Using powerful technique of stochastic time change, we introduce a new two-factor commodity price model, where one of the fundamental factors is the activity rate. This factor implicitly introduces stochastic volatility into the model. The model is developed under both physical and risk neutral probability measures, which allows for a wide range of applications ranging from derivatives pricing to risk management. We derive forward prices and forward curve evolution within the model's framework and develop an ingenious calibration procedure, which allows us to filter out the activity rate from daily observed price data. We apply the model to the rich dataset of daily crude oil and natural gas spot and futures prices and demonstrate its versatility and excellent fit to the historical forward curves.
{"title":"Commodity Forward Curves With Stochastic Time Change","authors":"S. Ladokhin, M. Schmeck, S. Borovkova","doi":"10.2139/ssrn.3871680","DOIUrl":"https://doi.org/10.2139/ssrn.3871680","url":null,"abstract":"Using powerful technique of stochastic time change, we introduce a new two-factor commodity price model, where one of the fundamental factors is the activity rate. This factor implicitly introduces stochastic volatility into the model. The model is developed under both physical and risk neutral probability measures, which allows for a wide range of applications ranging from derivatives pricing to risk management. We derive forward prices and forward curve evolution within the model's framework and develop an ingenious calibration procedure, which allows us to filter out the activity rate from daily observed price data. We apply the model to the rich dataset of daily crude oil and natural gas spot and futures prices and demonstrate its versatility and excellent fit to the historical forward curves.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114386393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Although negative interest rates have been a phenomenon observed in capital markets for years, little research has been done on the impact of negative interest rates on stock option valuations. This paper shows that the fundamental assumption of equivalence between American and European call options at negative riskless interest rates is no longer universal. The findings are illustrated by means of an example. Furthermore, there are implications for practical trading with options.
{"title":"Early Exercise of American Call Options under Negative Interest Rates","authors":"Jochen Schneider, Nils Helms","doi":"10.2139/ssrn.3854489","DOIUrl":"https://doi.org/10.2139/ssrn.3854489","url":null,"abstract":"Although negative interest rates have been a phenomenon observed in capital markets for years, little research has been done on the impact of negative interest rates on stock option valuations. This paper shows that the fundamental assumption of equivalence between American and European call options at negative riskless interest rates is no longer universal. The findings are illustrated by means of an example. Furthermore, there are implications for practical trading with options.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121571202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In an empirical study of Standard & Poor's 500 index options, this paper analyses the predictability of future market excess returns by means of decomposed higher-moment risk premiums. The study proposes a new measure of kurtosis risk premium and suggests a decomposition of higher-moment risk premiums up to the fourth moment into downside and upside premiums. Thereby, the paper enhances the understanding of higher-moment risk premiums. The decomposition uncovers valuable information for return forecasts, as decomposed higher-moment risk premiums deliver improved in-sample predictions. In an out-of-sample study, the predictive power of decomposed higher-moment risk premiums is shown to be particularly driven by downside higher-moment risk premiums.
{"title":"Decomposed Higher-Moment Risk Premiums and Market Return Predictability","authors":"Julian Dörries","doi":"10.2139/ssrn.3784496","DOIUrl":"https://doi.org/10.2139/ssrn.3784496","url":null,"abstract":"In an empirical study of Standard & Poor's 500 index options, this paper analyses the predictability of future market excess returns by means of decomposed higher-moment risk premiums. The study proposes a new measure of kurtosis risk premium and suggests a decomposition of higher-moment risk premiums up to the fourth moment into downside and upside premiums. Thereby, the paper enhances the understanding of higher-moment risk premiums. The decomposition uncovers valuable information for return forecasts, as decomposed higher-moment risk premiums deliver improved in-sample predictions. In an out-of-sample study, the predictive power of decomposed higher-moment risk premiums is shown to be particularly driven by downside higher-moment risk premiums.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122688193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Classic option pricing theory values a derivative contract via dynamic replication, and views the derivative as redundant relative to the replicating portfolio. In practice, while dynamic replication proves to be highly effective in drastically reducing the risks in derivative investments, the remaining risks can still be large and significant due to practical limits of arbitrage. Because of these limits, derivative securities can play primary roles in risk allocation and investors can demand risk premiums for taking these primary risks. This paper documents the effectiveness of delta hedging on U.S. stock options under practical situations, examines the cross-sectional and intertemporal variation of investment returns from writing options on different stocks, and attributes the return variation to variations in primary risk exposures in the delta-hedged option investments.
