Dynamic Programming
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In this paper, we provide a parameterization of multinomial lattice random walks which take cumulants into account. In the binomial and trinomial lattice cases, it reduces to standard results. Additionally, we show that higher order cumulants may be taken into account by using multinomial lattices with four or more branches. Finally, we outline two synthesis methods which take advantage of the multinomial lattice formulation. One is mean square optimal hedging in an incomplete market and the other involves pricing under "implied volatility" and "implied kurtosis".