Category: volatility arbitrage

Correlation is an important component of portfolio and risk management. However, unlike volatility, which has received significant attention and for which numerous models have been developed, correlations have received considerably less attention from a modeling perspective.

In this post, we give correlations the attention they deserve and examine their role in volatility dynamics, portfolio construction, and trading.

State-Dependent Correlation Between the S&P 500 and the VIX

It is well known that the correlation between the S&P 500 and the VIX index is usually negative. However, practitioners also recognize that this correlation can break down and even turn positive.

Reference [1] addresses this issue by examining the time-varying nature of the correlation. The paper first posits and demonstrates that volatility-of-volatility (VoV) risk is distinct from volatility (Vol) risk, and then classifies the market into four regimes:

1. Low VOL risk and low VOV risk (LVOL–LVOV)
2. High VOL risk and low VOV risk (HVOL–LVOV)
3. Low VOL risk and high VOV risk (LVOL–HVOV)
4. High VOL risk and high VOV risk (HVOL–HVOV).

Findings

-The authors distinguish between two types of uncertainty: volatility (VOL) risk in the S&P 500 and volatility-of-volatility (VOV) risk in the VIX.

-They propose a theoretical framework in which the correlation between the S&P 500 and VIX varies across different combinations of VOL and VOV risk states.

-The authors develop a state-dependent regime-switching DCC model that identifies four states: high/low VOL risk combined with high/low VOV risk.

-The empirical results indicate that shocks driving VOL risk are not necessarily the same as those driving VOV risk, and the two risks respond differently to market shocks.

-The strongest negative correlation between the S&P 500 and VIX occurs in the high-VOL/high-VOV state, while the weakest negative correlation occurs in the high-VOL/low-VOV state.

-The authors find that their state-dependent correlation model provides greater portfolio risk reduction than conventional time-dependent correlation models.

In short, the paper develops a regime-switching model to show that the relationship between the S&P 500 and the VIX varies across distinct volatility (VOL) and volatility-of-volatility (VOV) risk states. It identifies a four-state system and demonstrates that correlation strength depends on the joint VOL–VOV regime.

This is an important contribution, as it quantifies the dynamic relationship between the S&P 500 and the VIX, thereby providing useful guidance for refining hedging and arbitrage strategies.

Reference

[1] Leon Li, Carl R. Chen, Volatility Risk and Volatility-of-Volatility Risk: State-Dependent Correlations Between VIX and the S&P 500 Stock Index and Hedging Effectiveness, Journal of Futures Markets, 2025; 1–20

Predicting Volatility Risk Premium Through Sector Implied Correlation

Despite the frequent use of the volatility risk premium (VRP), it appears, however, that there exists no reliable method for forecasting the VRP. In the literature, we have seen methods ranging from a simple one such as IV-HV (historical volatility) to a more sophisticated method such as GARCH.

Article [2] examines the volatility risk premium through implied volatilities and the correlation of sector ETFs. It concludes, surprisingly, that correlation premium is a good predictor of the volatility risk premium,

Findings

-The authors examine options on sector ETFs within the S&P 500 and find predictable variation in sector implied volatilities associated with unusually high or low implied correlations.

-They argue that abnormal short-term option pricing can be identified more effectively through implied correlation than through implied volatility alone.

-The authors develop a sector-specific correlation premium that adjusts for aggregate correlation levels and isolates the idiosyncratic component of sector correlation.

-The sector-specific correlation premium predicts future changes in sector implied volatility more reliably than conventional volatility- or correlation-premium measures.

-One-day reversals in sector implied volatility are associated with reversals in the sector-specific correlation premium.

-The authors find that information extracted from individual-stock implied volatilities has little or no predictive ability for sector implied volatility.

-Trading signals based on the abnormal sector-specific correlation premium are profitable and outperform conventional strategies based on sector volatility premiums.

Reference

[2] Koticha, Apoorva and Li, Chen and Marks, Joseph M., Abnormal Sector Option Correlation Premiums and Predictable Changes in Implied Volatility, SSRN 3862777

Closing Thoughts

Together, these studies highlight the importance of correlations in understanding volatility dynamics and improving investment decisions. The first shows that correlations between the S&P 500 and VIX vary across different volatility and volatility-of-volatility regimes, with state-dependent correlations improving portfolio risk management. The second demonstrates that implied correlations contain information about future sector implied volatility beyond conventional volatility measures and can form the basis for profitable trading signals.

Overall, both studies suggest that explicitly modeling correlations can provide valuable information for volatility forecasting, portfolio construction, and trading.

Reflexivity is the process through which market participants’ actions influence the very market variables on which their decisions are based, creating feedback effects. In this post, we discuss how reflexivity manifests itself in options hedging and how it can be explicitly incorporated into option pricing models.

Feedback Effect in the Foreign Exchange Market

The Black–Scholes-Merton (BSM) model is the most frequently used option pricing model in the financial industry. However, it makes use of assumptions that are not always realistic. A crucial assumption of the BSM model is a frictionless and elastic market. In other words, it assumes that there are no barriers to buying or selling securities and that investors can instantly buy or sell an asset at the prevailing market price.

