Default Risk and Option Returns

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Xiao Xiao examines how default risk affects stock option returns. She demonstrates a cross-sectional and time-series relationship between default risk and option returns.

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Xiao Xiao examines how default risk affects stock option returns. She demonstrates a cross-sectional and time-series relationship between default risk and option returns.

Image courtesy of interviewee. January 19, 2024

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This video video discusses significant growth in the US equity options market over the past five years, particularly among retail investors, due in part to the availability of zero-commission trading platforms. The author aims to investigate the expected returns of holding options and the risk factors influencing these returns, noting a gap in existing research regarding the risk-based determinants of option returns.

Early studies highlighted the relationship between options pricing and default risk, but there remains limited understanding of how default risk affects option returns. The author proposes a theoretical model—a compound option model with jump risk—to analyze these relationships, emphasizing “delta hedged” option returns, which isolate the return primarily influenced by volatility rather than the underlying stock's price movements.

Key findings from the theoretical model indicate that higher leverage and asset variance lead to lower expected returns on delta hedged options. Empirical testing using data from 1996 to 2016 supports these findings, revealing a negative correlation between default risk and delta hedged option returns. Results from both cross-sectional and time series analyses confirm that as default risk increases, option returns tend to decrease, and this trend is consistent around credit rating changes.

The findings also connect default risk with various anomalies in the options market, suggesting that these anomalies may be better understood in the context of default risk. Overall, investors seem willing to pay a premium for options in high-default-risk firms, accepting lower expected returns in exchange for the hedging benefits against volatility and default risk. Ultimately, the research enhances understanding of the determinants of equity option returns, highlighting the critical role of firm-level risks.

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So we have seen, a huge growth in terms

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of the trading volume in the US

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equity option market in the past five years, right?

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So we have seen retail investors,

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retail option traders are more

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and more interested in this market

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because now we have some trading platform with zero,

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commission fee.

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So this market has become a more

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and more important market, in general, right?

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So we're interested in, for example,

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if we hold a option, how much,

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expected return, we expect

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from holding this kind of position.

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So, this could be considered as our motivation.

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And, before our study, there are not so much,

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studies on the risk,

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risk based determinants of option returns.

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There are some early research in 1980 and 1990.

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So for example, there are two papers, Johnson

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and Stu, 90 87, Holland White, 1995,

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very early research on the pricing of options

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with, default risk.

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So they show that the pricing of option,

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depends on the level of default risk,

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of the underlying firm.

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And later on, there are also recent studies, on

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how credit, credit spread and option prices are related.

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So, for example, grammar

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and 2008, they try to explain the level

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of credit spread using option implied jump risk premium

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in a firm value model,

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and then another paper 2018 by cop zawa.

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And, they,

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they constructed the option based credit spread.

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So it seems option price and the credit, spread

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or default risk, is closely related, right?

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Um, but we know very little about the relationship

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between default risk and option returns.

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So that's why we decided to study on this topic.

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We have a theoretical model,

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and we also have, protocol evidence.

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Um, so the model we consider compound option model

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with jump risk.

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So what is compound option model?

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Basically, we have two layers of optionality.

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So the first layer of optionality comes from the,

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capital structure of the firm.

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So the equity of the firm is a

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tion on firm's assets.

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And the second layer of optionality

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Is that, so

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because we considered equity options,

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so equity option is an option on firms equity.

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So there are two levels of optionality in the,

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and in, in this model.

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And in this model, we derive the,

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expected return of a delta hatched, option position.

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So maybe I should explain a little bit,

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why we're consider the delta hatched option return, right?

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Um, so option return, is like stock return.

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So, it's a return we hold,

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we're holding a security, right?

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Um, but when we're considering option return,

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we're not interested, interested in the part

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that can be explained by stock return.

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So that's why we would like to remove the exposure

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of the option to the underlying stock.

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So that's why we do the data hedging every day

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to remove the exposure, to the stock price movement.

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And then the clean part is the data hedge option return,

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which is mainly sensitive to the volatility movement.

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So that's a special part of, stock, option return.

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Here, we, we may also contract the difference

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between stock return and, option return.

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So stock return mainly reflects the equity risk premium,

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which is normally positive.

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So we expect to earn something from holding the

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stock, right?

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But for option return,

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it reflect something called the variance risk premium.

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So the variance is the second moment

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of the stock distribution.

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And the variance risk premium is in general,

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on average negative.

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So this is, a quite a, quite, counterintuitive

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in the sense that we actually lose money

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by holding this kind of contract on average.

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But why, do we do that?

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This is because holding a debt hedge adoption

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position can provide a hedge against the

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volatility movement.

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So when the volatility increases during the crisis,

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crisis or market crash,

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or when the firm

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firms default risk becomes higher and higher.

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So during this kind of negative situations,

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this delta hedge,

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the option position could provide the investor

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a positive return.

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So that's why, the investors, they prefer this kind

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of, this kind of position

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as a hedge against volatility, risk.

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So this can be a hedge of their portfolio.

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Um, they're willing to pay a high price

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and they're willing to accept

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A low and negative return.

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So that's the difference between stock return

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and option return.

