Internal-rate-of-return (IRR) settled swaptions are the main interest rate volatility instruments in the European interest rate markets. Industry practice is to use an approximation formula to price IRR swaptions based on Black model, which is not arbitrage-free. We formulate a unified market model to incorporate both swaptions and constant maturity swaps (CMS) pricing under a single, self-consistent framework. We demonstrate that the model is able to calibrate to market quotes well, and is also able to efficiently price both IRR-settled and swap-settled swaptions, along with CMS products. We use the model to illustrate the difference in implied volatilities for IRR-settled payer and receiver swaptions, the pricing of zero-wide collars and in-the-money (ITM) swaptions, the implication on put-call parity, and the issue of negative vega. These findings offer important insights to the ongoing reform in the European swaption market.
{"title":"A Unified Market Model for Swaptions and Constant Maturity Swaps","authors":"C. W. Tee, Jeroen Kerkhof","doi":"10.2139/ssrn.3441544","DOIUrl":"https://doi.org/10.2139/ssrn.3441544","url":null,"abstract":"Internal-rate-of-return (IRR) settled swaptions are the main interest rate volatility instruments in the European interest rate markets. Industry practice is to use an approximation formula to price IRR swaptions based on Black model, which is not arbitrage-free. We formulate a unified market model to incorporate both swaptions and constant maturity swaps (CMS) pricing under a single, self-consistent framework. We demonstrate that the model is able to calibrate to market quotes well, and is also able to efficiently price both IRR-settled and swap-settled swaptions, along with CMS products. We use the model to illustrate the difference in implied volatilities for IRR-settled payer and receiver swaptions, the pricing of zero-wide collars and in-the-money (ITM) swaptions, the implication on put-call parity, and the issue of negative vega. These findings offer important insights to the ongoing reform in the European swaption market.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129845861","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}
Benjamin Hood, J. Huss, R. Israelov, Matthew J. Klein
There have been increasingly frequent claims that risk parity strategies are hiding an implicit short volatility exposure or behave as though they are short volatility. In order to test the veracity of these claims, we simulate stylized versions of three-asset-class (equity, fixed income, and commodities) risk parity and short volatility strategies, and we compare the trading behavior and returns of each. We conclude that the two strategies’ similarities are overstated, and we find no empirical evidence to support the claimed hidden exposure. Even with conservative assumptions designed to heighten the similarity of the two strategies, their trades are uncorrelated (or even slightly negative correlated) at almost any horizon. Though their returns are moderately correlated, the correlation is explained by common exposure to equities and bonds, not by common exposure to gamma or other forms of convexity. Controlling for these static underlying exposures, we find that the returns of the two strategies are almost orthogonal, with short volatility explaining less than one percent of the total variance of risk parity returns. We extend our analysis to consider equity and fixed income asset classes in isolation, where we observe very similar results.
{"title":"Risk Parity is Not Short Volatility (Not That There's Anything Wrong with Short Volatility)","authors":"Benjamin Hood, J. Huss, R. Israelov, Matthew J. Klein","doi":"10.2139/ssrn.3432438","DOIUrl":"https://doi.org/10.2139/ssrn.3432438","url":null,"abstract":"There have been increasingly frequent claims that risk parity strategies are hiding an implicit short volatility exposure or behave as though they are short volatility. In order to test the veracity of these claims, we simulate stylized versions of three-asset-class (equity, fixed income, and commodities) risk parity and short volatility strategies, and we compare the trading behavior and returns of each. We conclude that the two strategies’ similarities are overstated, and we find no empirical evidence to support the claimed hidden exposure. Even with conservative assumptions designed to heighten the similarity of the two strategies, their trades are uncorrelated (or even slightly negative correlated) at almost any horizon. Though their returns are moderately correlated, the correlation is explained by common exposure to equities and bonds, not by common exposure to gamma or other forms of convexity. Controlling for these static underlying exposures, we find that the returns of the two strategies are almost orthogonal, with short volatility explaining less than one percent of the total variance of risk parity returns. We extend our analysis to consider equity and fixed income asset classes in isolation, where we observe very similar results.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116687621","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}
Using short-time expansion techniques, we obtain analytic implied volatilities for European and forward starting options for a family of stochastic volatility models with arbitrary local volatility component and time dependent (piecewise constant) parameters. The formulas can be used to efficiently calibrate the model to European options at two expiries and to calculate the spanning forward starting option price.
