مدل استوار بهينه سازي سبد مالي داراي اختيار معامله

ROBUST OPTIMIZATION OF A PORTFOLIO WHICH INCLUDES OPTIONS

در اين نوشتار دو مدل استوار براي مسايل بهينه‌سازي سبد ماليِ داراي اختيار معامله توسعه داده مي‌شود. در مدل اول با توجه به رابطه‌ي غير‌خطي (شكسته‌ي خطي) بين داده‌هاي غير‌قطعي (ارزش سهام و اختيار معامله)، يك مدل همتاي استوار بيش‌محافظه‌كارانه ارايه مي‌دهيم؛ در مدل دوم نيز با روشي متفاوت همتاي استوار با درجه محافظه‌كاري قابل كنترل ارايه مي‌شود. خصوصيت اصلي مدل‌هاي استوار ارايه شده در اين پژوهش، نحوه‌ي برخورد آن ها با روابط غيرخطي پارامترهاي داراي عدم قطعيت در مدل است. براي تحليل دو مدل مذكور سه مسيله با تعداد 100 نوع سهام و حدود 400 اختيار معامله حل شده و نتايج آن مورد بررسي قرار مي گيرد.

In this paper, we introduce two robust models to create a portfolio with the best combination of risk free assets, stocks and options within multi-periods. Many input parameters of the model, such as stock or option prices, are uncertain. If they are estimated as deterministic values, portfolio optimization may result in an infeasible solution. Among different approaches for handling uncertainties, we adopt robust optimization. First, as a basic model, we formulate the problem with deterministic data. The decision variables are the amount invested in the free risk asset, and the number of stocks, as well as options (both call and put), to include in the desired portfolio. The objective is to determine the optimal combination of assets, in order to maximize the total return at the end of the last period. The return of call and put options are: and , respectively. The constraints represent limitations on the initial investment budget in the first period; the available budget at the end of each period, which includes new investment opportunity for the next period plus the return of existing investment, and, finally, the constraint that calculates the final return of the portfolio. No shortselling is allowed. For developing the robust model, stock price uncertainty at the end of each investment period is presented within linear intervals, defined as follows: where, represents the mean of the stock price , and represents the maximum variation of the stock price from its mean value. In the first model, we develop a robust counterpart, which results in the best solution, if the worst situation within the intervals occurs. It is a conservative counterpart robust model, in which relations between the uncertain parameters, such as the value of stocks and options, are nonlinear (linear piecewise). In the second model, each constraint has a special set of uncertain parameters, which work independently from the other constraints. Therefore, for controlling the feasibility of the solution, each one must be considered separately. This model is a robust counterpart of multi-period and option based portfolio optimization problems. The optimal solution of this model is feasible for each combination of stock values for future periods. The optimal solution shows the best solution of the model for the worst combination of stock values for future periods. However, we consider an upper limit for the protection level of uncertainties of all constraints. In fact, for dealing with this problem and improving our model, we introduce a controlling factor for investor risk acceptance in investment. It is clear that the chances of realizing the worst case stock value combination at each period are very low. In the second model, we adopt a different approach to develop a counterpart model that is controllable, as far as its conservative degree is concerned. To handle uncertain nonlinear parameters, we introduce a new approach to develop the proposed modes. To solve the models, they are reformulated as dual problems. In this way, the optimal solution can be obtained analytically. To analyze the models, we solve three problems with 100 stocks and 400 options and study the results.