Pub Date : 2020-11-01DOI: 10.1080/1350486X.2021.1898998
Richard Biegler-König
ABSTRACT Hedging on commodity markets is usually done by applying either the rolling intrinsic strategy or the canonical delta hedge strategy. In this paper we introduce, compare and discuss both hedging strategies in the context of virtual power plants (VPP). We formulate the precise relationship of the two strategies mathematically. Our main result is that they are not only very similar regarding hedge construction but also that both strategies are equal in expectation. The proof involves some stochastic calculus and the Brownian local time. We illustrate our findings with simulated data as well as in prototypical market scenarios. These studies show that the rolling intrinsic hedge comes with a riskier profile than the delta hedge.
{"title":"Hedging Strategies in Commodity Markets – Rolling Intrinsic and Delta Hedging for Virtual Power Plants","authors":"Richard Biegler-König","doi":"10.1080/1350486X.2021.1898998","DOIUrl":"https://doi.org/10.1080/1350486X.2021.1898998","url":null,"abstract":"ABSTRACT Hedging on commodity markets is usually done by applying either the rolling intrinsic strategy or the canonical delta hedge strategy. In this paper we introduce, compare and discuss both hedging strategies in the context of virtual power plants (VPP). We formulate the precise relationship of the two strategies mathematically. Our main result is that they are not only very similar regarding hedge construction but also that both strategies are equal in expectation. The proof involves some stochastic calculus and the Brownian local time. We illustrate our findings with simulated data as well as in prototypical market scenarios. These studies show that the rolling intrinsic hedge comes with a riskier profile than the delta hedge.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"311 1","pages":"550 - 582"},"PeriodicalIF":0.0,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79991146","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 : 2020-11-01DOI: 10.1080/1350486X.2021.1893196
P. Carr, Gianna Figá-Talamanca
ABSTRACT An alternative volatility index called SPIKES has been recently introduced. Like VIX, SPIKES aims to forecast S&P 500 volatility over a 30-day horizon and both indexes are based on the same theoretical formula; yet, they differ in several ways. While some differences are introduced in response to the controversy surrounding possible VIX manipulation, others are due to the choice of the S&P500 exchange-traded fund (ETF), named SPY, as a substitute for the S&P500 (SPX) Index itself. Indeed, options on the SPX, used for VIX computation, are European-style, whereas options on the SPY ETF, used for SPIKES computation, are American-style. Overall, the difference is mainly due to the early exercise premium of the component options and the dividend timing of the underlying SPY versus SPX and we assess the magnitude of these separate contributions under the benchmark Black, Merton and Scholes setting. By applying both the finite difference method and newly-derived approximation formulas we show that the new SPIKES index will track the VIX index as long as 30-day US interest rates and annualized dividend yields continue to be range-bound between 0 and 10% per year. Hence, after more that 20 years of supremacy, VIX may have found its first competitor.
{"title":"Spiking the Volatility Punch","authors":"P. Carr, Gianna Figá-Talamanca","doi":"10.1080/1350486X.2021.1893196","DOIUrl":"https://doi.org/10.1080/1350486X.2021.1893196","url":null,"abstract":"ABSTRACT An alternative volatility index called SPIKES has been recently introduced. Like VIX, SPIKES aims to forecast S&P 500 volatility over a 30-day horizon and both indexes are based on the same theoretical formula; yet, they differ in several ways. While some differences are introduced in response to the controversy surrounding possible VIX manipulation, others are due to the choice of the S&P500 exchange-traded fund (ETF), named SPY, as a substitute for the S&P500 (SPX) Index itself. Indeed, options on the SPX, used for VIX computation, are European-style, whereas options on the SPY ETF, used for SPIKES computation, are American-style. Overall, the difference is mainly due to the early exercise premium of the component options and the dividend timing of the underlying SPY versus SPX and we assess the magnitude of these separate contributions under the benchmark Black, Merton and Scholes setting. By applying both the finite difference method and newly-derived approximation formulas we show that the new SPIKES index will track the VIX index as long as 30-day US interest rates and annualized dividend yields continue to be range-bound between 0 and 10% per year. Hence, after more that 20 years of supremacy, VIX may have found its first competitor.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"20 1","pages":"495 - 520"},"PeriodicalIF":0.0,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87137431","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 : 2020-10-16DOI: 10.1080/1350486X.2022.2039250
M. Ludkovski, Y. Saporito
We investigate a machine learning approach to option Greeks approximation based on Gaussian Process (GP) surrogates. Our motivation is to implement Delta hedging in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. The proposed method takes in noisily observed option prices, fits a non-parametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Thus, a single surrogate yields multiple self-consistent Greeks. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise. We moreover connect the quality of the Delta approximation to the resulting discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matérn kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.
