Random trinomial tree models and vanilla options

In this paper we introduce and study random trinomial model. The usual trinomial model is prescribed by triple of numbers (u, d, m). We call the triple (u, d, m) an environment of the trinomial model. A triple (U_n, D_n, M_n), where {U_n}, {D_n} and {M_n} are the sequences of independent, identically distributed random variables with 0 < D_n < 1 < U_n and M_n = 1 for all n, is called a random environment and trinomial tree model with random environment is called random trinomial model. The random trinomial model is considered to produce more accurate results than the random binomial model or usual trinomial model.