options trading strategies – Relative Value Arbitrage

Exploring Credit Risk: Its Influence on Equity Strategies and Risk Management

Credit risk, also known as default risk, is the likelihood of loss when a borrower or counterparty fails to meet its obligations. A lot of research has been conducted on credit risk, and an emerging line of study explores the connection between the equity and credit markets. In this post, we’ll discuss how credit risk impacts investment strategies in the equity market and how equity options can be used to hedge credit risk.

Understanding Credit Risk and Its Impact on Investment Strategies

Credit risk, also known as default risk, is the likelihood of loss from a borrower or counterparty not meeting its obligations. The Merton model, developed by Robert Merton, is a widely used model to measure a company’s credit risk, utilizing quantitative parameters. Reference [1] examined how credit risk impacts momentum and contrarian strategies in the equity markets.

Findings

-Credit risk is measured using default risk, specifically the distance to default (DD) from the Kealhofer, McQuown, and Vasicek (KMV) model.

– High credit risk firms, when subjected to momentum and contrarian strategies, can generate excess returns.

– Medium credit risk firms also offer opportunities for excessive returns with these strategies.

– Low credit risk firms do not show significant relationships with momentum and contrarian returns.

– Investors should consider credit risk when implementing momentum and contrarian investment strategies.

– The applicability of these findings to the US and other developed markets is suggested for further research.

Reference

[1] Ahmed Imran Hunjra, Tahar Tayachi, Rashid Mehmood, Sidra Malik and Zoya Malik, Impact of Credit Risk on Momentum and Contrarian Strategies: Evidence from South Asian Markets, Risks 2020, 8(2), 37

Using Equity Options to Hedge Credit Risks

Using credit derivatives, such as credit default swaps, to manage credit risks is a common practice in the financial industry. Reference [2] proposed an approach that uses equity derivatives to partially hedge credit risks.

The author generalized the Merton structural model, where a company’s equity is viewed as a call option on its assets. However, instead of using the total debt level as the default trigger, the author proposed an alternative default threshold where default is determined by the stock price’s initial crossing of a predefined level. The credit loss then resembles the payoff of a digital put option.

Findings

– Building on Merton’s model, the paper defines default as the event where the stock price S_T falls below a set barrier, B.

– By establishing a link between this default model and the probability P(S_T < B) at time T, the study shows that hedging with a European put option can reduce the capital required for projected losses.

– An optimization problem is formulated to find the optimal strike price for the put option, minimizing risk based on a specific measure.

– Numerical analysis indicates that this method reduces the Solvency Capital Requirement (SCR) in both jump and non-jump markets, providing insurance companies with an effective way to reduce losses within their existing risk management structures.

Reference

[2] Constantin Siggelkow, Partial hedging in credit markets with structured derivatives: a quantitative approach using put options, Journal of Derivatives and Quantitative Studies, 2024

Closing Thoughts

Credit risk remains a critical component shaping financial markets, with significant implications for equity investments. The growing research linking credit and equity markets highlights the importance of integrating credit risk considerations into investment strategies. Utilizing equity options for hedging provides a valuable approach to managing these risks effectively. As research in this area evolves, leveraging credit risk insights can enhance portfolio resilience and improve risk-adjusted returns.

Hedging Efficiently: How Optimization Improves Tail Risk Protection

Tail risk hedging aims to protect portfolios from extreme market downturns by using strategies such as out-of-the-money options or volatility products. While effective in mitigating large losses, the challenge lies in balancing cost and long-term returns. In this post, we’ll discuss tail risk hedging and whether it can be done at a reasonable cost.

Tail Risk Hedging Strategies: Are They Effective?

Tail risk hedging involves purchasing put options to protect the portfolio either partially or fully. Reference [1] presents a study of different tail risk hedging strategies. It explores the effectiveness of put option monetization strategies in protecting equity portfolios and enhancing returns.

Findings

– Eight different monetization strategies were applied using S&P 500 put options and the S&P 500 Total Return index from 1996 to 2020.

