An Application of Dynamic Programming Principle for Pricing Multi-assets Barrier Options

This paper is concerned with the stochastic control problem arising in pricing barrier option consisting of two or more assets. We refer to a barrier option where the volatilities of the underlying assets are stochastically moving within specified interval. The interval can be the maximum or the minimum value of the volatilities during the life of the contract. These values of the volatilities may correspond to the best and the worst case scenarios of the future positions in the portfolio of the options. The concept of suprehedging strategies in pricing an option is applied. Furthermore, in this case, the strategy is considered as a certain exit time control problem. First, we prove that the control u is lower semicontinuous. Then, under certain assumptions, we show the value function is bounded and nonnegative. Next, by applying probability methods, we prove that the value function of the exit control problem is continuous on the boundary. Finally, we prove that superhedging prices of multi-asset barrier options can be represented in the dynamic programming principle (DPP) for an exit control problem.