SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 1-25, March 2024. Abstract. This article presents a Hawkes process model with Markovian baseline intensities for high-frequency order book data modeling. We classified intraday order book trading events into a range of categories based on their order types and the price change after their arrivals. In order to capture the stimulating effects between multiple types of order book events, we use a multivariate Hawkes process to model the self-exciting and mutually exciting event arrivals. We also integrate Markovian baseline intensities into the event arrival dynamic, by including the impacts of order book liquidity state and time factor on the baseline intensity. A regression-based nonparametric estimation procedure is adopted to estimate the model parameters in our Hawkes+Markovian model. To eliminate redundant model parameters, LASSO regularization is incorporated into the estimation procedure. Besides, a model selection method based on Akaike information criteria is applied to evaluate the effect of each part of the proposed model. An implementation example based on real limit order book data is provided. Through the example we studied the empirical shapes of Hawkes excitement functions, the effects of liquidity as well as time factors, the LASSO variable selection, and the explanation power of Hawkes and Markovian elements to the dynamics of order book.
{"title":"Order Book Queue Hawkes Markovian Modeling","authors":"Philip E. Protter, Qianfan Wu, Shihao Yang","doi":"10.1137/22m1470815","DOIUrl":"https://doi.org/10.1137/22m1470815","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 1-25, March 2024. <br/> Abstract. This article presents a Hawkes process model with Markovian baseline intensities for high-frequency order book data modeling. We classified intraday order book trading events into a range of categories based on their order types and the price change after their arrivals. In order to capture the stimulating effects between multiple types of order book events, we use a multivariate Hawkes process to model the self-exciting and mutually exciting event arrivals. We also integrate Markovian baseline intensities into the event arrival dynamic, by including the impacts of order book liquidity state and time factor on the baseline intensity. A regression-based nonparametric estimation procedure is adopted to estimate the model parameters in our Hawkes+Markovian model. To eliminate redundant model parameters, LASSO regularization is incorporated into the estimation procedure. Besides, a model selection method based on Akaike information criteria is applied to evaluate the effect of each part of the proposed model. An implementation example based on real limit order book data is provided. Through the example we studied the empirical shapes of Hawkes excitement functions, the effects of liquidity as well as time factors, the LASSO variable selection, and the explanation power of Hawkes and Markovian elements to the dynamics of order book.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"227 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645156","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}
Alessandro Doldi, Marco Frittelli, Emanuela Rosazza Gianin
SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page SC1-SC14, March 2024. Abstract. Shortfall systemic (multivariate) risk measures [math] defined through an [math]-dimensional multivariate utility function [math] and random allocations can be represented as classical (1-dimensional) shortfall risk measures associated to an explicitly determined 1-dimensional function constructed from [math]. This finding allows for simplifying the study of several properties of [math], such as dual representations, law invariance, and stability.
{"title":"Short Communication: Are Shortfall Systemic Risk Measures One Dimensional?","authors":"Alessandro Doldi, Marco Frittelli, Emanuela Rosazza Gianin","doi":"10.1137/23m1580413","DOIUrl":"https://doi.org/10.1137/23m1580413","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page SC1-SC14, March 2024. <br/> Abstract. Shortfall systemic (multivariate) risk measures [math] defined through an [math]-dimensional multivariate utility function [math] and random allocations can be represented as classical (1-dimensional) shortfall risk measures associated to an explicitly determined 1-dimensional function constructed from [math]. This finding allows for simplifying the study of several properties of [math], such as dual representations, law invariance, and stability.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"100 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105370","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}
Francesca Biagini, Andrea Mazzon, Thilo Meyer-Brandis, Katharina Oberpriller
SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page 1304-1342, December 2023. Abstract. In this paper we study the evolution of asset price bubbles driven by contagion effects spreading among investors via a random matching mechanism in a discrete-time version of the liquidity based model of [R. A. Jarrow, P. Protter, and A. F. Roch, Quant. Finance, 12 (2012), pp. 1339–1349]. To this scope, we extend the Markov conditionally independent dynamic directed random matching of [D. Duffie, L. Qiao, and Y. Sun, J. Econ. Theory, 174 (2018), pp. 124–183] to a stochastic setting to include stochastic exogenous factors in the model. We derive conditions guaranteeing that the financial market model is arbitrage-free and present some numerical simulation illustrating our approach.
