This paper investigates optimal portfolio strategies in a market where the drift is driven by an unobserved Markov chain. Information on the state of this chain is obtained from stock prices and expert opinions in the form of signals at random discrete time points. As in Frey et al. (2012), Int. J. Theor. Appl. Finance, 15, No. 1, we use stochastic filtering to transform the original problem into an optimization problem under full information where the state variable is the filter for the Markov chain. The dynamic programming equation for this problem is studied with viscosity-solution techniques and with regularization arguments.
本文研究了漂移由不可观察马尔可夫链驱动的市场中的最优投资组合策略。这条链的状态信息以随机离散时间点的信号形式从股票价格和专家意见中获得。如Frey et al. (2012)， Int。j理论的。达成。在Finance, 15, No. 1中，我们使用随机滤波将原始问题转化为全信息下的优化问题，其中状态变量为马尔可夫链的滤波器。利用粘解技术和正则化参数，研究了该问题的动态规划方程。
{"title":"Portfolio Optimization under Partial Information with Expert Opinions: a Dynamic Programming Approach","authors":"R. Frey, A. Gabih, R. Wunderlich","doi":"10.31390/COSA.8.1.04","DOIUrl":"https://doi.org/10.31390/COSA.8.1.04","url":null,"abstract":"This paper investigates optimal portfolio strategies in a market where the drift is driven by an unobserved Markov chain. Information on the state of this chain is obtained from stock prices and expert opinions in the form of signals at random discrete time points. As in Frey et al. (2012), Int. J. Theor. Appl. Finance, 15, No. 1, we use stochastic filtering to transform the original problem into an optimization problem under full information where the state variable is the filter for the Markov chain. The dynamic programming equation for this problem is studied with viscosity-solution techniques and with regularization arguments.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123284110","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 this paper, we derive a new handy integral equation for the free-boundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion $X$. The new integral equation allows to explicitly find the free-boundary $b(cdot)$ in some so far unsolved cases, as when the operating profit function is not multiplicatively separable and $X$ is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we first show that $b(X(t))=l^*(t)$, with $l^*$ the unique optional solution of a representation problem in the spirit of Bank-El Karoui [Ann. Probab. 32 (2004) 1030-1067]; then, thanks to such an identification and the fact that $l^*$ uniquely solves a backward stochastic equation, we find the integral problem for the free-boundary.
{"title":"On an integral equation for the free-boundary of stochastic, irreversible investment problems","authors":"Giorgio Ferrari","doi":"10.1214/13-AAP991","DOIUrl":"https://doi.org/10.1214/13-AAP991","url":null,"abstract":"In this paper, we derive a new handy integral equation for the free-boundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion $X$. The new integral equation allows to explicitly find the free-boundary $b(cdot)$ in some so far unsolved cases, as when the operating profit function is not multiplicatively separable and $X$ is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we first show that $b(X(t))=l^*(t)$, with $l^*$ the unique optional solution of a representation problem in the spirit of Bank-El Karoui [Ann. Probab. 32 (2004) 1030-1067]; then, thanks to such an identification and the fact that $l^*$ uniquely solves a backward stochastic equation, we find the integral problem for the free-boundary.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129139236","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}
J. D. Lataillade, C. Deremble, M. Potters, J. Bouchaud
We consider the problem of the optimal trading strategy in the presence of linear costs, and with a strict cap on the allowed position in the market. Using Bellman's backward recursion method, we show that the optimal strategy is to switch between the maximum allowed long position and the maximum allowed short position, whenever the predictor exceeds a threshold value, for which we establish an exact equation. This equation can be solved explicitely in the case of a discrete Ornstein-Uhlenbeck predictor. We discuss in detail the dependence of this threshold value on the transaction costs. Finally, we establish a strong connection between our problem and the case of a quadratic risk penalty, where our threshold becomes the size of the optimal non-trading band.
