M. Calvo-Garrido, Matthias Ehrhardt, Carlos Vázquez Cendón
In this paper, we consider the numerical valuation of swing options in electricity markets based on a two-factor model. These kinds of contracts are modeled as path dependent options with multiple exercise rights. From a mathematical point of view, the valuation of these products is posed as a sequence of free boundary problems, where two exercise rights are separated by a time period. In order to solve the pricing problem, we propose appropriate numerical methods based on a Crank-Nicolson semi-Lagrangian method combined with biquadratic Lagrange finite elements for the discretization of the partial differential equation. In addition, we use an augmented Lagrangian active set method to cope with the early exercise feature when it appears. Moreover, we derive appropriate artificial boundary conditions to treat the unbounded domain numerically. Finally, we present some numerical results to illustrate the proper behavior of the numerical schemes.
{"title":"Pricing Swing Options in Electricity Markets with Two Stochastic Factors Using a Partial Differential Equation Approach","authors":"M. Calvo-Garrido, Matthias Ehrhardt, Carlos Vázquez Cendón","doi":"10.21314/JCF.2016.317","DOIUrl":"https://doi.org/10.21314/JCF.2016.317","url":null,"abstract":"In this paper, we consider the numerical valuation of swing options in electricity markets based on a two-factor model. These kinds of contracts are modeled as path dependent options with multiple exercise rights. From a mathematical point of view, the valuation of these products is posed as a sequence of free boundary problems, where two exercise rights are separated by a time period. In order to solve the pricing problem, we propose appropriate numerical methods based on a Crank-Nicolson semi-Lagrangian method combined with biquadratic Lagrange finite elements for the discretization of the partial differential equation. In addition, we use an augmented Lagrangian active set method to cope with the early exercise feature when it appears. Moreover, we derive appropriate artificial boundary conditions to treat the unbounded domain numerically. Finally, we present some numerical results to illustrate the proper behavior of the numerical schemes.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p><strong>Purpose: </strong>Image-guided cryotherapy of renal cancer is an emerging alternative to surgical nephrectomy, particularly for those who cannot sustain the physical burden of surgery. It is well known that the outcome of this therapy depends on the accurate placement of the cryotherapy probe. Therefore, a robotic instrument guide may help physicians aim the cryotherapy probe precisely to maximize the efficacy of the treatment and avoid damage to critical surrounding structures. The objective of this paper was to propose a robotic instrument guide for orienting cryotherapy probes in image-guided cryotherapy of renal cancers. The authors propose a body-mounted robotic guide that is expected to be less susceptible to guidance errors caused by the patient's whole body motion.</p><p><strong>Methods: </strong>Keeping the device's minimal footprint in mind, the authors developed and validated a body-mounted, robotic instrument guide that can maintain the geometrical relationship between the device and the patient's body, even in the presence of the patient's frequent body motions. The guide can orient the cryotherapy probe with the skin incision point as the remote-center-of-motion. The authors' validation studies included an evaluation of the mechanical accuracy and position repeatability of the robotic instrument guide. The authors also performed a mock MRI-guided cryotherapy procedure with a phantom to compare the advantage of robotically assisted probe replacements over a free-hand approach, by introducing organ motions to investigate their effects on the accurate placement of the cryotherapy probe. Measurements collected for performance analysis included accuracy and time taken for probe placements. Multivariate analysis was performed to assess if either or both organ motion and the robotic guide impacted these measurements.</p><p><strong>Results: </strong>The mechanical accuracy and position repeatability of the probe placement using the robotic instrument guide were 0.3 and 0.1 mm, respectively, at a depth of 80 mm. The phantom test indicated that the accuracy of probe placement was significantly better with the robotic instrument guide (4.1 mm) than without the guide (6.3 mm, p<0.001), even in the presence of body motion. When independent organ motion was artificially added, in addition to body motion, the advantage of accurate probe placement using the robotic instrument guide disappeared statistically [i.e., 6.0 mm with the robotic guide and 5.9 mm without the robotic guide (p = 0.906)]. When the robotic instrument guide was used, the total time required to complete the procedure was reduced from 19.6 to 12.7 min (p<0.001). Multivariable analysis indicated that the robotic instrument guide, not the organ motion, was the cause of statistical significance. The statistical power the authors obtained was 88% in accuracy assessment and 99% higher in duration measurement.</p><p><strong>Conclusions: </strong>The body-mounted robotic instr
