Building heavily on the recent nice paper [Weinan E-al (2017)], we introduce a primal-dual method for solving BSDEs based on the use of neural networks, stochastic gradient descent and a dual formulation of stochastic control problems. Our algorithm is illustrated with two examples relevant in Mathematical Finance: the pricing of counterparty risk and the computation of initial margin.
{"title":"Deep Primal-Dual Algorithm for BSDEs: Applications of Machine Learning to CVA and IM","authors":"P. Henry-Labordère","doi":"10.2139/ssrn.3071506","DOIUrl":"https://doi.org/10.2139/ssrn.3071506","url":null,"abstract":"Building heavily on the recent nice paper [Weinan E-al (2017)], we introduce a primal-dual method for solving BSDEs based on the use of neural networks, stochastic gradient descent and a dual formulation of stochastic control problems. Our algorithm is illustrated with two examples relevant in Mathematical Finance: the pricing of counterparty risk and the computation of initial margin.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117327143","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}
Matthew Baron, Jonathan Brogaard, Björn Hagströmer, A. Kirilenko
We study performance and competition among firms engaging in high-frequency trading (HFT). We construct measures of latency and find that differences in relative latency account for large differences in HFT firms’ trading performance. HFT firms that improve their latency rank due to colocation upgrades see improved trading performance. The stronger performance associated with speed comes through both the short-lived information channel and the risk management channel, and speed is useful for various strategies, including market making and cross-market arbitrage. We find empirical support for many predictions regarding relative latency competition.
{"title":"Risk and Return in High-Frequency Trading","authors":"Matthew Baron, Jonathan Brogaard, Björn Hagströmer, A. Kirilenko","doi":"10.2139/ssrn.2433118","DOIUrl":"https://doi.org/10.2139/ssrn.2433118","url":null,"abstract":"We study performance and competition among firms engaging in high-frequency trading (HFT). We construct measures of latency and find that differences in relative latency account for large differences in HFT firms’ trading performance. HFT firms that improve their latency rank due to colocation upgrades see improved trading performance. The stronger performance associated with speed comes through both the short-lived information channel and the risk management channel, and speed is useful for various strategies, including market making and cross-market arbitrage. We find empirical support for many predictions regarding relative latency competition.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114873786","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 study proposes and tests a risk-free rate model that simultaneously lets the risk-free rate migrate between rating categories as risk-free rate ranges, and follow a random walk within rating categories as risk-free rate ranges. Although the study arbitrarily assigned rating categories, and risk-free rate ranges to the rating categories, empirical research can clarify this, by examining the relationship between the risk-free rate and risk-free rate volatility, and by examining the relationship between sovereign credit ratings and risk-free rate ranges as well as risk-free rate volatility. Firstly, comparable risk-free rates should illustrate comparable risk-free rate volatility, and risk-free rates should cluster in terms of their risk-free rate volatility characteristics. Secondly, sovereign credit ratings should demonstrate risk-free rate ranges and risk-free rate volatility characteristics. To test the model, a risk-free bond portfolio, together with a risk-free rate rating migration matrix were simulated. The rating migration matrix governs the migration between risk-free rate rating categories. It is shown that the original migration matrix can again be decomposed with adequate accuracy, given that the appropriate constraints are used. It indicates that the model can be applied to empirical markets. Possible refinements to the model are noted.
{"title":"Sovereign Credit Rating, Rating Migration, and the Risk-Free Rate: A Joint Markov Process and Random Walk Modelling of the Risk-Free Rate","authors":"B. Barnard","doi":"10.2139/ssrn.3034284","DOIUrl":"https://doi.org/10.2139/ssrn.3034284","url":null,"abstract":"The study proposes and tests a risk-free rate model that simultaneously lets the risk-free rate migrate between rating categories as risk-free rate ranges, and follow a random walk within rating categories as risk-free rate ranges. Although the study arbitrarily assigned rating categories, and risk-free rate ranges to the rating categories, empirical research can clarify this, by examining the relationship between the risk-free rate and risk-free rate volatility, and by examining the relationship between sovereign credit ratings and risk-free rate ranges as well as risk-free rate volatility. Firstly, comparable risk-free rates should illustrate comparable risk-free rate volatility, and risk-free rates should cluster in terms of their risk-free rate volatility characteristics. Secondly, sovereign credit ratings should demonstrate risk-free rate ranges and risk-free rate volatility characteristics. To test the model, a risk-free bond portfolio, together with a risk-free rate rating migration matrix were simulated. The rating migration matrix governs the migration between risk-free rate rating categories. It is shown that the original migration matrix can again be decomposed with adequate accuracy, given that the appropriate constraints are used. It indicates that the model can be applied to empirical markets. Possible refinements to the model are noted.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130341525","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 tests the long run risk and valuation risk model using a robust estimation procedure. The persistent long run component of consumption growth process is proxied by a news based index that is created using a random forest algorithm. This news index is shown to predict aggregate long term consumption growth with an R-square of 57% and is robust to inclusion of other commonly used predictors. I theoretically derive an estimatable bias term in adjusted Euler equation of the model that arises due to measurement error in consumption data and show that this bias term is non-zero. Using a three pass estimation procedure that accounts for this bias, I show that the long run risk and valuation risk model fails to explain cross section of equity returns. This contrasts to the results from regular two pass Fama-MacBeth estimation procedure that implies that the same model explains the cross section of asset returns with statistically significant risk premia estimates.
