Semiparametric Option-Implied Information and Median-Variance Approach: A Game-Changer in Integrating Sustainable Practices in Portfolio Optimization

Portfolio optimization plays a crucial role in investment decision-making. Traditionally, mean-variance and minimum-variance strategies, pioneered by Markowitz [1], have been widely used. However, these methods suffer from a critical flaw: they assume normally distributed asset returns. Consequently, its effectiveness may diminish in the face of non-normal return distributions and uncertain parameter estimates. This assumption often crumbles in the face of real-world market complexities, characterized by fat tails, skewness, and heavy volatility clusters [18, 19]. As a result, they lead to biased estimations and suboptimal performance, as highlighted by Demiguel et al. [2] and Jacobs et al. [3] in their out-of-sample studies.

Recognizing this limitation, researchers have proposed diverse approaches for enhanced portfolio optimization. Some researchers, like Ledoit and Wolf [20] and Fang and Post [21], have focused on improved parameter estimation through shrinkage estimators or structured data generating processes inspired by Sharp [22] and Çela et al. [23].

Shrinkage estimators, proposed by researchers such as Ledoit and Wolf [20], aim to strike a balance between sample estimates and prior information. By incorporating external knowledge, these estimators mitigate the bias inherent in purely data-driven estimates. Despite the advancements, these methods still lack comprehensive empirical validation across various market conditions. Similarly, Fang and Post's improved Shrinkage Estimators with robustness to outliers require further empirical validation across different asset classes to ascertain their effectiveness. Their robustness across diverse market conditions remains a critical concern.

Studies such as Jacobs et al. [5] have highlighted the limitations of mean-variance strategies in real-world scenarios. However, there remains a gap in validation across different market regimes, limiting the generalizability of their findings.

Inspired by Sharp [22] and Çela et al. [23], structured data generating processes, on the other hand, offer an alternative to traditional parameter estimation. These approaches leverage additional information beyond historical returns, potentially improving parameter accuracy. Yet, identifying the optimal structure remains challenging. How sensitive are these processes to model misspecification? Can they adapt dynamically to changing market regimes? These questions highlight the need for rigorous empirical testing.

Researchers like Golosnoy and Gribisch [24] advocate for incorporating specific risk measures into portfolio construction. Metrics such as Value at Risk (VaR) and Conditional Value at Risk (CVaR) capture tail risk and extreme events. However, their performance during market stress and across different asset classes warrants scrutiny. It is unclear if they are strong enough to manage a variety of portfolios, particularly ones with intricate dependencies.

Others, like Golosnoy and Gribisch [24], have incorporated specific risk measures, while studies [25-30] have catered strategies to different investor risk preferences. These studies emphasize accounting for tail risk and skewness. These non-normal features significantly impact portfolio behavior.

Bayesian approaches inject uncertainty into portfolio optimization. Barry [31] and Chen and Brown [32] pioneered the use of Bayesian methodologies to introduce uncertainty and enhance risk incorporation. However, the sensitivity of portfolio allocations to the choice of prior distributions remains a central concern. Moreover, scalability to high-dimensional problems—common in real-world portfolios—requires careful consideration.

Many prior studies in portfolio optimization have encountered challenges stemming from a lack of thorough empirical validation across various market conditions, especially when dealing with non-normal distributions. These limitations have hindered the effectiveness of existing methodologies in accurately capturing the complexities of real-world financial markets. In response to these gaps in the literature, this study seeks to overcome these challenges by introducing a novel framework known as the Extended Generalized Leland (EGL) models.

The EGL models represent a synthesis of existing approaches, combining elements from both parametric and non-parametric frameworks to offer a more robust and flexible solution to portfolio optimization. By leveraging option-implied information derived from EGL models, this study aims to enhance asset allocation strategies and improve portfolio performance.

