Month: December 2025

Risk, Leverage, and Optimal Betting in Financial Markets

Most research in portfolio management focuses on alpha generation; however, another critical component of portfolio construction is position sizing. In this post, we examine key considerations in position sizing, including the Kelly criterion and the martingale betting system.

Does Kelly Portfolio Outperform the Market?

A method for capital allocation and position sizing is to employ the Kelly criterion. The Kelly criterion aims to optimize the expected growth rate of capital, maximizing the anticipated value of the logarithm of wealth. This strategy is rooted in John Kelly’s paper, “A New Interpretation of Information Rate.” According to Kelly, in repeated bets, a bettor should act to maximize the expected growth rate of capital, thus maximizing expected wealth at the end.

Reference [1] applies Thorp’s approach, as outlined in “The Kelly Criterion in Blackjack, Sports Betting and the Stock Market,” [2]  to construct a portfolio in the Norwegian stock market. The formula computes the optimal investment fraction in a set of assets, considering the expected excess returns of the assets and the inverse of the variance-covariance matrix.

Findings

-The study evaluates the performance of a growth-optimal Kelly portfolio in the Norwegian stock market over the period February 2003 to December 2022.

-It assesses abnormal performance using the CAPM, Fama–French three-factor model, and Carhart four-factor model.

-The Kelly portfolio achieves a higher compound annual growth rate (14.1%) and higher ending wealth than the benchmark index, which grows at 12%.

-It also outperforms a Markowitz portfolio, which delivers lower growth and final wealth.

-The Kelly portfolio and the benchmark exhibit similar Sharpe ratios (0.58), while the Kelly portfolio attains a higher Sortino ratio (0.95).

-Factor regressions indicate an annualized alpha of 16.8% for the Kelly portfolio, statistically significant at the 1% level before transaction costs.

-However, the factor models display very low explanatory power, suggesting that the estimated alpha may be overstated.

-Once transaction costs are incorporated, the Kelly portfolio no longer outperforms the benchmark in terms of final wealth.

-After costs, the alpha remains only marginally significant at the 10% level, implying limited real-world risk-adjusted excess returns.

This paper presents several interesting findings,

-First, the correlation of the Kelly portfolio with the market is nearly zero.

-Second, the performance is sensitive to transaction costs. We believe that with lower transaction costs, the Kelly portfolio has the potential to outperform the market and display zero correlation with it.

-Third, the Kelly portfolio surpasses the Markowitz mean-variance portfolio in performance.

We also concur with the author that the utilization of options can further enhance the risk-adjusted return.

Reference

[1] Jon Endresen and Erik Grødem, The Kelly criterion, an empricial study of the growth optimal Kelly portfolio, backtested on the Oslo Stock Exchange, 2023, Norwegian School of Economics.

[2] Thorp, E. O., The Kelly Criterion in Blackjack Sports Betting and the Stock Market, in: Zenios, S.A. & Ziemba, W.T., Handbook of Asset and Liability Management, Volume 1, 387–428, 2006

Enhanced Martingale Betting System with Stop Policy

The martingale betting system is a popular gambling strategy that involves doubling one’s wager after each loss in the pursuit of recovering previous losses and securing a profit equal to the original bet. The underlying idea is that, statistically, a win will eventually occur, allowing the player to recoup losses and gain a net profit equal to the initial stake. While simple in concept, the martingale system carries inherent risks, as it assumes unlimited funds for doubling bets and disregards the fact that losing streaks can persist longer than expected. Thus, this system will eventually result in bankruptcy.

Reference [3] however argues that different perspectives exist regarding whether stock price movements adhere strictly to a random walk, often modeled as a geometric Brownian motion. This suggests a potential for enhancement in the martingale betting system. The author has subsequently introduced an enhanced martingale betting system that includes a stop policy.

Findings

-The paper proposes an Improved Martingale Betting System (IMBS) by modifying the traditional martingale strategy with a stop policy and adapting it from casino gambling to intraday trading.

-The IMBS is empirically tested using TAIEX (TX) futures across three intraday trading strategies.

-Results show that the IMBS delivers strong performance and is applicable to TX intraday trading and related markets.

-The study finds that returns increase with leverage up to a certain threshold, beyond which traditional martingale strategies face a high probability of bankruptcy.

-By controlling key parameters—specifically leverage scaling (a), the number of steps (n), and total leverage—the IMBS significantly outperforms both the Equal-Weight Betting System (EWBS) and the traditional Martingale Betting System (MBS).

-The inclusion of a stop-loss mechanism further improves performance and risk control.

-Empirical tests indicate that IMBS performs particularly well when combined with price breakout strategies, which are identified as the most profitable approach for TX intraday trading.

In short, after testing on real data, the article concludes that

-The conventional martingale betting system inevitably leads to bankruptcy,

-With the integration of a stop policy, the new and improved martingale betting system demonstrates enhanced efficacy.

Reference

[3] Ting-Yuan Chen, and Szu-Lang Liao, Improved Martingale Betting System for Intraday Trading in Index Futures—Evidence of TAIEX Futures, Asian Journal of Economics and Business, Year:2023, Vol.4 (2), PP.339-366

Closing Thoughts

Taken together, the two studies highlight the trade-off between growth maximization and risk control in position sizing. The Kelly-based approach demonstrates strong theoretical and empirical growth performance, but its apparent alpha weakens once transaction costs and model limitations are accounted for, raising questions about real-world applicability. By contrast, the Improved Martingale Betting System shows that disciplined leverage control and stop policies can materially improve intraday trading outcomes relative to naive martingale schemes, especially when combined with breakout strategies. Overall, both strands of research suggest that position sizing is as critical as signal generation, and that practical constraints, parameter calibration, and market frictions ultimately determine whether theoretically attractive sizing rules translate into sustainable performance.

