We provide an efficient and unbiased Monte-Carlo simulation for the computation of bond prices in a structural default model with jumps. The algorithm requires the evaluation of integrals with the density of the firstpassage time of a Brownian bridge as the integrand. Metwally and Atiya (2002) suggest an approximation of these integrals. We improve this approximation in terms of precision. From a modeler's point of view, we show that a structural model with jumps is able to endogenously generate stochastic recovery rates. It is well known that allowing a sudden default by a jump results in a positive limit of credit spreads at the short end of the term structure. We provide an explicit formula for this limit, depending only on the Levy measure of the logarithm of the firm-value process, the recovery rate, and the distance to default.
{"title":"Pricing corporate bonds in an arbitrary jump-diffusion model based on an improved Brownian-bridge algorithm","authors":"J. Ruf, M. Scherer","doi":"10.21314/JCF.2011.235","DOIUrl":"https://doi.org/10.21314/JCF.2011.235","url":null,"abstract":"We provide an efficient and unbiased Monte-Carlo simulation for the computation of bond prices in a structural default model with jumps. The algorithm requires the evaluation of integrals with the density of the firstpassage time of a Brownian bridge as the integrand. Metwally and Atiya (2002) suggest an approximation of these integrals. We improve this approximation in terms of precision. From a modeler's point of view, we show that a structural model with jumps is able to endogenously generate stochastic recovery rates. It is well known that allowing a sudden default by a jump results in a positive limit of credit spreads at the short end of the term structure. We provide an explicit formula for this limit, depending only on the Levy measure of the logarithm of the firm-value process, the recovery rate, and the distance to default.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"127-145"},"PeriodicalIF":0.9,"publicationDate":"2011-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701146","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 introduce a new approach for pricing energy derivatives known as tolling agreement contracts. The pricing problem is reduced to a linear program. We prove that the optimal operating strategy for a power plant can be expressed through optimal exercise boundaries (similar to the exercise boundaries for American options). We find the boundaries as a byproduct of the pricing algorithm. The suggested approach can incorporate various real world power plant operational constraints. We demonstrate computational efficiency of the algorithm by pricing 1and 10-year tolling agreement contracts.
{"title":"Pricing Energy Derivatives by Linear Programming: Tolling Agreement Contracts","authors":"Valeriy Ryabchenko, S. Uryasev","doi":"10.21314/JCF.2011.236","DOIUrl":"https://doi.org/10.21314/JCF.2011.236","url":null,"abstract":"We introduce a new approach for pricing energy derivatives known as tolling agreement contracts. The pricing problem is reduced to a linear program. We prove that the optimal operating strategy for a power plant can be expressed through optimal exercise boundaries (similar to the exercise boundaries for American options). We find the boundaries as a byproduct of the pricing algorithm. The suggested approach can incorporate various real world power plant operational constraints. We demonstrate computational efficiency of the algorithm by pricing 1and 10-year tolling agreement contracts.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"73-126"},"PeriodicalIF":0.9,"publicationDate":"2011-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67701154","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 conventional control variate method proposed by Kemna and Vorst (1990) to evaluate Asian options under the Black-Scholes model can be interpreted as a particular selection of linear martingale controls. We generalize the constant control parameter into a control process to gain more reduction on variance. By means of an option price approximation, we construct a martingale control variate method, which outperforms the conventional control variate method. It is straightforward to extend such linear control to a nonlinear situation such as the American Asian option problem. From the variance analysis of martingales, the performance of control variate methods depends on the distance between the approximate martingale and the optimal martingale. This measure becomes helpful for the design of control variate methods for complex problems such as Asian option under stochastic volatility models. We demonstrate multiple choices of controls and test them under MC/QMC (Monte Carlo/ Quasi Monte Carlo)- simulations. QMC methods work signicantly well after adding a control, the variance reduction ratios increase to 260 times for randomized QMC compared with 60 times for MC simulations with a control.