Table 2. Synthesis of existing approaches

To provide clarity on the integration of EGL models and their implications for portfolio optimization, a summary of the synthesis of existing approaches is presented in Table 2. This table serves as a reference point for understanding how the EGL models address the limitations of previous studies and contribute to advancing the field of portfolio optimization.

While these advancements from existing studies represent considerable progress, this study pushes the boundaries even further by examining two key innovations: (1) option-implied information and (2) median-variance approach. The option-implied information, extracted through the EGL semiparametric model, leverages both implied volatility and risk premium for informed stock selection [4], potentially leading to more robust decision-making. Additionally, the median-variance approach offers a robust alternative as it bypasses the normality assumption and exhibits resilience to outliers, characteristics crucial for navigating real-world data [33-35]. Existing studies often focus solely on historical data, but our approach bridges the gap between past observations and future expectations.

This study not only addresses traditional method limitations but also explores the potential synergies between these innovative approaches and realistic constraints, such as short-selling and zero-correlation, building upon the framework proposed by Jagannathan and Ma [36]. By doing so, this research seeks to achieve improved returns and reduced risks for investors, particularly in unpredictable market environments. This focus aligns with recent studies [37-39], which highlight the importance of realistic constraints and robust risk estimation for optimal portfolio construction.

Ultimately, the findings of this research have the potential to offer valuable insights for portfolio optimization, empowering investors to make informed decisions and navigate complex market dynamics. This pursuit resonates with the broader goals of research by Zhang et al. [40] and Wang et al. [41], which emphasize the need for data-driven and market-adaptive approaches to achieve sustainable investment success.

This study explores the potential of option-implied information, extracted through the Extended Generalized Leland (EGL) models, to enhance sustainable stock portfolio performance. We investigate whether incorporating market expectations gleaned from option pricing, while accounting for risk premium, can lead to more resilient and rewarding portfolios within the context of sustainable investing.

Our analysis spans three portfolio categories: classical (unconstrained), short-selling, and zero-correlation. The short-selling portfolio incorporates the constraint of allowing positions to fall below zero, while the zero-correlation portfolio enforces zero correlation between assets. Within each category, we further assess three fundamental portfolio construction approaches: mean-variance, minimum-variance, and median-variance, resulting in a total of 36 distinct strategies for evaluation.

These strategies were tested using real data obtained from the options market for the DJIA index. Historical option prices and relevant market parameters were used to construct and simulate the portfolios over the specified time period.

The inclusion of real data ensures that our analysis reflects actual market conditions and dynamics, allowing for more meaningful insights and conclusions. By testing the portfolio strategies on real data, we aimed to provide practical and actionable recommendations for investors seeking to optimize their portfolios in real-world scenarios.

To provide a baseline for comparison, we utilize the naïve 1/N portfolio, where equal weight is assigned to each asset. To assess performance, we employ three key metrics: portfolio return, portfolio standard deviation (risk), and the Sharpe Ratio (risk-adjusted return).

Portfolios constructed using the EGL-derived adjusted information are designated as Portfolios with Option-implied Adjusted Information (POAI), while those without it are labeled Portfolios with No Option-implied Adjusted Information (PNOAI). A detailed breakdown of these portfolio variants and the overall methodological process is provided in Section 4.1 of the paper, further illustrated by the process chart presented in Figure 1.

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Figure 1. Process flow of asset allocation strategies

4.1 The Extended Generalized Leland function

This study proposes the Extended Generalized Leland (EGL) models, encompassing both the Generalized Leland-Infused (GLI) and the model-free implied Leland (MFIL) approaches. Both address the limitation of existing model-free frameworks by incorporating transaction costs. The EGL model acts as a bridge between parametric (Leland models) and non-parametric (Bakshi-Kapadia-Madan) frameworks, offering a semiparametric solution. Derived from both Leland models and the framework by Bakshi et al. [16], it integrates features like transaction costs and rebalancing intervals.

