Approximation and asymptotics in the superhedging problem for binary options

This paper considers Kolokoltsov’s multiplicative model of market price dynamics witout trading constraints. Under general assumptions and monotonic payoff functions, we show that the guaranteed deterministic approach, having a game-theoretic interpretation, yields the same result in the superhedging problem as in the probabilistic approach. We analyze in detail the superhedging problem for a special monotonic payoff function, i.e., a European-style binary option, within the guaranteed deterministic approach (GDA). Unlike the probabilistic counterpart, GDA allows a direct description of the most unfavorable mixed market strategy. We obtain some interesting analytical properties of the solutions of the corresponding Bellman–Isaacs equations, providing the minimal required reserves (also called the superhedging price) to cover the option payoff at the expiration time. The price process with the conditional distributions corresponding to the most unfavorable market scenarios can be approximated on a logarithmic scale by a random walk with two absorbing barriers. We also prove that, under an appropriate normalization, the price process weakly converges to the geometric Brownian motion with one absorbing barrier at the strike price when the discrete-time model number of steps tends to infinity.