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

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 aHyperbolic Absolute Risk Aversion (HARA) utility function. The optimal investment-consumption strategies aboutHARA utility function are also derived. Finally, an example and some discussions illustrating these results are also presented.