{"title":"Optimal Trading with Linear Costs","authors":"J. D. Lataillade, C. Deremble, M. Potters, J. Bouchaud","doi":"10.21314/JOIS.2012.005","DOIUrl":"https://doi.org/10.21314/JOIS.2012.005","url":null,"abstract":"We consider the problem of the optimal trading strategy in the presence of linear costs, and with a strict cap on the allowed position in the market. Using Bellman's backward recursion method, we show that the optimal strategy is to switch between the maximum allowed long position and the maximum allowed short position, whenever the predictor exceeds a threshold value, for which we establish an exact equation. This equation can be solved explicitely in the case of a discrete Ornstein-Uhlenbeck predictor. We discuss in detail the dependence of this threshold value on the transaction costs. Finally, we establish a strong connection between our problem and the case of a quadratic risk penalty, where our threshold becomes the size of the optimal non-trading band.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114307669","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}
Santiago Moreno-Bromberg, T. Pirvu, Anthony R'eveillac
This paper studies the problem of optimal investment with CRRA (constant, relative risk aversion) preferences, subject to dynamic risk constraints on trading strategies. The market model considered is continuous in time and incomplete; furthermore, financial assets are modeled by ItA´ processes. The dynamic risk constraints (time, state dependent) are generated by risk measures. The optimal trading strategy is characterized by a quadratic BSDE. Special risk measures (Value-at-Risk, Tail Value-at-Risk and Limited Expected Loss ) are considered and a three-fund separation result is established in these cases. Numerical results emphasize the effect of imposing risk constraints on trading.
{"title":"CRRA utility maximization under risk constraints","authors":"Santiago Moreno-Bromberg, T. Pirvu, Anthony R'eveillac","doi":"10.18452/4331","DOIUrl":"https://doi.org/10.18452/4331","url":null,"abstract":"This paper studies the problem of optimal investment with CRRA (constant, relative risk aversion) preferences, subject to dynamic risk constraints on trading strategies. The market model considered is continuous in time and incomplete; furthermore, financial assets are modeled by ItA´ processes. The dynamic risk constraints (time, state dependent) are generated by risk measures. The optimal trading strategy is characterized by a quadratic BSDE. Special risk measures (Value-at-Risk, Tail Value-at-Risk and Limited Expected Loss ) are considered and a three-fund separation result is established in these cases. Numerical results emphasize the effect of imposing risk constraints on trading.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121913195","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 considers the mean variance portfolio management problem. We examine portfolios which contain both primary and derivative securities. The challenge in this context is due to portfolio's nonlinearities. The delta-gamma approximation is employed to overcome it. Thus, the optimization problem is reduced to a well posed quadratic program. The methodology developed in this paper can be also applied to pricing and hedging in incomplete markets.
{"title":"On Mean-Variance Analysis","authors":"Yang Li, T. Pirvu","doi":"10.2307/j.ctvcm4j4m.6","DOIUrl":"https://doi.org/10.2307/j.ctvcm4j4m.6","url":null,"abstract":"This paper considers the mean variance portfolio management problem. We examine portfolios which contain both primary and derivative securities. The challenge in this context is due to portfolio's nonlinearities. The delta-gamma approximation is employed to overcome it. Thus, the optimization problem is reduced to a well posed quadratic program. The methodology developed in this paper can be also applied to pricing and hedging in incomplete markets.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122057717","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 : 2010-10-10DOI: 10.1007/978-3-319-02069-3_16
C. Kardaras
{"title":"A Time Before Which Insiders Would not Undertake Risk","authors":"C. Kardaras","doi":"10.1007/978-3-319-02069-3_16","DOIUrl":"https://doi.org/10.1007/978-3-319-02069-3_16","url":null,"abstract":"","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121789236","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 this short report, we discuss how coordinate-wise descent algorithms can be used to solve minimum variance portfolio (MVP) problems in which the portfolio weights are constrained by $l_{q}$ norms, where $1leq q leq 2$. A portfolio which weights are regularised by such norms is called a sparse portfolio (Brodie et al.), since these constraints facilitate sparsity (zero components) of the weight vector. We first consider a case when the portfolio weights are regularised by a weighted $l_{1}$ and squared $l_{2}$ norm. Then two benchmark data sets (Fama and French 48 industries and 100 size and BM ratio portfolios) are used to examine performances of the sparse portfolios. When the sample size is not relatively large to the number of assets, sparse portfolios tend to have lower out-of-sample portfolio variances, turnover rates, active assets, short-sale positions, but higher Sharpe ratios than the unregularised MVP. We then show some possible extensions; particularly we derive an efficient algorithm for solving an MVP problem in which assets are allowed to be chosen grouply.
