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Volume 2017 (2017), Article ID 8734235, 11 pages
Research Article

Closed-Form Optimal Strategies of Continuous-Time Options with Stochastic Differential Equations

Research Institute of Petroleum Exploration and Development, PetroChina, Beijing 100083, China

Correspondence should be addressed to Wei Yan

Received 18 October 2016; Revised 15 March 2017; Accepted 24 April 2017; Published 2 July 2017

Academic Editor: Pietro De Lellis

Copyright © 2017 Wei Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A continuous-time portfolio selection with options based on risk aversion utility function in financial market is studied. The different price between sale and purchase of options is introduced in this paper. The optimal investment-consumption problem is formulated as a continuous-time mathematical model with stochastic differential equations. The prices processes follow jump-diffusion processes (Weiner process and Poisson process). Then the corresponding Hamilton-Jacobi-Bellman (HJB) equation of the problem is represented and its solution is obtained in different conditions. The above results are applied to a special case under a Hyperbolic Absolute Risk Aversion (HARA) utility function. The optimal investment-consumption strategies about HARA utility function are also derived. Finally, an example and some discussions illustrating these results are also presented.