Ying-Ying Zhang, Hong-Kui Pang, Liming Feng, X. Jin
We consider option pricing problems in the stochastic volatility jump diffusion model with correlated and contemporaneous jumps in both the return and the variance processes (SVCJ). The option value function solves a partial integro-differential equation (PIDE). We discretize this PIDE in space by the quadratic FE method and integrate the resulting ordinary differential equation in time by an implicit-explicit Euler based extrapolation scheme. The coefficient matrix of the resulting linear systems is block penta-diagonal with penta-diagonal blocks. The preconditioned bi-conjugate gradient stabilized (PBiCGSTAB) method is used to solve the linear systems. According to the structure of the coefficient matrix, several preconditioners are implemented and compared. The performance of preconditioning techniques for solving block-tridiagonal systems resulting from the linear FE discretization of the PIDE is also investigated. The combination of the quadratic FE for spatial discretization, the extrapolation scheme for time discretization, and the PBiCGSTAB method with an appropriate preconditioner is found to be very efficient for solving the option pricing problems in the SVCJ model. Compared to the standard second order linear finite element method combined with the popular successive over-relaxation (SOR) linear system solver, the proposed method reduces computational time by about twenty times at the accuracy level of 1 cent and more than fifty times at the accuracy level of 0.1 cent for the barrier option example tested in the paper.
{"title":"Quadratic finite element and preconditioning methods for options pricing in the SVCJ model","authors":"Ying-Ying Zhang, Hong-Kui Pang, Liming Feng, X. Jin","doi":"10.21314/JCF.2014.287","DOIUrl":"https://doi.org/10.21314/JCF.2014.287","url":null,"abstract":"We consider option pricing problems in the stochastic volatility jump diffusion model with correlated and contemporaneous jumps in both the return and the variance processes (SVCJ). The option value function solves a partial integro-differential equation (PIDE). We discretize this PIDE in space by the quadratic FE method and integrate the resulting ordinary differential equation in time by an implicit-explicit Euler based extrapolation scheme. The coefficient matrix of the resulting linear systems is block penta-diagonal with penta-diagonal blocks. The preconditioned bi-conjugate gradient stabilized (PBiCGSTAB) method is used to solve the linear systems. According to the structure of the coefficient matrix, several preconditioners are implemented and compared. The performance of preconditioning techniques for solving block-tridiagonal systems resulting from the linear FE discretization of the PIDE is also investigated. The combination of the quadratic FE for spatial discretization, the extrapolation scheme for time discretization, and the PBiCGSTAB method with an appropriate preconditioner is found to be very efficient for solving the option pricing problems in the SVCJ model. Compared to the standard second order linear finite element method combined with the popular successive over-relaxation (SOR) linear system solver, the proposed method reduces computational time by about twenty times at the accuracy level of 1 cent and more than fifty times at the accuracy level of 0.1 cent for the barrier option example tested in the paper.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"3-30"},"PeriodicalIF":0.9,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702497","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":"TR-BDF2 for fast stable American option pricing","authors":"Fabien Le Floc’h","doi":"10.21314/JCF.2014.289","DOIUrl":"https://doi.org/10.21314/JCF.2014.289","url":null,"abstract":"","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"31-56"},"PeriodicalIF":0.9,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702525","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 develops a new scheme for improving an approximation method of a probability density function, which is inspired by the idea in best approximation in an inner product space. Moreover, we applies “Dykstra’s cyclic projections algorithm” for its implementation. Numerical examples for application to an asymptotic expansion method in option pricing demonstrate the effectiveness of our scheme under Black-Scholes and SABR models.
{"title":"A New Improvement Scheme on Approximation Methods for Probability Density Functions","authors":"Akihiko Takahashi, Yukihiro Tsuzuki","doi":"10.2139/ssrn.2205662","DOIUrl":"https://doi.org/10.2139/ssrn.2205662","url":null,"abstract":"This paper develops a new scheme for improving an approximation method of a probability density function, which is inspired by the idea in best approximation in an inner product space. Moreover, we applies “Dykstra’s cyclic projections algorithm” for its implementation. Numerical examples for application to an asymptotic expansion method in option pricing demonstrate the effectiveness of our scheme under Black-Scholes and SABR models.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67986400","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 approximate a Levy process by either truncating its small jumps or replacing them by a Brownian motion with the same variance. Then we derive the errors resulting from these approximations for some exotic options (Asian, barrier, lookback and American). We also propose a simple method to evaluate these options using the approximated Levy process.