– Results compared against an unhedged index position and a constant volatility strategy on the same underlying index.

– Tail risk hedging, in this study, yielded inferior results in terms of risk-adjusted and total returns compared to an unhedged index position.

– Over a 25-year period, all strategies’ total returns and Sharpe ratios were worse than the unhedged position.

– Buying puts involves paying for the volatility risk premium, contributing to less favorable results.

– The results are sensitive to choices of time to expiry and moneyness of purchased options in tested strategies.

– The authors suggest the possibility of minimizing hedging costs by optimizing for strikes and maturities.

Reference:

[1] C.V. Bendiksby, MOJ. Eriksson, Tail-Risk Hedging An Empirical Study, Copenhagen Business School

How Can Put Options Be Used in Tail Risk Hedging?

The effectiveness of using put options to hedge the tail risks depends on the cost of acquiring put options, which can eat into investment returns. Reference [2] proposes a mixed risk-return optimization framework for selecting long put options to hedge S&P 500 tail risk. It constructs hypothetical portfolios that continuously roll put options for a tractable formulation.

Findings

– The article discusses the effectiveness of tail risk hedging. It highlights that the premium paid for put options can be substantial, especially when continuously renewing them to maintain protection. This cost can significantly impact investment returns and overall portfolio performance.

– The article introduces an optimization-based approach to tail-risk hedging, using dynamic programming with variance and CVaR as risk measures. This approach involves constructing portfolios that constantly roll over put options, providing protection without losing significant long-term returns.

– Contrary to previous research, the article suggests that an effective tail-risk hedging strategy can be designed using this optimization-based approach, potentially overcoming the drawbacks of traditional protective put strategies.

-The proposed hedging strategy overcame traditional drawbacks of protective put strategies. It outperforms both direct investments in the S&P 500 and static long put option positions.

Reference

[2] Yuehuan He and Roy Kwon, Optimization-based tail risk hedging of the S&P 500 index, THE ENGINEERING ECONOMIST, 2023

Closing Thoughts

Tail risk hedging is expensive. While the first paper demonstrated that tail risk hedging leads to inferior returns, it suggested that results could be improved by optimizing strike prices and maturities. The second paper built on this idea and proposed a hedging scheme based on optimization. The proposed strategy outperforms both direct investments in the S&P 500 and static long put option positions.

The Predictive Power of Dividend Yield in Equity Markets

Dividend yield has long been a cornerstone of equity valuation. In this post, we explore how dividend yield predicts stock returns, its impact on stock volatility, and why it holds unique significance for mature, dividend-paying firms.

Relationship Between Implied Volatility and Dividend Yield

Reference [1] explores the relationship between implied volatility (IV) and dividend yield. It investigates how the dividend yield impacts the implied volatility. The study supports the bird-in-hand theory rather than the dividend irrelevance theory. Results show that there exists a negative relationship between dividend yield and IV, and this relationship is stronger for puts than calls.

Findings

– This thesis examines the link between implied volatility and dividend yield in the options market, comparing the Bird-in-Hand theory and the Dividend Irrelevancy theory.

– Results show that dividend yield significantly impacts implied volatility, with a stronger and consistent negative relationship observed in put options, aligning with the Bird-in-Hand theory.

– The relationship in put options suggests a stronger and more consistent impact of dividend yield, aligning with the Bird-in-hand theory.

– The findings support the hypothesis that an increase in a firm’s dividend yield tends to decrease future volatility.

– This effect was particularly pronounced in put option models but also observed in call option models.

– The study emphasizes the need for alternative methodologies, larger sample sizes, and additional variables to deepen the understanding of option pricing dynamics.

Reference

[1] Jonathan Nestenborg, Gustav Sjöberg, Option Implied Volatility and Dividend Yields, Linnaeus University, 2024

The Impact of Dividend Yield on Stock Performance

Dividend yield is a reliable predictor of future stock returns, particularly during periods of heightened volatility. This article [2] explores the connection between dividend yield, stock volatility, and expected returns.

