Recent developments and results concerned with pricing and hedging
derivative securities in markets with proportional transaction costs represented as bid-ask spreads are reported. The focus is on a constructive approach to representing and computing the prices and hedging portfolios, leading to efficient numerical algorithms. A wide range of derivative securities, from plain vanilla European options to basket options, American and Bermudan type derivatives to game options are covered. New results are presented for hedging and pricing derivatives under deferred solvency conditions as well as game options.