The Relationship Between Leverage, Investor Behavior, and Returns

Margin lending and borrowing are integral parts of a well-functioning financial system. However, excessive use of margin, or leverage, can create significant risks. In this context, Reference [1] examines the returns and trading behavior of retail investors using margin financing in the Chinese stock market.

The study uses transaction data from nearly five million brokerage accounts between May and July 2015, covering China's equity market bubble and crash. The authors report an average assets-to-equity ratio of 1.4 for brokerage margin accounts and 6.4 for shadow accounts. They then construct a distance-to-margin-call measure (Z), which estimates how many standard deviations of portfolio losses would bring an account to its liquidation threshold.

The paper  pointed out,

We find that margin traders tend to hold stocks with more scope for “informed trading” and lottery-like features, though they do not outperform their unlevered counterparts. Our results thus suggest that lottery preferences, partly driven by overconfidence, can help explain why investors are willing to pay an annual interest rate of up to 25% in order to lever up.

Both shadow and brokerage margin investors heavily sell their holdings when their account-level leverage edges toward their Pingcang Lines (the maximum leverage limit, or maintenance margin), controlling for stock-date and account fixed effects. In other words, we find strong empirical support for both forced and preemptive margin-induced trading. The fact that investors react asymmetrically to positive and negative return shocks suggests a leverage constraint channel in addition to a rebalancing motive. We exploit the discontinuity around the eligibility of brokerage margin trading and the variation in Pingcang Lines across certain shadow accounts to provide unique evidence that leverage constraints induce selling. Moreover, we find that margin investors are relatively more likely to sell non-lottery stocks as the leverage constraint binds in a down market.

Aggregating trading behavior across margin investors, we find a significant return spillover in the near future: a stock’s return can be strongly forecasted by a portfolio of stocks with which it shares common margin-investor ownership. This pattern is subsequently reversed, and is present only in market downturns.

In short, the paper finds that,

• Retail margin borrowing is associated with lottery preferences and possible overconfidence.
• Margin-induced liquidation creates a feedback loop between declining asset prices and forced selling, transmitting shocks across otherwise unrelated stocks.

Although the study focuses on leveraged trading in the Chinese stock market, its conclusions may also be applicable to developed markets. We would be interested to hear from readers who are aware of similar studies in developed markets.

Another interesting contribution is the adaptation of the Merton credit risk model to leveraged trading, providing a framework for analyzing margin-related risks.

Let us know what you think in the comments below or in the discussion forum.

References

[1]  Jiangze Bian, Zhi Da, Zhiguo He, Dong Lou, Kelly Shue, and Hao Zhou, The Drivers and Implications of Retail Margin Trading, Journal of Finance, 2026.

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Revisiting Gambler’s Ruin: Trading Edge, Drawdowns, and Survival

The gambler's ruin problem is a classical probability problem that examines the likelihood of a gambler reaching a specified wealth target or losing all available capital. Reference [1] revisits this problem. Specifically, it seeks to answer the following questions:

Suppose a gambler enters a casino with finite capital,

• What is the probability of reaching a target wealth level or going bankrupt, and
• What is the expected duration of play?

The author first derives analytical formulas to address these questions, then uses Monte Carlo simulations to validate the theoretical results. Finally, the findings are applied to a real-world investment problem using daily S&P 500 data from January 1990 through September 2026.

The author pointed out,

Applied to daily S&P 500 returns, the model estimates a win rate of p = 0.5369 and finds that the sign sequence is statistically distinguishable from an independent one while being, in magnitude, very nearly independent. The more serious failure is elsewhere. When the fixed unit stake is replaced by a bootstrap of the actual return distribution, holding the win rate and the mean absolute move constant, the projected probability of ruin rises from 5.0% to 42.7%. The model’s weakness is not that markets are predictable but that they do not move in equal steps. That is a useful thing to know about a model, and it is the kind of thing that only becomes visible when the mathematics is exact enough that the discrepancy cannot be blamed on the algebra.

A central finding of the paper is that risk models based solely on the frequency of winning/losing days can substantially underestimate the risk of ruin when actual return magnitudes are unequal and heavy-tailed. In other words, the size of losses relative to gains, as well as the presence of fat tails, is critical in determining ruin probabilities.