{"title":"On the Forward Smile","authors":"T. Roos","doi":"10.2139/ssrn.3429050","DOIUrl":"https://doi.org/10.2139/ssrn.3429050","url":null,"abstract":"Using short-time expansion techniques, we obtain analytic implied volatilities for European and forward starting options for a family of stochastic volatility models with arbitrary local volatility component and time dependent (piecewise constant) parameters. The formulas can be used to efficiently calibrate the model to European options at two expiries and to calculate the spanning forward starting option price.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121521489","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 propose a new parsimonious model for valuating portfolio credit derivatives dependent on aggregate loss. The starting point is the loss distribution, which is constructed to be time dependent. We let the loss be beta distributed, and, by implication, the loss process becomes a stochastic jump process, where a jump corresponds to losses appearing simultaneously. The model matches empirical loss data well with only two parameters in addition to expected loss. The size of the jump is controlled by the clustering parameter, and the temporal correlation of jumps is controlled by the autocorrelation parameter. The full model is relatively efficient to implement, as we use a Monte Carlo at portfolio level. We derive analytical expressions for valuating tranches and for calculating regulatory capital. We provide examples of credit default swap index tranche pricing, including forward starting tranches. Comparisons are made with the one-factor Gaussian copula default time model, which fits historical loss data badly and has a deficient loss volatility term structure.
{"title":"An Efficient Portfolio Loss Model","authors":"C. Fenger","doi":"10.21314/JCR.2019.252","DOIUrl":"https://doi.org/10.21314/JCR.2019.252","url":null,"abstract":"We propose a new parsimonious model for valuating portfolio credit derivatives dependent on aggregate loss. The starting point is the loss distribution, which is constructed to be time dependent. We let the loss be beta distributed, and, by implication, the loss process becomes a stochastic jump process, where a jump corresponds to losses appearing simultaneously. The model matches empirical loss data well with only two parameters in addition to expected loss. The size of the jump is controlled by the clustering parameter, and the temporal correlation of jumps is controlled by the autocorrelation parameter. The full model is relatively efficient to implement, as we use a Monte Carlo at portfolio level. We derive analytical expressions for valuating tranches and for calculating regulatory capital. We provide examples of credit default swap index tranche pricing, including forward starting tranches. Comparisons are made with the one-factor Gaussian copula default time model, which fits historical loss data badly and has a deficient loss volatility term structure.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129564465","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}
Karol Binkowski, Peilun He, N. Kordzakhia, P. Shevchenko
{"title":"On the Parameter Estimation in the Schwartz-Smith’s Two-Factor Model","authors":"Karol Binkowski, Peilun He, N. Kordzakhia, P. Shevchenko","doi":"10.1007/978-9","DOIUrl":"https://doi.org/10.1007/978-9","url":null,"abstract":"","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116293622","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 examines an alternative avenue through which trading in options can expand investors' opportunity sets, unrelated to private information, differing opinions, endowments, or trading restrictions in the stock market. Investors can synthetically replicate the return profile of optionable stocks using options for a fraction of the cost of holding the underlying securities, which makes diversification more cost-efficient. We find that the option to stock volume ratio increases when stock price, idiosyncratic risk, stock illiquidity, borrowing cost, and market risk aversion are high. In addition, institutional holdings and option trading have a U-shaped relation.
