Pub Date : 2011-10-27DOI: 10.4310/MAA.2012.V19.N2.A3
P. Lescot
We determine the algebra of isovectors for the Black--Scholes equation. As a consequence, we obtain some previously unknown families of transformations on the solutions.
我们确定了Black- Scholes方程的等向量代数。因此，我们得到了解上一些以前未知的变换族。
{"title":"Symmetries of the Black-Scholes equation","authors":"P. Lescot","doi":"10.4310/MAA.2012.V19.N2.A3","DOIUrl":"https://doi.org/10.4310/MAA.2012.V19.N2.A3","url":null,"abstract":"We determine the algebra of isovectors for the Black--Scholes equation. As a consequence, we obtain some previously unknown families of transformations on the solutions.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131071459","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 maximizing the discounted utility of dividend payments of an insurance company whose reserves are modeled as a classical Cram'er-Lundberg risk process. We investigate this optimization problem under the constraint that dividend rate is bounded. We prove that the value function fulfills the Hamilton-Jacobi-Bellman equation and we identify the optimal dividend strategy.
{"title":"Optimizing expected utility of dividend payments for a Cram'er-Lundberg risk proces","authors":"Z. Palmowski, Sebastian Baran","doi":"10.4064/AM2333-5-2017","DOIUrl":"https://doi.org/10.4064/AM2333-5-2017","url":null,"abstract":"We consider the problem of maximizing the discounted utility of dividend payments of an insurance company whose reserves are modeled as a classical Cram'er-Lundberg risk process. We investigate this optimization problem under the constraint that dividend rate is bounded. We prove that the value function fulfills the Hamilton-Jacobi-Bellman equation and we identify the optimal dividend strategy.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"136 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114544788","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 LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a L'evy-driven LIBOR model and aim at developing accurate and efficient log-L'evy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps, swaptions and sticky ratchet caps show that the approximations perform very well. In addition, we also consider the log-L'evy approximation of annuities, which offers good approximations for high volatility regimes.
{"title":"Efficient and accurate log-L'evy approximations to L'evy driven LIBOR models","authors":"A. Papapantoleon, J. Schoenmakers, D. Skovmand","doi":"10.21314/JCF.2012.250","DOIUrl":"https://doi.org/10.21314/JCF.2012.250","url":null,"abstract":"The LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a L'evy-driven LIBOR model and aim at developing accurate and efficient log-L'evy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps, swaptions and sticky ratchet caps show that the approximations perform very well. In addition, we also consider the log-L'evy approximation of annuities, which offers good approximations for high volatility regimes.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130214003","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 : 2011-01-05DOI: 10.1007/978-3-642-25746-9_12
Marie Bernhart, H. Pham, P. Tankov, X. Warin
{"title":"Swing Options Valuation: a BSDE with Constrained Jumps Approach","authors":"Marie Bernhart, H. Pham, P. Tankov, X. Warin","doi":"10.1007/978-3-642-25746-9_12","DOIUrl":"https://doi.org/10.1007/978-3-642-25746-9_12","url":null,"abstract":"","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133673317","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 a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.
{"title":"On optimal arbitrage","authors":"Daniel Fernholz, I. Karatzas","doi":"10.1214/09-AAP642","DOIUrl":"https://doi.org/10.1214/09-AAP642","url":null,"abstract":"In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129535469","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 proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension of some useful techniques from these two fields to a much wider class of problems. In particular, combining certain finite-dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance, we are able to close the duality gap in some situations where traditional topological arguments fail.
{"title":"Convex duality in stochastic programming and mathematical finance","authors":"T. Pennanen","doi":"10.18452/3036","DOIUrl":"https://doi.org/10.18452/3036","url":null,"abstract":"This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension of some useful techniques from these two fields to a much wider class of problems. In particular, combining certain finite-dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance, we are able to close the duality gap in some situations where traditional topological arguments fail.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129274560","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}
Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed.