SIAM 金融数学期刊》，第 14 卷第 4 期，第 1304-1342 页，2023 年 12 月。 摘要本文研究了基于流动性模型的离散时间版本 [R. A. Jarrow, P. Protter, and A. F. Ro.A. Jarrow, P. Protter, and A. F. Roch, Quant.金融》，12 (2012)，第 1339-1349 页]。在此范围内，我们将[D. Duffie, L. Qiao, and Y. Sun, J. Econ. Theory, 174 (2018), pp.我们推导了保证金融市场模型无套利的条件，并给出了一些数值模拟来说明我们的方法。
{"title":"Liquidity Based Modeling of Asset Price Bubbles via Random Matching","authors":"Francesca Biagini, Andrea Mazzon, Thilo Meyer-Brandis, Katharina Oberpriller","doi":"10.1137/22m1531580","DOIUrl":"https://doi.org/10.1137/22m1531580","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page 1304-1342, December 2023. <br/> Abstract. In this paper we study the evolution of asset price bubbles driven by contagion effects spreading among investors via a random matching mechanism in a discrete-time version of the liquidity based model of [R. A. Jarrow, P. Protter, and A. F. Roch, Quant. Finance, 12 (2012), pp. 1339–1349]. To this scope, we extend the Markov conditionally independent dynamic directed random matching of [D. Duffie, L. Qiao, and Y. Sun, J. Econ. Theory, 174 (2018), pp. 124–183] to a stochastic setting to include stochastic exogenous factors in the model. We derive conditions guaranteeing that the financial market model is arbitrage-free and present some numerical simulation illustrating our approach.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683778","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}
SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page SC60-SC71, December 2023. Abstract. For time-inconsistent stochastic controls in discrete time and finite horizon, an open problem in Björk and Murgoci [Finance Stoch., 18 (2014), pp. 545–592] is the existence of an equilibrium control. A nonrandomized Borel measurable Markov equilibrium policy exists if the objective is inf-compact in every time step. We provide a sufficient condition for the inf-compactness and thus existence with costs that are lower semicontinuous and bounded from below and transition kernels that are continuous in controls under given states. The control spaces need not to be compact.
{"title":"Short Communication: Existence of Markov Equilibrium Control in Discrete Time","authors":"Erhan Bayraktar, Bingyan Han","doi":"10.1137/23m1594121","DOIUrl":"https://doi.org/10.1137/23m1594121","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page SC60-SC71, December 2023. <br/> Abstract. For time-inconsistent stochastic controls in discrete time and finite horizon, an open problem in Björk and Murgoci [Finance Stoch., 18 (2014), pp. 545–592] is the existence of an equilibrium control. A nonrandomized Borel measurable Markov equilibrium policy exists if the objective is inf-compact in every time step. We provide a sufficient condition for the inf-compactness and thus existence with costs that are lower semicontinuous and bounded from below and transition kernels that are continuous in controls under given states. The control spaces need not to be compact.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"41 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555298","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}
SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page 1290-1303, December 2023. Abstract. The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [J. Han, A. Jentzen, and W. E, Proc. Natl. Acad. Sci. USA, 115 (2018), pp. 8505–8510] has shown great power in solving high-dimensional forward-backward stochastic differential equations and has inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which cannot be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [H. Wang et al., Deep Learning-Based BSDE Solver for LIBOR Market Model with Application to Bermudan Swaption Pricing and Hedging, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide a rigorous theory for the backward deep BSDE method. Specifically, (1) we derive the a posteriori error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and (2) we give an upper bound of the loss function, which can be sufficiently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory.
金融数学学报，第14卷，第4期，1290-1303页，2023年12月。摘要。最优止损问题是金融市场的核心问题之一，在美国期权和百慕大期权的定价中有着广泛的应用。深度BSDE方法[J]。Han, A. Jentzen和W. E, Proc. Natl。学会科学。美国，115 (2018)，pp. 8505-8510]在求解高维正反向随机微分方程方面显示出巨大的力量，并激发了许多应用。然而，该方法以正向方式求解倒向随机微分方程(BSDEs)，不能用于通常需要倒向运行BSDEs的最优停止问题。为了克服这个困难，最近的一篇论文[H。Wang et al.，基于深度学习的LIBOR市场模型的BSDE求解器及其在berberan互换定价和套期保值中的应用，[j] . vol . 7: 187.06622, 2018]提出了求解最优停止问题的反向深度BSDE方法。本文为后向深度BSDE方法提供了一个严密的理论。具体来说，(1)我们导出了后验误差估计，即数值解的误差可以用训练损失函数有界;(2)我们给出了损失函数的上界，这个上界在普适近似下可以足够小。给出了两个数值算例，结果与所证明的理论一致。
{"title":"Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems","authors":"Chengfan Gao, Siping Gao, Ruimeng Hu, Zimu Zhu","doi":"10.1137/22m1539952","DOIUrl":"https://doi.org/10.1137/22m1539952","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page 1290-1303, December 2023. <br/> Abstract. The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [J. Han, A. Jentzen, and W. E, Proc. Natl. Acad. Sci. USA, 115 (2018), pp. 8505–8510] has shown great power in solving high-dimensional forward-backward stochastic differential equations and has inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which cannot be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [H. Wang et al., Deep Learning-Based BSDE Solver for LIBOR Market Model with Application to Bermudan Swaption Pricing and Hedging, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide a rigorous theory for the backward deep BSDE method. Specifically, (1) we derive the a posteriori error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and (2) we give an upper bound of the loss function, which can be sufficiently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"40 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514687","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}
SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page SC49-SC59, December 2023. Abstract. Il significato soggettivo della probabilità (1931) by B. de Finetti is unanimously considered the rise of “subjectivism,” a notion which strongly influenced both probability and decision theory. What is less acknowledged is that in 1931 de Finetti posed the foundations of modern arbitrage theory. In this paper we aim at examining how de Finetti’s contribution should be considered as the precursor of asset pricing theory and we show how his findings relate to recent developments in robust finance.