The Effectiveness of Collar Structures in Equity and Commodity Markets

There are several popular options strategies frequently discussed in the trading and investing literature, as well as on social media. In a previous post, we examined the effectiveness of the covered call strategy, which has gained wide adoption among retail investors. In this edition, we extend our critical evaluation to another widely used approach—the options collar, a strategy employed by both retail traders and institutional investors.

Assessing the Effectiveness of Zero-Cost Collar in Different Markets

The zero-cost collar strategy is an options trading strategy that involves the simultaneous purchase of a put option and the sale of a call option. The options are usually of the same maturity, and the transaction results in a zero or small credit to the trader’s account.

This strategy is often used by investors who are bullish on a stock but want to protect themselves against a potential drop in the price. By buying the put option, they have the right to sell the stock at a predetermined price (the strike price). If the stock price falls below the strike price, they can sell the stock and offset any losses.

The sale of the call option helps to offset the cost of the put option and results in a zero or small credit to the trader’s account. This strategy is sometimes referred to as a “zero cost” collar because the net cost of the trade is zero.

Reference [1] examined the effectiveness of the zero-cost collar strategy in the developed and developing markets.

Findings

-The paper’s objective is to provide investors with a continuously implemented trading strategy that can effectively handle turbulent market periods such as the Dotcom bubble, the 2008–2009 financial crisis, and the COVID-19 pandemic.

-The study analyzes stock indices from six countries across both developed and developing economies to assess how extreme market events affect performance.

-Zero-cost collars are proposed as a costless option-based protection strategy, created by equating the cost of the long and short option components.

-Prior literature has not evaluated zero-cost collars across different rebalancing frequencies or tested their outcomes in both turbulent and stable markets.

-Results show that zero-cost collars generate strong returns when market volatility is moderate, and the underlying indices perform well, especially when the put strike is set at a higher level.

-The strategy performs respectably during severe downturns as well as during trending or declining markets.

– Its effectiveness depends on market conditions and the choice of strike levels.

Overall, the paper contributes a practical trading strategy that helps investors manage turbulent market conditions through the continuous application of zero-cost collars.

Reference

[1] Lj Basson, Suné Ferreira-Schenk and Zandri Dickason-Koekemoer, The performance of zero-cost option derivative strategies during turbulent market conditions in developing and developed countries, Cogent Economics & Finance, Volume 10, 2022 – Issue 1

How the Airlines Hedge Fuel Costs

The recent rise in the cost of airline tickets can be attributed in part to the escalating fuel prices, which significantly affect operating expenses for airlines. To counter the adverse impact of fuel price volatility, airlines often adopt a strategic approach known as fuel hedging. This practice involves entering into financial contracts to secure future fuel purchases at predetermined prices, mitigating the vulnerability to sudden spikes in fuel costs. Fuel hedging provides airlines with a degree of price certainty, offering a measure of stability in budgeting and operational planning while allowing them to better manage the economic challenges posed by fluctuating fuel prices.

Amongst the US airlines, Southwest Airlines distinguishes itself for its efficient execution of the hedging strategy. It has maintained a record of profitability since 1973, an accomplishment that sets it apart in the US airline sector. Expert observers attribute Southwest’s sustained financial success to its proficient utilization of derivatives for the purposes of hedging. An analysis of Southwest Airlines’ financial statements across multiple years reveals a distinct trend: the share of jet fuel expenses is consistently lower compared to the industry norm. This achievement can be directly attributed to the precise implementation of their jet fuel hedging strategy, a practice that effectively shields the airline from fluctuations in fuel prices.

Reference [2] examined the fuel hedging strategy of Southwest Airlines in detail.

Findings

-The paper analyzes why hedging jet fuel is critical for airlines, given the high volatility of oil prices, and highlights how Southwest Airlines’ long-term low-cost strategy is closely tied to its effective fuel-hedging program.

-It examines Southwest’s financial background and stock performance as a foundation for evaluating its hedging approach.

-The study identifies four key hedging strategies used by Southwest Airlines: call options, collar structures, call spreads, and put spreads.

-By combining these four strategies, Southwest effectively mitigated jet-fuel price risk without engaging in speculative derivative positions.

-A comparison with other airlines shows that Southwest’s disciplined, non-speculative approach contributed significantly to its hedging success and cost stability.

-The paper also evaluates how COVID-19 and oil-price movements related to production-cut agreements adversely affected Southwest, leading to over-hedging and substantial losses in 2020.

In short, the paper discussed the intricacies of Southwest Airlines’ hedging strategies and demonstrated their value-added effect. It is apparent that the deployment of hedging by Southwest Airlines yields advantages. However, it is important to underscore the necessity of a well-designed hedging program, one that avoids the potential pitfall of over-hedging.

Reference

[2] Xiao Han, Hedging Strategy Analysis of Southwest Airlines, 2023, Proceedings of the 6th International Conference on Economic Management and Green Development

Closing Thoughts

Taken together, the two studies illustrate how collar-based strategies can play an important role in managing volatile market conditions, whether for broad equity portfolios or for highly fuel-sensitive industries such as airlines. The first study shows that zero-cost collars can deliver respectable risk-adjusted outcomes during major market disruptions, particularly when volatility is moderate, and strike selection is calibrated appropriately. The second study highlights how Southwest Airlines effectively applied options structures, including collars, to hedge jet-fuel exposure, while also underscoring the risks of over-hedging during periods such as COVID-19. Collectively, the evidence reinforces that collar strategies can be valuable risk-management tools, but their effectiveness depends critically on market regime, implementation frequency, and disciplined design.