{"title":"Calculation of aggregate loss distributions","authors":"P. Shevchenko","doi":"10.21314/JOP.2010.077","DOIUrl":"https://doi.org/10.21314/JOP.2010.077","url":null,"abstract":"Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115482988","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 : 2010-04-13DOI: 10.1007/978-3-642-03479-4_9
P. Imkeller, Gonccalo dos Reis, Jianing Zhang
{"title":"Results on Numerics for FBSDE with Drivers of Quadratic Growth","authors":"P. Imkeller, Gonccalo dos Reis, Jianing Zhang","doi":"10.1007/978-3-642-03479-4_9","DOIUrl":"https://doi.org/10.1007/978-3-642-03479-4_9","url":null,"abstract":"","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117046762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new Quantization algorithm for the approximation of inhomogeneous random walks, which are the key terms for the valuation of CDO-tranches in latent factor models. This approach is based on a dual quantization operator which posses an intrinsic stationarity and therefore automatically leads to a second order error bound for the weak approximation. We illustrate the numerical performance of our methods in case of the approximation of the conditional tranche function of synthetic CDO products and draw comparisons to the approximations achieved by the saddlepoint method and Stein's method.
{"title":"Dual Quantization for random walks with application to credit derivatives","authors":"G. Pagès, B. Wilbertz","doi":"10.21314/JCF.2012.239","DOIUrl":"https://doi.org/10.21314/JCF.2012.239","url":null,"abstract":"We propose a new Quantization algorithm for the approximation of inhomogeneous random walks, which are the key terms for the valuation of CDO-tranches in latent factor models. This approach is based on a dual quantization operator which posses an intrinsic stationarity and therefore automatically leads to a second order error bound for the weak approximation. We illustrate the numerical performance of our methods in case of the approximation of the conditional tranche function of synthetic CDO products and draw comparisons to the approximations achieved by the saddlepoint method and Stein's method.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123962299","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 : 2008-02-29DOI: 10.1142/S0219024913500143
S. Malham, Anke Wiese
The transition probability of a Cox-Ingersoll-Ross process can be represented by a non-central chi-square density. First we prove a new representation for the central chi-square density based on sums of powers of generalized Gaussian random variables. Second we prove Marsaglia's polar method extends to this distribution, providing a simple, exact, robust and efficient acceptance-rejection method for generalized Gaussian sampling and thus central chi-square sampling. Third we derive a simple, high-accuracy, robust and efficient direct inversion method for generalized Gaussian sampling based on the Beasley-Springer-Moro method. Indeed the accuracy of the approximation to the inverse cumulative distribution function is to the tenth decimal place. We then apply our methods to non-central chi-square variance sampling in the Heston model. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. Using the additivity property of the chi-square distribution, our methods apply in all parameter regimes.
{"title":"Chi-square simulation of the CIR process and the Heston model","authors":"S. Malham, Anke Wiese","doi":"10.1142/S0219024913500143","DOIUrl":"https://doi.org/10.1142/S0219024913500143","url":null,"abstract":"The transition probability of a Cox-Ingersoll-Ross process can be represented by a non-central chi-square density. First we prove a new representation for the central chi-square density based on sums of powers of generalized Gaussian random variables. Second we prove Marsaglia's polar method extends to this distribution, providing a simple, exact, robust and efficient acceptance-rejection method for generalized Gaussian sampling and thus central chi-square sampling. Third we derive a simple, high-accuracy, robust and efficient direct inversion method for generalized Gaussian sampling based on the Beasley-Springer-Moro method. Indeed the accuracy of the approximation to the inverse cumulative distribution function is to the tenth decimal place. We then apply our methods to non-central chi-square variance sampling in the Heston model. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. Using the additivity property of the chi-square distribution, our methods apply in all parameter regimes.","PeriodicalId":197400,"journal":{"name":"arXiv: Computational Finance","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128262744","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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