Another interesting finding is that a relatively small increase in the probability of winning can dramatically reduce the probability of ruin. For example, at a 51% win rate, the probability of ruin is 44.93%, whereas at a 56% win rate, it falls to just 0.8%.

This paper makes an important contribution by formally confirming two principles already familiar to practitioners, namely

• Keeping losses and drawdowns small, and
• Increasing the trading edge whenever possible.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Jasman Mander, Random Walks and the Gambler’s Ruin Problem: Exact Results, Independent Derivations, and an Empirical Application to Equity Index Returns, Working Paper, 2026

Article Source Here: Revisiting Gambler’s Ruin: Trading Edge, Drawdowns, and Survival

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Market Timing Strategies Using the Implied Volatility Surface

Market timing is typically conducted using technical, macroeconomic, and volatility indicators. However, there is a growing trend in the research community toward using options-implied volatility to extract information for constructing market-timing strategies. For example, we have discussed how option-implied distributions can be used to time the market. Along the same line, Reference [1] develops an approach that uses the implied-volatility surface (IVS) to identify market entry points.

Using the IVS, the authors develop two groups of indicators,

1. Directional-expectation indicators: Directional Sentiment Bias (DSB) and Relative Upside Conviction (RUC).
2. Risk-stability indicators: Tail Risk Concentration (TRC) and Intertemporal Uncertainty Spread (IUS).

These indicators are then combined into three bullish structural regimes:

1. Extreme Hedging Reversal: panic/hedging conditions remain elevated, but directional sentiment has already turned bullish.
2. Trend-Stabilizing Correction: short-term stress has subsided relative to longer-term uncertainty while bullish sentiment remains.
3. Persistent Trend Potential: low short-term stress and longer-term uncertainty combined with bullish conviction, interpreted as conditions favorable to continued trends.

The authors test their approach on AAPL, TSLA, NVDA, and SPY. They generate an entry signal when IVS structural indicators cross dynamically calibrated thresholds into a defined regime, using only the first day of each structural transition as the entry point.

The paper pointed out,

The timely identification of effective market entry points is critical for capturing favorable risk-return opportunities while mitigating downside exposure. To address this challenge, the proposed framework leverages the structural information contained in the IVS through an economically motivated segmentation into the Central Sentiment and Peripheral Risk domains. Structural factors extracted from these domains disentangle directional sentiment from extreme risk pricing, while regime-specific thresholds identify distinct structural market regimes. By focusing on IVS state transitions rather than continuous price movements, this approach identifies discrete, effective entry points. Experimental results show that the framework outperforms conventional price-based and volatility-level baselines in directional accuracy and risk-adjusted returns. These findings highlight the practical value of structurally informed IVS analysis for precise, early-phase entry point detection and tactical decision-making in financial markets.

In short, the authors find that the IVS strategy generally delivers higher risk-adjusted performance, although the magnitude varies materially across assets. Their interpretation is that combining directional information with structural risk conditions improves entry timing.

A noteworthy conclusion is that this approach produced more precise and robust early-phase entry points than the technical and sentiment benchmarks tested, particularly because it uses economically interpretable information from different regions of the IV surface rather than continuously forecasting prices. Consequently, the approach can be used as a standalone regime-detection framework.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Yi-Chieh Sung, Shiou-Chi Li, and Jen-Wei Huang, Market Timing via Structural Transition Detection in the Options Implied Volatility Surface, in Advances in Computational Collective Intelligence, Springer 2026

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Statistical Arbitrage in Prediction Markets

Prediction markets are markets in which participants trade contracts based on the outcomes of future events, and they have become increasingly popular in recent years. Given their rising popularity, an important question is how to develop quantitative betting strategies.

Reference [1] proposes an interesting approach based on an adaptation of the classic pairs-trading strategy. Specifically, the paper adapts financial-market pairs trading and mean reversion to U.S. presidential betting markets. But instead of modeling the price spread between two stocks, it models the sum of the implied probabilities of the presidential nominees.

