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Stock Investment: The p-index Approach

This paper introduces the p-index, a European put option-based risk measure, to evaluate investment strategies across Chinese and US markets from 2018 to 2023, revealing that sector-specific fair price strategies and momentum/contrarian approaches yield varying returns depending on market sentiment and the distinct behavioral patterns of efficient versus inefficient stocks in each region.

Original authors: Xinzhao Xie, Bopei Nie, Kuo-Ping Chang

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xinzhao Xie, Bopei Nie, Kuo-Ping Chang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a gardener trying to decide which plants to water and which to prune. In the world of stocks, most people use a ruler called "Beta" to measure how wild or risky a plant is. But this paper argues that ruler is a bit old-fashioned and sometimes misleading. Instead, the authors introduce a new tool called the p-index.

Here is a simple breakdown of what they did and what they found, using everyday analogies.

The New Tool: The "Insurance Fee" (The p-index)

Think of a stock like a car. You want to drive it, but you're worried it might crash (lose value).

The Old Way (Beta): Measures how bumpy the road usually is compared to the whole highway.
The New Way (p-index): The authors ask a different question: "How much would I have to pay for insurance to guarantee that my car doesn't lose value, but actually gains a little bit?"

The p-index is essentially the price of that insurance policy per dollar.

High p-index: The insurance is expensive. This means the "car" is risky; it's likely to crash or fail to meet your expectations.
Low p-index: The insurance is cheap. This means the "car" is safe and likely to do what you promised it would.

They also created a p-ratio, which is like a "bang-for-your-buck" score. It tells you how much profit you are getting for every dollar of risk (insurance cost) you are taking on.

The Experiment: Two Different Gardens

The researchers tested this new tool in two very different "gardens" (stock markets) over five years (2018–2023):

China's SSE 50: A garden of 50 giant, famous Chinese companies.
The US S&P 500: A garden of 500 large American companies.

They tried different strategies, like "Fair Price" (buying when the stock is cheap compared to its true value) and "Momentum/Contrarian" (betting on winners to keep winning or losers to bounce back).

What They Found in China (The SSE 50)

In the Chinese garden, the plants behaved in a surprising way that was the opposite of what traditional finance usually predicts.

The "Winners" Stopped Winning: Usually, people think if a stock is doing great (efficient), it will keep doing great (momentum). But in China, the "winning" stocks actually stopped their momentum. They didn't keep climbing.
The "Losers" Didn't Bounce Back: Usually, people think if a stock is doing poorly, it will eventually recover (mean reversion). But in China, the "losing" stocks just kept struggling; they didn't bounce back.
The Best Strategy: Because of this weird behavior, the best way to make money was to bet against the winners (sell them) and bet on the losers (buy them), but only if you used their new "p-index" to pick them.
- Analogy: Imagine a race where the person in first place suddenly gets tired and slows down, while the person in last place just keeps plodding along. The smart bet isn't to cheer for the leader; it's to bet on the leader to slow down and the loser to catch up.
The "Mood" Factor: They found that when the market was quiet and people were calm (low volume), betting against the winners worked best. When the market was noisy and excited (high volume), betting on the losers to keep losing worked best.
Sector Surprise: The "Materials" sector (companies making raw stuff like chemicals and metals) was the star performer, growing faster than tech or finance.

What They Found in the US (The S&P 500)

In the American garden, the plants behaved more like traditional finance theory predicts.

The "Winners" Kept Winning: The stocks that were doing well (efficient) kept their momentum. They kept climbing.
The "Losers" Bounced Back: The stocks that were doing poorly eventually recovered and started climbing again.
The Best Strategy: Here, the best strategy was the opposite of China. You should bet on the winners to keep winning and bet on the losers to bounce back.
The Result: The "Momentum" strategy (buying the winners) using the p-index was the most profitable, though the returns were more modest (around 3.69%) compared to the wild swings in China.

The Big Takeaway

The paper concludes that the p-index is a better "risk ruler" than the old "Beta" ruler.

In China, the market is tricky: Winners tire out, and losers don't recover easily. You make money by betting against the current trend.
In the US, the market is more predictable: Winners keep winning, and losers recover. You make money by following the trend.

The authors suggest that by understanding the specific "insurance cost" (p-index) of a stock, investors can navigate these different market personalities much better than by using old-school methods. They didn't test this on medical treatments or future technologies; they strictly tested how to make money in these two specific stock markets using this new math.

Technical Summary: Stock Investment: The p-index Approach

Problem Statement
The paper addresses the limitations of traditional risk metrics, particularly the Capital Asset Pricing Model (CAPM) and its reliance on beta, in explaining asset returns. The authors note that empirical evidence (e.g., Banz, 1981; Fama and French, 1992, 1993, 1996) suggests market risk is not the sole determinant of returns, with factors like size, value, liquidity, and momentum playing significant roles. Furthermore, multi-factor models face challenges regarding the economic interpretation of factors and the time-decay of factor premiums, leading to concerns about overfitting and long-term robustness. The paper seeks to propose an alternative risk measure derived from option pricing theory to evaluate investment strategies more effectively.

