning your portfolio ahead of major shifts. In this post, we feature two articles: one that analyzes indicators signaling a potential market correction, and another that examines recession forecasting models based on macroeconomic data.

Predicting Recessions Using The Volatility Index And The Yield Curve

The yield curve is a graphical representation of the relationship between the yields of bonds with different maturities. The yield curve has been inverted before every recession in the United States since 1971, so it is often used as a predictor of recessions.

A study [1] shows that the co-movement between the yield-curve spread and the VIX index, a measure of implied volatility in S&P500 index options, offers improvements in predicting U.S. recessions over the information in the yield-curve spread alone.

Findings

-The VIX index measures implied volatility in S&P 500 index options and reflects investor sentiment and market uncertainty.

-A counterclockwise pattern (cycle) between the VIX index and the yield-curve spread aligns closely with the business cycle.

-A cycle indicator based on the VIX-yield curve co-movement significantly outperforms the yield-curve spread alone in predicting recessions.

-This improved forecasting performance holds true for both in-sample and out-of-sample data using static and dynamic probit models.

-The predictive strength comes from the interaction between monetary policy and financial market corrections, not from economic policy uncertainty.

-Shadow rate analysis confirms the cycle indicator’s effectiveness, even during periods of unconventional monetary policy and flattened yield curves.

-The findings suggest a new framework for macroeconomic forecasting, with the potential to enhance early detection of financial instability.

-The VIX-yield curve cycle adds value beyond existing leading indicators and may help in anticipating major economic disruptions like the subprime crisis.

In short, the study concludes that the co-movement between the yield curve spread and the VIX index, which is a measure of implied volatility in S&P 500 index options, provides an improved prediction for U.S. recessions over any information available from just considering the yield-curve spreads alone.

This new research will have implications for how macroeconomists forecast future economic conditions and could even change how we predict periods of high financial instability like the subprime crisis.

Reference

[1] Hansen, Anne Lundgaard, Predicting Recessions Using VIX-Yield-Curve Cycles (2021). SSRN 3943982

Can We Predict a Market Correction?

A correction in the equity market refers to a downward movement in stock prices after a sustained period of growth. Market corrections can be triggered by various factors such as economic conditions, changes in investor sentiment, or geopolitical events. During a correction, stock prices may decline by a certain percentage from their recent peak, signaling a temporary pause or reversal in the upward trend.

Reference [2] examines whether a correction in the equity market can be predicted. It defines a correction as a 4% decrease in the SP500 index. It utilizes logistic regression to examine the predictability of several technical and macroeconomic indicators.

Findings

-Eight technical, macroeconomic, and options-based indicators were selected based on prior research.

-Volatility Smirk (skew), Open Interest Difference, and Bond-Stock Earnings Yield Differential (BSEYD) are statistically significant predictors of market corrections.

-These three predictors were significant at the 1% level, indicating strong reliability in forecasting corrections.

-TED Spread, Bid-Offer Spread, Term Spread, Baltic Dry Index, and S&P GSCI Commodity Index did not show consistent predictive power.

-The best-performing model used a 3% correction threshold and achieved 77% accuracy in in-sample prediction.

-Out-of-sample testing showed 59% precision in identifying correction events, offering an advantage over random prediction.

-The results highlight inefficiencies in the market and support the presence of a lead-lag effect between option and equity markets.

-The research provides valuable tools for risk management and identifying early signs of downturns in equity markets.

In short, the following indicators are good predictors of a market correction,

-Volatility Smirk (i.e. skew),

-Open Interest Difference, and

-Bond-Stock Earnings Yield Differential (BSEYD)

The following indicators are not good predictors,

-The TED Spread,

-Bid-Offer Spread,

-Term Spread,

-Baltic Dry Index, and

-S&P GSCI Commodity Index

This is an important research subject, as it allows investors to manage risks effectively and take advantage of market corrections.

Reference

[2] Elias Keskinen, Predicting a Stock Market Correction, Evidence from the S&P 500 Index, University of VAASA

Closing Thoughts

This research underscores the growing value of combining traditional financial indicators with options market metrics to improve market correction and recession forecasts. Tools like the VIX-yield curve cycle, Volatility Smirk, and BSEYD offer a more refined understanding of market risks. As financial markets evolve, integrating diverse data sources will be key to staying ahead of economic and market shifts.

