Path independence of exotic options and convergence of binomial approximations

The analysis of the convergence of tree methods for pricing barrier and lookback options has been the subject of numerous publications aiming at describing, quantifying, and improving the slow and oscillatory convergence in such methods. For barrier and lookback options, we find path-independent options whose price is exactly that of the original path-dependent option. The usual binomial models converge at a speed of order 1∕√𝑛 to the Black-Scholes price. Our new path-independent approach yields convergence of order 1∕𝑛. Furthermore, we derive a closed form formula for the coefficient of 1∕𝑛 in the expansion of the error of our path-independent pricing when the underlying is approximated by the Cox, Ross, and Rubinstein (CRR) model. Using this we obtain a corrected model with a convergence of order 𝑛⁻³/² to the price of barrier and lookback options in the Black-Scholes model. Our results are supported and illustrated by numerical examples.