Findings

– This study shows that dividend yield predicts returns for dividend-paying firms more effectively than alternative pricing factors, challenging previous research.

– Using the most recent declaration date to calculate dividend yield significantly improves return predictability compared to using the trailing yield.

– Asset pricing strategies tend to underperform within mature, profitable firms that pay dividends, highlighting a unique pattern.

– Cross-sectional tests suggest dividend yield predicts returns because investors value receiving dividends rather than as an indicator of future earnings.

– Dividend yield is concluded to be a valuable valuation metric for mature, easier-to-value firms that typically pay dividends.

– Volatility, measured as the trailing twelve-month average of monthly high and low prices, impacts return predictability.

– Excessively volatile prices drive predictability, with dividend yield strategies generating around 1.5% per month.

– During heightened volatility periods, dividend yield strategies yield significant returns.

– Cross-sectionally, dividend yield is a more accurate predictor for returns in volatile firms.

Reference:

[2] Ahn, Seong Jin and Ham, Charles and Kaplan, Zachary and Milbourn, Todd T., Volatility, dividend yield and stock returns (2023). SSRN

Closing Thoughts

Dividend yield is shown to be a useful valuation metric, particularly for mature and easily valued firms that consistently pay dividends. Furthermore, the research emphasizes that investors prioritize the receipt of dividends over their informational value regarding future earnings. These insights reaffirm the importance of dividend yield in understanding market dynamics and developing effective investment strategies.

Monte Carlo Simulations: Pricing Weather Derivatives and Convertible Bonds

Monte Carlo simulations are widely used in science, engineering, and finance. They are an effective method capable of addressing a wide range of problems. In finance, they are applied to derivative pricing, risk management, and strategy design. In this post, we discuss the use of Monte Carlo simulations in pricing complex derivatives.

Pricing of Weather Derivatives Using Monte Carlo Simulations

Weather derivatives are a particular class of financial instruments that individuals or companies can use in support of risk management in relation to unpredictable or adverse weather conditions. There is no standard model for valuing weather derivatives similar to the Black–Scholes formula. This is primarily due to the non-tradeable nature of the underlying asset, which violates several assumptions of the Black–Scholes model.

Reference [1] presented a valuation method for pricing an exotic wind power option using Monte Carlo simulations.

Findings

– Wind power generators are exposed to risks stemming from fluctuations in market prices and variability in power production, primarily influenced by their dependency on wind speed.

– The research focuses on designing and pricing an up-and-in European wind put barrier option using Monte Carlo simulation.

– In the presence of a structured weather market, wind producers can mitigate fluctuations by purchasing this option, thereby safeguarding their investments and optimizing profits.

– The wind speed index serves as the underlying asset for the barrier option, effectively capturing the risks associated with wind power generation.

– Autoregressive Fractionally Integrated Moving Average (ARFIMA) is utilized to model wind speed dynamics.

– The study applies this methodology within the Colombian electricity market context, which is vulnerable to phenomena like El Niño.

– During El Niño events, wind generators find it advantageous to sell energy to the system because their costs, including the put option, are lower than prevailing power prices.

– The research aims to advocate for policy initiatives promoting renewable energy sources and the establishment of a financial market for trading options, thereby enhancing resilience against climate-induced uncertainties in the electrical grid.

Reference

[1] Y.E. Rodríguez, M.A. Pérez-Uribe, J. Contreras, Wind Put Barrier Options Pricing Based on the Nordix Index, Energies 2021, 14, 1177

Pricing Convertible Bonds Using Monte Carlo Simulations

The Chinese convertible bonds (CCB) have a special feature, which is a downward adjustment clause. Essentially, this clause states that when the underlying stock price remains below a pre-set level for a pre-defined number of days over the past consecutive trading days, issuers can lower the conversion price to make the conversion value higher and more attractive to investors.

Reference [2] utilized the Monte Carlo simulation approach to account for this feature and to price the convertible bond.