{"title":"KrigHedge: Gaussian Process Surrogates for Delta Hedging","authors":"M. Ludkovski, Y. Saporito","doi":"10.1080/1350486X.2022.2039250","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2039250","url":null,"abstract":"We investigate a machine learning approach to option Greeks approximation based on Gaussian Process (GP) surrogates. Our motivation is to implement Delta hedging in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. The proposed method takes in noisily observed option prices, fits a non-parametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Thus, a single surrogate yields multiple self-consistent Greeks. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise. We moreover connect the quality of the Delta approximation to the resulting discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matérn kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"97 1","pages":"330 - 360"},"PeriodicalIF":0.0,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81413411","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 : 2020-09-02DOI: 10.1080/1350486X.2021.1891554
Chanaka Edirisinghe, Yonggan Zhao
ABSTRACT Index funds that track a benchmark, such as the market cap-weighted S&P 500 index, tend to have portfolio holdings biased towards slower-growth large-cap equities that result in the fund’s under-performance, especially in economic downturns. We develop a rigorous quantitative framework that allows dynamic-rebalancing of the allocations such that portfolio exposure in a market segment can change periodically based on economic activity, measured via a set of macro-economic and financial indicators. The method incorporates potential shifts in the economic state, and the likelihood thereof, to determine the fund’s risk orientation optimally in tracking or not tracking the benchmark index. That is, the greater the likelihood of a stronger economic state, the higher the degree of tracking the market index; however, a lack of confidence in the economic state results in a more index-neutral portfolio composition. The proposed smart indexing optimal strategy generates superior risk-adjusted returns consistently in out-of-sample testing, relative to (pure) index tracking. We test several variants and present sensitivity analyses that support our actively-managed smart indexing approach.
{"title":"Smart Indexing Under Regime-Switching Economic States","authors":"Chanaka Edirisinghe, Yonggan Zhao","doi":"10.1080/1350486X.2021.1891554","DOIUrl":"https://doi.org/10.1080/1350486X.2021.1891554","url":null,"abstract":"ABSTRACT Index funds that track a benchmark, such as the market cap-weighted S&P 500 index, tend to have portfolio holdings biased towards slower-growth large-cap equities that result in the fund’s under-performance, especially in economic downturns. We develop a rigorous quantitative framework that allows dynamic-rebalancing of the allocations such that portfolio exposure in a market segment can change periodically based on economic activity, measured via a set of macro-economic and financial indicators. The method incorporates potential shifts in the economic state, and the likelihood thereof, to determine the fund’s risk orientation optimally in tracking or not tracking the benchmark index. That is, the greater the likelihood of a stronger economic state, the higher the degree of tracking the market index; however, a lack of confidence in the economic state results in a more index-neutral portfolio composition. The proposed smart indexing optimal strategy generates superior risk-adjusted returns consistently in out-of-sample testing, relative to (pure) index tracking. We test several variants and present sensitivity analyses that support our actively-managed smart indexing approach.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"74 1","pages":"422 - 456"},"PeriodicalIF":0.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80004454","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 : 2020-09-02DOI: 10.1080/1350486X.2021.1877559
E. Eberlein, Christoph Gerhart, E. Lütkebohmert
ABSTRACT In this paper, we develop an arbitrage-free multiple curve model through the specification of forward swap rates. Two sets of assets are chosen as fundamentals: OIS zero-coupon bonds and forward rate agreements. This is a very natural approach since, on the one hand, OIS bonds represent the class of risk-free discount bonds and, on the other hand, the mid and long maturity part of the interest rate term structure is bootstrapped from quotes of swap rates that can be represented by FRA rates and OIS bond prices in the multiple curve setting. We construct the rates via a backward induction along the tenor structure on the basis of the forward swap measures. Time-inhomogeneous Lévy processes are used as drivers of the dynamics. As an application, we derive an approximative Fourier-based valuation formula for swaptions. The model is implemented and calibrated by using generalized hyperbolic Lévy processes as drivers.
