Enhanced indexation using both equity assets and index options

This paper proposes an enhanced indexation framework that treats index options as artificial assets through defined trading rules, demonstrating via S&P 500 data (2017–2025) and second-order stochastic dominance that integrating these option strategies significantly improves out-of-sample portfolio performance.

The Gist

Imagine you are trying to build the perfect investment portfolio. Your goal isn't just to match the performance of the stock market (the "Index"), but to beat it. This is called Enhanced Indexation.

Usually, investors do this by picking a basket of individual stocks (like Apple, Microsoft, or Tesla) and hoping they do better than the average.

The Problem: The "Short-Lived" Tool
The authors of this paper asked: Why not use financial "options" to beat the market too?

An option is like a coupon or a bet. It gives you the right to buy or sell something at a specific price later. But here's the catch: Options have expiration dates. They are like fresh milk; they have a short shelf life. If you buy a stock, you can hold it for 100 years. If you buy an option, it might expire in 30 days.

If you try to build a portfolio based on past data, you run into a glitch: The options you bought in the past might have already expired by the time you try to use that strategy in the future. You can't just say, "I'll keep holding this expired coupon."

The Solution: The "Option Strategy" (The Recipe)
The authors came up with a clever fix. Instead of treating an option as a single asset, they treat an Option Strategy as the asset.

Think of it this way:

• Buying a single option is like buying a single, perishable apple. It rots quickly.
• Creating an Option Strategy is like writing a recipe for an apple pie.

The recipe (the strategy) says:

1. If the market drops 5% in a month, buy a "safety net" option (a Put).
2. If the market is calm, sell that option and buy a new one.
3. If the option gets close to expiring, swap it for a fresh one.

Even though the individual "apples" (options) expire, the recipe (the strategy) lives forever. You can follow the recipe today, tomorrow, and next year. This turns a "short-lived" tool into a "long-lived" asset that you can put in your portfolio.

The Experiment: The Race
The authors tested this idea using the S&P 500 (the US stock market index) from 2017 to 2025. They used a mathematical method called Second-Order Stochastic Dominance (SSD).

• The Analogy: Imagine two runners. One runs on a flat track (the Index). The other runs on a track with hills and valleys (the Portfolio).
• SSD is a way of measuring if the second runner is consistently better at avoiding the deep valleys (big losses) while still reaching the high peaks (big gains), even if their average speed looks similar.

They tested three scenarios:

1. The Index: Just buying the S&P 500.
2. The Stock Portfolio: Using math to pick the best mix of stocks to beat the index.
3. The Super Portfolio: Using the best mix of stocks PLUS the "Option Strategies" (the recipes).

The Results: The Safety Net Pays Off
The results were clear: The Super Portfolio won.

• In Good Times (Bull Markets): The portfolio with options did almost as well as the stocks alone, but it was smarter about risk.
• In Bad Times (Bear Markets): This is where the magic happened. When the market crashed, the "Option Strategies" acted like a shock absorber or a safety net. They prevented the portfolio from falling as hard as the rest of the market.
• The Outcome: Over the long run, the portfolio with options made more money and lost less money during crashes than the portfolio with just stocks.

Why This Matters
Usually, people think options are too complicated or risky for regular investing. This paper shows that if you use a systematic set of rules (the strategy) rather than gambling on a single bet, options become a powerful tool.

The Takeaway
Think of the stock market as a car ride.

• Stocks are the engine.
• Options Strategies are the suspension and airbags.

You can drive fast with just an engine, but when you hit a pothole (a market crash), the car without suspension gets damaged. By adding the "suspension" (the option strategies), the ride is smoother, safer, and ultimately gets you to your destination with more of your money intact.

The authors proved that by following a smart "recipe" for trading options, you can build a portfolio that is tougher and more profitable than just buying stocks alone.

Technical Summary

Here is a detailed technical summary of the paper "Enhanced indexation using both equity assets and index options" by Cristiano Arbex Valle and John E. Beasley.

1. Problem Statement

Enhanced indexation aims to construct a portfolio that outperforms a specific market index (e.g., S&P 500) while maintaining a similar risk profile. Traditional approaches rely on equity assets (stocks) or Exchange-Traded Funds (ETFs).

The core problem addressed in this paper is the mismatch in asset lifespans:

• Equity Assets: Have a "long life" and exist indefinitely, making them straightforward to include in optimization models based on historical data.
• Options: Have a "short life" (specific expiry dates). An option purchased during the in-sample (historical) period will likely expire before the out-of-sample (future) period begins. This creates a fundamental issue: an asset selected based on past performance may not exist in the future, rendering standard portfolio optimization invalid for options.

The authors ask: How can we incorporate options into enhanced indexation portfolios without violating the assumption that assets selected in-sample will be available out-of-sample?

2. Methodology

A. The Core Concept: Option Strategies

The authors introduce the concept of an "Option Strategy" to bridge the lifespan gap.

• Definition: An option strategy is a predefined set of rules dictating how a specific type of option is traded (bought, sold, held, or rolled over) based on evolving market conditions.
• Transformation: By applying these rules, a "short-life" individual option is converted into a "long-life" synthetic asset.
• Mechanism:
  • Rules: The strategy defines triggers for buying (e.g., if 30-day return < -5%), selling (e.g., if 30-day return > -5%), holding, or rolling over (replacing an expiring option with a new one of the same type).
  • Evaluation: When applied to historical data, the strategy generates a time series of returns just like a stock.
  • Optimization: These strategies are treated as distinct assets in the optimization model. If a strategy is selected for the portfolio, the same rules are applied to the future market data.