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And if we go back to our theoretical model, we derive

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this, expected return of the other hedge,

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the option position,

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and we find that, there are several,

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firm level determinants.

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So basically it's related to the variance with the premium

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of firm's asset,

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and it's also related to

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firm's equity elasticity.

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So what is the firm's equity elasticity?

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Uh, if the firm, if the firm's asset value increases

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by 1%, how much

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the firm's equity value increases?

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So it's like, leverage is related to leverage.

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The higher the leverage, the higher the elasticity,

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the higher the asset variance, also the high, higher the,

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elasticity.

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Here, we, we, we find two firm level determinants,

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firm, leverage and, asset variance.

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So from the theoretical part, we, we, we find,

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based on some reasonable assumptions,

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the higher the leverage, the lower, like slightly return

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of delta hedge doctrine position.

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And the higher the asset varies also, the lower the,

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the lower the, the expected return.

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We also link it to the default risk

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or default probability of the firm.

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So from the simulations, we find,

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the default probability

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and the expected return of a data hedge position,

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it's negative.

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So they have the negative relationship.

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This is, main result from the theoretical part.

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And then we, we try to, test this hypothesis

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using the data, from 1996

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to 2016, in the United States.

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So we have several parts of result,

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to study the relationship between the two.

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We have cross-sectional analysis,

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we also have time series analysis.

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So in terms of cross-section,

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we do a single sort.

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So basically it would sort all of the firms

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by their credit rating

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or by their default probability.

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So credit rating comes from sp, s

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and p and, default probability we calculated

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by ourself based, based on, merchant type

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of capital structure model.

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So we have a formula, to calculate this

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Probability, and we sort all the firms

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by these two measures of prob, default risk measures.

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Uh, and then, we find the higher the default risk,

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the lower the data hedge adoption return

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and the long short, return

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and long, long short return is about, minus

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0.7% per month.

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And the level is similar for the two measures.

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So this is the first, crosssectional analysis.

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And in the regression with more control variables, we find,

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similar results.

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So basically, this rela relationship cannot be explained

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by other form characteristics.

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So this is a cross-section part

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and, in the, time service part.

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Um, so intuitively,

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if the firm has a higher default probability

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in the future over time, then we would expect this firm

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to have a lower option return in future, right?

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So, we consider event, which is credit rating,

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downgrade, and, credit rating upgrade.

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So we look at the option return before and after downgrade

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and the upgrade.

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So we find after the credit rating downgrade.

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Um, so the data hedge,

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the option return significantly decrease,

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and for the upgrade, the option return increase.

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So that's consistent with our hypothesis.

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Uh, with our theoretical model, we find a negative relation

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between default risk and the delta hedge option return.

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So both theoretically and, and empirically.

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Um, so and empirically we

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find the supportive evidence in the time series using

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portfolio sorting and also in the time series when

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after credit rating downgrades and upgrades.

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And, besides that, we also find,

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the relationship between, default risk

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and other anomalies in the option market.

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So in the literature, people find,

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some other anomalies, for example, they find

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option return is related to firm size, momentum

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and other characteristics.

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So we consider 10 other anomalies in the option market,

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and we find that these anomalies are closely

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related to the do,

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The relation we study in this paper.

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So what we find is, the,

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the other anomalies are the strongest

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in the high default risk group.

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So five out of the 10 anomalies only exist

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in the high default risk group.

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So it means, other anomalies,

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in adoption market may be potentially explained

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by our rational explanation.

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So there should be, rational relationship

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between default risk and option return.

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After considering this relationship, we may explain

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a quite large portion of other anoma in the market.

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So that's another part of our analysis.

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So in general,

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the intuition is that investors are willing

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to pay more

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to buy the options on the high default risk companies,

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and they're willing to accept a very low return on

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this options because holding this position could provide

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a hedge against, volatility risk and, default risk.

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So that's why we're observing this,

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negative relationship.

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So overall, the results in this paper

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help us understand the, determinants

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firm level determinants of equity option return.

Growth of US Equity Options Market

00:00:00 - So we have seen, a huge growth in terms of the trading volume in the US equity option market in the past five years, right? So we have seen retail investors, retail option traders are more and more interested in this market because now we have some trading platform with zero, commission fee. So this market has become a more and more important market, in general, right? So we're interested in, for example,
00:00:38 - expected return, we expect from holding this kind of position. So, this could be considered as our motivation. And, before our study, there are not so much, studies on the risk, risk based determinants of option returns. There are some early research in 1980 and 1990.
00:01:10 - and Stu, 90 87, Holland White, 1995, very early research on the pricing of options with, default risk. So they show that the pricing of option, depends on the level of default risk, of the underlying firm. And later on, there are also recent studies, on how credit, credit spread and option prices are related.
00:01:43 - and 2008, they try to explain the level of credit spread using option implied jump risk premium in a firm value model, and then another paper 2018 by cop zawa. And, they, they constructed the option based credit spread. So it seems option price and the credit, spread
00:02:17 - but we know very little about the relationship between default risk and option returns. So that's why we decided to study on this topic. We have a theoretical model, and we also have, protocol evidence. so the model we consider compound option model with jump risk. So what is compound option model? Basically, we have two layers of optionality.
00:02:48 - capital structure of the firm. So the equity of the firm is a tion on firm's assets. And the second layer of optionality Is that, so because we considered equity options, so equity option is an option on firms equity. So there are two levels of optionality in the, and in, in this model.