{"title":"A Multiple Curve Lévy Swap Market Model","authors":"E. Eberlein, Christoph Gerhart, E. Lütkebohmert","doi":"10.1080/1350486X.2021.1877559","DOIUrl":"https://doi.org/10.1080/1350486X.2021.1877559","url":null,"abstract":"ABSTRACT In this paper, we develop an arbitrage-free multiple curve model through the specification of forward swap rates. Two sets of assets are chosen as fundamentals: OIS zero-coupon bonds and forward rate agreements. This is a very natural approach since, on the one hand, OIS bonds represent the class of risk-free discount bonds and, on the other hand, the mid and long maturity part of the interest rate term structure is bootstrapped from quotes of swap rates that can be represented by FRA rates and OIS bond prices in the multiple curve setting. We construct the rates via a backward induction along the tenor structure on the basis of the forward swap measures. Time-inhomogeneous Lévy processes are used as drivers of the dynamics. As an application, we derive an approximative Fourier-based valuation formula for swaptions. The model is implemented and calibrated by using generalized hyperbolic Lévy processes as drivers.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"20 1","pages":"396 - 421"},"PeriodicalIF":0.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80460761","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 : 2020-08-21DOI: 10.1080/1350486X.2020.1846573
Samuel N. Cohen, C. Reisinger, Sheng Wang
ABSTRACT Option price data are used as inputs for model calibration, risk-neutral density estimation and many other financial applications. The presence of arbitrage in option price data can lead to poor performance or even failure of these tasks, making pre-processing of the data to eliminate arbitrage necessary. Most attention in the relevant literature has been devoted to arbitrage-free smoothing and filtering (i.e., removing) of data. In contrast to smoothing, which typically changes nearly all data, or filtering, which truncates data, we propose to repair data by only necessary and minimal changes. We formulate the data repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimize prices’ changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real-world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from prices data by our repair method can improve model calibration with enhanced robustness and reduced calibration error.
{"title":"Detecting and Repairing Arbitrage in Traded Option Prices","authors":"Samuel N. Cohen, C. Reisinger, Sheng Wang","doi":"10.1080/1350486X.2020.1846573","DOIUrl":"https://doi.org/10.1080/1350486X.2020.1846573","url":null,"abstract":"ABSTRACT Option price data are used as inputs for model calibration, risk-neutral density estimation and many other financial applications. The presence of arbitrage in option price data can lead to poor performance or even failure of these tasks, making pre-processing of the data to eliminate arbitrage necessary. Most attention in the relevant literature has been devoted to arbitrage-free smoothing and filtering (i.e., removing) of data. In contrast to smoothing, which typically changes nearly all data, or filtering, which truncates data, we propose to repair data by only necessary and minimal changes. We formulate the data repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimize prices’ changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real-world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from prices data by our repair method can improve model calibration with enhanced robustness and reduced calibration error.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"199 1","pages":"345 - 373"},"PeriodicalIF":0.0,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81079854","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 consider the problem of maximizing portfolio value when an agent has a subjective view on asset value which differs from the traded market price. The agent’s trades will have a price impact which affects the price at which the asset is traded. In addition to the agent’s trades affecting the market price, the agent may change his view on the asset’s value if its difference from the market price persists. We also consider a situation of several agents interacting and trading simultaneously when they have a subjective view of the asset value. Two cases of the subjective views of agents are considered: one in which they all share the same information, and one in which they all have an individual signal correlated with price innovations. To study the large agent problem we take a mean-field game approach which remains tractable. After classifying the mean-field equilibrium we compute the cross-sectional distribution of agents’ inventories and the dependence of price distribution on the amount of shared information among the agents.
