Imprecise Volatility and Option Replication and Hedging

Option pricing is usually done within the risk-neutral world. The justification is the ability to set up a replication portfolio - the portfolio, consisting of the underlying asset and the riskless investment, which will replicate the target payoff of the option for all states of the world. If the portfolio exists, it can be used to set up hedged, and therefore riskless, portfolio. It means, that the real world drift does not play any role and we can use the riskless rate in order to both, model the future payoff and discount it back to the beginning. The only problem which remains is how to specify the underlying asset price risk - the volatility of its returns. In this paper, we reformulate the binomial model for the case of unknown volatility. We propose formulas for both, replication and hedging portfolio.