{"title":"A Note on Sparse Minimum Variance Portfolios and Coordinate-Wise Descent Algorithms","authors":"Yu-Min Yen","doi":"10.2139/ssrn.1604093","DOIUrl":"https://doi.org/10.2139/ssrn.1604093","url":null,"abstract":"In this short report, we discuss how coordinate-wise descent algorithms can be used to solve minimum variance portfolio (MVP) problems in which the portfolio weights are constrained by $l_{q}$ norms, where $1leq q leq 2$. A portfolio which weights are regularised by such norms is called a sparse portfolio (Brodie et al.), since these constraints facilitate sparsity (zero components) of the weight vector. We first consider a case when the portfolio weights are regularised by a weighted $l_{1}$ and squared $l_{2}$ norm. Then two benchmark data sets (Fama and French 48 industries and 100 size and BM ratio portfolios) are used to examine performances of the sparse portfolios. When the sample size is not relatively large to the number of assets, sparse portfolios tend to have lower out-of-sample portfolio variances, turnover rates, active assets, short-sale positions, but higher Sharpe ratios than the unregularised MVP. We then show some possible extensions; particularly we derive an efficient algorithm for solving an MVP problem in which assets are allowed to be chosen grouply.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115300409","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 : 2010-04-20DOI: 10.1007/978-3-0348-0545-2_16
S. Cawston, L. Vostrikova
{"title":"f-Divergence Minimal Equivalent Martingale Measures and Optimal Portfolios for Exponential Lévy Models with a Change-point","authors":"S. Cawston, L. Vostrikova","doi":"10.1007/978-3-0348-0545-2_16","DOIUrl":"https://doi.org/10.1007/978-3-0348-0545-2_16","url":null,"abstract":"","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130229722","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 : 2010-03-12DOI: 10.1142/9789814304078_0001
Mark H. A. Davis, Sébastien Lleo
In this paper, we extend the jump-diffusion model proposed by Davis and Lleo to include jumps in asset prices as well as valuation factors. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance.) In this setting, the Hamilton- Jacobi-Bellman equation is a partial integro-differential PDE. The main result of the paper is to show that the value function of the control problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.
{"title":"Risk Sensitive Investment Management with Affine Processes: a Viscosity Approach","authors":"Mark H. A. Davis, Sébastien Lleo","doi":"10.1142/9789814304078_0001","DOIUrl":"https://doi.org/10.1142/9789814304078_0001","url":null,"abstract":"In this paper, we extend the jump-diffusion model proposed by Davis and Lleo to include jumps in asset prices as well as valuation factors. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance.) In this setting, the Hamilton- Jacobi-Bellman equation is a partial integro-differential PDE. The main result of the paper is to show that the value function of the control problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132968819","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 : 2010-02-19DOI: 10.1007/978-3-319-02069-3_6
Bénamar Chouaf, S. Pergamenchtchikov
{"title":"Optimal investment with bounded VaR for power utility functions","authors":"Bénamar Chouaf, S. Pergamenchtchikov","doi":"10.1007/978-3-319-02069-3_6","DOIUrl":"https://doi.org/10.1007/978-3-319-02069-3_6","url":null,"abstract":"","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128840959","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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