{"title":"Body-mounted robotic instrument guide for image-guided cryotherapy of renal cancer.","authors":"Nobuhiko Hata, Sang-Eun Song, Olutayo Olubiyi, Yasumichi Arimitsu, Kosuke Fujimoto, Takahisa Kato, Kemal Tuncali, Soichiro Tani, Junichi Tokuda","doi":"10.1118/1.4939875","DOIUrl":"10.1118/1.4939875","url":null,"abstract":"<p><strong>Purpose: </strong>Image-guided cryotherapy of renal cancer is an emerging alternative to surgical nephrectomy, particularly for those who cannot sustain the physical burden of surgery. It is well known that the outcome of this therapy depends on the accurate placement of the cryotherapy probe. Therefore, a robotic instrument guide may help physicians aim the cryotherapy probe precisely to maximize the efficacy of the treatment and avoid damage to critical surrounding structures. The objective of this paper was to propose a robotic instrument guide for orienting cryotherapy probes in image-guided cryotherapy of renal cancers. The authors propose a body-mounted robotic guide that is expected to be less susceptible to guidance errors caused by the patient's whole body motion.</p><p><strong>Methods: </strong>Keeping the device's minimal footprint in mind, the authors developed and validated a body-mounted, robotic instrument guide that can maintain the geometrical relationship between the device and the patient's body, even in the presence of the patient's frequent body motions. The guide can orient the cryotherapy probe with the skin incision point as the remote-center-of-motion. The authors' validation studies included an evaluation of the mechanical accuracy and position repeatability of the robotic instrument guide. The authors also performed a mock MRI-guided cryotherapy procedure with a phantom to compare the advantage of robotically assisted probe replacements over a free-hand approach, by introducing organ motions to investigate their effects on the accurate placement of the cryotherapy probe. Measurements collected for performance analysis included accuracy and time taken for probe placements. Multivariate analysis was performed to assess if either or both organ motion and the robotic guide impacted these measurements.</p><p><strong>Results: </strong>The mechanical accuracy and position repeatability of the probe placement using the robotic instrument guide were 0.3 and 0.1 mm, respectively, at a depth of 80 mm. The phantom test indicated that the accuracy of probe placement was significantly better with the robotic instrument guide (4.1 mm) than without the guide (6.3 mm, p<0.001), even in the presence of body motion. When independent organ motion was artificially added, in addition to body motion, the advantage of accurate probe placement using the robotic instrument guide disappeared statistically [i.e., 6.0 mm with the robotic guide and 5.9 mm without the robotic guide (p = 0.906)]. When the robotic instrument guide was used, the total time required to complete the procedure was reduced from 19.6 to 12.7 min (p<0.001). Multivariable analysis indicated that the robotic instrument guide, not the organ motion, was the cause of statistical significance. The statistical power the authors obtained was 88% in accuracy assessment and 99% higher in duration measurement.</p><p><strong>Conclusions: </strong>The body-mounted robotic instr","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"6 1","pages":"843-53"},"PeriodicalIF":3.8,"publicationDate":"2016-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1118/1.4939875","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89816824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new efficient and robust framework for European option pricing under continuous time asset models from the family of exponential semimartingale processes. We introduce B-spline interpolation theory to derivative pricing to provide an accurate closed-form representation of the option price under an inverse Fourier transform.We compare our method with some state-of-the-art option pricing methods, and demonstrate that it is extremely fast and accurate. This suggests a wide range of applications, including the use of more realistic asset models in high frequency trading. Examples considered in the paper include option pricing under asset models, including stochastic volatility and jumps, computation of the Greeks, and the inverse problem of cross-sectional calibration.