{"title":"Limits of Arbitrage and Primary Risk Taking in Derivative Securities","authors":"Meng Tian, Liuren Wu","doi":"10.2139/ssrn.3779350","DOIUrl":"https://doi.org/10.2139/ssrn.3779350","url":null,"abstract":"Classic option pricing theory values a derivative contract via dynamic replication, and views the derivative as redundant relative to the replicating portfolio. In practice, while dynamic replication proves to be highly effective in drastically reducing the risks in derivative investments, the remaining risks can still be large and significant due to practical limits of arbitrage. Because of these limits, derivative securities can play primary roles in risk allocation and investors can demand risk premiums for taking these primary risks. This paper documents the effectiveness of delta hedging on U.S. stock options under practical situations, examines the cross-sectional and intertemporal variation of investment returns from writing options on different stocks, and attributes the return variation to variations in primary risk exposures in the delta-hedged option investments.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126845333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Since the collapse of the Metallgesellschaft AG due to hedging losses in 1993, energy practitioners have been concerned with the ability to hedge long-dated linear and non-linear oil liabilities with short-dated futures and options. This paper identifies a model-free non-parametric approach to extrapolating futures prices and implied volatilities. When we expand the analysis to implementing hedge portfolios for long-dated futures or option contracts over the time period 2007–2017, we utilize the useful benchmark of hedge ratios arising from Schwartz and Smith. With respect to the empirical consequences of hedging long-dated futures and options with their short-dated counterparts, we find that the long-term tracking errors are, on average, quite close to zero, but there is increasing risk entailed in attempting to do so, as the hedge-tracking errors for both futures and option contracts increase with time-to-maturity.
{"title":"Hedging Long-Dated Oil Futures and Options Using Short-Dated Securities — Revisiting Metallgesellschaft","authors":"James S. Doran, Ehud I. Ronn","doi":"10.2139/ssrn.3773257","DOIUrl":"https://doi.org/10.2139/ssrn.3773257","url":null,"abstract":"Since the collapse of the Metallgesellschaft AG due to hedging losses in 1993, energy practitioners have been concerned with the ability to hedge long-dated linear and non-linear oil liabilities with short-dated futures and options. This paper identifies a model-free non-parametric approach to extrapolating futures prices and implied volatilities. When we expand the analysis to implementing hedge portfolios for long-dated futures or option contracts over the time period 2007–2017, we utilize the useful benchmark of hedge ratios arising from Schwartz and Smith. With respect to the empirical consequences of hedging long-dated futures and options with their short-dated counterparts, we find that the long-term tracking errors are, on average, quite close to zero, but there is increasing risk entailed in attempting to do so, as the hedge-tracking errors for both futures and option contracts increase with time-to-maturity.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133697836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper compares two approaches to options: (1) Risk-Aware Approach, and (2) Risk-Neutral Approach. The risk-aware approach requires a probabilistic specification of the underlying’s returns, addressing higher than second moments, as hedging errors are singularly dependent on the excess kurtosis of the returns. Becoming risk-aware requires explicitly assessing hedge slippage of a hedging strategy to attempt option replication. In contrast, the risk-neutral tautology sets the option price equal to an expectation of option payoff under a risk-neutral probability that is inferred from option prices and under which the asset does not expect to accrete/deplete wealth. In the presence of irreducible risks, while a risk-neutral probability measure may be fit to observed option prices, it does not inform about the partitioning between expected attempted replication costs and compensation for irreducible risks. In segmented option markets with distinct risk premiums such a risk-neutral probability measure fails to exist.
{"title":"Irreducible Risks: Fallacy of Risk-Neutral Approach to Options","authors":"M. Sundberg, Jake Freeman, V. Kapoor","doi":"10.2139/ssrn.3761304","DOIUrl":"https://doi.org/10.2139/ssrn.3761304","url":null,"abstract":"This paper compares two approaches to options: (1) Risk-Aware Approach, and (2) Risk-Neutral Approach. The risk-aware approach requires a probabilistic specification of the underlying’s returns, addressing higher than second moments, as hedging errors are singularly dependent on the excess kurtosis of the returns. Becoming risk-aware requires explicitly assessing hedge slippage of a hedging strategy to attempt option replication. In contrast, the risk-neutral tautology sets the option price equal to an expectation of option payoff under a risk-neutral probability that is inferred from option prices and under which the asset does not expect to accrete/deplete wealth. In the presence of irreducible risks, while a risk-neutral probability measure may be fit to observed option prices, it does not inform about the partitioning between expected attempted replication costs and compensation for irreducible risks. In segmented option markets with distinct risk premiums such a risk-neutral probability measure fails to exist.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"158 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125781809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper provides a characterization of the dynamic interactions in the VIX complex, composed of the VIX itself, the term structure of VIX futures, and VIX ETPs. I investigate a model that summarizes the VIX futures term structure using latent factors (level, slope, and curvature) and expand it with the VIX and VIX futures demand stemming from VIX ETPs. I find evidence of VIX ETPs impacting the VIX futures term structure, but no evidence of any impacts on the VIX.