《金融数学学报》，第14卷，第4期，第49- 59页，2023年12月。摘要。B. de Finetti的《Il significato soggettivo della probabilitom》(1931)被一致认为是“主观主义”的兴起，这一概念强烈地影响了概率论和决策理论。不太为人所知的是，德菲内蒂在1931年提出了现代套利理论的基础。在本文中，我们旨在研究如何将德菲内蒂的贡献视为资产定价理论的先驱，并展示他的发现如何与稳健金融的最新发展联系起来。
{"title":"Short Communication: The Birth of (a Robust) Arbitrage Theory in de Finetti’s Early Contributions","authors":"Marco Maggis","doi":"10.1137/23m1604096","DOIUrl":"https://doi.org/10.1137/23m1604096","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 14, Issue 4, Page SC49-SC59, December 2023. <br/> Abstract. Il significato soggettivo della probabilità (1931) by B. de Finetti is unanimously considered the rise of “subjectivism,” a notion which strongly influenced both probability and decision theory. What is less acknowledged is that in 1931 de Finetti posed the foundations of modern arbitrage theory. In this paper we aim at examining how de Finetti’s contribution should be considered as the precursor of asset pricing theory and we show how his findings relate to recent developments in robust finance.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"59 3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514738","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}
Anthony Coache, Sebastian Jaimungal, Álvaro Cartea
We propose a novel framework to solve risk-sensitive reinforcement learning problems where the agent optimizes time-consistent dynamic spectral risk measures. Based on the notion of conditional elicitability, our methodology constructs (strictly consistent) scoring functions that are used as penalizers in the estimation procedure. Our contribution is threefold: we (i) devise an efficient approach to estimate a class of dynamic spectral risk measures with deep neural networks, (ii) prove that these dynamic spectral risk measures may be approximated to any arbitrary accuracy using deep neural networks, and (iii) develop a risk-sensitive actor-critic algorithm that uses full episodes and does not require any additional nested transitions. We compare our conceptually improved reinforcement learning algorithm with the nested simulation approach and illustrate its performance in two settings: statistical arbitrage and portfolio allocation on both simulated and real data.
{"title":"Conditionally Elicitable Dynamic Risk Measures for Deep Reinforcement Learning","authors":"Anthony Coache, Sebastian Jaimungal, Álvaro Cartea","doi":"10.1137/22m1527209","DOIUrl":"https://doi.org/10.1137/22m1527209","url":null,"abstract":"We propose a novel framework to solve risk-sensitive reinforcement learning problems where the agent optimizes time-consistent dynamic spectral risk measures. Based on the notion of conditional elicitability, our methodology constructs (strictly consistent) scoring functions that are used as penalizers in the estimation procedure. Our contribution is threefold: we (i) devise an efficient approach to estimate a class of dynamic spectral risk measures with deep neural networks, (ii) prove that these dynamic spectral risk measures may be approximated to any arbitrary accuracy using deep neural networks, and (iii) develop a risk-sensitive actor-critic algorithm that uses full episodes and does not require any additional nested transitions. We compare our conceptually improved reinforcement learning algorithm with the nested simulation approach and illustrate its performance in two settings: statistical arbitrage and portfolio allocation on both simulated and real data.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"25 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134953967","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}
Bastien Baldacci, Philippe Bergault, Joffrey Derchu, Mathieu Rosenbaum
The tick size, which is the smallest increment between two consecutive prices for a given asset, is a key parameter of market microstructure. In particular, the behavior of high frequency market makers is highly related to its value. We take the point of view of an exchange and investigate the relevance of having different tick sizes on the bid and ask sides of the order book. Using an approach based on the model with uncertainty zones, we show that when side-specific tick sizes are suitably chosen, it enables the exchange to improve the quality of liquidity provision.