The markets are, however, not frictionless, and the buying and selling of securities can have an impact on the markets. This is often called a feedback loop. Reference [1] examined how delta hedging in the foreign exchange market can have an impact on market volatility,

Findings

-The model considers an aggregate option market maker (OMM) and an aggregate option market taker (OMT), whose exposures represent outstanding option positions across the market.

-Differences in the hedge ratios of market makers and market takers generate net delta-hedging activity in the spot market, creating market friction and affecting volatility.

-The theoretical model predicts that negative market-maker gamma exposure increases spot volatility, while positive gamma exposure decreases it.

-The magnitude of the volatility effect depends on the amount of net delta hedging executed in the spot market.

-Using trade repository data, the authors reconstruct aggregate market-maker gamma exposure and find that market makers have negative gamma exposure during the sample period.

-The empirical results strongly support the theoretical model, with market-maker gamma exposure being highly significant in explaining spot-market volatility.

-A negative gamma exposure of approximately −$1 trillion is estimated to increase volatility by 0.7% in EUR/USD and 0.9% in USD/JPY.

Overall, the results indicate that market-maker delta hedging can create a feedback effect in which short-gamma positioning amplifies volatility in the underlying foreign exchange market.

The findings are consistent with our experience, as it’s widely acknowledged that a feedback effect (or reflexivity) exists in the financial markets. This article provided an empirical proof. It would be interesting to see a similar study on leveraged Exchange-Traded Funds.

Reference

[1] B. Anderegga, F.Ulmann, D. Sornette, The impact of option hedging on the spot market volatility, Journal of International Money and Finance, 124, 2022, 102627

Incorporating Reflexivity into the Black–Scholes-Merton Framework

While the first paper studies reflexivity and its impact on financial market dynamics, Reference [2] explicitly incorporates reflexivity into an option pricing model, extending the Black-Scholes-Merton framework to account for feedback effects generated by model-based trading.

Specifically, the author proposes three modifications,

1. Predictability of Brownian motion: Trader dependence on predictive models induces path dependence, breaking the martingale property of the Wiener process (Wt).
2. Drift coefficient deviation from risk-free rate: Large-scale institutional model use shifts the drift away from the risk-free rate by a reflexivity-induced term.
3. Volatility endogeneity: The volatility at time t depends on lagged volatility and hedging intensity.

These modifications were implemented by introducing a variable α, which represents the proportion of traders who rely on mathematical models for investment. The remaining (1–α) do not use such models and instead depend on trends or other factors. As a result, the stochastic differential equation for the stock price is adjusted.

Findings

-The framework challenges the assumption that financial-market dynamics are exogenous, arguing that widespread use of mathematical trading models can affect the same variables used as inputs to those models.

-The paper reformulates the stochastic differential equation for asset prices by modifying the drift, diffusion, and volatility terms to incorporate these feedback effects.

-The framework challenges several assumptions of classical financial theory, including risk-neutral drift equal to the risk-free rate, exogenous volatility, and market completeness.

-The author argues that collective use of mathematical models can create feedback mechanisms that reshape asset-price and option-pricing dynamics.

-The framework provides a formal mathematical structure linking George Soros’s concept of market reflexivity with option-pricing theory.

-The author suggests that reflexivity can amplify systemic risk and market fragility, potentially helping explain extreme market events.

-The paper highlights implications for volatility estimation, systemic risk, and the limitations of financial models that assume their inputs are unaffected by widespread model-based trading.

In short, this paper presents a promising extension of the BSM model. The next step is to calibrate it with extensive real-world data and to develop and test strategies based on it.

Reference

[2] Atharva Singh Rathore, Reflexivity in Financial Markets: A Theoretical Framework Extending the Black–Scholes Model, 2025

Closing Thoughts

Together, these studies highlight reflexivity as an important feature of modern financial markets. The first demonstrates how market-maker delta hedging can feed back into the underlying market, with short-gamma exposure amplifying spot volatility. The second approaches the same phenomenon theoretically by explicitly incorporating reflexivity into option pricing, allowing model-based trading itself to influence asset-price dynamics and volatility.

Both studies challenge the conventional assumption that financial models operate on market variables that are independent of their own use.

The Black-Scholes-Merton model is one of the cornerstones of modern quantitative finance. Despite its elegance and widespread use, its simplifying assumptions limit its ability to capture many features of real financial markets. As a result, researchers continue to extend the model to make it more realistic and applicable in practice.

In this post, we discuss two such extensions. The first incorporates a stochastic volatility model and intraday momentum into the option pricing framework. The second applies stochastic volatility models under the real-world measure to portfolio construction and volatility targeting, highlighting their practical use in risk management beyond derivative pricing.

