We conduct an empirical analysis of the term structure in the volatility risk premium in the fixed income market by constructing long-short combinations of two at-the-money straddles for the four major swaption markets (USD, JPY, EUR and GBP). Our findings are consistent with a concave, upward-sloping maturity structure for all markets, with the largest negative premium for the shortest term maturity. The fact that both delta–vega and delta–gamma neutral straddle combinations earn positive returns that seem uncorrelated suggests that the term structure is affected by both jump risk and volatility risk. The results seem robust for macroeconomic announcements and the specific model choice to estimate the risk exposures for hedging.
{"title":"Riding the Swaption Curve","authors":"Johan Duyvesteyn, Gerben J. de Zwart","doi":"10.2139/ssrn.2008841","DOIUrl":"https://doi.org/10.2139/ssrn.2008841","url":null,"abstract":"We conduct an empirical analysis of the term structure in the volatility risk premium in the fixed income market by constructing long-short combinations of two at-the-money straddles for the four major swaption markets (USD, JPY, EUR and GBP). Our findings are consistent with a concave, upward-sloping maturity structure for all markets, with the largest negative premium for the shortest term maturity. The fact that both delta–vega and delta–gamma neutral straddle combinations earn positive returns that seem uncorrelated suggests that the term structure is affected by both jump risk and volatility risk. The results seem robust for macroeconomic announcements and the specific model choice to estimate the risk exposures for hedging.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116445192","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 paper, we price covariance and correlation swaps for financial markets with Markov-modulated volatilities. As an example, we consider stochastic volatility driven by a two-state continuous Markov chain. In this case, numerical examples are presented for VIX and VXN volatility indices (S&P 500 and NASDAQ-100, from January 2004 to June 2012). We also use VIX (January 2004 to June 2012) to price variance and volatility swaps for the two-state Markov-modulated volatility, and we present a numerical result in this case.
{"title":"Covariance and Correlation Swaps for Financial Markets with Markov-Modulated Volatilities","authors":"Giovanni Salvi, A. Swishchuk","doi":"10.2139/ssrn.2103304","DOIUrl":"https://doi.org/10.2139/ssrn.2103304","url":null,"abstract":"In this paper, we price covariance and correlation swaps for financial markets with Markov-modulated volatilities. As an example, we consider stochastic volatility driven by a two-state continuous Markov chain. In this case, numerical examples are presented for VIX and VXN volatility indices (S&P 500 and NASDAQ-100, from January 2004 to June 2012). We also use VIX (January 2004 to June 2012) to price variance and volatility swaps for the two-state Markov-modulated volatility, and we present a numerical result in this case.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116997742","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 describe a single parametric model for the entire volatility surface with interpolation and extrapolation technique generating a smooth and robust implied volatility surface without arbitrage in space and time. It is used for marking option prices on indices and single stocks as well as for computing analytically a proper local volatility with smooth risk-neutral density. Greeks and stress scenarios are calculated analytically in the parametric model without recalibration of the model parameters. We perform a simple expansion of the parametric model obtaining an analytic representation of its implied volatility surface along its cone of diffusion. In view of adding control to the generated volatility surface, we modify the model by adding three new parameters producing, in an independent way, a parallel shift, skew shift and curvature shift of that surface along its cone of diffusion. These parameters can be used manually to modify the entire shape of the volatility surface, and can also be used to generate analytically the new local volatility surface when computing the vegas of an option. Then, in view of defining the best possible volatility surface for non-liquid stocks where only few brokers quotes exist, we describe a method combining historical model parameters of the implied volatility surface together with parameters from other liquid stocks observed on the market.
{"title":"From Implied to Local Volatility Surface","authors":"D. Bloch","doi":"10.2139/SSRN.2091117","DOIUrl":"https://doi.org/10.2139/SSRN.2091117","url":null,"abstract":"We describe a single parametric model for the entire volatility surface with interpolation and extrapolation technique generating a smooth and robust implied volatility surface without arbitrage in space and time. It is used for marking option prices on indices and single stocks as well as for computing analytically a proper local volatility with smooth risk-neutral density. Greeks and stress scenarios are calculated analytically in the parametric model without recalibration of the model parameters. We perform a simple expansion of the parametric model obtaining an analytic representation of its implied volatility surface along its cone of diffusion. In view of adding control to the generated volatility surface, we modify the model by adding three new parameters producing, in an independent way, a parallel shift, skew shift and curvature shift of that surface along its cone of diffusion. These parameters can be used manually to modify the entire shape of the volatility surface, and can also be used to generate analytically the new local volatility surface when computing the vegas of an option. Then, in view of defining the best possible volatility surface for non-liquid stocks where only few brokers quotes exist, we describe a method combining historical model parameters of the implied volatility surface together with parameters from other liquid stocks observed on the market.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129633460","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 paper we investigate the impact of collateral posting on derivative prices. We build a complete discounting framework from vanilla swap pricing to single currency exotic option pricing. We show how to extract initial discount and forecast curves from the market of OIS and IR Vanilla swaps. We extend our considerations to the case of collateralisation in a currency other than the transaction denomination currency.