{"title":"A Novel Fourier Transform B-spline Method for Option Pricing","authors":"Gareth Gordon Haslip, V. Kaishev","doi":"10.2139/SSRN.2269370","DOIUrl":"https://doi.org/10.2139/SSRN.2269370","url":null,"abstract":"We present a new efficient and robust framework for European option pricing under continuous time asset models from the family of exponential semimartingale processes. We introduce B-spline interpolation theory to derivative pricing to provide an accurate closed-form representation of the option price under an inverse Fourier transform.We compare our method with some state-of-the-art option pricing methods, and demonstrate that it is extremely fast and accurate. This suggests a wide range of applications, including the use of more realistic asset models in high frequency trading. Examples considered in the paper include option pricing under asset models, including stochastic volatility and jumps, computation of the Greeks, and the inverse problem of cross-sectional calibration.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"19 1","pages":"41-74"},"PeriodicalIF":0.9,"publicationDate":"2015-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68048268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the derivation of a residual-based a posteriori error estimator and mesh-adaptation strategies for the space-time finite element approximation of parabolic problems with irregular data. Typical applications arise in the field of mathematical finance, where the Black–Scholes equation is used for modeling the pricing of European options. A conforming finite element discretization in space is combined with second-order time discretization by a damped Crank–Nicolson scheme for coping with data irregularities in the model. The a posteriori error analysis is developed within the general framework of the dual weighted residual method for sensitivity-based, goal-oriented error estimation and mesh optimization. In particular, the correct form of the dual problem with damping is considered.
{"title":"The Damped Crank–Nicolson Time-Marching Scheme for the Adaptive Solution of the Black–Scholes Equation","authors":"C. Goll, R. Rannacher, W. Wollner","doi":"10.21314/JCF.2015.301","DOIUrl":"https://doi.org/10.21314/JCF.2015.301","url":null,"abstract":"This paper is concerned with the derivation of a residual-based a posteriori error estimator and mesh-adaptation strategies for the space-time finite element approximation of parabolic problems with irregular data. Typical applications arise in the field of mathematical finance, where the Black–Scholes equation is used for modeling the pricing of European options. A conforming finite element discretization in space is combined with second-order time discretization by a damped Crank–Nicolson scheme for coping with data irregularities in the model. The a posteriori error analysis is developed within the general framework of the dual weighted residual method for sensitivity-based, goal-oriented error estimation and mesh optimization. In particular, the correct form of the dual problem with damping is considered.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2015-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fabian Crocce, Juho Häppölä, Jonas Kiessling, R. Tempone
We provide a bound for the error committed when using a Fourier method to price European options when the underlying follows an exponential levy dynamic. The price of the option is described by a partial integro-differential equation (PIDE). Applying a Fourier transformation to the PIDE yields an ordinary differential equation that can be solved analytically in terms of the characteristic exponent of the levy process. Then, a numerical inverse Fourier transform allows us to obtain the option price. We present a novel bound for the error and use this bound to set the parameters for the numerical method. We analyse the properties of the bound for a dissipative and pure-jump example. The bound presented is independent of the asymptotic behaviour of option prices at extreme asset prices. The error bound can be decomposed into a product of terms resulting from the dynamics and the option payoff, respectively. The analysis is supplemented by numerical examples that demonstrate results comparable to and superior to the existing literature.
{"title":"Error Analysis in Fourier Methods for Option Pricing","authors":"Fabian Crocce, Juho Häppölä, Jonas Kiessling, R. Tempone","doi":"10.21314/JCF.2016.327","DOIUrl":"https://doi.org/10.21314/JCF.2016.327","url":null,"abstract":"We provide a bound for the error committed when using a Fourier method to price European options when the underlying follows an exponential levy dynamic. The price of the option is described by a partial integro-differential equation (PIDE). Applying a Fourier transformation to the PIDE yields an ordinary differential equation that can be solved analytically in terms of the characteristic exponent of the levy process. Then, a numerical inverse Fourier transform allows us to obtain the option price. We present a novel bound for the error and use this bound to set the parameters for the numerical method. We analyse the properties of the bound for a dissipative and pure-jump example. The bound presented is independent of the asymptotic behaviour of option prices at extreme asset prices. The error bound can be decomposed into a product of terms resulting from the dynamics and the option payoff, respectively. The analysis is supplemented by numerical examples that demonstrate results comparable to and superior to the existing literature.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2015-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we clarify the relationships among popular methods for pricing European options based on the Fourier expansion of the payoff function (iFT method) and the simlified trapezoid rule. We suggest new variations that allow us to decrease the number of terms by a factor of between five and ten (when the iFT requires several dozen terms), or even by a factor of several dozen or a hundred (when the iFT may need thousands or millions of terms). We also give efficient recommendations for an (approximately) optimal choice of parameters for each numerical scheme.