{"title":"On Bid and Ask Side-Specific Tick Sizes","authors":"Bastien Baldacci, Philippe Bergault, Joffrey Derchu, Mathieu Rosenbaum","doi":"10.1137/21m146065x","DOIUrl":"https://doi.org/10.1137/21m146065x","url":null,"abstract":"The tick size, which is the smallest increment between two consecutive prices for a given asset, is a key parameter of market microstructure. In particular, the behavior of high frequency market makers is highly related to its value. We take the point of view of an exchange and investigate the relevance of having different tick sizes on the bid and ask sides of the order book. Using an approach based on the model with uncertainty zones, we show that when side-specific tick sizes are suitably chosen, it enables the exchange to improve the quality of liquidity provision.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":" 26","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191523","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 study the problem of active portfolio management where an investor aims to outperform a benchmark strategy’s risk profile while not deviating too far from it. Specifically, an investor considers alternative strategies whose terminal wealth lies within a Wasserstein ball surrounding a benchmark’s terminal wealth—being distributionally close—and that have a specified dependence/copula tying state-by-state outcomes to it. The investor then chooses the alternative strategy that minimizes a distortion risk measure of terminal wealth. In a general complete market model, we prove that an optimal dynamic strategy exists and provide its characterization through the notion of isotonic projections. We further propose a simulation approach to calculate the optimal strategy’s terminal wealth, making our approach applicable to a wide range of market models. Finally, we illustrate how investors with different copula and risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped, and lower- and upper-tail distortion risk measures as examples. We find that investors’ optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving the benchmark’s structure.
{"title":"Portfolio Optimization within a Wasserstein Ball","authors":"Silvana M. Pesenti, Sebastian Jaimungal","doi":"10.1137/22m1496803","DOIUrl":"https://doi.org/10.1137/22m1496803","url":null,"abstract":"We study the problem of active portfolio management where an investor aims to outperform a benchmark strategy’s risk profile while not deviating too far from it. Specifically, an investor considers alternative strategies whose terminal wealth lies within a Wasserstein ball surrounding a benchmark’s terminal wealth—being distributionally close—and that have a specified dependence/copula tying state-by-state outcomes to it. The investor then chooses the alternative strategy that minimizes a distortion risk measure of terminal wealth. In a general complete market model, we prove that an optimal dynamic strategy exists and provide its characterization through the notion of isotonic projections. We further propose a simulation approach to calculate the optimal strategy’s terminal wealth, making our approach applicable to a wide range of market models. Finally, we illustrate how investors with different copula and risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped, and lower- and upper-tail distortion risk measures as examples. We find that investors’ optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving the benchmark’s structure.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"9 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135821495","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 studies a continuous-time optimal portfolio selection problem in the complete market for a behavioral investor whose preference is of the prospect type with probability distortion. The investor concerns about the terminal relative growth rate (log-return) instead of absolute capital value. This model can be regarded as an extension of the classical growth optimal problem to the behavioral framework. It leads to a new type of M-shaped utility maximization problem under nonlinear Choquet expectation. Due to the presence of probability distortion, the classical stochastic control methods are not applicable. By the martingale method, concavification and quantile optimization techniques, we derive the closed-form optimal growth rate. We find that the benchmark growth rate has a significant impact on investment behaviors. Compared to Zhang et al where the same preference measure is applied to the terminal relative wealth, we find a new phenomenon when the investor's risk tolerance level is high and the market states are bad. In addition, our optimal wealth in every scenario is less sensitive to the pricing kernel and thus more stable than theirs.
{"title":"Relative Growth Rate Optimization Under Behavioral Criterion","authors":"Jing Peng, Pengyu Wei, Zuo Quan Xu","doi":"10.1137/22m1496943","DOIUrl":"https://doi.org/10.1137/22m1496943","url":null,"abstract":"This paper studies a continuous-time optimal portfolio selection problem in the complete market for a behavioral investor whose preference is of the prospect type with probability distortion. The investor concerns about the terminal relative growth rate (log-return) instead of absolute capital value. This model can be regarded as an extension of the classical growth optimal problem to the behavioral framework. It leads to a new type of M-shaped utility maximization problem under nonlinear Choquet expectation. Due to the presence of probability distortion, the classical stochastic control methods are not applicable. By the martingale method, concavification and quantile optimization techniques, we derive the closed-form optimal growth rate. We find that the benchmark growth rate has a significant impact on investment behaviors. Compared to Zhang et al where the same preference measure is applied to the terminal relative wealth, we find a new phenomenon when the investor's risk tolerance level is high and the market states are bad. In addition, our optimal wealth in every scenario is less sensitive to the pricing kernel and thus more stable than theirs.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"24 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135113619","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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