Understanding Delta Hedged Option Returns

00:03:19 - expected return of a delta hatched, option position. So maybe I should explain a little bit, why we're consider the delta hatched option return, right? so option return, is like stock return. So, it's a return we hold, we're holding a security, right? but when we're considering option return, we're not interested, interested in the part
00:03:50 - So that's why we would like to remove the exposure of the option to the underlying stock. So that's why we do the data hedging every day to remove the exposure, to the stock price movement. And then the clean part is the data hedge option return, which is mainly sensitive to the volatility movement. So that's a special part of, stock, option return.
00:04:21 - between stock return and, option return. So stock return mainly reflects the equity risk premium, which is normally positive. So we expect to earn something from holding the stock, right? But for option return, it reflect something called the variance risk premium. So the variance is the second moment of the stock distribution. And the variance risk premium is in general,
00:04:52 - So this is, a quite a, quite, counterintuitive in the sense that we actually lose money by holding this kind of contract on average. But why, do we do that? This is because holding a debt hedge adoption position can provide a hedge against the volatility movement. So when the volatility increases during the crisis, crisis or market crash,
00:05:25 - firms default risk becomes higher and higher. So during this kind of negative situations, this delta hedge, the option position could provide the investor a positive return. So that's why, the investors, they prefer this kind of, this kind of position as a hedge against volatility, risk. So this can be a hedge of their portfolio. they're willing to pay a high price
00:05:57 - A low and negative return. So that's the difference between stock return and option return. And if we go back to our theoretical model, we derive this, expected return of the other hedge, the option position, and we find that, there are several, firm level determinants. So basically it's related to the variance with the premium of firm's asset,

Exploring Firm Leverage and Equity Elasticity

00:06:29 - firm's equity elasticity. So what is the firm's equity elasticity? if the firm, if the firm's asset value increases by 1%, how much the firm's equity value increases? So it's like, leverage is related to leverage. The higher the leverage, the higher the elasticity, the higher the asset variance, also the high, higher the,
00:07:00 - Here, we, we, we find two firm level determinants, firm, leverage and, asset variance. So from the theoretical part, we, we, we find, based on some reasonable assumptions, the higher the leverage, the lower, like slightly return of delta hedge doctrine position. And the higher the asset varies also, the lower the,
00:07:32 - We also link it to the default risk or default probability of the firm. So from the simulations, we find, the default probability and the expected return of a data hedge position, it's negative. So they have the negative relationship. This is, main result from the theoretical part. And then we, we try to, test this hypothesis
00:08:04 - to 2016, in the United States. So we have several parts of result, to study the relationship between the two. We have cross-sectional analysis, we also have time series analysis. So in terms of cross-section, we do a single sort. So basically it would sort all of the firms by their credit rating
00:08:35 - So credit rating comes from sp, s and p and, default probability we calculated by ourself based, based on, merchant type of capital structure model. So we have a formula, to calculate this Probability, and we sort all the firms by these two measures of prob, default risk measures. and then, we find the higher the default risk,
00:09:11 - and the long short, return and long, long short return is about, minus 0.7% per month. And the level is similar for the two measures. So this is the first, crosssectional analysis. And in the regression with more control variables, we find, similar results.

Impact of Default Risk on Option Returns

00:09:41 - by other form characteristics. So this is a cross-section part and, in the, time service part. so intuitively, if the firm has a higher default probability in the future over time, then we would expect this firm to have a lower option return in future, right?
00:10:16 - downgrade, and, credit rating upgrade. So we look at the option return before and after downgrade and the upgrade. So we find after the credit rating downgrade. so the data hedge, the option return significantly decrease, and for the upgrade, the option return increase.
00:10:48 - with our theoretical model, we find a negative relation between default risk and the delta hedge option return. So both theoretically and, and empirically. so and empirically we find the supportive evidence in the time series using portfolio sorting and also in the time series when after credit rating downgrades and upgrades.
00:11:23 - the relationship between, default risk and other anomalies in the option market. So in the literature, people find, some other anomalies, for example, they find option return is related to firm size, momentum and other characteristics. So we consider 10 other anomalies in the option market,
00:11:51 - related to the do, The relation we study in this paper. So what we find is, the, the other anomalies are the strongest in the high default risk group. So five out of the 10 anomalies only exist in the high default risk group. So it means, other anomalies, in adoption market may be potentially explained
00:12:25 - So there should be, rational relationship between default risk and option return. After considering this relationship, we may explain a quite large portion of other anoma in the market. So that's another part of our analysis. So in general, the intuition is that investors are willing

Impact of Default Risk on Option Returns

00:12:55 - to buy the options on the high default risk companies, and they're willing to accept a very low return on this options because holding this position could provide a hedge against, volatility risk and, default risk. So that's why we're observing this, negative relationship. So overall, the results in this paper
00:13:28 - firm level determinants of equity option return.

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