A leveraged Exchanged Traded Fund (LETF) is a financial instrument designed to deliver a multiple of the daily return of an underlying index. Despite criticism, LETFs are frequently used by institutional investors. In this post, I discuss the practicality of LETFs and show that they are not as risky as they may seem.

Information Content of Leveraged ETFs Options

Leveraged ETFs, or exchange-traded funds, are investment funds designed to amplify the returns of an underlying index or asset class through the use of financial derivatives and debt. These ETFs aim to achieve returns that are a multiple of the performance of the index they track, typically two or three times (2x, 3x) the daily performance.

There is evidence that 1x ETF options provide an indication of the future return of the underlying 1x ETF. Reference [1] goes further and postulates that options on leveraged ETFs provide an even stronger indication of the 1x ETF future return.

Findings

-Options on leveraged ETFs provide stronger predictive signals for future ETF returns compared to standard ETF options, showing higher economic and statistical significance.

-The study uses unexpected changes in implied volatility from call and put options on leveraged ETFs to identify signals of informed trading activity.

-Leveraged ETF option signals consistently outperform unleveraged signals in predicting future returns of the underlying ETFs across various market conditions.

-Sophisticated investors often trade leveraged ETFs for exposure and rely on their options markets to hedge or speculate based on market expectations.

-A $1 investment in SPY based on leveraged option signals would have generated $27.59 in net returns from 2009 to 2021 after transaction costs.

-The predictive power of leveraged ETF option signals is especially strong during economic downturns, making them useful in volatile or declining markets.

-Inverse leveraged ETFs provide particularly strong predictive signals, especially when markets are trending downward or experiencing negative momentum.

-A trading strategy based on leveraged ETF option signals produced average abnormal returns of 1.13% per month, even after accounting for transaction costs.

-The findings suggest that options on leveraged ETFs play a key role in market efficiency and price discovery by reflecting informed investor activity.

-Both leveraged and unleveraged ETF options contain return-predictive information, but the economic impact is far greater when using leveraged ETF option signals.

In short, by using the difference in implied volatility innovations between calls and puts of leveraged ETFs as a trading signal, one can gain excess returns.

Reference

[1] Collin Gilstrap, Alex Petkevich, Pavel Teterin, Kainan Wang, Lever up! An analysis of options trading in leveraged ETFs, J Futures Markets. 2024, 1–17

Leveraged Exchange Traded Funds Revisited: Enhancing Returns or Adding Risk?

LETFs have received a lot of criticism. Despite the controversy, they remain popular among institutional investors. Reference [2] revisited the use of LETFs in portfolio allocation.

Findings

-LETFs aim to deliver amplified daily returns using derivatives and debt, making them suitable for short-term tactical strategies but requiring careful risk management.

-The study shows LETFs exhibit call option–like payoff characteristics, suggesting they can offer inexpensive leverage with built-in downside protection in certain scenarios.

-Under ideal conditions like continuous rebalancing and no constraints, the authors derived a closed-form information ratio–optimal strategy that followed a contrarian investment approach

-In realistic market conditions, including quarterly trading and margin constraints, a neural network approach was used to identify performance-optimized LETF allocation strategies.

-Results showed that unleveraged strategies using LETFs outperform benchmarks more frequently than leveraged strategies using standard (vanilla) ETFs on the same index.

-These unleveraged LETF strategies also showed partial stochastic dominance over both the benchmark and vanilla ETF-based strategies in terms of terminal wealth outcomes.

-The neural network–based strategy, trained on historical market data, further supports the practical value of including LETFs in actively managed portfolios.

-The findings challenge the common belief that LETFs only serve short-term speculation, revealing potential for long-term, dynamically optimized investment use.

-Overall, incorporating LETFs through informed strategies can enhance risk-adjusted returns, outperform traditional benchmarks, and improve the robustness of portfolio performance.

An interesting finding of this study is that, through a closed-form solution and numerical simulations, the authors demonstrated that LETFs behave like call options. Based on this, it is intuitive that if LETFs are part of a portfolio, they can enhance risk-adjusted returns.

Reference

[2] Pieter van Staden, Peter Forsyth, Yuying Li, Smart leverage? Rethinking the role of Leveraged Exchange Traded Funds in constructing portfolios to beat a benchmark, 2024, arXiv:2412.05431

Closing Thoughts

In conclusion, both studies provide compelling evidence that leveraged ETFs and their options hold significant value beyond short-term speculation. Leveraged ETF options offer strong predictive signals that can enhance trading strategies and market insight, while actively managed LETF allocations can improve long-term portfolio performance. When used thoughtfully, these instruments can deliver meaningful returns, manage risk, and contribute to price discovery.

