On the convergence scheme in the CRR model
We investigate the convergence scheme from the discrete to the continuous time model in the binomial tree model of Cox-Ross-Rubinstein (CRR). We introduce the notion of $\bar\Sigma $-decomposition and we classify the financial payoffs according to their representation in terms of $\bar\Sigma $ functions in the CRR model. We find the exact convergence rate for a single $\bar\Sigma $ function and we obtain the general convergence scheme for their linear combinations. This way, we get a universal and convenient tool to investigate the payoffs in the continuous time model by reducing the problem to the discrete time model. An illustration of this method is retrieving the formula for the double barrier option price in the Black–Scholes model, in the form of the sum of an infinite series, derived by N. Kunitomo and M. Ikeda in 1992. To the best of our knowledge, it is the first time that the double barrier options payoff is obtained by convergence from the discrete time model. Finally, we show that an alternative form of the results obtained corresponds to P. Biane’s research in the area of the Riemann function.