The combined probability is modelled as a latent Ornstein-Uhlenbeck (OU) process with a time-varying mean plus additive observation noise. The OU component represents short-term mean reversion, while the moving equilibrium represents gradual changes in the election market.

The authors utilize political betting odds obtained through BetData. The 2020 election (Trump/Biden) is used for model construction and training, while the 2024 election (Trump/Harris) is used for out-of-sample testing. They pointed out,

This study demonstrates the feasibility of adapting a pairs-trading framework to political prediction markets. We model the combined implied probability of the two major-party nominees using a latent time-varying-mean Ornstein-Uhlenbeck process with additive observation noise, use bootstrap-based prediction bounds to identify trading signals, and apply a Bradley-Terry pairwise comparison model to select the candidate-specific odds. In the 2024 out-of-sample evaluation, the strategy generated 130 completed trades, with a mean realized odds-price return of 1.86% and an unannualized per-trade Sharpe-type ratio of 1.12. These results illustrate the potential of the integrated signal-generation and candidate-selection framework in electoral markets dominated by two leading candidates.

In short, the authors present the framework as a proof of concept showing that a pairs-trading-style mean-reversion model can be adapted to two-candidate political betting markets. They point to the 2024 130-trade out-of-sample result, +1.86% mean synthetic odds-price return and 1.12 per-trade Sharpe-type ratio as evidence for the integrated signal-generation/candidate-selection framework.

This is an interesting paper, as it provides a foundation for developing quantitative strategies in prediction markets.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Haoyu Liu, Len Thomas, Benjamin Baer, and Carl Donovan, Adapting Pairs Trading to Gambling Markets: A Case Study of the U.S. Presidential Election, arXiv 2609.22639v1, 2026

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Understanding P&L in Risk Reversal Strategies

Option P&L attribution is an important component of portfolio and risk management. We have previously discussed related topics, including how option P&L is path-dependent. Reference [1] examines P&L attribution and, specifically, the sources of P&L in a risk-reversal position, a structure commonly used to express a view on volatility skew.

The author investigates the problem both within a formal framework and empirically, using 886 SPY risk reversal trades from 2020–2026, marked daily. An equal-delta risk reversal consists of a long position in one OTM wing and a short position in the other at equal absolute deltas, with the overall position delta-hedged.

The author pointed out,

…The frame is natural: the risk reversal is the canonical vanna-bearing structure, and vanna is how the vanna–volga pricing tradition prices the smile’s slope. This paper argues the frame inverts the economics of the delta-matched, hedged version of the trade. Deriving the structure’s Greeks from first principles and reconciling a five-year, 886-trade tape’s P&L into Greek buckets—exactly, so the decomposition is confirmed rather than fitted—we find that the money is made by differential vega applied to skew mean-reversion, that the vanna term is a structural cost whose sign is pinned by the leverage effect, and that the two facts are connected by an impossibility result: the vanna cannot be removed with the structure’s own legs, at any positive weighting.

In short, the paper finds that the P&L comes from differential vega applied to skew mean reversion, while vanna is a structural cost. Furthermore, vanna cannot be eliminated using just the two option legs.

This is an interesting paper that [glossary_exclude]warrants [/glossary_exclude]further investigation. Several questions remain worth exploring. For example,

• Can other structures be used in which vanna is not a cost?
• And what happens when the spot-volatility correlation changes sign, as it has done recently?

Let us know what you think in the comments below or in the discussion forum.

References

[1] Charlie Yan, A Formal Theory of the Delta-Matched Risk Reversal: Slope Vega, Irreducible Vanna, and a Dollar-Reconciling Attribution, Working paper 2026.

Originally Published Here: Understanding P&L in Risk Reversal Strategies

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Credit Spreads as Predictors of Equity Market Returns

Credit risk is the risk of financial loss arising from a borrower’s failure to meet its debt obligations. However, credit risk not only affects credit markets but can also affect equity markets, for example, through options volatility and returns. We have previously discussed how credit risk affects the returns of momentum strategies.

