Linear-time option pricing algorithms by combinatorics

Options are popular financial derivatives that play essential roles in financial markets. How to price them efficiently and accurately is very important both in theory and practice. Options are often priced by the lattice model. Although the prices computed by the lattice converge to the theoretical option value under the continuous-time model, they may converge slowly. Worse, for some options like barrier options, the prices can even oscillate wildly. For such options, huge amounts of computational time are required to achieve acceptable accuracy. Combinatorial techniques have been used to improve the performance in pricing a wide variety of options. This paper uses vanilla options, power options, single-barrier options, double-barrier options, and lookback options as examples to show how combinatorics can help us to derive linear-time pricing algorithms. These algorithms compare favorably against popular lattice methods, which take at least quadratic time.