Incorporating Momentum into Option Pricing Models

The Black–Scholes–Merton (BSM) model is a cornerstone of derivative pricing; however, it is not without limitations, and researchers continue to extend it. Reference [1] proposes an extension by incorporating intraday momentum into the BSM framework. This is achieved by introducing a drift term that represents intraday momentum, measured using a simple moving average of returns.

The model also adopts a modified Heston-type structure in which volatility follows a mean-reverting square-root process, allowing it to capture volatility clustering and remain consistent with empirical features such as volatility smiles. The momentum-driven drift adjustment influences the expected price path, while the stochastic volatility process models uncertainty around that path.

Findings

-The study extends the BSM option pricing framework by incorporating intraday momentum into the drift term of a stochastic volatility-modified model.

-It models time-varying volatility using a Heston-type stochastic volatility model and derives the momentum term from recent relative price changes.

-The study analyzes the impact of intraday momentum on stock prices, volatility, and option valuations, with particular attention to high-momentum scenarios.

-Numerical simulations show that positive momentum increases option valuations, while negative momentum decreases them.

-The study finds that the proposed model converges to the classical Black-Scholes model under low-volatility or low-momentum conditions.

-It concludes that incorporating momentum provides a theoretical framework for evaluating momentum-driven effects in derivative pricing and establishes quantitative metrics for empirical testing.

In short, the paper introduces a momentum term based on recent price changes to dynamically adjust the drift, capturing short-term intraday effects. Numerical results show that strong positive or negative momentum leads to substantial deviations from standard BSM prices, indicating that momentum is an important factor in option pricing.

This represents an interesting and potentially useful extension of the BSM model for traders and risk managers. However, as noted by the authors, the findings are based on simulated results rather than empirical data, and it would be valuable to see the model tested on real market data.

Reference

[1] Hossain, M.S., Yuan, X. & Sultan, S. Momentum-Driven Option Pricing: Integrating Intraday Trends into Financial Derivative Models. Comput Econ (2025).

Use of the Real-World Measure in Portfolio Management

In the realm of finance, the risk-neutral measure takes precedence in pricing financial derivatives. However, the real-world measure remains valuable and indispensable across various domains. It plays an important role in risk management and asset-liability applications, facilitating comprehensive evaluation and mitigation of risks.

Real-world measures are useful for simulation-based analyses of trading and investment strategies, offering insights into the practical implications of decisions in complex market environments. Reference [2] undertakes the calibration of stochastic volatility models as a means to estimate the real-world measure.

Employing the efficient method of moments (EMM), the authors perform calibration on the Heston and Bates SVJ models. Subsequently, the calibrated models are used to explore and analyze the risk and returns associated with volatility-target strategies.

Findings

-The study shows how a real-world stochastic volatility model can be applied to test a simple volatility targeting strategy.

-The results suggest that both stochastic volatility and jumps are required to characterize equity returns.

-The results indicate that volatility targeting reduces the likelihood of extreme returns and lowers the volatility of volatility.

-The study finds that portfolio risk and return both increase as the volatility target increases.

-The 10% volatility target produces the lowest risk, measured by both the mean of volatility and the volatility of volatility, but also the lowest return.

-An equity-only strategy produces the highest risk and the highest expected return.

-The study states that volatility targeting provides an effective way to manage portfolio downside risk while limiting upside potential.

This article serves to exemplify the practical utility of the real-world measure by demonstrating its application in assessing investment strategies. Specifically, the study underscores the effectiveness of volatility targeting as a strategic approach that empowers investors to effectively manage and mitigate the downside risk inherent in portfolio management.

Reference

[2] Alexis Levendis and Eben Mare, On the calibration of stochastic volatility models to estimate the real-world measure used in option pricing, Orion, Volume 39(1), pp. 65 – 91

Closing Thoughts

Both papers highlight the practical importance of stochastic volatility models in real-world applications. While the first study extends the classical Black-Scholes framework by incorporating momentum into a Heston-type stochastic volatility model, the second demonstrates how stochastic volatility models calibrated under the real-world measure can be used to implement a practical volatility targeting strategy. Together, they illustrate how stochastic volatility models continue to evolve beyond theoretical option pricing and provide useful tools for derivative valuation and portfolio risk management.

Retail trading, especially in the options market, which has traditionally been the domain of institutional traders, has received relatively little attention. However, with the rapid growth of educational content, AI, social media, and commission-free trading platforms, this is no longer the case.

Today, retail investors account for a significant share of options market volume and are changing market dynamics. In this post, we examine the characteristics of retail options trading and how it is reshaping the options market.

How Retail Investors Trade Options

Options trading is often thought of as a professional’s domain. However, with the advent of online trading platforms, retail traders now have access to the same tools and information as professional traders. This has changed the dynamics of the options market, as retail traders can now trade options on a level playing field with professionals.

However, a question remains to be answered: do retail options traders have the same knowledge, experience, and discipline as the professionals? Reference [1] examined this question

Findings

-The paper documents a rapid increase in retail participation in the U.S. options market in recent years.

-It also finds a sharp rise in payment for order flow (PFOF) paid by wholesalers to retail brokerages for executing customer option orders.