The key difference between the GLI and MFIL models lies in their treatment of initial trading costs. While GLI focuses on post-trade costs like transaction fees and rebalancing, MFIL additionally factors in the initial investment cost. Both models are applied to "All-Cash" and "All-Stock" scenarios, leading to four variants designated as MFIL All-Cash, MFIL All-Stock, GLI All-Cash and GLI All-Stock models.

In our study, the EGL models were applied to extract option-implied information by incorporating transaction costs and rebalancing intervals into the calculation of implied volatilities. The GLI model directly calculates calls and puts option prices using Leland's adjusted function, resulting in new option-implied volatilities. On the other hand, the MFIL approach utilizes call and put option prices directly from Leland models and feeds them into the semiparametric framework.

To ensure accuracy and reliability, the EGL models were calibrated and validated using historical options data. This involved estimating model parameters, such as transaction costs and rebalancing intervals, and testing the models' performance against observed market prices and implied volatilities.

The function of extended generalized model-free implied volatility was utilized in this research based on the design developed by Harun and Abdullah [48] as follows:

$E G=\frac{-k}{\sqrt{2 \pi \cdot \Delta \tau}}+\sqrt{\frac{k^2}{\pi \cdot \Delta \tau}-2\left(\mu^2-e^{r \tau} \cdot V\right)}$      (1)

The notations are defined as:

V is the variance contract;

k is the round-trip transaction cost rate per unit dollar of transaction;

∆τ is the time between hedging adjustment, i.e., the rebalancing interval.

To address the limitations of the existing MFBKM model, the semiparametric model was modified by incorporating several additional parameters: transaction cost rate and rebalancing interval. Notably, the transaction cost function from the Leland models was integrated into this framework.

4.2 Portfolio selection strategies

This study investigates the effectiveness of incorporating option-implied information in portfolio optimization, focusing on portfolios that comprise both risk-free and risky assets. Portfolio weights for each asset, denoted by w_i, are optimized using various models described in Table 3. The naïve 1/N portfolio serves as the benchmark.

Table 3. Asset allocation model

Specifically, the mean-variance approach follows Markowitz [1] by formulating a quadratic optimization problem. This model balances two key factors: maximizing expected return (Eq. (2)) while minimizing portfolio risk (Eq. (3)), subject to constraints Eq. (4).

$\operatorname{Min} \sigma_p^2=\sum_{i=1}^n \sum_{j=1}^n w_i \cdot w_j \cdot \operatorname{Cov}_{i, j}$    (2)

$\sum_{i=1}^n w_i=1, w_i \geq 0$        (3)

$R_p=\sum_{i=1}^n w_i r_j$      (4)

in which, R_p is a portfolio return and $\sigma_p^2$ is a portfolio risk. In addition, in each asset i, w_i is the investment weight and r_i is the average rate of return. In any given asset i and j, Cov_i,j is the covariance between asset i and asset j. The number of assets in the portfolio was represented by n. The mean-variance portfolio can be reduced into the minimum-variance if the assumption of equal mean returns of portfolio, R_p, on all assets was imposed.

In contrast, the median-variance strategy departs from the mean-variance approach by substituting the mean with the median in Eq. (4). This change aims to better reflect the realities of the stock market where skewed return distributions and outliers are more common [32]. The median-variance model is described in Eqs. (5)-(7).

$\operatorname{Min} \sigma_p^2=\sum_{i=1}^n \sum_{j=1}^n w_i \cdot w_j \cdot \operatorname{Cov}_{i, j}$   (5)

$\tilde{R}_p=\sum_{i=1}^n w_i \cdot \bar{r}_j$     (6)

$\sum_{i=1}^n w_i=1, w_i \geq 0$     (7)

where, $\bar{r}_j$ is the median return rate of asset i. The only difference between mean-variance and median-variance is the portfolio return equation, $\tilde{R}_p$, explained by Eq. (6).