{"title":"Robust Test of Long Run Risk and Valuation Risk Model","authors":"G. Gopalakrishna","doi":"10.2139/ssrn.3043821","DOIUrl":"https://doi.org/10.2139/ssrn.3043821","url":null,"abstract":"This paper tests the long run risk and valuation risk model using a robust estimation procedure. The persistent long run component of consumption growth process is proxied by a news based index that is created using a random forest algorithm. This news index is shown to predict aggregate long term consumption growth with an R-square of 57% and is robust to inclusion of other commonly used predictors. I theoretically derive an estimatable bias term in adjusted Euler equation of the model that arises due to measurement error in consumption data and show that this bias term is non-zero. Using a three pass estimation procedure that accounts for this bias, I show that the long run risk and valuation risk model fails to explain cross section of equity returns. This contrasts to the results from regular two pass Fama-MacBeth estimation procedure that implies that the same model explains the cross section of asset returns with statistically significant risk premia estimates.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129077923","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}
Abstract We develop a model to characterize and quantify the effects of stock, option, and fixed compensation on a manager’s risk-taking incentive and investment choice. We find the average chief executive officer’s (CEO) compensation contract incentivizes overinvestment by 1.3 percentage points per year, with significant variation across firms and over time. We estimate a value of CEO effort implied by compensation contracts and find it to be strongly related to firm intangibility. Finally, we assess the effects on investment of FAS 123R and a hypothetical ban on option grants and find heterogeneous responses that depend on firm volatility and the prior structure of compensation.
{"title":"Idiosyncratic Risk and the Manager","authors":"Brent Glover, Oliver Levine","doi":"10.2139/ssrn.2024384","DOIUrl":"https://doi.org/10.2139/ssrn.2024384","url":null,"abstract":"Abstract We develop a model to characterize and quantify the effects of stock, option, and fixed compensation on a manager’s risk-taking incentive and investment choice. We find the average chief executive officer’s (CEO) compensation contract incentivizes overinvestment by 1.3 percentage points per year, with significant variation across firms and over time. We estimate a value of CEO effort implied by compensation contracts and find it to be strongly related to firm intangibility. Finally, we assess the effects on investment of FAS 123R and a hypothetical ban on option grants and find heterogeneous responses that depend on firm volatility and the prior structure of compensation.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121572451","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 aim at constructing a global risk model using the term structure from major bond-issuing countries. The goal is twofold: first this allows quantifying global interest rate risk (level, slope and curvature effects), providing insights on global risks at play. Secondly, such information could be used in order to design sovereign bond indexes in a risk parity framework where each country's sensitivity to global interest risk is accounted for. More specifically, we propose two innovative indexing schemes, a first one where we equalize contribution to global level risk exposures across countries, and a second one where we turn to level, slope and curvature risk exposures within a country. Indeed at the country level, only parallel (level) risk matters, while when turning to maturity buckets within a country, non parallel risks (slope and curvature) have to be accounted for. Finally, we demonstrate that the conjunctive use of these two approaches allows to efficiently tackle exposure to global interest rate risk while providing appealing improvements in the risk-return profile.