{"title":"Efficient Variations of the Fourier Transform in Applications to Option Pricing","authors":"S. Boyarchenko, S. Levendorskii","doi":"10.21314/JCF.2014.277","DOIUrl":"https://doi.org/10.21314/JCF.2014.277","url":null,"abstract":"In this paper, we clarify the relationships among popular methods for pricing European options based on the Fourier expansion of the payoff function (iFT method) and the simlified trapezoid rule. We suggest new variations that allow us to decrease the number of terms by a factor of between five and ten (when the iFT requires several dozen terms), or even by a factor of several dozen or a hundred (when the iFT may need thousands or millions of terms). We also give efficient recommendations for an (approximately) optimal choice of parameters for each numerical scheme.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2014-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) and enables us to choose the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time for the initial condition are introduced. Finally, we present a novel reduced basis method that is able to use the initial condition of the parabolic partial differential equation as a parameter (function). The corresponding numerical results are shown.
{"title":"A Reduced Basis Method for Parabolic Partial Differential Equations with Parameter Functions and Application to Option Pricing","authors":"A. Mayerhofer, K. Urban","doi":"10.21314/JCF.2016.323","DOIUrl":"https://doi.org/10.21314/JCF.2016.323","url":null,"abstract":"We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) and enables us to choose the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time for the initial condition are introduced. Finally, we present a novel reduced basis method that is able to use the initial condition of the parabolic partial differential equation as a parameter (function). The corresponding numerical results are shown.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2014-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a general calibration problem for derivative pricing models, which we reformulate into a Bayesian framework to attain posterior distributions for model parameters. It is then shown how the posterior distribution can be used to estimate prices for exotic options. We apply the procedure to a discrete local volatility model and work in great detail through numerical examples to clarify the construction of Bayesian estimators and their robustness to the model specification, number of calibration products, noisy data and misspecification of the prior.
{"title":"Robust calibration of financial models using Bayesian estimators","authors":"A. Gupta, C. Reisinger","doi":"10.21314/JCF.2014.285","DOIUrl":"https://doi.org/10.21314/JCF.2014.285","url":null,"abstract":"We consider a general calibration problem for derivative pricing models, which we reformulate into a Bayesian framework to attain posterior distributions for model parameters. It is then shown how the posterior distribution can be used to estimate prices for exotic options. We apply the procedure to a discrete local volatility model and work in great detail through numerical examples to clarify the construction of Bayesian estimators and their robustness to the model specification, number of calibration products, noisy data and misspecification of the prior.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"3-36"},"PeriodicalIF":0.9,"publicationDate":"2014-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michalis Kapsos, Steve Zymler, Nicos Christofides, B. Rustem
The Omega Ratio is a recent performance measure. It captures both, the downside and upside potential of the constructed portfolio, while remaining consistent with utility maximization. In this paper, a new approach to compute the maximum Omega Ratio as a linear program is derived. While the Omega ratio is considered to be a non-convex function, we show an exact formulation in terms of a convex optimization problem, and transform it as a linear program. The convex reformulation for the Omega Ratio maximization is a direct analogue to mean-variance framework and the Sharpe Ratio maximization.
{"title":"Optimizing the Omega ratio using linear programming","authors":"Michalis Kapsos, Steve Zymler, Nicos Christofides, B. Rustem","doi":"10.21314/JCF.2014.283","DOIUrl":"https://doi.org/10.21314/JCF.2014.283","url":null,"abstract":"The Omega Ratio is a recent performance measure. It captures both, the downside and upside potential of the constructed portfolio, while remaining consistent with utility maximization. In this paper, a new approach to compute the maximum Omega Ratio as a linear program is derived. While the Omega ratio is considered to be a non-convex function, we show an exact formulation in terms of a convex optimization problem, and transform it as a linear program. The convex reformulation for the Omega Ratio maximization is a direct analogue to mean-variance framework and the Sharpe Ratio maximization.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"49-57"},"PeriodicalIF":0.9,"publicationDate":"2014-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Robust and accurate Monte Carlo simulation of (cross-) Gammas for Bermudan swaptions in the LIBOR market model","authors":"R. Korn, Qian Liang","doi":"10.21314/JCF.2014.290","DOIUrl":"https://doi.org/10.21314/JCF.2014.290","url":null,"abstract":"","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"87-110"},"PeriodicalIF":0.9,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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