Option price intervals based on bellman dynamic programming principle

The assumption of constant underlying´s volatility in Black-Scholes formula cannot be satisfied in financiap market. In this paper, we get the option price intervals assuming the stock volatility lies within a given interval. First we transform this financial problem to a stochastic optimal control problem, then obtain options´ maximum and minimum price models through dynamic programming principle. We solve the nonlinear PDE model and narrow the price interval through optimal static hedging. We conclude this paper by giving its applications in U.S.A option market, get the MCD options intervals, comparing with Black-scholes, and find a way to identify arbitrage opportunity in option markets.