{"title":"Introducing Global Term Structure in a Risk Parity Framework","authors":"Lauren Stagnol","doi":"10.2139/ssrn.2956337","DOIUrl":"https://doi.org/10.2139/ssrn.2956337","url":null,"abstract":"In this paper, we aim at constructing a global risk model using the term structure from major bond-issuing countries. The goal is twofold: first this allows quantifying global interest rate risk (level, slope and curvature effects), providing insights on global risks at play. Secondly, such information could be used in order to design sovereign bond indexes in a risk parity framework where each country's sensitivity to global interest risk is accounted for. More specifically, we propose two innovative indexing schemes, a first one where we equalize contribution to global level risk exposures across countries, and a second one where we turn to level, slope and curvature risk exposures within a country. Indeed at the country level, only parallel (level) risk matters, while when turning to maturity buckets within a country, non parallel risks (slope and curvature) have to be accounted for. Finally, we demonstrate that the conjunctive use of these two approaches allows to efficiently tackle exposure to global interest rate risk while providing appealing improvements in the risk-return profile.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121741385","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 risk neutral measure is identified as a symmetric location-scale family of distribution in the local regime of the λ model. A partial differential equation is derived as the transformation between the implied volatility surface and such risk neutral probability. Given a well-interpolated volatility surface from market data, the risk neutral probability and the implied rate of growth can be extracted in a model-free manner. On the other hand, assuming a priori knowledge on the rate of growth and the risk neutral probability being an symmetric λ distribution, the closed form solution of the local volatility function be derived from Fokker-Planck equation. Based on such solution, I discuss possible forms of diffusion process, implement a leptokurtic extension of Weiner process, and discover a mean-reverting process that bridges between the Ornstein-Uhlenbeck process and Bessel process.
{"title":"From Volatility Smile to Risk Neutral Probability and Closed Form Solution of Local Volatility Function","authors":"Stephen H-T. Lihn","doi":"10.2139/ssrn.2906522","DOIUrl":"https://doi.org/10.2139/ssrn.2906522","url":null,"abstract":"The risk neutral measure is identified as a symmetric location-scale family of distribution in the local regime of the λ model. A partial differential equation is derived as the transformation between the implied volatility surface and such risk neutral probability. Given a well-interpolated volatility surface from market data, the risk neutral probability and the implied rate of growth can be extracted in a model-free manner. On the other hand, assuming a priori knowledge on the rate of growth and the risk neutral probability being an symmetric λ distribution, the closed form solution of the local volatility function be derived from Fokker-Planck equation. Based on such solution, I discuss possible forms of diffusion process, implement a leptokurtic extension of Weiner process, and discover a mean-reverting process that bridges between the Ornstein-Uhlenbeck process and Bessel process.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130731373","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 study examines rating migration, and default probability term structures obtained from rating migration matrices. It expands on the use of rating migration matrices with reduced form bond valuation models, by formally delineating the probability of default according to the likely rating paths of a bond, as implied by the rating migration matrix. Further, two alternatives are also considered. First, the cost of default is stipulated as the recovery of par according to the exit rating upon default. Also, in addition to stating the value of a bond in terms of expected cash flows, when considering the probability of default, the value of a bond is alternatively stated as the present value of all likely rating paths of the bond, discounted against the market risk-bearing bond forward rates of the different rating categories. The impact of term structure volatility and rating migration uncertainty on bond valuation is also considered.It is shown that the relationship between rating migration and default probability is complex, and the default probabilities of different rating categories are time-dependent and not isolated from each other. Also, rating migration resembles a delayed default process that influences default probabilities of subsequent intervals. The implications of a rating migration matrix may perhaps only be fully understood through simulation. This form one of the first points by which to evaluate rating migration matrices. The results of the valuation model show that historical rating migration matrices may not be optimal for pricing bonds ahistorically. A principal premise of the study is the dichotomy between historical values and ahistorical estimates, particularly with regards to rating migration. It is argued that historical estimates face two key shortcomings: they must be able to accurately forecast future rating migration and rating category intensities as a result, and they must specify a method to include rating migration uncertainty. An optimization model is delineated to extract ahistorical rating migration matrices from market prices. This too has implications that should be considered. In light of the above, reduced form models may have an advantage over structural models, in their ability to portray a far more sophisticated default process.