While both mean-variance and median-variance optimization aim to balance expected return and risk, they differ in how they handle potential outliers and deviations from normality. The mean-variance approach uses the mean return in its calculations, making it susceptible to extreme values that might not accurately represent the typical investment experience. On the other hand, the median-variance approach utilizes the median return, which is less sensitive to outliers and offers a more robust measure of central tendency for skewed data, often encountered in real-world markets. While seemingly a subtle change, replacing the mean with the median in the portfolio return Eq. (6) leads to fundamentally different risk-return profiles and portfolio compositions. This seemingly simple modification significantly alters the optimization process, resulting in potentially diverse portfolio constructions across the two methods.

4.3 Historical Volatility Risk Premium (HVRP)

This study utilizes the Historical Volatility Risk Premium (HVRP) to quantify the market's compensation for holding risky assets. HVRP estimation is conducted monthly, aligning with the fixed 30-day option maturity period established at the outset of the research. This choice of maturity follows Demiguel et al. [49] and Harun and Abdullah [48].

For this objective measure, we begin by assuming a proportional relationship between the volatility risk premium magnitude and the overall volatility level. Consequently, the HVRP is estimated as the ratio of average implied volatilities to realized volatilities for each individual stock.

The HVRP was estimated over the T + Δt trading days, in which:

$H V R P_t=\frac{\sum_{t-T-\Delta t+1}^{T-\Delta t} M F I A V_{i, i+\Delta t}}{\sum_{t-T-\Delta t+1}^{T-\Delta t} R V_{i, i+\Delta t}}$    (8)

MFIAV indicated the option-implied volatility adjusted from the EGL models, whereas the realized volatility was denoted by RV.

Having estimated the volatility risk premium (HVRP) as suggested by Demiguel et al. [49] and Harun and Abdullah [50], we applied it to correct the option-implied volatility derived from the EGL models. This correction aims to account for the market's compensation for bearing volatility risk, ensuring our analysis reflects realistic investor expectations. We hypothesize that incorporating this refined volatility measure into the portfolio optimization strategy will lead to superior risk-adjusted returns. The successive realized volatility can be best represented by the risk-premium-corrected implied adjusted volatility as follows:

$\widehat{R V}_{t, t+\Delta t}=\frac{M F I A V_{t, t+\Delta t}}{H V R P_t}$        (9)

4.4 Portfolio performance evaluation measures

In this section, we considered the following three performance metrics in order to measure the performance of each portfolio: (1) portfolio return (Rp), (2) portfolio volatility, denoted as standard deviation (SD), and (3) Sharpe Ratio (SR). Beyond the classical (unconstrained) portfolios, we additionally investigate two types of constrained portfolios:

(1) Short-Selling Portfolios: These portfolios allow for positions below zero, enabling strategic shorting of assets.

(2) Zero-Correlation Portfolios: Enforcing zero correlation between all assets, these portfolios aim for diversification through uncorrelated holdings.

This comprehensive evaluation encompasses a total of 36 distinct portfolio strategies across different configurations. Calculating the portfolio return relies on the weighted average return of each individual asset within the portfolio. For instance, for any portfolio with i number of assets, the portfolio returns were derived as:

$R_P=w_1 r_1+w_2 r_2+w_3 r_3+\cdots+w_i r_i$          (10)

in which w_i denoted the weighted average returns of the i-th asset in the portfolio. This metric reflects the average gain or loss experienced by the portfolio over the investment period.

Second, we evaluated portfolio risk through its standard deviation (SD). Intuitively, this measure captures the overall volatility of the portfolio's returns, indicating the potential for fluctuations in value. For n-period returns of a portfolio, the portfolio standard deviation was formulated as:

$S D_P=\sqrt{\sum_{i=1}^n w_1^2 \sigma_1^2+\sum_{i=1}^n \sum_{i=1}^n w_i w_j \operatorname{Cov}_{i j}}$        (11)

For n number of assets, $\sigma_i^2$ is the variance of its returns, r, over a period of time and Cov_ij is the covariance between the rates of return for asset i and j.