{"title":"Simulation of Lévy processes and option pricing","authors":"E. H. A. Dia","doi":"10.21314/JCF.2013.260","DOIUrl":"https://doi.org/10.21314/JCF.2013.260","url":null,"abstract":"We approximate a Levy process by either truncating its small jumps or replacing them by a Brownian motion with the same variance. Then we derive the errors resulting from these approximations for some exotic options (Asian, barrier, lookback and American). We also propose a simple method to evaluate these options using the approximated Levy process.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"41-69"},"PeriodicalIF":0.9,"publicationDate":"2013-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702257","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}
A compound option (the mother option) gives the holder the right, but not the obligation, to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we consider the problem of pricing American-type compound options when the underlying dynamics follow Heston’s stochastic volatility and with stochastic interest rate driven by Cox–Ingersoll–Ross processes. We use a partial differential equation (PDE) approach to obtain a numerical solution. The problem is formulated as the solution to a two-pass free-boundary PDE problem, which is solved via a sparse grid approach and is found to be accurate and efficient compared with the results from a benchmark solution based on a least-squares Monte Carlo simulation combined with the projected successive over-relaxation method.
{"title":"The evaluation of American compound option prices under stochastic volatility and stochastic interest rates","authors":"C. Chiarella, Boda Kang","doi":"10.21314/JCF.2013.264","DOIUrl":"https://doi.org/10.21314/JCF.2013.264","url":null,"abstract":"A compound option (the mother option) gives the holder the right, but not the obligation, to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we consider the problem of pricing American-type compound options when the underlying dynamics follow Heston’s stochastic volatility and with stochastic interest rate driven by Cox–Ingersoll–Ross processes. We use a partial differential equation (PDE) approach to obtain a numerical solution. The problem is formulated as the solution to a two-pass free-boundary PDE problem, which is solved via a sparse grid approach and is found to be accurate and efficient compared with the results from a benchmark solution based on a least-squares Monte Carlo simulation combined with the projected successive over-relaxation method.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"71-92"},"PeriodicalIF":0.9,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701823","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 pricing of a special kind of options, the so-called autocallables, which may terminate prior to maturity due to a barrier condition on one or several underlyings. Standard Monte Carlo (MC) algorithms work well for pricing these options but they do not behave stable with respect to numerical differentiation. Hence, to calculate sensitivities, one would typically resort to regularized differentiation schemes or derive an algorithm for directly calculating the derivative. In this work we present an alternative solution and show how to adapt a MC algorithm in such a way that its results can be stably differentiated by simple finite differences. Our main tool is the one-step survival idea of Glasserman and Staum which we combine with a technique known as GHK Importance Sampling for treating multiple underlyings. Besides the stability with respect to differentiation our new algorithm also possesses a significantly reduced variance and does not require evaluations of multivariate cumulative normal distributions.
