In a market with frictions, bid and ask prices are described by sublinear pricing functionals, which can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model with transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of American options and compute an implied liquidity to describe the bid–ask spread. This method describes liquidity over time very well, compared to the classical approach of describing bid and ask prices by quoting bid and ask implied volatilities.