{"title":"Discounting Consistently with Collateral Posting","authors":"Messaoud Chibane, Yi-Chen Huang, JayaPrakash Selvaraj","doi":"10.2139/ssrn.2084452","DOIUrl":"https://doi.org/10.2139/ssrn.2084452","url":null,"abstract":"In this paper we investigate the impact of collateral posting on derivative prices. We build a complete discounting framework from vanilla swap pricing to single currency exotic option pricing. We show how to extract initial discount and forecast curves from the market of OIS and IR Vanilla swaps. We extend our considerations to the case of collateralisation in a currency other than the transaction denomination currency.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129160908","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 empirical evidence that jumps in the underlying stock price process are superfluous for European option pricing in time changed Levy models. We introduce a model with a.s. continuous sample paths and a parsimonious description in terms of free parameters. The conducted in- and out-of-sample analysis show almost no di fference concerning calibration quality to the German DAX index between the continuous model and pure jump alternatives. The jump models in fact show signs of over-parameterization displayed by dramatically varying parameter estimates over time. An analysis of the resulting fitting errors also reveals no benefi t from including jumps in the underlying process.
{"title":"Are Jumps in Time Changed L évy Models Superfluous? An Empirical Investigation","authors":"K. Herrmann, R. Schoebel","doi":"10.2139/ssrn.2082302","DOIUrl":"https://doi.org/10.2139/ssrn.2082302","url":null,"abstract":"This paper provides empirical evidence that jumps in the underlying stock price process are superfluous for European option pricing in time changed L\u0013evy models. We introduce a model with a.s. continuous sample paths and a parsimonious description in terms of free parameters. The conducted in- and out-of-sample analysis show almost no di fference concerning calibration quality to the German DAX index between the continuous model and pure jump alternatives. The jump models in fact show signs of over-parameterization displayed by dramatically varying parameter estimates over time. An analysis of the resulting fitting errors also reveals no benefi t from including jumps in the underlying process.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126715984","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 consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential L'evy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors -- one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of citet*{fpss} to models of the exponential L'evy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility. To illustrate the flexibility of our modeling framework we extend five exponential L'evy processes to include stochastic volatility and jump-intensity. For each of the extended models, using a single fast-varying factor of volatility and jump-intensity, we perform a calibration to the S&P500 implied volatility surface. Our results show decisively that the extended framework provides a significantly better fit to implied volatility than both the traditional exponential L'evy models and the fast mean-reverting stochastic volatility models of citet{fpss}.
{"title":"Exponential Lévy Models with Stochastic Volatility and Stochastic Jump-Intensity","authors":"Matthew J. Lorig","doi":"10.2139/ssrn.2055939","DOIUrl":"https://doi.org/10.2139/ssrn.2055939","url":null,"abstract":"We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential L'evy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors -- one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of citet*{fpss} to models of the exponential L'evy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility. To illustrate the flexibility of our modeling framework we extend five exponential L'evy processes to include stochastic volatility and jump-intensity. For each of the extended models, using a single fast-varying factor of volatility and jump-intensity, we perform a calibration to the S&P500 implied volatility surface. Our results show decisively that the extended framework provides a significantly better fit to implied volatility than both the traditional exponential L'evy models and the fast mean-reverting stochastic volatility models of citet{fpss}.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114892295","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 framework where interest rate futures pricing do not require convexity adjustment. The adjustment depends on the definition of curves and we build them in such a way that no adjustment is necessary. The framework is theoretically as acceptable as the standard (current) approach and may prove in some circumstances simpler to work with in practice.
