Abstract:In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics of returns. Thus, the description of underlying asset price and the pricing of options are more accurate. Moreover, using the martingale method, we obtain closed form solutions for geometric average Asian options. Furthermore, the numerical analysis shows that the model can avoid underestimating risks relative to the Black-Scholes model.

Global optimization, especially on a large scale, is challenging to solve due to its nonlinearity and multimodality. In this paper, in order to enhance the global searching ability of the firefly algorithm (FA) inspired by bionics, a novel hybrid meta-heuristic algorithm is proposed by embedding the cross-entropy (CE) method into the firefly algorithm. With adaptive smoothing and co-evolution, the proposed method fully absorbs the ergodicity, adaptability and robustness of the cross-entropy method. The new hybrid algorithm achieves an effective balance between exploration and exploitation to avoid falling into a local optimum, enhance its global searching ability, and improve its convergence rate. The results of numeral experiments show that the new hybrid algorithm possesses more powerful global search capacity, higher optimization precision, and stronger robustness.

The deficiencies of keeping population diversity, prematurity and low success rate of searching the global optimal solution are the shortcomings of genetic algorithm (GA). Based on the bias of samples in the uniform design sampling (UDS) point set, the crossover operation in GA is redesigned. Using the concentrations of antibodies in artificial immune system (AIS), the chromosomes concentration in GA is defined and the clonal selection strategy is designed. In order to solve the maximum clique problem (MCP), an new immune GA (UIGA) is presented based on the clonal selection strategy and UDS. The simulation results show that the UIGA provides superior solution quality, convergence rate, and other various indices to those of the simple and good point GA when solving MCPs.

To describe the movement of asset prices accurately, we employ the non-extensive statistical mechanics and the semi-Markov process to establish an asset price model. The model can depict the peak and fat tail characteristics of returns and the regime-switching phenomenon of macroeconomic system. Moreover, we use the risk-minimizing method to study the hedging problem of contingent claims and obtain the explicit solutions of the optimal hedging strategies.

Let f be a continuous map from a compact metric space X to itself. In this paper, We introduce two concepts of upper density one sensitivity and positive lower density sensitivity, and prove that (1) if f is a topologically strongly ergodic map, then it is upper density one sensitive;(2) if f is a sensitive map satisfying the large deviations theorem, then f is positive lower density sensitive.