Findings

– The downward adjustment provision presents a significant challenge in pricing Chinese Convertible Bonds (CCBs).

– The triggering of the downward adjustment is treated as a probabilistic event related to the activation of the put option.

– The Least Squares method is employed to regress the continuation value at each exercise time, demonstrating the existence of a unique solution.

– The downward adjustment clause is integrated with the put provision as a probabilistic event to simplify the model.

-When the condition for the put provision is met, the downward adjustment occurs with 80% probability, and the conversion price is adjusted to the maximum of the average of the underlying stock prices over the previous 20 trading days and the last trading day.

Reference

[2] Yu Liu, Gongqiu Zhang, Valuation Model of Chinese Convertible Bonds Based on Monte Carlo Simulation, arXiv:2409.06496

Closing Thoughts

We have explored advanced applications of Monte Carlo simulations in pricing weather derivatives and complex convertible bonds. This versatile method demonstrates its broad applicability across various areas of finance and trading.

Option Pricing Models and Strategies for Crude Oil Markets

Financial models and strategies are usually universal and can be applied across different asset classes. However, in some cases, they must be adapted to the unique characteristics of the underlying asset. In this post, I’m going to discuss option pricing models and trading strategies in commodities, specifically in the crude oil market.

Volatility Smile in the Commodity Market

Paper [1] investigates the volatility smile in the crude oil market and demonstrates how it differs from the smile observed in the equity market. It proposes to use the new method developed by Carr and Wu in order to study the volatility smile of commodities. Specifically, the authors examine the volatility smile of the United States Oil ETF, USO.

Findings

– This paper examines the information derived from the no-arbitrage Carr and Wu formula within a new option pricing framework in the USO (United States Oil Fund) options market.

– The study investigates the predictability of this information in forecasting future USO returns.

– Using the no-arbitrage formula, risk-neutral variance, and covariance estimates are obtained under the new framework.

– The research identifies the term structure and dynamics of these risk-neutral estimates.

– The findings reveal a “U”-shaped implied volatility smile with a positive curvature in the USO options market.

Usually, an equity index such S&P 500 exhibits a downward-sloping implied volatility pattern, i.e. a negative implied volatility skew. Oil, on the other hand, possesses a different volatility smile. This is because while equities are typically associated with crash risks, oil prices exhibit both sharp spikes and crashes, leading to a different implied volatility pattern. This highlights the importance of considering the specific characteristics and dynamics of different asset classes when analyzing and interpreting implied volatility patterns.

Reference

[1] Xiaolan Jia, Xinfeng Ruan, Jin E. Zhang, Carr and Wu’s (2020) framework in the oil ETF option market, Journal of Commodity Markets, Volume 31, September 2023, 100334

Statistical Arbitrage in the Crude Oil Markets

Reference [2] directly applies statistical arbitrage techniques, commonly used in equity markets, to the crude oil market. It utilizes cointegration to construct a statistical arbitrage portfolio. Various methods are then used to test for stationarity and mean reversion: the Quandt likelihood ratio (QLR), augmented Dickey-Fuller (ADF) test, autocorrelations, and the variance ratio. The constructed strategy performed well both in- and out-of-sample.

Findings

– This paper introduces the concept of statistical arbitrage through a trading strategy known as the mispricing portfolio.

– It focuses specifically on mean-reverting strategies designed to exploit persistent anomalies observed in financial markets.

– Empirical evidence is presented to demonstrate the effectiveness of statistical arbitrage in the crude oil markets.

– The mispricing portfolio is constructed using cointegration regression, establishing long-term pricing relationships between WTI crude oil futures and a replication portfolio composed of Brent and Dubai crude oils.

-Mispricing dynamics revert to equilibrium with predictable behaviour. Trading rules, which are commonly used in equity markets, are then applied to the crude oil market to exploit this pattern.

Reference

[2] Viviana Fanelli, Mean-Reverting Statistical Arbitrage Strategies in Crude Oil Markets, Risks 2024, 12, 106.