-The authors develop a novel measure of retail options trading using transaction-level data and new regulatory reporting requirements.

-The measure closely tracks other proxies for retail trading activity and declines significantly during brokerage outages and trading restrictions.

-The study finds that retail investors strongly prefer inexpensive weekly options.

-These options have very wide quoted bid-ask spreads, averaging approximately 12%, making them costly to trade.

-The paper finds that retail investors frequently fail to exercise call options optimally before ex-dividend dates.

-Market makers and arbitrageurs profit from these mistakes through nearly risk-free “dividend play” arbitrage strategies.

-The study reports that retail trading now accounts for over 60% of total U.S. options trading volume, with nearly 90% of PFOF originating from three major wholesalers.

The findings are very interesting. In the next paper, we’ll look at how retail traders have changed the volatility term structure and dynamics of the option market.

Reference

[1] S. Bryzgalova, A. Pavlova, T. Sikorskaya, Retail Trading in Options and the Rise of the Big Three Wholesalers, SSRN 4065019

The Impact of Retail Options Trading on the Implied Volatility Surface

Retail options trading is rising rapidly, driven by factors such as the growth of retail brokers, the popularity of social media, and more flexible working hours. Alongside this trend, there has been an increased interest in research on retail options trading behavior.

Reference [2] examines how retail trading reshapes the implied volatility (IV) surface dynamics. The authors utilize OPRA and Nasdaq data for this study. To isolate the effect of retail options trading, they apply a difference-in-differences approach around retail broker outages, 82 events from 2019 to 2021, comparing implied volatility between high-retail and low-retail stocks, during versus pre-outage periods.

Findings

-The paper documents a sharp increase in option trading activity driven by retail investors.

-It finds that retail trading is concentrated in call options, short-dated options, and out-of-the-money call options.

-The authors use brokerage outages as exogenous shocks to identify the impact of retail trading on option markets.

-Retail buying volume falls significantly during outages for the option contracts most favored by retail investors.

-In contrast, buying volume for long-dated options increases during outages, consistent with retail investors typically being net sellers of these contracts.

-The study finds that retail demand has a significant impact on option implied volatility.

-Implied volatility declines during brokerage outages, particularly for call, short-dated, and out-of-the-money options.

-Implied volatility increases for long-dated options during outages, reflecting reduced retail option-writing activity.

-The findings suggest that retail demand influences not only the level of implied volatility but also the term structure, moneyness curve, and call-put spread of the implied volatility surface.

-Robustness tests confirm that these effects are specific to brokerage outages and are not driven by a small subset of actively traded options or by the choice of trading dataset.

In short, retail investors systematically buy short-dated, out-of-the-money (especially calls) and sell long-dated options, creating predictable pressure across the surface. When retail trading activity is reduced during brokerage outages, IV falls for short-dated and OTM options but rises for long-dated options.

This paper contributes to a better understanding of retail options trading and shows how retail traders can materially affect the implied volatility surface.

Reference

[2] Eaton, Gregory W., T. Clifton Green, Brian S. Roseman, and Yanbin Wu (2025). Retail Option Traders and the Implied Volatility Surface. SSRN 4104788

Closing Thoughts

Taken together, these two papers show that retail investors have become a major force in the options market, influencing not only trading volume but also option pricing. Their preference for short-dated, out-of-the-money call options has measurable effects on implied volatility, while payment for order flow and trading behavior have reshaped market microstructure. For practitioners, understanding retail option flows is becoming increasingly important, as they now represent a significant driver of option prices and volatility dynamics.

Hedging is a fundamental risk management tool. The most common hedging instruments are futures and options associated with a given underlying asset, when available. For equity exposure, index options are also widely used for hedging.

However, hedging can be done not only through equity index options, but also through volatility derivatives, although the latter are considerably more complex and nuanced. In this post, we discuss the evolving dynamics of VIX futures and volatility ETPs, including lead-lag relationships, price discovery, and how hedging flows can influence volatility markets across different regimes and trading periods.

Lead-Lag Relationship Between the VIX Index and VIX Futures

The volatility index, VIX, is a measure of the stock market’s expectation of volatility over the next 30 days. The VIX index is calculated by taking a weighted average of the prices of put and call options on the S&P 500 index. The VIX is sometimes referred to as the “fear index” because it tends to spike when investors are worried about a sudden drop in the stock market.

VIX futures are derivative contracts that allow investors to bet on the direction of the VIX.  They are traded on the Chicago Board Options Exchange (CBOE). VIX futures were first introduced in 2004, and they are now one of the most popular derivatives contracts. VIX futures are traded in monthly contracts, and each contract represents a bet on the direction of the VIX index at the end of the contract month.

Reference [1] examined the lead-lag relationship between the VIX index and VIX futures. It utilized the symmetric thermal optimal path (TOPS) method that can handle non-stationary time series.

Findings

-The study examines the dynamic lead-lag relationship between the VIX and VIX futures markets using the symmetric thermal optimal path (TOPS) method.