Methodology
The core of the methodology is the construction of the p-index, a risk metric derived from European put options and put-call parity.

Theoretical Foundation:
- The authors define risk as the probability of failing to meet a stated obligation.
- Using the put-call parity equation ( $c + \frac{K}{1+r} = S_0 + p$ ), they interpret the put option ( $p$ ) as an insurance fee.
- The p-index is defined as the insurance fee per insured dollar ( $\frac{p_i}{K_i}$ ) required to guarantee that an asset achieves at least a $\delta$ rate of return on a future date $T$ . Here, $K_i = S_{0i}(1+\delta)$ .
- A higher p-index indicates a higher risk level (lower likelihood) of achieving the target return $\delta$ .
- The p-ratio is subsequently defined as the risk-adjusted excess return: $\frac{r_i - r}{p-index}$ , analogous to the Treynor ratio.
Binomial Model Application:
- The paper derives the p-index within a binomial asset pricing model. It demonstrates that the p-index is inversely related to the "down" move ( $d$ ) of the asset and directly related to the target return ( $\delta$ ).
- The model assumes a risk-free rate $r$ and calculates risk-neutral probabilities ( $\pi$ ) based on the up ( $u$ ) and down ( $d$ ) movements of the asset price.
Empirical Implementation:
- Data: Daily data for 50 stocks in China's SSE 50 index (2018–2023) and 500 stocks in the US S&P 500 index (2018–2023).
- Fair Price Strategy: The authors calculate a theoretical "fair" Monday closing price for each stock using the binomial model (Eq. 8). If the actual price is below the fair price, the stock is bought; if above, it is shorted.
- Momentum and Contrarian Strategies:
  - Stocks are classified as "efficient" (top 10% or 30% by p-ratio or efficiency score) or "inefficient" (bottom 10% or 30%).
  - Momentum: Buy efficient/inefficient stocks in Week 2 and sell in Week 3 (or short the opposite).
  - Contrarian: Short efficient/inefficient stocks in Week 2 and buy back in Week 3 (or buy the opposite).
- Efficiency Measurement: The study employs the Minimum Convex Input Requirement Set (MCIRS) method (a non-parametric Data Envelopment Analysis approach) to evaluate stock efficiency. Inputs include capital, labor, and either beta or the p-index; the output is the rate of return.

Key Results

China (SSE 50 Index):
- Fair Price Strategy: One-week and one-month holding periods outperformed two-week and two-month periods. The Materials sector generated the highest annualized returns (11.04% for one-week, 11.93% for two-week, 10.18% for one-month).
- Risk-Return Relationship: High-yield firms exhibited significantly lower p-indexes (lower risk) compared to low-yield firms.
- Strategy Performance:
  - The p-ratio-efficient-contrarian strategy yielded the highest annualized return (9.97%).
  - The p-index-inefficient-momentum strategy followed (9.01%).
  - The p-index-efficient-contrarian strategy yielded 6.48%.
  - Efficient stocks failed to sustain momentum, and inefficient stocks showed no mean reversion.
- Method Comparison: The MCIRS method using the p-index consistently outperformed the beta-based approach.
- Market Sentiment: The p-index-efficient-contrarian strategy outperformed during low-sentiment (low-volume) periods, while the p-index-inefficient-momentum strategy outperformed during high-sentiment (high-volume) periods.
United States (S&P 500 Index):
- Behavioral Difference: Contrary to the Chinese market, efficient stocks sustained their momentum, while inefficient stocks exhibited mean reversion.
- Strategy Performance:
  - The p-index-efficient-momentum strategy produced the highest annualized return (3.69%).
  - The p-ratio-inefficient-contrarian strategy followed (3.67%).
  - The beta-efficient-momentum strategy yielded 3.48%.

Significance and Claims
The paper claims that the p-index offers a superior risk measure compared to beta, as evidenced by the higher returns generated by strategies utilizing the p-index within the MCIRS framework. The study highlights distinct market behaviors between China and the US:

Market Specificity: In China's A-share market, outperforming stocks do not sustain momentum, and underperforming stocks do not revert, suggesting a unique market dynamic where contrarian strategies on efficient stocks and momentum strategies on inefficient stocks are profitable.
Risk Measurement: The p-index, grounded in option pricing theory, effectively captures the insurance cost required to guarantee a minimum return, providing a more intuitive and robust measure of risk than variance or beta.
Sentiment Interaction: The performance of these strategies is contingent on market sentiment (proxied by trading volume), with specific strategies thriving in low-volume versus high-volume regimes.

The authors conclude that the p-index approach provides a viable alternative for constructing investment portfolios that outperform traditional beta-based strategies, particularly in the context of the Chinese market, while also offering insights into the divergent momentum/reversion behaviors of efficient and inefficient stocks across different markets.