{"title":"Generalized Control Variate Methods for Pricing Asian Options","authors":"Chuan-Hsiang Han, Yongzeng Lai","doi":"10.21314/JCF.2010.212","DOIUrl":"https://doi.org/10.21314/JCF.2010.212","url":null,"abstract":"The conventional control variate method proposed by Kemna and Vorst (1990) to evaluate Asian options under the Black-Scholes model can be interpreted as a particular selection of linear martingale controls. We generalize the constant control parameter into a control process to gain more reduction on variance. By means of an option price approximation, we construct a martingale control variate method, which outperforms the conventional control variate method. It is straightforward to extend such linear control to a nonlinear situation such as the American Asian option problem. From the variance analysis of martingales, the performance of control variate methods depends on the distance between the approximate martingale and the optimal martingale. This measure becomes helpful for the design of control variate methods for complex problems such as Asian option under stochastic volatility models. We demonstrate multiple choices of controls and test them under MC/QMC (Monte Carlo/ Quasi Monte Carlo)- simulations. QMC methods work signicantly well after adding a control, the variance reduction ratios increase to 260 times for randomized QMC compared with 60 times for MC simulations with a control.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"87-118"},"PeriodicalIF":0.9,"publicationDate":"2010-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67700579","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 develop methods for estimating price sensitivities by simulation for Levydriven models. The methods combine pathwise derivatives and likelihood ratio method estimators with alternative approaches to approximating and simulating Levy processes. We develop estimators based on exact sampling of increments, time-change representations of Levy processes, saddlepoint approximations to the score functions of the increments, compound Poisson approximations and compound Poisson approximations with Brownian approximations to small jumps. We discuss the relative merits of these various alternatives, both in theory and in practice, and we illustrate their use through examples.
{"title":"Estimating Greeks in simulating Lévy-driven models","authors":"P. Glasserman, Zongjian Liu","doi":"10.21314/JCF.2010.210","DOIUrl":"https://doi.org/10.21314/JCF.2010.210","url":null,"abstract":"We develop methods for estimating price sensitivities by simulation for Levydriven models. The methods combine pathwise derivatives and likelihood ratio method estimators with alternative approaches to approximating and simulating Levy processes. We develop estimators based on exact sampling of increments, time-change representations of Levy processes, saddlepoint approximations to the score functions of the increments, compound Poisson approximations and compound Poisson approximations with Brownian approximations to small jumps. We discuss the relative merits of these various alternatives, both in theory and in practice, and we illustrate their use through examples.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"3-56"},"PeriodicalIF":0.9,"publicationDate":"2010-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67699899","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":"Potential Future Exposure Calculations of Multi-Asset Exotic Products using the Stochastic Mesh Method","authors":"Leslie Ng, Dave Peterson, A. Rodriguez","doi":"10.21314/JCF.2010.211","DOIUrl":"https://doi.org/10.21314/JCF.2010.211","url":null,"abstract":"","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"119-153"},"PeriodicalIF":0.9,"publicationDate":"2010-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67700477","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}
Marginal risk that represents the risk contribution of an individual asset is an important criterion in portfolio selection and risk management. In the literature, however, the measure of marginal risk has been only employed in ex post analysis of a portfolio policy, and the control of marginal risk is achieved usually via position diversification that simply imposes upper bounds on the portfolio position without considering the effect of correlations of asset returns in risk diversification. We investigate in this paper a new optimal portfolio selection problem with direct (relative) marginal risk control in the mean-variance framework, accounting for the correlations of asset returns. The resulting optimization model, however, is a notorious non-convex quadratically constrained quadratic programming problem. By exploiting the special structure of the problems, we propose an efficient branch-and-bound solution method to achieve a global optimality in which convex quadratic relaxation subproblems with second-order cone constraints are formulated to generate a tight lower bound. Empirical study shows that the model with marginal risk control is a suitable analytical tool for active portfolio risk management and demonstrates several preferable features of this new model to the traditional mean-variance model in risk management. The method is tested and compared with the commercial global optimization solver BARON for portfolio optimization problems with up to hundreds of assets and tens of marginal risk constraints.
{"title":"Portfolio selection with marginal risk control","authors":"Shushang Zhu, Duan Li, Xiaoling Sun","doi":"10.21314/JCF.2010.213","DOIUrl":"https://doi.org/10.21314/JCF.2010.213","url":null,"abstract":"Marginal risk that represents the risk contribution of an individual asset is an important criterion in portfolio selection and risk management. In the literature, however, the measure of marginal risk has been only employed in ex post analysis of a portfolio policy, and the control of marginal risk is achieved usually via position diversification that simply imposes upper bounds on the portfolio position without considering the effect of correlations of asset returns in risk diversification. We investigate in this paper a new optimal portfolio selection problem with direct (relative) marginal risk control in the mean-variance framework, accounting for the correlations of asset returns. The resulting optimization model, however, is a notorious non-convex quadratically constrained quadratic programming problem. By exploiting the special structure of the problems, we propose an efficient branch-and-bound solution method to achieve a global optimality in which convex quadratic relaxation subproblems with second-order cone constraints are formulated to generate a tight lower bound. Empirical study shows that the model with marginal risk control is a suitable analytical tool for active portfolio risk management and demonstrates several preferable features of this new model to the traditional mean-variance model in risk management. The method is tested and compared with the commercial global optimization solver BARON for portfolio optimization problems with up to hundreds of assets and tens of marginal risk constraints.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"3-28"},"PeriodicalIF":0.9,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67700654","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}
Random sampling from a multivariate normal distribution is essential for Monte Carlo simulations in many credit risk models. For a portfolio of N obligors, standard methods usually require O(N) calculations to get one random sample. In many applications, the correlation matrix has a block structure that, as we show, can be converted to a “quasi-factor” model. As a result, the cost to get one sample can be reduced to O(N). Such a conversion also enables us to check whether a user-defined “correlation” matrix is positive semidefinite and “fix” it if necessary in an efficient manner. Disclaimer: The models and analyses presented here are exclusively part of a quantitative research effort intended to improve the computation time of Monte Carlo simulations when we deal with a correlation matrix that has a block structure. The views expressed in this paper are the authors’ own and do not necessarily represent the views of Standard & Poor’s. Furthermore, no inferences should be made with regard to Standard & Poor’s credit ratings or any current or future criteria or models used in the ratings process for credit portfolios or any type of financial security.