{"title":"Optimal Trading with Differing Trade Signals","authors":"Ryan Francis Donnelly, Matthew J. Lorig","doi":"10.2139/ssrn.3634629","DOIUrl":"https://doi.org/10.2139/ssrn.3634629","url":null,"abstract":"ABSTRACT We consider the problem of maximizing portfolio value when an agent has a subjective view on asset value which differs from the traded market price. The agent’s trades will have a price impact which affects the price at which the asset is traded. In addition to the agent’s trades affecting the market price, the agent may change his view on the asset’s value if its difference from the market price persists. We also consider a situation of several agents interacting and trading simultaneously when they have a subjective view of the asset value. Two cases of the subjective views of agents are considered: one in which they all share the same information, and one in which they all have an individual signal correlated with price innovations. To study the large agent problem we take a mean-field game approach which remains tractable. After classifying the mean-field equilibrium we compute the cross-sectional distribution of agents’ inventories and the dependence of price distribution on the amount of shared information among the agents.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"15 1","pages":"317 - 344"},"PeriodicalIF":0.0,"publicationDate":"2020-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88253301","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 : 2020-06-23DOI: 10.1080/1350486x.2020.1819831
G. Bouzianis, L. Hughston
ABSTRACT We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a Lévy-Ito market, where assets are driven jointly by an n-dimensional Brownian motion and an independent Poisson random measure on an n-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of a least expected squared error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.
{"title":"Optimal Hedging in Incomplete Markets","authors":"G. Bouzianis, L. Hughston","doi":"10.1080/1350486x.2020.1819831","DOIUrl":"https://doi.org/10.1080/1350486x.2020.1819831","url":null,"abstract":"ABSTRACT We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a Lévy-Ito market, where assets are driven jointly by an n-dimensional Brownian motion and an independent Poisson random measure on an n-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of a least expected squared error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"11 1","pages":"265 - 287"},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74921395","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 : 2020-05-03DOI: 10.1080/1350486X.2020.1825968
Shi Qiu
ABSTRACT In this paper, we show that the double optimal stopping boundaries for American strangle options with finite horizon can be characterized as the unique pair of solution to a system of two nonlinear integral equations arising from the early exercise premium (EEP) representation. The proof of EEP representation is based on the change-of-variable formula with local time on curves. After comparing the return of the alternative portfolio including an American call and an American put option, we find that it is more preferable for an investor to select American strangle options to hedge an underlying asset with high volatility.
{"title":"American Strangle Options","authors":"Shi Qiu","doi":"10.1080/1350486X.2020.1825968","DOIUrl":"https://doi.org/10.1080/1350486X.2020.1825968","url":null,"abstract":"ABSTRACT In this paper, we show that the double optimal stopping boundaries for American strangle options with finite horizon can be characterized as the unique pair of solution to a system of two nonlinear integral equations arising from the early exercise premium (EEP) representation. The proof of EEP representation is based on the change-of-variable formula with local time on curves. After comparing the return of the alternative portfolio including an American call and an American put option, we find that it is more preferable for an investor to select American strangle options to hedge an underlying asset with high volatility.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"62 1","pages":"228 - 263"},"PeriodicalIF":0.0,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83257394","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 : 2020-05-03DOI: 10.1080/1350486x.2020.1805337
Simon Schnürch, A. Wagner
ABSTRACT This paper employs machine learning algorithms to forecast German electricity spot market prices. The forecasts utilize in particular bid and ask order book data from the spot market but also fundamental market data like renewable infeed and expected total demand. Appropriate feature extraction for the order book data is developed proceeding from existing literature. Using cross-validation to optimize hyperparameters, neural networks and random forests are fit to the data. Their in-sample and out-of-sample performance is compared to statistical reference models. The machine learning models outperform traditional approaches.
{"title":"Electricity Price Forecasting with Neural Networks on EPEX Order Books","authors":"Simon Schnürch, A. Wagner","doi":"10.1080/1350486x.2020.1805337","DOIUrl":"https://doi.org/10.1080/1350486x.2020.1805337","url":null,"abstract":"ABSTRACT This paper employs machine learning algorithms to forecast German electricity spot market prices. The forecasts utilize in particular bid and ask order book data from the spot market but also fundamental market data like renewable infeed and expected total demand. Appropriate feature extraction for the order book data is developed proceeding from existing literature. Using cross-validation to optimize hyperparameters, neural networks and random forests are fit to the data. Their in-sample and out-of-sample performance is compared to statistical reference models. The machine learning models outperform traditional approaches.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"40 1","pages":"189 - 206"},"PeriodicalIF":0.0,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73262295","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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