{"title":"A Monte Carlo pricing algorithm for autocallables that allows for stable differentiation","authors":"T. Alm, B. Harrach, Daphne Harrach, Marco Keller","doi":"10.21314/JCF.2013.265","DOIUrl":"https://doi.org/10.21314/JCF.2013.265","url":null,"abstract":"We consider the pricing of a special kind of options, the so-called autocallables, which may terminate prior to maturity due to a barrier condition on one or several underlyings. Standard Monte Carlo (MC) algorithms work well for pricing these options but they do not behave stable with respect to numerical differentiation. Hence, to calculate sensitivities, one would typically resort to regularized differentiation schemes or derive an algorithm for directly calculating the derivative. In this work we present an alternative solution and show how to adapt a MC algorithm in such a way that its results can be stably differentiated by simple finite differences. Our main tool is the one-step survival idea of Glasserman and Staum which we combine with a technique known as GHK Importance Sampling for treating multiple underlyings. Besides the stability with respect to differentiation our new algorithm also possesses a significantly reduced variance and does not require evaluations of multivariate cumulative normal distributions.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"17 1","pages":"43-70"},"PeriodicalIF":0.9,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701827","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 the execution cost problem, an investor wants to minimize the total expected cost and risk in the execution of a portfolio of risky assets to achieve desired positions. A major source of the execution cost comes from price impacts of both the investor’s own trades and other concurrent institutional trades. Indeed price impact of large trades have been considered as one of the main reasons for fat tails of the short term return’s probability distribution function. However, current models in the literature on the execution cost problem typically assume normal distributions. This assumption fails to capture the characteristics of tail distributions due to institutional trades. In this paper we provide arguments that compound jump diffusion processes naturally model uncertain price impact of other large trades. This jump diffusion model includes two compound Poisson processes where random jump amplitudes capture uncertain permanent price impact of other large buy and sell trades. Using stochastic dynamic programming, we derive analytical solutions for minimizing the expected execution cost under discrete jump diffusion models. Our results indicate that, when the expected market price change is nonzero, likely due to large trades, assumptions on the market price model, and values of mean and covariance of the market price change can have significant impact on the optimal execution strategy. Using simulations, we computationally illustrate minimum CVaR execution strategies under different models. Furthermore, we analyze qualitative and quantitative differences of the expected execution cost and risk between optimal execution strategies, determined under a multiplicative jump diffusion model and an additive jump diffusion model.
{"title":"Optimal Execution Under Jump Models For Uncertain Price Impact","authors":"S. Moazeni, T. Coleman, Yuying Li","doi":"10.21314/JCF.2013.267","DOIUrl":"https://doi.org/10.21314/JCF.2013.267","url":null,"abstract":"In the execution cost problem, an investor wants to minimize the total expected cost and risk in the execution of a portfolio of risky assets to achieve desired positions. A major source of the execution cost comes from price impacts of both the investor’s own trades and other concurrent institutional trades. Indeed price impact of large trades have been considered as one of the main reasons for fat tails of the short term return’s probability distribution function. However, current models in the literature on the execution cost problem typically assume normal distributions. This assumption fails to capture the characteristics of tail distributions due to institutional trades. In this paper we provide arguments that compound jump diffusion processes naturally model uncertain price impact of other large trades. This jump diffusion model includes two compound Poisson processes where random jump amplitudes capture uncertain permanent price impact of other large buy and sell trades. Using stochastic dynamic programming, we derive analytical solutions for minimizing the expected execution cost under discrete jump diffusion models. Our results indicate that, when the expected market price change is nonzero, likely due to large trades, assumptions on the market price model, and values of mean and covariance of the market price change can have significant impact on the optimal execution strategy. Using simulations, we computationally illustrate minimum CVaR execution strategies under different models. Furthermore, we analyze qualitative and quantitative differences of the expected execution cost and risk between optimal execution strategies, determined under a multiplicative jump diffusion model and an additive jump diffusion model.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"16 1","pages":"35-78"},"PeriodicalIF":0.9,"publicationDate":"2013-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701834","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}
The focus of this work is on the problem of tracking parameters describing both the stochastic discount factor and the objective / real-world measure dynamically, with the aim of monitoring value at risk or other related diagnostics of interest. The methodology presented incorporates information from derivative prices as well as from the underlying instrument’s price over time in order to perform on-line parameter inference. We construct a parametric model of the stochastic discount factor which is introduced based on empirical results in the literature (Aı̈t-Sahalia and Lo, 2000; Jackwerth, 2000; Rosenberg and Engle, 2002, for example). This is used in a sequential Monte Carlo algorithm for tracking the parameters of this and of an objective density over time. Further, two new techniques for pricing European options in the framework are discussed. In applying this approach to price data, Variance Gamma and Normal Inverse Gaussian models of the underlying price process have been discussed. These are for illustrative purposes and other models could easily be also considered. Both models appear to track realistically; detailed results are presented for the Variance Gamma model. These cover the value at risk estimates, expected price change estimates and parameter estimates.
