This note contains two main results. (1) (Discrete time) Suppose S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose the initial option prices agree with the Cox–Ross–Rubinstein binomial tree model.) Then S is a geometric simple random walk. (2) (Continuous time) Suppose S = S0eσX−σ 〈X〉/2 is a cont...

Financial derivatives are contracts concerning rights and obligations to engage in future transactions on some underlying financial instrument. A major concern in financial markets is to compute an expected value of such contracts as a basis for trading decisions. The Cox, Ross and Rubinstein (CRR) binomial tree model is a popular discrete approach to such computations, which requires time quad...

Tuenter considered centered binomial sums of the form Sr(n) = 2n ∑

The. theory of reduced incidence algebras of binomial posets furnishes a unified treatment of several types of generating functions that arise in enumerative combinatorics. Using this theory as a tool, we study ‘reduced covering algebras’ of binomial lattices and show that they are isomorphic to various algebras of q-binomial generating functions for certain modular binomial lattices.

Binomial trees are very popular in both theory and applications of option pricing. As they often suffer from an irregular convergence behavior, improving this is an important task. We build upon a new version of the Edgeworth expansion for lattice models to construct new and quickly converging binomial schemes with a particular application to barrier options.

A general method to construct recombinant tree approximations for stochastic volatility models is developed and applied to the Heston model for stock price dynamics. In this application, the resulting approximation is a four tuple Markov process. The first two components are related to the stock and volatility processes and take values in a two-dimensional binomial tree. The other two component...

A new binomial approximation to the Black–Scholes model is introduced. It is shown that for digital options and vanilla European call and put options that a complete asymptotic expansion of the error in powers of n−1 exists. This is the first binomial tree for which such an asymptotic expansion has been shown to exist.

* We are grateful for helpful comments from Salih Neftci. The comments of an anonymous referee were enormously beneficial. We are grateful for support from the Schweger Fund. Remaining errors are our own.

The beta-binomial distribution is resulted when the probability of success per trial in the binomial distribution varies in successive trials and the mixing distribution is from the beta family. For experiments with binary outcomes, often it may happen that observations exhibit some extra binomial variation and occur in clusters. In such experiments the beta-binomial distribution can generally ...