The ratio of gold prices to other asset classes has been shown to be a useful predictor of stock market returns. In this post, we discussed several gold-based ratios and how they can be used to forecast equity market performance.

Gold Oil Price Ratio As a Predictor of Stock Market Returns

Analyzing intermarket relationships between assets can help identify trends and predict returns. Traditionally, analysts use commodity, currency, and interest-rate data to predict the direction of the stock market. In this regard, Reference [1] brings a fresh new perspective. It utilizes price ratios of gold over other assets in order to forecast stock market returns.

Findings

-The gold-oil price ratio (GO) is shown to be a strong predictor of future stock market returns.

-Researchers created ten different gold price ratios by comparing gold to various assets like oil, silver, CPI, corn, copper, and several financial indicators.

-They used statistical models (univariate and bivariate regressions) to test how well these ratios could predict U.S. stock returns.

-Among all the ratios tested, the gold-oil ratio (GO) had the highest predictive power.

-A one standard deviation increase in the GO ratio is linked to a 6.60% rise in annual excess stock returns for the following month.

-The GO ratio performs better than traditional forecasting methods, including the historical average model.

-It also offers meaningful economic benefits for investors who use mean-variance strategies.

-The study concludes that the predictive ability of the GO ratio is both statistically reliable and economically useful.

In summary, the gold-oil price ratio is identified as a robust predictor of stock market returns, outperforming traditional predictors and other gold price ratios. A one standard deviation increase in GO is associated with a significant 6.60% increase in annual excess returns for the next month.

Reference

[1] T. Fang, Z. Su, and L Yin, Gold price ratios and aggregate stock returns, SSRN 3950940

The Bitcoin-Gold Ratio as a Predictor of Stock Market Returns

The ratio of gold prices to other asset classes has been shown to be a useful predictor of stock market returns. The previous article discussed how the gold-oil ratio serves as one such indicator.

Continuing this line of inquiry, Reference [2] examines the informational value of the Bitcoin-gold (BG) price ratio. The logic behind this metric is that Bitcoin represents a high-risk asset, whereas gold is traditionally viewed as a safe haven. Therefore, a rising BG ratio may signal increased investor risk appetite. It may also reflect growing optimism and interest in technological innovation, which boosts demand for Bitcoin. As a result, a higher BG ratio can indicate a tech-driven risk appetite that translates into stronger stock returns.

Findings

-The Bitcoin-Gold (BG) ratio is positively linked to U.S. stock market returns, especially during and after the COVID-19 pandemic.

-A rising BG ratio suggests increased investor risk appetite, as Bitcoin is seen as high-risk and gold as a safe haven.

-The effect of the BG ratio on stock returns remains strong even when using Ethereum instead of Bitcoin, showing broader crypto-gold relevance.

-The positive impact of the BG ratio also applies to the European stock market, not just the U.S., indicating global relevance.

-The main channel through which the BG ratio affects stock returns is investor risk aversion or appetite.

-The study uses various economic controls, like volatility, inflation, and liquidity, and still finds the results hold strong.

-There was no significant impact of the BG ratio on stock returns before the pandemic, suggesting this relationship is more recent.

-The BG ratio reflects shifts in market sentiment and offers a new tool for gauging investor behavior.

-Investors can use the BG ratio as a signal to adjust their equity exposure based on prevailing market conditions.

In summary, the paper makes a novel contribution by introducing crypto-gold ratios as reliable indicators of stock market direction across multiple regions.

Reference

[2] Elie Bouri, Ender Demir, Bitcoin-to-gold ratio and stock market returns, Finance Research Letters (2025) 107456

Closing Thoughts

Both studies show that gold price ratios can offer valuable insights into stock market returns. The gold-oil ratio (GO) stands out as a strong, traditional predictor, while the Bitcoin-gold ratio (BG) brings a modern twist by capturing shifts in investor risk appetite. Together, these findings suggest that combining safe-haven and risk assets in a ratio form can help investors better understand and respond to changing market conditions.

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.

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.

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.

Cryptocurrencies are becoming mainstream. In this post, I feature some strategies for trading and managing risks in cryptocurrencies.

