No abstract

In this paper, we apply an improved version of Monte Carlo methods to pricing barrier options. This kind of options may match with risk hedging needs more closely than standard options. Barrier options behave like a plain vanilla option with one exception. A zero payoff may occur before expiry, if the option ceases to exist; accordingly, barrier options are cheaper than similar standard vanilla options. We apply a new Monte Carlo method to compute the prices of single and double barrier options written on stocks. The basic idea of the new method is to use uniformly distributed random numbers and an exit probability in order to perform a robust estimation of the first time the stock price hits the barrier. Using uniformly distributed random numbers decreases the estimation of first hitting time error in comparison with standard Monte Carlo or similar methods. It is numerically shown that the answer of our method is closer to the exact value and the first hitting time error is reduced.

This paper presents a neural network model for solving two models for portfolio selection in which the securities are assumed to be uncertain variables. The main idea is to replace the portfolio selection models with linear programming (LP) problems. According to the convex optimization theory and some concepts of ordinary differential equations, a neural network model for solving LP problems is presented. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns. Two illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.

A new Monte Carlo method is presented to compute the prices of digital barrier options on stocks. The main idea of the new approach is to use an exceedance probability and uniformly distributed random numbers in order to efficiently estimate the first hitting time of barriers. It is numerically shown that the answer of this method is closer to the exact value and the first hitting time error of the modified Monte Carlo method decreases much faster than of the standard Monte Carlo methods.