-The results show that the VIX dominated VIX futures during the early years, particularly before the introduction of VIX options.

-In most periods, the relationship alternates rather than showing persistent dominance by one market.

-During the initial phase, VIX futures typically lagged the VIX by less than five days.

-The weaker role of VIX futures in the early period is attributed to lower trading volume.

-The importance of VIX futures in price discovery increases over time, especially after the launch of VIX options in 2006 and VIX ETPs in 2009.

-Since 2006, the lead-lag relationship has alternated, with the VIX sometimes leading futures and futures sometimes leading the VIX.

-The growth of VIX derivatives markets appears to have increased the informational efficiency of VIX futures.

Briefly, in the early days, the VIX index led its futures. However, the dynamics have changed; VIX futures now sometimes lead the spot market. This could be explained by the launch of VIX options and Exchange-Traded Notes.

Reference

[1] Yan-Hong Yang and Ying-Hui Shao, Time-dependent lead-lag relationships between the VIX and VIX futures markets, 2019, arXiv:1910.13729

Intraday Elasticity Between VIX Futures and Volatility ETPs

Reference [2] analyzes the sensitivity of VIX ETPs to movements in VIX futures. Specifically, the authors investigate the intraday price dynamics of the SPVXSTR, along with three VIX ETNs (VXX, XIV, TVIX) and three ETFs (VIXY, SVXY, UVXY), all linked to that index. Rather than relying on standard OLS regression, the study employs quantile regression, which minimizes a weighted sum of absolute errors and allows for asymmetric penalties on over- and under-predictions.

Findings

-The study analyzes the elasticity of VIX futures to volatility ETP prices using decile regressions on the S&P 500 VIX Short-Term Total Return Index (SPVXSTR).

-The results show that elasticity is lower near market close but higher during intraday trading, likely reflecting liquidity differences.

-Elasticity increases at the extreme ends of the return distribution near the close.

-VXX exhibits significantly higher elasticity than VIXY, attributed to its dominant and largely unhedged note structure.

-XIV and SVXY display similar elasticity patterns, while TVIX shows roughly half the elasticity of UVXY due to its lower leverage.

-The findings suggest that intraday liquidity amplifies the responsiveness of VIX futures to ETP price movements.

-VIX futures are found to be more sensitive to VXX than to TVIX or XIV during most trading periods.

-Sensitivity to XIV increases throughout the trading day in higher-return environments, likely reflecting increased hedging demand.

-The study highlights that VIX futures may overreact to ETP flows during stress periods and volatile market closes.

In short, the results show that VIX futures (SPVXSTR) are generally more sensitive to VXX than to TVIX or XIV, with the exception of the late-afternoon window (3:45–4:15 p.m.). Intraday elasticity is elevated—especially near the close and in the tails—implying that VIX futures can overreact to ETP price changes, which creates potential trading opportunities and important considerations for hedging under stress.

Reference

[2] Michael O’Neill, Gulasekaran Rajaguru, Elasticity dynamics between VIX futures and ETPs: a quantile regression analysis of intraday and closing market behavior, Journal of Accounting Literature (2025) 47 (5): 694–701.

Closing Thoughts

Taken together, these studies highlight the evolving dynamics of volatility markets and the growing importance of VIX derivatives and ETPs in price discovery and market behavior. The evidence suggests that lead-lag relationships between the VIX and VIX futures are time-dependent and increasingly influenced by derivative products and hedging flows.

At the same time, the elasticity of VIX futures to ETP activity varies across volatility regimes and intraday periods, implying that liquidity conditions and dealer positioning can materially affect market dynamics. These findings are particularly relevant for volatility traders, portfolio managers, and risk managers operating in increasingly complex derivatives markets.

The decomposition of risks and returns into overnight and intraday components is an emerging area of research. In this post, we examine how these components differ in terms of volatility clustering and the variance risk premium, and what this implies for forecasting, risk management, and strategy design.

Breaking Down the Volatility Risk Premium: Overnight vs. Intraday Returns

The decomposition of the volatility risk premium (VRP) into overnight and intraday components is an emerging area of research. Most studies indicate that the VRP serves as compensation for investors bearing overnight risks.

Reference [1] continues this line of research, with its main contribution being the decomposition of the variance risk premium into overnight and intraday components using a variance swap approach. The study also tests the predictive ability of these components and examines the seasonality (day-of-week effects) of the VRP.

Findings

-The paper decomposes the variance risk premium into overnight and intraday components across the US, Europe, and Asia.

-It finds that the variance risk premium is significantly negative during the overnight non-trading period.

-During the intraday trading period, the variance risk premium becomes positive and often insignificant.

-The results show that the overall negative variance risk premium documented in prior studies is largely driven by the overnight component.

-The study uses the P&L of a hypothetical variance swap to analyze these components.

-The intraday variance risk premium captures short-term risk and has predictive power over 1 to 3-month horizons.

-The overnight variance risk premium reflects longer-term risk and shows predictive ability over 6 to 12-month horizons.