Along the same line, Reference [1] examines whether movements in corporate credit spreads reliably predict future U.S. equity-market performance. The hypothesis is that widening IG/HY spreads precede lower S&P 500 returns, narrowing spreads precede higher returns, and predictive strength varies across market regimes.

The author uses monthly data from January 2000 to July 2025, comprising approximately 305 observations. The dataset includes ICE BofA investment-grade (IG) and high-yield (HY) option-adjusted spreads (OAS), 10-year Treasury and 3-month T-bill yields, and the S&P 500 Total Return Index. The analysis employs OLS regressions with Newey-West HAC standard errors, with additional specifications controlling for the Treasury term spread and lagged equity returns.

The paper pointed out,

The results indicate that corporate credit spreads, especially changes in Investment-Grade (IG) spreads, carry meaningful predictive information about equity market performance. The consistently negative regression coefficients show that widening spreads are typically followed by weaker S&P 500 returns, supporting the hypothesis that tightening credit conditions signal rising risk aversion and slower economic growth. This relationship was strongest during 2008-2015, when financial stress was elevated, suggesting that the predictive power of spreads depends on market conditions and becomes more useful in periods of heightened uncertainty. The later 2016–2025 period shows a different pattern, with some predictors displaying positive coefficients. This reflects the unusual macroeconomic environment following the COVID-19 shock, where aggressive monetary intervention and rapid market recoveries caused spreads to widen during rebounds rather than deteriorations. This suggests that the credit-equity link is not constant, but shifts depending on the dominant macro regime.

These findings reinforce the idea that credit markets often reprice risk before equities. Credit spreads reflect how investors view the overall level of risk in the market, including the chances of default, the availability of liquidity, and expectations about the broader economy. When they widen, it reflects a deterioration in market confidence that later translates into weaker corporate earnings and stock performance. This result supports earlier studies suggesting that movements in credit spreads can signal where the economy and financial markets are headed, reflecting shifts in overall risk.

In short, the results indicate that corporate credit spreads, particularly changes in investment-grade spreads, contain predictive information about equity returns. Widening spreads generally precede weaker S&P 500 returns, although the relationship varies across market regimes and was strongest during 2008–2015. The findings suggest that credit markets can reprice risk before equities, making credit spreads potentially useful indicators of subsequent equity-market performance.

These findings provide further evidence that developments in credit markets can affect equity markets. The results can be used directly as trading signals, or the variables can be incorporated into a regime-detection algorithm.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Vishwa Kalal, Corporate Credit Spreads as Predictors of Equity Market Performance, AlgoGators Capstone Project, 2025.

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Machine Learning and Macroeconomic Indicators for Stock Return Prediction

Financial markets are nonstationary, and predicting asset prices is a nontrivial task. In the past, practitioners often relied primarily on technical indicators, which are derived from prices and trading volume, to forecast asset prices.

More recently, research has increasingly incorporated additional exogenous variables that may contain predictive information. Reference [1] follows this direction by combining technical indicators with volatility measures and macroeconomic variables using deep-learning models and hybrid feature-selection methods.

The authors forecast next-day individual-stock log returns rather than price levels. If the predicted next-day return is positive, the strategy holds the stock; otherwise, it remains out of the market. The study uses daily data from January 2005 to May 30, 2024 for AAPL, XOM, Goldman Sachs, Pfizer, and Ford, selected to represent different industries and company sizes and to cover both the Global Financial Crisis and COVID-19 periods.

In addition to traditional technical indicators, the authors incorporate the VIX, DXY, gold, oil, the 13-week Treasury rate, 10-year Treasury yield, term spread, and dividend yield.  They pointed out,

Identifying the most significant features to enhance stock price prediction has become increasingly complex and challenging. In this study, we demonstrated the impact of using the complete feature set on model performance… The results reveal that combining the filter and wrapper FS methods with DL models yields outstanding results and significantly improves the trading profitability across all the models and stocks.

… The results of this study demonstrate an important and consistent finding: low forecast error does not necessarily improve trading profitability. Empirical results revealed that although LR and RF achieved the lowest RMSE and MAE performance, Bi-GRU consistently produced the highest economic returns. These results not only question the practice of using statistical accuracy (MSE, RMSE, and MAE) as the main criterion for model evaluation but also highlight economic significance analysis, such as trading returns and Sharpe ratios, as a crucial and complementary performance evaluation framework.

