غلامحسین یاری

As we are looking for knowledge of stock future returns in portfolio optimization, we are practically faced with two principal concepts: Uncertainty and Information about variables. This paper attempts to introduce a pragmatic bi-objective investment model based on uncertainty, instead of probability space and information theory, instead of variance and other moments as a risk measure for portfolio optimization. Not only is uncertainty space expected to be more in line with investment theory, but also, applying and learning this approach seems more straightforward and practical for novice investors. The proposed model simultaneously maximizes the uncertain mean of stock returns and minimizes uncertain entropy as a measure of portfolio risk. The uncertain zigzag distribution has been used for variables to avoid the complexity of fitting distributions for data. This uncertain mean-entropy portfolio optimization (UMEPO) has been solved by three meta-heuristic methods of multi-objective optimization: NSGA-II, MOPS, and MOICA. Finally, it was observed that the optimal portfolio obtained from the proposed model has a higher return and a lower entropy as a risk measure compared to the same model in the probability space.

This paper focuses on two main issues that are based on two important concepts: exponential Levy process and minimal entropy martingale measure. First, we intend to obtain risk measurement such as value-at-risk (VaR) and conditional value-at-risk (CvaR) using Monte-Carlo methodunder minimal entropy martingale measure (MEMM) for exponential Levy process. This Martingale measure is used for the exponential type of the processes such as exponential Levy process. Also, it can be said MEMM is a kind of important sampling method where the probability measure with minimal relative entropy replaces the main probability. Then we are going to obtain VaR and CVaR by Monte-Carlo simulation. For this purpose, we have to calculate option price, implied volatility and returns under MEMM and then obtain risk measurement by proposed algorithm. Finally, this model is simulated for exponential variance gamma process. Next, we intend to develop two theorems for implied volatility under minimal entropy martingale measure by examining the conditions. These theorems consider the asymptotic implied volatility for the case that time to maturity tends to zero and infinity.