# Pricing derivatives on multiple assets: recombining multinomial trees based on Pascal’s simplex

## Abstract

In this paper a direct generalisation of the recombining binomial tree model by Cox et al. (J Financ Econ 7:229–263, 1979) based on the Pascal’s simplex is constructed. This discrete model can be used to approximate the prices of derivatives on multiple assets in a Black–Scholes market environment. The generalisation keeps most aspects of the binomial model intact, of which the following are the most important: The direct link to the Pascal’s simplex (which specialises to Pascal’s triangle in the binomial case); the matching of moments of the (log-transformed) process; convergence to the correct option prices both for European and American options, when the time step length goes to zero and the completeness of the model, at least for sufficiently small time step. The goal of this paper is to present basic theoretical aspects of this approach. However, we also illustrate the approach by a number of example calculations. Further possible developments of this approach are discussed in a final section.

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The Von Mises–Fisher distribution was used to draw orthogonal matrices uniformly.

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The Matlab function randfixedsum.m by Roger Stafford was used to draw the probability vectors.

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Correspondence to Dirk Sierag.

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