{"title":"Tracking value-at-risk through derivative prices","authors":"S. I. Hill","doi":"10.21314/JCF.2013.269","DOIUrl":"https://doi.org/10.21314/JCF.2013.269","url":null,"abstract":"The focus of this work is on the problem of tracking parameters describing both the stochastic discount factor and the objective / real-world measure dynamically, with the aim of monitoring value at risk or other related diagnostics of interest. The methodology presented incorporates information from derivative prices as well as from the underlying instrument’s price over time in order to perform on-line parameter inference. We construct a parametric model of the stochastic discount factor which is introduced based on empirical results in the literature (Aı̈t-Sahalia and Lo, 2000; Jackwerth, 2000; Rosenberg and Engle, 2002, for example). This is used in a sequential Monte Carlo algorithm for tracking the parameters of this and of an objective density over time. Further, two new techniques for pricing European options in the framework are discussed. In applying this approach to price data, Variance Gamma and Normal Inverse Gaussian models of the underlying price process have been discussed. These are for illustrative purposes and other models could easily be also considered. Both models appear to track realistically; detailed results are presented for the Variance Gamma model. These cover the value at risk estimates, expected price change estimates and parameter estimates.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"44 1","pages":"79-121"},"PeriodicalIF":0.9,"publicationDate":"2013-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701889","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":"Pricing synthetic collateralized debt obligations based on exponential approximations to the payoff function","authors":"I. Iscoe, K. Jackson, A. Kreinin, Xiofang Ma","doi":"10.21314/JCF.2013.271","DOIUrl":"https://doi.org/10.21314/JCF.2013.271","url":null,"abstract":"","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"16 1","pages":"127-150"},"PeriodicalIF":0.9,"publicationDate":"2013-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701950","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}
The derivatives of option prices with respect to underlying parameters are commonly referred to as Greeks, and they measure the sensitivities of option prices to these parameters. When the closed-form solutions for option prices do not exist and the discounted payoff functions of the options are not sufficiently smooth, estimating Greeks is computationally challenging and could be a burdensome task for high-dimensional problems in particular. The aim of this paper is to develop a new method for estimating option Greeks by using random parameters and least-squares regression. Our approach has several attractive features. First, just like the finite-difference method it is easy to implement and does not require explicit knowledge of the probability density function and the pathwise derivative of the underlying stochastic model. Second, it can be applied to options with discontinuous discounted payoffs as well as options with continuous discounted payoffs. Third, and most importantly, we can estimate multiple derivatives simultaneously. The performance of our approach is illustrated for a variety of examples with up to fifty Greeks estimated simultaneously. The algorithm is able to produce computationally efficient results with good accuracy.
{"title":"Estimating multiple option Greeks simultaneously using random parameter regression","authors":"Haifeng Fu, Xing Jin, G. Pan, Yanrong Yang","doi":"10.21314/JCF.2012.241","DOIUrl":"https://doi.org/10.21314/JCF.2012.241","url":null,"abstract":"The derivatives of option prices with respect to underlying parameters are commonly referred to as Greeks, and they measure the sensitivities of option prices to these parameters. When the closed-form solutions for option prices do not exist and the discounted payoff functions of the options are not sufficiently smooth, estimating Greeks is computationally challenging and could be a burdensome task for high-dimensional problems in particular. The aim of this paper is to develop a new method for estimating option Greeks by using random parameters and least-squares regression. Our approach has several attractive features. First, just like the finite-difference method it is easy to implement and does not require explicit knowledge of the probability density function and the pathwise derivative of the underlying stochastic model. Second, it can be applied to options with discontinuous discounted payoffs as well as options with continuous discounted payoffs. Third, and most importantly, we can estimate multiple derivatives simultaneously. The performance of our approach is illustrated for a variety of examples with up to fifty Greeks estimated simultaneously. The algorithm is able to produce computationally efficient results with good accuracy.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"16 1","pages":"85-118"},"PeriodicalIF":0.9,"publicationDate":"2012-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701205","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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