Closing Thoughts

As we’ve seen, techniques and models utilized in the equity market can sometimes be applied directly to the crude oil market, while other times they need to be adapted to the unique characteristics of the crude oil market. In any case, strong domain knowledge is essential.

Educational Video

In this webinar, Quantitative Trading in the Oil Market, Dr Ilia Bouchouev delivers an interesting and insightful presentation on algorithmic trading in the oil market. He also encourages viewers to apply the techniques discussed for the oil market to other markets, such as equities.

When Correlations Break or Hold: Strategies for Effective Hedging and Trading

It’s well known that there is a negative relationship between an equity’s price and its volatility. This can be explained by leverage or, alternatively, by volatility feedback effects. In this post, I’ll discuss practical applications to exploit this negative correlation between equity prices and their volatility.

A Trading Strategy Based on the Correlation Between the VIX and S&P500 Indices

This paper [1] examines the strong correlation in the S&P 500 and identifies trading opportunities when this correlation weakens or breaks down.

Findings

-The study covers the period from January 1995 to October 2020, utilizing 6,488 daily observations of the VIX and S&P500 indexes.

– In scenarios where the options market indicates increased drawdown risk with higher implied volatility but negative returns have not yet occurred, consider shorting the market.

– The signal to short the market occurs when the negative correlation between the S&P 500 and VIX is broken, and they start exhibiting a positive correlation.

– The test setup involves identifying one or two consecutive days with positive co-movement between the VIX and S&P 500, then setting the transaction date for the day after or at the close of the chosen date.

– Empirical results show that the strategy outperforms the S&P500 index over the 25-year period, achieving higher returns, lower systematic risk, and reduced volatility.

-The findings provide evidence that excess returns can be generated by timing the market using historical data, even after accounting for trading costs.

Reference

[1] Tuomas Lehtinen, Statistical arbitrage strategy based on VIX-to-market based signal, Hanken School of Economics

Optimal Hedging for Options Using Minimum-Variance Delta

Contrary to the first paper, Reference [2] focuses on the strong correlation between the S&P 500 and its volatility, designing an efficient scheme for hedging an options book.

The authors developed a so-called minimum variance (MV) delta. Essentially, the MV delta is the Black-Scholes delta with an additional adjustment term.

Findings

-Due to the negative relationship between price and volatility for equities, the minimum variance delta is consistently less than the practitioner Black-Scholes delta.

-Traders should under-hedge equity call options and over-hedge equity put options compared to the practitioner Black-Scholes delta.

-The study demonstrates that the minimum variance delta can be accurately estimated using the practitioner Black-Scholes delta and the historical relationship between implied volatilities and asset prices.

-The expected movement in implied volatility for stock index options can be approximated as a quadratic function of the practitioner Black-Scholes delta divided by the square root of time.

-A formula for converting the practitioner Black-Scholes delta to the minimum variance delta is provided, yielding good out-of-sample results for both European and American call options on stock indices.

-For S&P 500 options, the model outperforms stochastic volatility models and models based on the slope of the volatility smile.

-The model works less well for certain ETFs

Reference:

[2] John Hull and Alan White, Optimal Delta Hedging for Options, Journal of Banking and Finance, Vol. 82, Sept 2017: 180-190

Closing Thoughts

These two papers take opposing approaches: one exploits correlation breakdown, while the other capitalizes on the correlation remaining strong. However, they are not mutually exclusive. Combining insights from both can lead to a more efficient trading or hedging strategy.

Educational Video

This seminar by Prof. J. Hull delves into the second paper discussed above.

Abstract

The “practitioner Black-Scholes delta” for hedging equity options is a delta calculated from the Black-Scholes-Merton model with the volatility parameter set equal to the implied volatility. As has been pointed out by a number of researchers, this delta does not minimize the variance of a trader’s position. This is because there is a negative correlation between equity price movements and implied volatility movements. The minimum variance delta takes account of both the impact of price changes and the impact of the expected change in implied volatility conditional on a price change. In this paper, we use ten years of data on options on stock indices and individual stocks to investigate the relationship between the Black-Scholes delta and the minimum variance delta. Our approach is different from earlier research in that it is empirically-based. It does not require a stochastic volatility model to be specified. Joint work with Allan White.