{"title":"Are Options Redundant? The Benefits of Synthetic Diversification","authors":"Yifan Liu, Louis R. Piccotti","doi":"10.2139/ssrn.3392421","DOIUrl":"https://doi.org/10.2139/ssrn.3392421","url":null,"abstract":"This paper examines an alternative avenue through which trading in options can expand investors' opportunity sets, unrelated to private information, differing opinions, endowments, or trading restrictions in the stock market. Investors can synthetically replicate the return profile of optionable stocks using options for a fraction of the cost of holding the underlying securities, which makes diversification more cost-efficient. We find that the option to stock volume ratio increases when stock price, idiosyncratic risk, stock illiquidity, borrowing cost, and market risk aversion are high. In addition, institutional holdings and option trading have a U-shaped relation.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129592779","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 note is an answer to the consultation published by ISDA in May 2019 regarding the amendment of documentation to implement fallbacks for certain key IBORs.
本说明是对ISDA于2019年5月发布的关于修改文件以实施某些关键ibor的回退的咨询意见的回答。
{"title":"Answer to 'Supplemental Consultation on Spread and Term Adjustments for Fallbacks in Derivatives Referencing USD LIBOR, CDOR and HIBOR and Certain Aspects of Fallbacks for Derivatives Referencing SOR' issued by ISDA","authors":"Marc Henrard","doi":"10.2139/ssrn.3415930","DOIUrl":"https://doi.org/10.2139/ssrn.3415930","url":null,"abstract":"This note is an answer to the consultation published by ISDA in May 2019 regarding the amendment of documentation to implement fallbacks for certain key IBORs.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133225902","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 present a new formalism for option pricing that does not require an assumption on the stochastic process of the underlying asset price and yet produces remarkably accurate results versus the market. The new formalism applies for general Markovian stochastic behavior including continuous and discontinuous (jump) processes and in its broadest scheme contains all known models for Markovian option pricing and some new ones. The method is based on obtaining the risk neutral density function that satisfies a consistency condition, guaranteeing no arbitrage. For example, we show that when the underlying asset undergoes a continuous stochastic process with deterministic time dependent standard deviation the formalism produces the Black-Scholes-Merton formula without using a Wiener process. We show that in the general case the price of European options depends only on all the moments of the price return of the underlying asset. We offer a method to calculate the prices of European options when the volatility smile at maturity is independent of the term structure prior to the maturity, as observed in options markets. In the continuous case where only moments up to second order contribute to the price then any set of three option prices with the same maturity contains the information to determine the whole volatility smile for this maturity. In all the many examples we examined our method generates option prices that match the option markets prices very accurately in all asset classes. This confirms that the options market exhibits no-arbitrage. Moreover, using bootstrapping we demonstrate how to determine the conditional density function from inception to maturity, thus allowing the calculation of path dependent options. The new formalism also allows for the replication of ‘W-shape’ volatility smile that infrequently appears in some equity markets.