This study not only highlights the benefits of the proposed models in enhancing stock price prediction for understanding the behavior of different stock markets, providing significant insights for investors and financial analysts, but also demonstrates that integrating different features with hybrid FS methods and DL models provides a robust framework for forecasting stock closing prices, improving performance and reliability in predicting future stock prices.

In short, the results indicate that incorporating macroeconomic and volatility indicators adds predictive value.

Another important finding of the paper is that forecast-error metrics and trading performance can give very different model rankings; therefore, financial forecasting should evaluate economic outcomes (return/Sharpe) alongside RMSE/MAE, rather than treating statistical forecast accuracy as sufficient.

This is an important contribution, as it provides additional evidence that exogenous variables can improve return forecasting and that machine-learning techniques can further enhance predictive performance.

Let us know what you think in the comments below or in the discussion forum.

References

[1]  Nabil, Barakat, Aboelfetouh & Elhishi, Integrating macroeconomic and technical indicators into forecasting the stock market: a hybrid approach for efficient feature selection, Scientific Reports 16, 27553 (2026)

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Two Decades of Leveraged ETFs

Leveraged exchange-traded funds (LETFs) first appeared in 2006 and were initially viewed as revolutionary investment products. In Reference [1], the authors examined several important aspects of LETFs. Almost two decades later, following numerous market stress events, including the recent 2026 South Korea extreme-volatility episode, they published a follow-up article [2] revisiting the insights and conclusions of their earlier work.

The authors pointed out,

However, to some observers, these products are characterized by issues often cited as contributors to the global financial crisis including (1) inadequate investor education regarding product complexity, (2) systemic risk arising from rebalancing dynamics, and (3) lack of transparency around hidden frictions….

On the second point, there is now less concern about same-direction rebalancing on the broader market in the US given the relatively small footprint of LETPs relative to total AUM, less than 1%. We do note that LETPs have a much larger share of dollar trading volume, some 16%, and their AUM when factoring leverage is actually understated. A major concern lies with high leverage products on single stocks and niche indexes, where history suggests that value can be quickly destroyed in extreme events as in South Korea in July 2026…

Our formal model twenty years ago addressed key questions raised by investors, market practitioners, regulators, policymakers, and the press seeking to understand a new product. In doing so, it demonstrated the value of the scientific approach to market analysis championed by JOIM. This remains an active area of theoretical and empirical research. Important questions about path-dependent returns, volatility, and microstructure effects remain open and relevant to practitioners. Recent events, including South Korea’s market volatility in the summer of 2026, show that these concerns are not merely theoretical; they can have real consequences for markets and everyday investors.

Basically, the core insights from the original paper [1] still hold: LETPs are useful short-term trading tools, but long-horizon returns are path-dependent, daily rebalancing can amplify market moves, and hidden financing frictions can materially erode returns.

In addition, the paper also finds that

• Predictable rebalancing can act as a momentum accelerator near the close. The effect should be stronger in thin markets, highly leveraged products, and products with large AUM,
• LETPs should primarily be treated as tactical, short-term instruments, not simple long-term leveraged holdings.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Cheng, M. and A. Madhavan (2009), The Dynamics of Leveraged and Inverse Exchange-Traded Funds, Journal of Investment Management, 7(4), 43–62

[2] Cheng, M., & Madhavan, A. (2026), Twenty Years of Leveraged and Inverse Exchange-Traded Products: What Have We Learned? Working paper

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Tail-Risk Hedging with Deep Out-of-the-Money Puts

Tail-risk hedging is an important topic that has attracted considerable attention in the academic community. However, a fully satisfactory solution has yet to emerge. The main challenge is that hedging comes at a cost, creating a persistent drag on portfolio performance.

Reference [1] addresses the same issue of tail-risk hedging but examines whether using deep out-of-the-money (DOOM) puts can provide a more efficient approach. The authors consider a one-year long SPX exposure combined with European SPX puts, with option spending capped at 5% of initial wealth. Purchasing puts reduces the initial SPX allocation rather than introducing additional leverage.