B. Optimization Framework: Second-Order Stochastic Dominance (SSD)

The authors employ a Linear Programming (LP) approach based on Second-Order Stochastic Dominance (SSD) rather than Mixed-Integer Programming (MIP).

• Rationale: SSD is computationally more efficient than MIP, which is crucial because the authors consider a large number of potential option strategies (potentially more than the number of equities).
• Objective: Maximize the minimum scaled tail difference between the portfolio returns and the index returns.
• Constraints:
  • Tail Differences: The sum of the $s$ smallest portfolio returns must exceed the sum of the $s$ smallest index returns for all $s$.
  • Asset Allocation: Includes equities, risk-free assets, and the defined option strategies.
  • No Short Selling: Weights must be non-negative.
  • Overfitting Mitigation: Strategies that never actively trade options during the in-sample period are excluded from the optimization to prevent overfitting.

C. Data and Pricing

• Dataset: Daily S&P 500 stock prices from March 17, 2017, to August 1, 2025. The data is manually adjusted for survivorship bias (only stocks present in the index at a specific time are available for selection).
• Option Pricing: Since historical daily option prices were unavailable, the authors used the Black-Scholes (BS) model to price options.
  • Implied Volatility: Approximated using the VIX index.
  • Risk-Free Rate: Approximated using the IRX (13-week Treasury bill yield).
• Strategies Tested: 12 distinct option strategies were created, including:
  • Puts: Buying ATM or 3% OTM puts when momentum is negative.
  • Calls: Buying ATM or 3% OTM calls when negative momentum reverses.
  • Volatility Plays: Buying Straddles/Strangles when realized volatility is low (betting on a "storm after calm").
  • Rolling: Options are rolled over if lifetime $L < 20$ days or if moneyness deviates significantly from the target.

3. Key Contributions

1. Novel Concept: First paper to propose and mathematically formalize Option Strategies as "long-life" assets for portfolio optimization, solving the expiry mismatch problem.
2. First Application to Enhanced Indexation: While options have been used in other portfolio contexts, this is the first application specifically for enhanced indexation (beating an index).
3. Methodological Integration: Demonstrates how to integrate these synthetic assets into a standard SSD-based linear programming framework.
4. Public Dataset: The authors made the manually adjusted S&P 500 dataset (2017–2025) publicly available for future research.
5. Empirical Validation: Provided robust computational evidence that option strategies improve out-of-sample performance.

4. Computational Results

The study compared three main scenarios using a rolling window backtest (201-day in-sample, 20-day out-of-sample):

1. SSD with Equities only.
2. SSD with Equities + Option Strategies.
3. SSD with SPY (ETF) + Option Strategies.

Key Findings:

• Superior Out-of-Sample Performance: The strategy combining SSD with option strategies significantly outperformed both the "Equities Only" SSD and the raw S&P 500 index.
  • Final Value (FV): SSD + Options achieved a final value of 3.61 (starting from 1.0), compared to 2.89 for Equities-only and 2.31 for the S&P 500.
  • CAGR: 18.51% (with options) vs. 15.05% (without) vs. 11.74% (Index).
• Risk-Adjusted Returns:
  • Sharpe Ratio: Increased from 0.63 (Equities only) to 0.74 (with options).
  • Sortino Ratio: Increased significantly from 0.88 to 1.08, indicating that the additional volatility introduced by options was primarily "good volatility" (upside) rather than downside risk.
• Downside Protection:
  • Maximum Drawdown (MDD): Reduced from 29.98% (Equities only) to 21.43% (with options).
  • Bear Market Performance: In bearish regimes, the option-enhanced strategy significantly outperformed the index and the equity-only strategy in terms of both return preservation and risk mitigation.
• Volatility Trade-off: While volatility increased slightly (from 21.80% to 22.58%), the trade-off was favorable due to the substantial reduction in tail risk (CVaR) and drawdowns.
• Transaction Costs: An ex-post analysis suggested that the strategy remains profitable even with transaction costs up to 100 basis points (bps) per rebalance, well above typical institutional costs (15–30 bps).

5. Significance and Implications

• Paradigm Shift: The paper shifts the view of options from being merely hedging instruments or speculative tools to being core asset classes that can be systematically optimized alongside equities.
• Scalability: By using Linear Programming and SSD, the approach is computationally feasible even when considering hundreds of potential option strategies, avoiding the complexity of Mixed-Integer Programming.
• Practical Applicability: The results suggest that semi-passive investors (using ETFs like SPY) can significantly enhance their returns and risk profiles by dynamically allocating capital to rule-based option strategies.
• Future Research: The authors highlight that "Option Strategy Design" is a rich field for future study, potentially involving more complex rules, American options, and tailored strategies for specific market regimes (e.g., specific downside protection rules).

In conclusion, the paper successfully demonstrates that treating option strategies as synthetic assets within an SSD framework allows for the construction of enhanced index portfolios that achieve higher returns and better risk management than traditional equity-only approaches.