Arbitrage Trading in the Cryptocurrency Market

Arbitrage trading takes advantage of price differences in different markets and/or instruments. Reference [1] examined some common and unique arbitrage trading opportunities in cryptocurrency exchanges that are not discussed often in the literature. They are,

-Exchange futures contract funding rate arbitrage

-Exchange futures contract intertemporal arbitrage

-Triangular arbitrage

-Pairs trading

-Order book spread prediction arbitrage

I provide details about the funding rate arbitrage below. Other arbitrage strategies are described in the paper.

Cryptocurrency exchanges use the funding rate to ensure perpetual futures prices align with spot prices, improving liquidity and narrowing bid-ask spreads. This mechanism periodically compensates long or short traders based on price differences. For example, Binance settles funding payments every 8 hours to balance demand between buyers and sellers.

When the perpetual contract trades above the spot price, longs pay shorts, discouraging further price increases and encouraging shorts to push it back down. An arbitrage strategy involves shorting Bitcoin in the perpetual market while holding an equal amount in the spot market, earning funding payments with minimal exposure to price fluctuations—excluding exchange and market risks.

When the perpetual contract trades below the spot price, shorts pay longs, so we buy the futures and short the spot.

Findings

– Research on cryptocurrency price prediction focuses on both time-series and cross-sectional analysis.

– This paper explores arbitrage opportunities in cryptocurrency exchanges that are often overlooked in academic literature.

– These arbitrage strategies can generate high returns with minimal risk.

– However, real market conditions and exchange constraints can reduce their effectiveness in live trading compared to backtesting.

– Incorporating these arbitrage strategies into a portfolio can improve the Sharpe ratio compared to simply holding cryptocurrencies.

In short, arbitrage trading is possible and profitable in the crypto market. However, we note that,

-These trading strategies are not riskless. Drawdown can happen

-Diversification helps smooth out the equity curves greatly

Reference

[1] Tianyu Zhou, Semi Risk-free Arbitrages with Cryptocurrency, 2022 5th International Conference on Financial Management, Education and Social Science (FMESS 2022)

Detecting Trends and Risks in Crypto Using the Hurst Exponent

The Hurst exponent is a statistical measure used to assess the long-term memory and persistence of a time series. It quantifies the tendency of a system to revert to the mean, follow a random walk, or exhibit a trending behavior. A Hurst exponent (H) value between 0 and 0.5 indicates mean-reverting behavior, H = 0.5 suggests a purely random process, and H between 0.5 and 1 signals persistent, trending behavior.

Reference [2] utilized the Detrended Fluctuation Analysis technique to study the Hurst exponent of the five major cryptocurrencies. Its main novelty is the calculation of a weekly time series of the Hurst exponent and its usage.

Findings

-This study examines long-range correlations in the cryptocurrency market using Hurst exponents across multiple time scales. It analyzes the log-returns of the top five cryptocurrencies, covering over 70% of market capitalization from 2017 to 2023.

-Four out of five cryptocurrencies exhibit persistent long-range correlations, while XRP follows a random walk.

-Trend Monitoring: The Hurst exponent (H) can help detect trend continuation or reversal. Cryptocurrencies like XRP showed transitions from short-term persistence to long-term anti-persistence, which could signal trend changes.

-Dynamic Strategy Adjustments: Rolling-window DFA estimates can track shifts in market behavior, aiding in strategy adjustments by identifying when a market moves from trend-following (H>0.5) to mean-reverting (H<0.5).

-Asset-Specific Behavior: Different cryptocurrencies exhibit unique behavioral patterns, suggesting that H-based analysis can inform tailored trading strategies.

-Systemic Risk Monitoring: Synchronization of H values across multiple cryptocurrencies during extreme market events may indicate rising volatility or instability, helping traders implement defensive measures like diversification.

In short, the findings suggest opportunities for using Hurst exponents as tools to monitor trend continuation or reversal, develop asset-specific strategies, and detect systemic risks during extreme market conditions, offering valuable insights for traders and policymakers navigating the cryptocurrency market’s inherent volatility.

Reference

[2] Huy Quoc Bui, Christophe Schinckus and Hamdan Amer Ali Al-Jaifi, Long-Range Correlations in Cryptocurrency Markets: A Multi-Scale DFA Approach, Physica A: Statistical Mechanics and its Applications, (2025), j.physa.2025.130417

Closing Thoughts

We have shown that arbitrage strategies in the crypto market are both possible and profitable. Additionally, risk management, trend detection, and reversal identification can be improved using the Hurst exponent, offering traders a valuable tool to navigate market volatility more effectively.