{"title":"My Future is Not Convex","authors":"Marc Henrard","doi":"10.2139/ssrn.2053657","DOIUrl":"https://doi.org/10.2139/ssrn.2053657","url":null,"abstract":"We propose a framework where interest rate futures pricing do not require convexity adjustment. The adjustment depends on the definition of curves and we build them in such a way that no adjustment is necessary. The framework is theoretically as acceptable as the standard (current) approach and may prove in some circumstances simpler to work with in practice.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123348981","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}
Day-count conventions are a ubiquitous but often overlooked aspect of interest-bearing investments. While many market traded securities have adopted fixed or standard conventions, over-the-counter agreements such as interest rate swaps can and do use a wide variety of conventions, and many investors may not be aware of the effects of this choice on their future cash flows. Here, we show that the choice of day-count convention can have a surprisingly large effect on the market value of swap agreements. We highlight the importance of matching day-count conventions on obligations and accompanying swap agreements, and demonstrate various factors which influence the magnitude of day-count convention effects. As interest rate swaps are very common amongst municipal and other institutional investors, we urge investors to thoroughly understand these and other `fine print' terms in any potential agreements. In particular, we highlight the ability of financial intermediaries to effectively increase their fees substantially through their choice of day-count conventions.
{"title":"Isolating the Effect of Day-Count Conventions on the Market Value of Interest Rate Swaps","authors":"Geng Deng, Tim Dulaney, Tim Husson, C. McCann","doi":"10.2139/ssrn.2049501","DOIUrl":"https://doi.org/10.2139/ssrn.2049501","url":null,"abstract":"Day-count conventions are a ubiquitous but often overlooked aspect of interest-bearing investments. While many market traded securities have adopted fixed or standard conventions, over-the-counter agreements such as interest rate swaps can and do use a wide variety of conventions, and many investors may not be aware of the effects of this choice on their future cash flows. Here, we show that the choice of day-count convention can have a surprisingly large effect on the market value of swap agreements. We highlight the importance of matching day-count conventions on obligations and accompanying swap agreements, and demonstrate various factors which influence the magnitude of day-count convention effects. As interest rate swaps are very common amongst municipal and other institutional investors, we urge investors to thoroughly understand these and other `fine print' terms in any potential agreements. In particular, we highlight the ability of financial intermediaries to effectively increase their fees substantially through their choice of day-count conventions.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127088764","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}
Given a forwarding LIBOR-style curve F corresponding to a fixed tenor (e.g. 6m) and an exogenous discounting curve D (e.g. an OIS curve or cross-currency basis swap curve) we build on Bianchetti's results to propose dynamics for the forward LIBOR-style rate collateralized by D.In contrast with what other authors do (Bianchetti, Mercurio, Fujii, et al.) we do not assume that the collateralized forward rate is a martingale process under the corresponding forward risk neutral measure associated with the discount process. At time zero the collateralized forward rate is the forwarding curve rate multiplied by a quanto adjustment, but at reset time the expectation of the collateralized forward aligns with the forwarding curve rate.In order to calculate the quanto adjustment we show how to construct a deterministic drift, which can be computed with the information available at time zero by bootstrapping (under certain assumptions on the spot swap rates). We extend the result to forward swap rates in the context of swap market models.
{"title":"Non-Martingale Dynamics for Two Curve Derivatives Pricing","authors":"Mauricio Alvarez-Manilla","doi":"10.2139/ssrn.2040581","DOIUrl":"https://doi.org/10.2139/ssrn.2040581","url":null,"abstract":"Given a forwarding LIBOR-style curve F corresponding to a fixed tenor (e.g. 6m) and an exogenous discounting curve D (e.g. an OIS curve or cross-currency basis swap curve) we build on Bianchetti's results to propose dynamics for the forward LIBOR-style rate collateralized by D.In contrast with what other authors do (Bianchetti, Mercurio, Fujii, et al.) we do not assume that the collateralized forward rate is a martingale process under the corresponding forward risk neutral measure associated with the discount process. At time zero the collateralized forward rate is the forwarding curve rate multiplied by a quanto adjustment, but at reset time the expectation of the collateralized forward aligns with the forwarding curve rate.In order to calculate the quanto adjustment we show how to construct a deterministic drift, which can be computed with the information available at time zero by bootstrapping (under certain assumptions on the spot swap rates). We extend the result to forward swap rates in the context of swap market models.","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121442516","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}
Pub Date : 2012-04-12DOI: 10.1007/978-3-319-02069-3_8
Sébastien Darses, E. Lépinette
{"title":"Mean Square Error and Limit Theorem for the Modified Leland Hedging Strategy with a Constant Transaction Costs Coefficient","authors":"Sébastien Darses, E. Lépinette","doi":"10.1007/978-3-319-02069-3_8","DOIUrl":"https://doi.org/10.1007/978-3-319-02069-3_8","url":null,"abstract":"","PeriodicalId":280702,"journal":{"name":"ERN: Econometric Studies of Derivatives Markets (Topic)","volume":"2020 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127976510","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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