The Weekend Effect in The Market Indices

The weekend (or Monday) effect in the stock market refers to the phenomenon where stock returns exhibit different patterns on Mondays compared to the rest of the week. Historically, there has been a tendency for stock prices to be lower on Mondays. Various theories attempt to explain the weekend effect, including investor behaviour, news over the weekend, and the impact of events occurring during the weekend on market sentiment.

In this post, we’ll investigate the weekend effect in the market indices using data from Yahoo Finance spanning January 2001 to December 2023. Specifically we choose SPY, which tracks the SP500, and the volatility index, VIX.

Our strategy involves taking a long position in SPY at Friday’s close and exiting the position at Monday’s close, or the next business day’s close if Monday is a holiday. The figure below depicts the cumulative, non-compounded return of the strategy.

Cumulative return of holding SPY over the weekend

From the figure, we observe that holding SPY over the weekend resulted in negative returns during the GFC, Covid pandemic, and the recent 2022 bear market. The overall return is flat-ish, indicating a low reward/risk ratio for holding the SPY over the weekend.

Next, we analyze the change in the VIX index during the weekend. We compute the change in the VIX index from Friday’s close to the close of the next business day and plot the cumulative difference in the figure below. A noticeable upward trend is observed in the cumulative difference. This result indicates that maintaining a long vega/gamma position over the weekend would offer a favourable reward-to-risk trade.

Cumulative difference of the VIX index over the weekend

It’s important to note, however, that investing directly in the spot VIX is not possible. To confirm and capitalize on the weekend effect in the volatility index, one would have to:

Trade a volatility ETN, or
Trade a delta-hedged option position

Each of these approaches introduces additional risk factors, specifically 1- the roll yield and contango, and 2- PnL originating from gamma and theta. These issues will be addressed in the next installment.

Is Asset Dynamics Priced In Correctly by Black-Scholes-Merton Model?

A lot of research has been devoted to answering the question: do options price in the volatility risks correctly? The most noteworthy phenomenon (or bias) is called the volatility risk premium, i.e. options implied volatilities tend to overestimate future realized volatilities. Much less attention is paid, however, to the underlying asset dynamics, i.e. to answering the question: do options price in the asset dynamics correctly?

Note that within the usual BSM framework, the underlying asset is assumed to follow a GBM process. So to answer the above question, it’d be useful to use a different process to model the asset price.

We found an interesting article on this subject [1]. Instead of using GBM, the authors used a process where the asset returns are auto-correlated and then developed a closed-form formula to price the options. Specifically, they assumed that the underlying asset follows an MA(1) process,

volatility trading strategies mean reverting asset

where β represents the impact of past shocks and h is a small constant. We note that and in case β=0 the price dynamics becomes GBM.

After applying some standard pricing techniques, a closed-form option pricing formula is derived which is similar to BSM except that the variance (and volatility) contains the autocorrelation coefficient,

From the above equation, it can be seen that

When the underlying asset is mean reverting, i.e. β<0, which is often the case for equity indices, the MA(1) volatility becomes smaller. Therefore if we use BSM with σ as input for volatility, it will overestimate the option price.
Conversely, when the asset is trending, i.e. β>0, BSM underestimates the option price.
Time to maturity, τ, also affects the degree of over- underpricing. Longer-dated options will be affected more by the autocorrelation factor.

References

[1] Liao, S.L. and Chen, C.C. (2006), Journal of Futures Markets, 26, 85-102.

A Simple System For Hedging Long Portfolios

In this post, we are going to examine a trading system with the goal of using it as a hedge for long equity exposure. To this end, we test a simple, short-only momentum system. The rules are as follows,

Short at the close when Close of today < lowest Close of the last 10 days

Cover at the close when Close of today > lowest Close of the last 10 days

The Table below presents results for SPY from 1993 to the present. We performed the tests for 2 different volatility regimes: low (VIX<=20) and high (VIX>20). Note that we have tested other lookback periods and VIX filters, but obtained qualitatively the same results.