Portfolio allocation is an important research area. In this issue, we explore not only asset allocation but also the allocation of strategies. Specifically, I discuss tactical asset and trend-following strategy allocation.

Tactical Asset Allocation: From Simple to Advanced Strategies

Tactical Asset Allocation (TAA) is an active investment strategy that involves adjusting the allocation of assets in a portfolio to take advantage of short- to medium-term market opportunities. Unlike strategic asset allocation, which focuses on long-term asset allocation based on a fixed mix, TAA seeks to exploit market inefficiencies by overweighting or underweighting certain asset classes depending on market conditions, economic outlooks, or valuation anomalies. This approach allows investors to be more flexible and responsive to changing market environments, potentially improving returns while managing risk.

Reference [1] examines five approaches to tactical asset allocation. They are,

1. The SMA 200-day strategy, which uses the price of an asset relative to its 200-day moving average.
2. The SMA Plus strategy, which builds on the SMA 200-day by adding a volatility signal to the trend signal, dynamically adjusting allocations between risky assets and cash.
3. The Dynamic Tactical Asset Allocation (DTAA) strategy, which applies the same trend and volatility signals as SMA Plus but across the entire portfolio, rather than on individual assets.
4. The Risk Parity method, popularized by Ray Dalio’s All Weather Portfolio, equalizes the risk contributions of different asset classes.
5. The Maximum Diversification method, which aims to maximize the diversification ratio by balancing individual asset volatilities against overall portfolio volatility.

Findings

– The SMA strategy provides strong risk-adjusted returns by shifting to cash during downturns, though it may miss early recovery phases.

– SMA Plus builds on SMA by adding a more dynamic allocation approach, achieving higher returns but at a slightly increased risk level.

– The DTAA strategy yields the highest returns but experiences significant drawdowns due to aggressive equity exposure and limited risk management.

– Risk Parity and Maximum Diversification focus on stability, offering lower returns with minimal volatility, making them suitable for conservative investors.

In short, TAA based on a simple moving average still delivers the best risk-adjusted return.

This is an interesting and surprising result. Does this prove once again that simpler is better?

Reference

[1] Mohamed Aziz Zardi, Quantitative Methods of Dynamic Tactical Asset Allocation, HEC – Faculty of Business and Economics, University of Lausanne, 2024

Using Trends and Risk Premia in Portfolio Allocation

Trend-following strategies play a crucial role in portfolio management, but constructing an optimal portfolio based on these signals requires a solid theoretical foundation. Reference [2] builds on previous research to develop a unified framework that integrates an autocorrelation model with the covariance structure of trends and risk premia.

Findings

– The paper develops a theoretical framework to derive implementable solutions for trend-following portfolio allocation.

– The optimal portfolio is determined by the covariance matrix of returns, the covariance matrix of trends, and the risk premia.

– The study evaluates five well-established portfolio strategies: Agnostic Risk Parity (ARP), Markowitz, Equally Weighted, Risk Parity (RP), and Trend on Risk Parity (ToRP).

– Using daily futures market data from 1985 to 2020, covering 24 stock indexes, 14 bond indexes, and 9 FX pairs, the authors assess the performance of these portfolios.

– The optimal combination of the three best portfolios—ARP (19.5%), RP (51%), and ToRP (30%)—achieves a Sharpe ratio of 1.37, balancing traditional and alternative approaches.

– The RP portfolio, representing a traditional diversified approach, is a key driver of performance, aligning with recent literature.

– The combination of ARP and ToRP offers the best Sharpe ratio for trend-following strategies, as it minimizes asset correlation.

In the context of a portfolio optimization problem, the article solved the optimal allocation amongst a set of trend-following strategies. It utilized the covariance matrix of returns, trends, and risk premia in its optimization algorithm. The allocation scheme combined both traditional and alternative approaches, offering a better Sharpe ratio than each of the previous methods individually.

Reference

[2] Sébastien Valeyre, Optimal trend following portfolios, (2021), arXiv:2201.06635

Closing Thoughts

We have discussed both asset and strategy allocation, one advocating a relatively simple approach, while the other is more sophisticated. Each method has its advantages, depending on the investor’s objectives and risk tolerance. A well-balanced portfolio may benefit from integrating both approaches to achieve optimal performance and diversification.