Finally, we employed the Sharpe Ratio (SR) to assess the portfolio's risk-adjusted return. This ratio essentially measures the excess return earned by the portfolio over a risk-free rate, per unit of portfolio risk (represented by standard deviation). It thus serves as a valuable indicator of whether the portfolio generates sufficient return relative to the risk it exposes investors to. The measurement of the return and the risk were represented by the sample mean of excess returns, $\hat{\mu}$, and by the sample standard deviation of excess returns, $\hat{\sigma}$, correspondingly. As a result, in a risk-adjusted framework, we can denote the SR of a portfolio as below:

$S R_i=\frac{\hat{\mu}}{\hat{\sigma}}$       (12)

This suggested that the best-performed portfolio was not essentially the one with the maximum return, but those with the highest return-to-risk ratio. We utilized the p-values to investigate the statistical significance of the difference in the SR between particular portfolios against a benchmark portfolio. In this p-test, the null hypothesis was the POAI being evaluated performs no better than the corresponding PNOAI, in terms of the SR value.

This study proposes a novel, data-driven approach to portfolio optimization by incorporating option-implied volatility and employing the median-variance selection strategy. By addressing limitations in existing literature, our approach remains robust even when historical data is scarce or unreliable. The EGL methodology's ability to bridge the gap between parametric and non-parametric frameworks was reflected in the results. The integration of features from both Leland models and the framework by Bakshi et al. [16] allowed for a more comprehensive and flexible approach to portfolio optimization, leading to superior risk-adjusted returns.

By incorporating option-implied volatility and employing the median-variance selection strategy, we observed significant improvements in portfolio performance metrics such as Sharpe ratios and volatility. This demonstrates how our approach mitigates the shortcomings associated with traditional mean-variance and minimum-variance strategies, which often struggle to adapt to real-world market conditions.

Our methodology addresses a key limitation in existing portfolio optimization literature, which often relies exclusively on historical data analysis under a normal distribution setting. While historical data provides valuable insights into past market behavior, it may not fully capture the complexities of future market movements. By integrating option-implied information, our approach complements historical data analysis with forward-looking indicators, offering a more comprehensive and nuanced understanding of market dynamics. This enables investors to make more informed and adaptive portfolio decisions, enhancing their ability to navigate changing market conditions and achieve their investment objectives.

While this study utilized DJIA data for empirical validation, this may limit the generalizability of our findings to other asset classes or market indices. Future research could explore the applicability of our methodology across diverse asset classes and geographical regions to assess its broader effectiveness and robustness.

The use of a more recent dataset could offer additional insights into the performance of our methodology. The use of historical DJIA data spanning from January 2009 to December 2019 may not fully capture the dynamics of current market conditions.

Even so, the chosen period is observed to offer a stable foundation for evaluating the methodology's effectiveness. Option-adjusted volatility derived from EGL models significantly boosted performance, particularly with the median-variance strategy. Although minimum-variance portfolios minimize risk, median-variance strategies yielded superior Sharpe ratios, underlining the approach's advantage in optimizing portfolio selection.

Imposing realistic constraints like short-selling and zero-correlation demonstrated the method's robustness across diverse market conditions. Examining additional constrained portfolios could broaden the understanding of exploiting option-implied volatility within semiparametric pricing models, potentially refining future portfolio selection strategies.

Overall, this study contributes significantly to portfolio optimization research, highlighting the potential benefits for investors, particularly in emerging markets. As data availability and reliability in these markets improve, our methodology has the potential to become an increasingly crucial tool for optimizing portfolios and achieving investment goals.

However, further research is warranted to explore the generalizability of these results across diverse market conditions and asset classes. Additionally, examining the practical challenges and limitations of implementing this approach in real-world settings would be valuable for translating these findings into actionable investment strategies. By building upon this research and addressing these aspects, we can pave the way for more robust and data-driven investment strategies, ultimately empowering investors to make informed decisions and achieve their financial goals.