-The findings highlight the importance of non-trading periods in explaining the behavior of the variance risk premium.

In summary, the study reaffirms that the variance risk premium is significantly negative during the non-trading overnight period, while it becomes positive and often insignificant during the intraday trading period.

An interesting finding is the day-of-week seasonality. For instance, going long volatility at the open and closing the position at the close tends to be profitable on most days, except Fridays.

Reference

[1] Papagelis, Lucas and Dotsis, George, The Variance Risk Premium Over Trading and Non-Trading Periods (2024). SSRN 4954623

Intraday and Overnight Volatility Clustering Effect

Volatility clustering is a phenomenon observed in financial markets where periods of high volatility tend to cluster together, followed by periods of low volatility. This pattern suggests that extreme price movements are not randomly distributed over time but rather occur in clusters or groups.

Volatility clustering has undergone extensive study within the daily timeframe. Reference [2] delves into volatility clustering within intraday and overnight timeframes. It specifically investigates clustering within each timeframe and between them.

Findings

-The paper studies volatility clustering in intraday and overnight returns across 15 global equity markets.

-It finds that volatility clustering is present in both intraday and overnight returns across multiple time scales, from daily to long-term horizons.

-The results show that volatility clustering is generally stronger in overnight returns than in intraday returns.

-Cross clustering between intraday and overnight volatility is relatively weak within each market, especially at shorter time scales.

-The findings are consistent across both developed and emerging markets, indicating a universal pattern.

-The study highlights the importance of considering both short-term and long-term risks in equity markets.

-The results suggest that volatility dynamics differ between trading and non-trading periods.

-The paper provides implications for trading and risk management strategies based on volatility clustering behavior.

In short, the paper shows that volatility clustering is a universal feature of both intraday and overnight returns across multiple time scales. It also finds that clustering is stronger overnight, while cross-effects between intraday and overnight volatility remain weak, with consistent patterns across global markets.

Reference

[2] Xiaojun Zhao, Na Zhang, Yali Zhang, Chao Xu, Pengjian Shang, Equity markets volatility clustering: A multiscale analysis of intraday and overnight returns, Journal of Empirical Finance 77 (2024) 101487

Closing Thoughts

Taken together, these studies show that volatility dynamics differ significantly between intraday and overnight periods, both in terms of risk pricing and clustering behavior. The variance risk premium is largely driven by the overnight component, while intraday and overnight volatility exhibit distinct clustering patterns with limited interaction. These findings highlight the importance of separating trading and non-trading periods in both forecasting and risk management, as each captures different horizons and sources of risk, offering more refined inputs for portfolio construction and strategy design.

Momentum has been studied extensively across equities, commodities, and other asset classes, with well-documented evidence of cross-sectional and time-series continuation effects. More recently, an emerging line of research has shifted attention to momentum in option returns, examining whether derivative markets exhibit their own systematic return patterns.

In this post, we review the latest evidence on option return momentum across both monthly and intraday horizons and assess the economic mechanisms that may explain these persistent dynamics.

Momentum in the Option Market

In the financial market, momentum is the tendency for assets to continue moving in the same direction. It is a reflection of the underlying strength or weakness of an asset’s price action and can be used to identify trends. Momentum is one of the most pervasive market phenomena and can be observed in nearly all stock markets around the world.

Does this anomaly exist in other asset classes?

Reference [1] studied momentum in the options market. It examined the returns of delta-neutral straddles on individual equities.

Findings

-The study finds strong evidence of momentum in option returns, as options that performed well over the previous 6 to 36 months tend to generate high returns in the subsequent month.

-Momentum is observed under both cross-sectional and time-series definitions of past performance.

-The strategy is profitable across all five-year subsamples and carries significantly lower risk than short straddle positions on the S&P 500 Index or individual stocks.

-There is no evidence of momentum crashes in option returns, although the sample length may limit detection of such events.

-The authors find limited evidence of short-term cross-sectional reversal, where options that outperform in one month may underperform in the following month.

-There is no evidence of long-run reversal in option returns, in contrast to equities, and momentum persists even at 2- to 3-year horizons.

-Option momentum differs from stock momentum because the results are based on delta-hedged positions and remain robust after controlling for stock momentum effects.

-Momentum profits are unaffected by controls for implied versus historical volatility and other option characteristics, and remain significant after factor risk adjustments.

-The study shows that high historical-return options significantly outperform low-return options across multiple horizons, including when using out-of-the-money options or delta-hedged returns.

In short, like in equities, options also exhibit momentum. The options momentum is mean-reverting in the short term and trending in the long term.

Reference

[1] Heston, Steven L. and Jones, Christopher S. and Khorram, Mehdi and Li, Shuaiqi and Mo, Haitao, Option Momentum (2022), SSRN 4113680

Momentum in the Option Market, Intraday Case

While the previous article examines momentum in option returns across monthly horizons, Reference [2] extends this line of research by focusing on intraday option return dynamics.

Findings

-The paper documents novel seasonal patterns in intraday returns of individual stock option straddles.