	Number of Trades	Winner	Average trade PnL
All	455	24.8%	-0.30%
VIX<=20	217	23.5%	-0.23%
VIX>20	260	26.5%	-0.37%

It can be seen that the average PnL for all trades is -0.3%, so overall shorting SPY is a losing trade. This is not surprising, since in the short term the SP500 exhibits a strong mean reverting behavior, and in a long term it has a positive drift.

We still expected that when volatility is high, the SP500 would exhibit some momentum characteristics and short selling would be profitable. The result indicates the opposite. When VIX>20, the average trade PnL is -0.37%, which is higher (in absolute value) than the average trade PnL for the lower volatility regime and all trades combined (-0.23% and -0.3% respectively). This result implies that the mean reversion of the SP500 is even stronger when the VIX is high.

The average trade PnL, however, does not tell the whole story. We next look at the maximum favorable excursion (MFE). Table below summarizes the results

	Average	Median	Max
VIX<=20	0.83%	0.44%	10.59%
VIX>20	1.62%	0.73%	24.25%

Despite the fact that the short SPY trade has a negative expectancy, both the average and median MFEs are positive. This means that the short SPY trades often have large unrealized gains before they are exited at the close. Also, as volatility increases, the average, median and largest MFEs all increase. This is consistent with the fact that higher volatility means higher risks.

The above result implies that during a sell-off, a long equity portfolio can suffer a huge drawdown before the market stabilizes and reverts. Therefore, it’s prudent to hedge long equity exposure, especially when volatility is high.

An interesting, related question arises: should we use options or futures to hedge, which one is cheaper? Based on the average trade PnL of -0.37% and gamma rent derived from the lower bound of the VIX, a back of the envelope calculation indicated that hedging using futures appears to be cheaper.

Are Short Out-of-the-Money Put Options Risky? Part 2: Dynamic Case

This post is the continuation of the previous one on the riskiness of OTM vs. ATM short put options and the effect of leverage on the risk measures. In this installment we’re going to perform similar studies with the only exception that from inception until maturity the short options are dynamically hedged. The simulation methodology and parameters are the same as in the previous study.

As a reference, results for the static case are replicated here:

		ATM (K=100)			OTM (K=90)
Margin	Return	Variance	VaR	Return	Variance	VaR
100%	0.0171	0.0075	0.1940	0.0118	0.0031	0.1303
50%	0.0370	0.0292	0.3844	0.0206	0.0133	0.2783
15%	0.1317	0.3155	1.2589	0.0679	0.1502	0.9339

Table below summarizes the results for the dynamically hedged case

		ATM (K=100)			OTM (K=90)
Margin	Return	Variance	VaR	Return	Variance	VaR
100%	-0.0100	1.9171E-05	0.0073	-0.0059	1.4510E-05	0.0062
50%	-0.0199	7.6201E-05	0.0145	-0.0118	5.8016E-05	0.0121
15%	-0.0660	8.7943E-04	0.0480	-0.0400	6.5201E-04	0.0424

From the Table above, we observe that:

Similar to the static case, delta-hedged OTM put options are less risky than the ATM counterparts. However, the reduction in risk is less significant. This is probably due to the fact that delta hedging itself already reduces the risks considerably (see below).
Leverage also increases risks.

It is important to note that given the same notional amount, a delta-hedged position is less risky than a static position. For example, the VaR of a static, cash-secured (m=100%) short put position is 0.194, while the VaR of the corresponding dynamically-hedged position is only 0.0073. This explains why proprietary trading firms and hedge funds often engage in the practice of dynamic hedging.

Finally, we note that while Value at Risk takes into account the tail risks to some degree, it’s probably not the best measure of tail risks. Using other risk measures that better incorporate the tail risks can alter the results and lead to different conclusions.