{"title":"Correlation matrix with block structure and efficient sampling methods","authors":"Jinggang Huang, Liming Yang","doi":"10.21314/JCF.2010.216","DOIUrl":"https://doi.org/10.21314/JCF.2010.216","url":null,"abstract":"Random sampling from a multivariate normal distribution is essential for Monte Carlo simulations in many credit risk models. For a portfolio of N obligors, standard methods usually require O(N) calculations to get one random sample. In many applications, the correlation matrix has a block structure that, as we show, can be converted to a “quasi-factor” model. As a result, the cost to get one sample can be reduced to O(N). Such a conversion also enables us to check whether a user-defined “correlation” matrix is positive semidefinite and “fix” it if necessary in an efficient manner. Disclaimer: The models and analyses presented here are exclusively part of a quantitative research effort intended to improve the computation time of Monte Carlo simulations when we deal with a correlation matrix that has a block structure. The views expressed in this paper are the authors’ own and do not necessarily represent the views of Standard & Poor’s. Furthermore, no inferences should be made with regard to Standard & Poor’s credit ratings or any current or future criteria or models used in the ratings process for credit portfolios or any type of financial security.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"81-94"},"PeriodicalIF":0.9,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67700485","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 propose a model for jointly predicting stock price and volume at the tick-by-tick level. We model the investors’ preferences by a random utility model that incorporates several important behavioral biases such as the status quo bias, the disposition effect, and loss-aversion. Our model is a logistic regression model with incomplete information; consequently, we are unable to use the maximum likelihood estimation method and have to resort to Markov Chain Monte Carlo (MCMC) to estimate the model parameters. Moreover, the constraint that the volume predicted by the MCMC model exactly match the observed volume vt introduces serial correlation in the stock price. Thus, the standard MCMC methods for calibrating parameters do not work. We develop new modifications of the Metropolis-within-Gibbs method to estimate the parameters in our model. Our primary goal in developing this model is to predict the market impact function and VWAP (volume weighted average price) of individual stocks.
{"title":"A behavioural finance-based tick-by-tick model for price and volume","authors":"G. Iyengar, Alfred Ka, Chun-mei Ma","doi":"10.21314/JCF.2010.215","DOIUrl":"https://doi.org/10.21314/JCF.2010.215","url":null,"abstract":"We propose a model for jointly predicting stock price and volume at the tick-by-tick level. We model the investors’ preferences by a random utility model that incorporates several important behavioral biases such as the status quo bias, the disposition effect, and loss-aversion. Our model is a logistic regression model with incomplete information; consequently, we are unable to use the maximum likelihood estimation method and have to resort to Markov Chain Monte Carlo (MCMC) to estimate the model parameters. Moreover, the constraint that the volume predicted by the MCMC model exactly match the observed volume vt introduces serial correlation in the stock price. Thus, the standard MCMC methods for calibrating parameters do not work. We develop new modifications of the Metropolis-within-Gibbs method to estimate the parameters in our model. Our primary goal in developing this model is to predict the market impact function and VWAP (volume weighted average price) of individual stocks.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"14 1","pages":"57-80"},"PeriodicalIF":0.9,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67700398","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":"Unbiased Monte Carlo valuation of lookback, swing and barrier options with continuous monitoring under variance gamma models","authors":"Martin Becker","doi":"10.21314/JCF.2010.218","DOIUrl":"https://doi.org/10.21314/JCF.2010.218","url":null,"abstract":"","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"13 1","pages":"35-61"},"PeriodicalIF":0.9,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67700694","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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