They apply three pricing models: Black-Scholes-Merton, Merton jump-diffusion, and Heston stochastic volatility models, to calculate risk-neutral exercise probabilities, which are then used to select put-option strikes.

The authors pointed out,

The research question addressed by this thesis was: to what extent can deep out-of-the-money put options enhance downside protection for equity portfolios while preserving cost efficiency under different market regimes? The results show that DOOM puts can reduce downside losses, but only when the selected strike region matches the market regime. This alignment depends on the volatility level and downside skew priced in the option surface. These two factors determine the cost of protection across strikes. The hedge is cost-efficient when the selected strike is close enough to the loss region implied by the market regime, ensuring a sufficiently high probability of finishing in the money relative to the premium paid… The historical SPX backtest reveals that the best protection does not necessarily come from the deepest puts. Deep low-strike puts create a far-tail floor, but they protect only if losses reach that region. In some regimes, a higher-strike put bought in smaller quantity is more robust because it is closer to the relevant loss region and pays in a wider range of realized losses.

In short, the paper finds that DOOM puts can reduce downside losses, but their effectiveness is highly regime-, strike-, surface-, and model-dependent. The deepest puts are not automatically the most cost-efficient. Successful protection requires the strike region to line up with the losses actually being targeted and with the option-surface regime prevailing when protection is purchased.

The results are relatively intuitive to practitioners, and no precise optimization algorithm is presented. Overall, the paper highlights that considerable research is still needed in the area of tail-risk hedging.

An innovative proposal of this paper is the use of risk-neutral tail probabilities to select strikes instead of using moneyness. In the BSM framework, this would correspond to using delta for strike selection, an approach that has been employed by practitioners.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Roba, A. (2026). To what extent can deep out-of-the-money put options enhance downside protection for equity portfolios while preserving cost efficiency under different market regimes? Master’s dissertation, Louvain School of Management, UCLouvain.

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Dollar-Cost Averaging or Buy the Dip?

Dollar-cost averaging (DCA) is an investment approach in which a fixed amount of capital is invested at regular intervals, regardless of market conditions.

Reference [1] examines an interesting problem: suppose you have a steady stream of relatively small cash flows, such as a paycheque. Should you practice DCA, or set aside some or all of the cash and wait for a market dip?

To answer this question, the author utilizes daily VFINX total returns from January 1990 to August 2026 and 440 monthly contributions to model DCA, while evaluating 115 mechanical dip-buying rules as alternatives to immediate investment, that is, different forms of market timing.

The paper  pointed out,

No tradable cash-reserve overlay beats monthly DCA by an economically meaningful margin. Rules that wait for 10–20% peak drawdowns lose 3–17% of terminal wealth. A contribution-level test shows why: a dollar delayed until a 10% drawdown is invested, on average, at a 22% higher total-return price, not a lower one, because delay buys later in a rising market. An oracle that buys exact troughs of every ≥10% episode still loses over the full sample. The same conclusion holds for a separate “dip sleeve” of extra capital: the fair twin is to DCA that extra, and crash sleeves capture only 66–79% of the incremental wealth. For a pile already in hand, lump-sum investment dominates both DCA of the pile and waiting for a dip.

In short, the paper finds that, although some exceptions exist, for a long-horizon S&P 500 saver whose cash arrives over time, the historical evidence favors investing as the cash arrives rather than withholding it for a future dip.  The reason is that with a positive equity premium, delaying investment has an opportunity cost. A future 10% drawdown can occur at a price higher than today’s price because the market may rise substantially before falling.

This problem is also relevant to active traders. Suppose you receive additional capital from investors or accumulate capital gains, and assume that your trading strategies have positive expectancy. Should you immediately increase your allocation, or hold some of the capital in reserve and wait for a drawdown before deploying it?

Let us know what you think in the comments below or in the discussion forum.

References

[1] Dong, E. K. (2026). Does “Buy the Dip” Improve Dollar-Cost Averaging? Evidence from Fair-Capital Experiments in the S&P 500, 1990–2026. Working paper

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