-Despite being delta-neutral, straddle returns exhibit the same persistent intraday seasonality as their underlying stocks.

-Returns in a given half-hour interval predict returns in the same interval on the following trading day.

-This continuation effect is strongest at the market open and close, referred to as morning and afternoon momentum.

-Morning momentum is attributed to investors’ underreaction to volatility shocks.

-Afternoon momentum is driven by persistent inventory management behavior by option market makers.

In summary, it was shown that a straddle’s return during a particular 30-minute trading interval today positively predicts its return during the same interval on subsequent days. Morning momentum reflects a continued under-reaction to overnight volatility news. Afternoon momentum, on the other hand, is attributed to persistent price pressure caused by inventory management from option market makers.

Reference

[2] Da, Zhi and Goyenko, Ruslan and Zhang, Chengyu, Intraday Option Return: A Tale of Two Momentum (2024), SSRN 5018430

Closing Thoughts

Taken together, these studies show that option markets exhibit systematic return patterns across both monthly and intraday horizons. Momentum persists over 6 to 36 months without the long-run reversals observed in equities, while intraday straddle returns display predictable continuation at the market open and close.

The evidence suggests that option return dynamics are driven by distinct forces, including behavioral underreaction, inventory management by market makers, and structural features of volatility trading. Collectively, these findings reinforce the view that options are not merely derivatives of stocks, but markets with their own persistent and economically meaningful return patterns.

Delta hedging is a fundamental topic in portfolio and risk management. In this post, we discuss which volatility measure should be used in the delta hedging process, while a future edition will examine the appropriate hedging frequency and time horizon.

Which Free Lunch Would You Like Today Sir?

Reference [1] is a classic article on delta hedging that addresses the following question: if an investor has an accurate estimate of future realized volatility that differs from current implied volatility, a position can be initiated to exploit this discrepancy and then dynamically hedged—but which volatility should be used as the input in the hedging process?

Hedging with Actual Volatility

Pros

-Hedging with actual volatility guarantees the final profit at expiration, equal to the difference between theoretical option values under actual and implied volatility.

-The final profit has zero variance, making it attractive from a long-term, global risk–reward perspective.

-Expected profit is often insensitive to small errors in the volatility used for hedging, providing some robustness to estimation error.

Cons

-Mark-to-market P&L during the life of the option can fluctuate significantly, which is problematic for short-term risk management.

-Interim P&L depends on the true drift of the underlying asset, introducing uncertainty before expiration.

-In practice, traders are rarely confident in their estimate of actual volatility, weakening the appeal of this approach.

Hedging with Implied Volatility

Pros

-Mark-to-market P&L evolves smoothly with no random fluctuations, which is advantageous for daily risk monitoring.

-The trader only needs to be directionally correct about volatility (i.e., actual > implied or vice versa), not to estimate actual volatility precisely.

-Implied volatility is directly observable from the market, simplifying implementation.

Cons

-The final profit is path-dependent and therefore uncertain at inception.

-While profits are always positive in expectation, their magnitude cannot be known in advance.

-Profitability depends on the realized price path, particularly whether the underlying remains near regions of high gamma.

Reference

[1] R Ahmad, P Wilmott, Which Free Lunch Would You Like Today Sir?, Wilmott, 2005

Delta Hedging with Implied vs. Historical Volatility

Similar to the previous paper, Reference [2] examines the effectiveness of hedging using implied versus realized volatility. The study is based on empirical analysis using index ETF options, specifically the Nasdaq-100 ETF (QQQ).

Findings

-The study examines the role of volatility estimation in delta-neutral hedging, with a focus on short-term options trading and risk management.

-It empirically compares implied volatility (IV) and historical volatility (HV) using Nasdaq-100 ETF (QQQ) options over several months of daily data.

-The analysis evaluates hedging performance, return stability, transaction costs, hedging errors, and sensitivity under varying market volatility conditions.

-Results show that IV-based hedging delivers more stable returns, lower return volatility, and better risk mitigation, making it more suitable for conservative and risk-averse investors.

– IV-based strategies benefit from forward-looking, market-implied inputs, which improve delta accuracy, reduce rebalancing frequency, and lower transaction costs.

-HV-based hedging can generate higher potential returns but exhibits greater variability, larger hedging errors, and higher portfolio risk, particularly during volatile markets.

-Sensitivity tests confirm that IV adapts more effectively to changing market conditions than HV.

-The study highlights a clear trade-off between stability and return potential, emphasizing that volatility measure selection should depend on market conditions and risk preferences.

The findings provide practical guidance for traders and risk managers and contribute to the literature on optimal volatility modeling under real-world constraints. Though the paper has some limitations, notably the small sample size, this research direction is worth pursuing, particularly in establishing a delta band and determining the optimal hedging frequency.