{"title":"A General Theory of Option Pricing","authors":"D. Gershon","doi":"10.2139/ssrn.3291664","DOIUrl":"https://doi.org/10.2139/ssrn.3291664","url":null,"abstract":"We present a new formalism for option pricing that does not require an assumption on the stochastic process of the underlying asset price and yet produces remarkably accurate results versus the market. The new formalism applies for general Markovian stochastic behavior including continuous and discontinuous (jump) processes and in its broadest scheme contains all known models for Markovian option pricing and some new ones. The method is based on obtaining the risk neutral density function that satisfies a consistency condition, guaranteeing no arbitrage. For example, we show that when the underlying asset undergoes a continuous stochastic process with deterministic time dependent standard deviation the formalism produces the Black-Scholes-Merton formula without using a Wiener process. We show that in the general case the price of European options depends only on all the moments of the price return of the underlying asset. We offer a method to calculate the prices of European options when the volatility smile at maturity is independent of the term structure prior to the maturity, as observed in options markets. In the continuous case where only moments up to second order contribute to the price then any set of three option prices with the same maturity contains the information to determine the whole volatility smile for this maturity. In all the many examples we examined our method generates option prices that match the option markets prices very accurately in all asset classes. This confirms that the options market exhibits no-arbitrage. Moreover, using bootstrapping we demonstrate how to determine the conditional density function from inception to maturity, thus allowing the calculation of path dependent options. The new formalism also allows for the replication of ‘W-shape’ volatility smile that infrequently appears in some equity markets.<br>","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"165 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120935945","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 this article, we discuss the calibration of wrong way risk (WWR) model by using information from the credit default swap (CDS) market. A Quanto CDS provides credit protection against the default of a reference entity but is denominated in a non-domestic currency. The payoff of a Quanto CDS contract therefore reflects the market-implied interaction of FX risk and a credit event. This in turn, defines the cost of hedging WWR for a FX-sensitive portfolio. Our empirical evidence shows that the implied FX jump sizes are significant for a wide range of corporates. For systemic counterparties, the CVA WWR add-on could be 40% higher than the standard case, and choosing a proper jump-at-default WWR model is critical to capture the impact. In contrast, historical correlation gives the incorrect relationship (right-way risk) and cannot calibrate to the market prices in many cases, leading to the mispricing of CVA WWR.
{"title":"CVA Wrong Way Risk: Calibration Using Quanto CDS Basis","authors":"T. Chung, J. Gregory","doi":"10.2139/ssrn.3188793","DOIUrl":"https://doi.org/10.2139/ssrn.3188793","url":null,"abstract":"In this article, we discuss the calibration of wrong way risk (WWR) model by using information from the credit default swap (CDS) market. A Quanto CDS provides credit protection against the default of a reference entity but is denominated in a non-domestic currency. The payoff of a Quanto CDS contract therefore reflects the market-implied interaction of FX risk and a credit event. This in turn, defines the cost of hedging WWR for a FX-sensitive portfolio. Our empirical evidence shows that the implied FX jump sizes are significant for a wide range of corporates. For systemic counterparties, the CVA WWR add-on could be 40% higher than the standard case, and choosing a proper jump-at-default WWR model is critical to capture the impact. In contrast, historical correlation gives the incorrect relationship (right-way risk) and cannot calibrate to the market prices in many cases, leading to the mispricing of CVA WWR.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126562582","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 analyzes the effectiveness of hedging a defaultable bond, that may not be at par, with a credit default swap (CDS) by quantifying the variance of the hedging errors and determining the optimal hedge ratio. The static hedging framework uses bond recovery and time to default, which are correlated, to calculate the variance of the hedging errors and the optimal hedge ratio for the bond-CDS trade. The results show that there are irreducible risks when hedging a defaultable bond with a CDS; these irreducible risks increase with the magnitude of the premium/discount of the bond and decrease as the correlation between default time and recovery increase. The results also show that the optimal hedging ratio was closer to the bond price than the par value of the bond. This paper provides a framework distinct from the risk neutral framework by transparently showing the residual risks and their drivers.
{"title":"Irreducible Risks of Hedging a Bond with a Default Swap","authors":"V. Kapoor, Jake Freeman","doi":"10.2139/ssrn.3405367","DOIUrl":"https://doi.org/10.2139/ssrn.3405367","url":null,"abstract":"This paper analyzes the effectiveness of hedging a defaultable bond, that may not be at par, with a credit default swap (CDS) by quantifying the variance of the hedging errors and determining the optimal hedge ratio. The static hedging framework uses bond recovery and time to default, which are correlated, to calculate the variance of the hedging errors and the optimal hedge ratio for the bond-CDS trade. The results show that there are irreducible risks when hedging a defaultable bond with a CDS; these irreducible risks increase with the magnitude of the premium/discount of the bond and decrease as the correlation between default time and recovery increase. The results also show that the optimal hedging ratio was closer to the bond price than the par value of the bond. This paper provides a framework distinct from the risk neutral framework by transparently showing the residual risks and their drivers.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123147147","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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