M. T. V. Martínez-Palacios, A. Hernández-del-Valle, Ambrosio Ortiz-Ramírez
{"title":"On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach","authors":"M. T. V. Martínez-Palacios, A. Hernández-del-Valle, Ambrosio Ortiz-Ramírez","doi":"10.3934/jdg.2019004","DOIUrl":null,"url":null,"abstract":"In this work, through stochastic optimal control in continuous time the optimal decision making in consumption and investment is modeled by a rational economic agent, representative of an economy, who is a consumer and an investor adverse to risk; this in a finite time horizon of stochastic length. The assumptions of the model are: a consumption function of HARA type, a representative company that has a stochastic production process, the agent invests in a stock and an American-style Asian put option with floating strike equal to the geometric average subscribed on the stock, both modeled by controlled Markovian processes; as well as the investment of a principal in a bank account. The model is solved with dynamic programming in continuous time, particularly the Hamilton-Jacobi-Bellman PDE is obtained, and a function in separable variables is proposed as a solution to set the optimal trajectories of consumption and investment. In the solution analysis is determined: in equilibrium, the process of short interest rate that is driven by a square root process with reversion to the mean; and through a system of differential equations of risk premiums, a PDE is deduced equivalent to the Black-Scholes-Merton but to value an American-style Asian put option.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":" ","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Dynamics and Games","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/jdg.2019004","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, through stochastic optimal control in continuous time the optimal decision making in consumption and investment is modeled by a rational economic agent, representative of an economy, who is a consumer and an investor adverse to risk; this in a finite time horizon of stochastic length. The assumptions of the model are: a consumption function of HARA type, a representative company that has a stochastic production process, the agent invests in a stock and an American-style Asian put option with floating strike equal to the geometric average subscribed on the stock, both modeled by controlled Markovian processes; as well as the investment of a principal in a bank account. The model is solved with dynamic programming in continuous time, particularly the Hamilton-Jacobi-Bellman PDE is obtained, and a function in separable variables is proposed as a solution to set the optimal trajectories of consumption and investment. In the solution analysis is determined: in equilibrium, the process of short interest rate that is driven by a square root process with reversion to the mean; and through a system of differential equations of risk premiums, a PDE is deduced equivalent to the Black-Scholes-Merton but to value an American-style Asian put option.
期刊介绍:
The Journal of Dynamics and Games (JDG) is a pure and applied mathematical journal that publishes high quality peer-review and expository papers in all research areas of expertise of its editors. The main focus of JDG is in the interface of Dynamical Systems and Game Theory.