Reference

[2] Yimao Zhao, Implied Volatility vs. Historical Volatility: Evaluating the Effectiveness of Delta-Neutral Hedging Strategies, Proceedings of the 2025 5th International Conference on Enterprise Management and Economic Development (ICEMED 2025)

Closing Thoughts

Taken together, these two studies highlight that the choice of volatility input is an important decision in delta hedging, rather than a technical detail. Both papers show that implied volatility, with its forward-looking and market-based nature, generally delivers more stable hedging performance, lower tracking errors, and better risk control, particularly in short-term and actively rebalanced strategies.

Historical or realized volatility, while simpler and sometimes effective in calmer market regimes, tends to lag during volatility shifts and leads to larger hedging errors. The broader implication for practitioners is that effective delta hedging requires aligning the volatility measure with market conditions, risk tolerance, and trading horizon, rather than relying on a one-size-fits-all approach.

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.

Volatility and volatility of volatility are highly correlated and share many similar characteristics. However, there are subtle but important differences between them. In this post, we will examine some of these differences and explore an application of volatility of volatility in portfolio management.

Improving Portfolio Management with Volatility of Volatility

Managing portfolios using volatility has proven effective. Reference [1] builds on this research by proposing the use of volatility of volatility for portfolio management. The rationale behind using volatility of volatility is that it represents uncertainty.

Unlike risk, which refers to situations where future returns are unknown but follow a known distribution, uncertainty means that both the outcome and the distribution are unknown. Stocks may exhibit uncertainty when volatility or other return distribution characteristics vary unpredictably over time.

Practically, the author used a stock’s daily high and low prices to derive its volatility of volatility.

Findings

-The study investigates how volatility-managed investment strategies perform under different levels of uncertainty across stocks and over time.

-A new measure of volatility-of-volatility (vol-of-vol) is introduced as a proxy for uncertainty about risk, capturing a unique dimension distinct from traditional volatility.

-Results show that abnormal returns from volatility management are concentrated in stocks with low uncertainty and during periods of low aggregate uncertainty.

-The effectiveness of sentiment-based explanations for volatility-managed returns is conditional on the level of uncertainty.

-Cross-sectional differences in uncertainty help explain why volatility-managed factor portfolios perform unevenly across stocks and time.

-Theoretical analysis extends a biased belief model, showing that higher vol-of-vol reduces volatility predictability and belief persistence, weakening the benefits of volatility timing.

-The study hypothesizes that volatility management is most effective for low-uncertainty stocks and in low-uncertainty market environments.

-Empirical tests use realized vol-of-vol derived from intraday high and low prices as the measure of uncertainty.

-Consistent with prior literature, uncertainty is positively related to future returns and contains unique predictive information not explained by other stock characteristics.

-Volatility management significantly improves risk-adjusted performance in low-uncertainty stocks and during low aggregate uncertainty periods, while uncertainty also helps explain performance variation across asset pricing factor portfolios.

In short, using the volatility of volatility as a filter proves to be effective, particularly for low-uncertainty stocks.

We find it insightful that the author distinguishes between risk and uncertainty and utilizes the volatility of volatility to represent uncertainty.

Reference

[1] Harris, Richard D. F. and Li, Nan and Taylor, Nicholas, The Impact of Uncertainty on Volatility-Managed Investment Strategies (2024), SSRN 4951893

Beyond volatility of volatility

This section is written by Alpha in Academia

The Volatility of Volatility Index (VVIX) is a composite measure, driven by both short-term market panic and long-term risk expectations.

For years, the VVIX, often dubbed the “fear of fear” index, was treated primarily as a measure of the volatility of volatility (VOV), but new research reveals it contains a second, equally critical component: Long-Run Variance (LRV).

Using a sophisticated model and leveraging a novel technique involving risk-neutral cumulant data extracted from VIX options, researchers decomposed the VVIX dynamics. Their analysis reveals that the factors driving the index change dramatically depending on market conditions. Specifically, the short-term panic measure, VOV, significantly contributes only during acute periods of financial distress, which aligns with intuition. However, during stable or bull markets, the VVIX is primarily driven by the LRV component, reflecting persistent, underlying risk expectations.

In fact, when testing the explanatory power on market-neutral straddle portfolios using S&P 500 options, combining LRV and VOV produced an adjusted explanatory power up to three times greater than baseline models. The finding shows that the index provides “a clear answer to the question of the informational content of the VVIX, showing that it reflects not only the VOV but also an additional important component—the LRV”. Investors should thus view the VVIX not just as a fear gauge, but as a dual-sensor monitoring immediate market stress and long-term risk.

Reference

[2] Bacon, Étienne and Bégin, Jean-François and Gauthier, Geneviève, Beyond volatility of volatility: Decomposing the informational content of VVIX, 2025, SSRN 5611090

Closing Thoughts

In summary, both studies emphasize the role of volatility-of-volatility in understanding risk and market behavior. The first shows that volatility management is most effective in low-uncertainty environments, while the second reveals that the VVIX reflects not only short-term market stress but also long-term risk expectations. Together, they suggest that volatility-of-volatility offers deeper insight into both portfolio performance and the broader dynamics of market uncertainty.