{"title":"Dynamics in the VIX Complex","authors":"Anders Merrild Posselt","doi":"10.2139/ssrn.3723728","DOIUrl":"https://doi.org/10.2139/ssrn.3723728","url":null,"abstract":"This paper provides a characterization of the dynamic interactions in the VIX complex, composed of the VIX itself, the term structure of VIX futures, and VIX ETPs. I investigate a model that summarizes the VIX futures term structure using latent factors (level, slope, and curvature) and expand it with the VIX and VIX futures demand stemming from VIX ETPs. I find evidence of VIX ETPs impacting the VIX futures term structure, but no evidence of any impacts on the VIX.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121829903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By drawing on a nested logit model that unifies the conditional logit model and the Poisson regression, Brülhart and Schmidheiny (2015) have developed a method to estimate the degree of rivalness between options by a factor rho. This paper suggests that the Brülhart and Schmidheiny (2015) method involves a parameter normalisation of the nested logit model that is not choice consistent. Aside from being unsatisfactory from a theoretical point of view, this also implies that the estimated rivalness factor (rho) is time constant. A remedy for these caveats using the choice consistent normalisation of Herger and McCorriston (2013) is provided.
{"title":"Estimating Rivalness: The Role of Choice Consistent Parameter Normalisation","authors":"Nils Herger","doi":"10.2139/ssrn.3753331","DOIUrl":"https://doi.org/10.2139/ssrn.3753331","url":null,"abstract":"By drawing on a nested logit model that unifies the conditional logit model and the Poisson regression, Brülhart and Schmidheiny (2015) have developed a method to estimate the degree of rivalness between options by a factor rho. This paper suggests that the Brülhart and Schmidheiny (2015) method involves a parameter normalisation of the nested logit model that is not choice consistent. Aside from being unsatisfactory from a theoretical point of view, this also implies that the estimated rivalness factor (rho) is time constant. A remedy for these caveats using the choice consistent normalisation of Herger and McCorriston (2013) is provided.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122575491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze the VIX futures market with a focus on the exchange-traded notes written on such contracts, in particular, we investigate the VXX notes tracking the short-end part of the futures term structure. Inspired by recent developments in commodity smile modeling, we present a multi-factor stochastic local-volatility model able to jointly calibrate plain vanilla options both on VIX futures and VXX notes. We discuss numerical results on real market data by highlighting the impact of model parameters on implied volatilities.
{"title":"A General Framework for a Joint Calibration of VIX and VXX Options","authors":"M. Grasselli, Andrea Mazzoran, A. Pallavicini","doi":"10.2139/ssrn.3749995","DOIUrl":"https://doi.org/10.2139/ssrn.3749995","url":null,"abstract":"We analyze the VIX futures market with a focus on the exchange-traded notes written on such contracts, in particular, we investigate the VXX notes tracking the short-end part of the futures term structure. Inspired by recent developments in commodity smile modeling, we present a multi-factor stochastic local-volatility model able to jointly calibrate plain vanilla options both on VIX futures and VXX notes. We discuss numerical results on real market data by highlighting the impact of model parameters on implied volatilities.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116768907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Alternative Reference Rates Committee (ARRC) has set July 2021 the goal for creating a forward-looking indicative term SOFR. In this paper we present paradoxes that will result from publishing the indicative term SOFR: complexity versus transparency of the methodology, the true risk-free rate, which bears no market, credit or operational risk, versus a market driven rate, the hedging inefficiency between cash market versus derivative market and the outcome of rising systematic risk. In light of the paradoxes, we believe that the indicative term SOFR does not possess the same economic justification as Libor, nor will it provide the necessary incentives for trading. The following sections will discuss methodology for publishing indicative term SOFR, followed by detailed discussion of the paradoxes. It is our view that these conceptual paradoxes of forward-looking term SOFR give rise to significant drawbacks in the applications, thus posing significant risk for the Libor Transition.
{"title":"Forward-looking Forward Rates: The Term SOFR Paradoxes","authors":"Xi (Figo) Liu, Yu Bai","doi":"10.2139/ssrn.3747159","DOIUrl":"https://doi.org/10.2139/ssrn.3747159","url":null,"abstract":"The Alternative Reference Rates Committee (ARRC) has set July 2021 the goal for creating a forward-looking indicative term SOFR. In this paper we present paradoxes that will result from publishing the indicative term SOFR: complexity versus transparency of the methodology, the true risk-free rate, which bears no market, credit or operational risk, versus a market driven rate, the hedging inefficiency between cash market versus derivative market and the outcome of rising systematic risk. In light of the paradoxes, we believe that the indicative term SOFR does not possess the same economic justification as Libor, nor will it provide the necessary incentives for trading. The following sections will discuss methodology for publishing indicative term SOFR, followed by detailed discussion of the paradoxes. It is our view that these conceptual paradoxes of forward-looking term SOFR give rise to significant drawbacks in the applications, thus posing significant risk for the Libor Transition.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133732471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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