{"title":"Rating Migration and Bond Valuation: Towards Ahistorical Rating Migration Matrices and Default Probability Term Structures","authors":"B. Barnard","doi":"10.2139/ssrn.2893521","DOIUrl":"https://doi.org/10.2139/ssrn.2893521","url":null,"abstract":"The study examines rating migration, and default probability term structures obtained from rating migration matrices. It expands on the use of rating migration matrices with reduced form bond valuation models, by formally delineating the probability of default according to the likely rating paths of a bond, as implied by the rating migration matrix. Further, two alternatives are also considered. First, the cost of default is stipulated as the recovery of par according to the exit rating upon default. Also, in addition to stating the value of a bond in terms of expected cash flows, when considering the probability of default, the value of a bond is alternatively stated as the present value of all likely rating paths of the bond, discounted against the market risk-bearing bond forward rates of the different rating categories. The impact of term structure volatility and rating migration uncertainty on bond valuation is also considered.It is shown that the relationship between rating migration and default probability is complex, and the default probabilities of different rating categories are time-dependent and not isolated from each other. Also, rating migration resembles a delayed default process that influences default probabilities of subsequent intervals. The implications of a rating migration matrix may perhaps only be fully understood through simulation. This form one of the first points by which to evaluate rating migration matrices. The results of the valuation model show that historical rating migration matrices may not be optimal for pricing bonds ahistorically. A principal premise of the study is the dichotomy between historical values and ahistorical estimates, particularly with regards to rating migration. It is argued that historical estimates face two key shortcomings: they must be able to accurately forecast future rating migration and rating category intensities as a result, and they must specify a method to include rating migration uncertainty. An optimization model is delineated to extract ahistorical rating migration matrices from market prices. This too has implications that should be considered. In light of the above, reduced form models may have an advantage over structural models, in their ability to portray a far more sophisticated default process.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134342908","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}
I provide evidence that financial contagion risk is an important source of the equity risk premium. Banks’ contributions to aggregate financial contagion are estimated in a state space framework and linked to systemic risk. Greater bank connectedness today leads to increased systemic risk 3–12 months later. More contagious banks earn significantly greater risk-adjusted returns than less contagious ones and the tradable high contagion-minus-low contagion bank portfolio is priced in the cross-section of stock returns. Stocks that co-move more strongly with contagious banks have greater expected returns. These results are robust to factor model specification, test assets, and time period considered.
{"title":"Financial Contagion Risk and the Stochastic Discount Factor","authors":"Louis R. Piccotti","doi":"10.2139/ssrn.2411788","DOIUrl":"https://doi.org/10.2139/ssrn.2411788","url":null,"abstract":"I provide evidence that financial contagion risk is an important source of the equity risk premium. Banks’ contributions to aggregate financial contagion are estimated in a state space framework and linked to systemic risk. Greater bank connectedness today leads to increased systemic risk 3–12 months later. More contagious banks earn significantly greater risk-adjusted returns than less contagious ones and the tradable high contagion-minus-low contagion bank portfolio is priced in the cross-section of stock returns. Stocks that co-move more strongly with contagious banks have greater expected returns. These results are robust to factor model specification, test assets, and time period considered.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130651117","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 extend the definition of multi-year claims development results and quantification of multi-year non-life insurance risk to the bivariate chain ladder model as introduced by Braun in 2004. In this model, we assume two correlated loss portfolios each of which is underlying the classical chain ladder model. In accordance with standard literature, multi-year risk is defined through the variation of the multi-year claims development result and quantified in terms of the corresponding mean squared error of prediction. Following previous research on the univariate chain ladder model, for the first time we derive closed analytical expressions for the prediction error of the aggregate multi-year claims development result via first-order Taylor approximation. We reproduce well-known results for the ultimo view from literature. The goodness of our approximation is confirmed by a simulation study. Furthermore, a case study demonstrates the applicability of our analytical formulae.
{"title":"Multi-Year Non-Life Insurance Risk for Correlated Loss Portfolios Under Chain Ladder Model Assumptions","authors":"Marc Linde","doi":"10.2139/ssrn.2869217","DOIUrl":"https://doi.org/10.2139/ssrn.2869217","url":null,"abstract":"In this paper we extend the definition of multi-year claims development results and quantification of multi-year non-life insurance risk to the bivariate chain ladder model as introduced by Braun in 2004. In this model, we assume two correlated loss portfolios each of which is underlying the classical chain ladder model. In accordance with standard literature, multi-year risk is defined through the variation of the multi-year claims development result and quantified in terms of the corresponding mean squared error of prediction. Following previous research on the univariate chain ladder model, for the first time we derive closed analytical expressions for the prediction error of the aggregate multi-year claims development result via first-order Taylor approximation. We reproduce well-known results for the ultimo view from literature. The goodness of our approximation is confirmed by a simulation study. Furthermore, a case study demonstrates the applicability of our analytical formulae.","PeriodicalId":187811,"journal":{"name":"ERN: Other Econometric Modeling: Capital Markets - Risk (Topic)","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114938729","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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