Multinomial Lattices Research Articles

The continuous weak order

The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet Σ={x,y,z,…}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card(Σ)=2, as lattices of lattice paths. By interpreting the letters x,y,z,… as axes, these words can be interpreted as discrete increasing paths on a grid of a d-dimensional cube, with d=card(Σ).We show how to extend this ordering to images of continuous monotone functions from the unit interval to a d-dimensional cube and prove that this ordering is a lattice, denoted by L(Id). This construction relies on a few algebraic properties of the quantale of join-continuous functions from the unit interval of the reals to itself: it is cyclic ⋆-autonomous and it satisfies the mix rule.We investigate structural properties of these lattices, which are self-dual and not distributive. We characterize join-irreducible elements and show that these lattices are generated under infinite joins from their join-irreducible elements, they have no completely join-irreducible elements nor compact elements. We study then embeddings of the d-dimensional multinomial lattices into L(Id). We show that these embeddings arise functorially from subdivisions of the unit interval and observe that L(Id) is the Dedekind-MacNeille completion of the colimit of these embeddings. Yet, if we restrict to embeddings that take rational values and if d>2, then every element of L(Id) is only a join of meets of elements from the colimit of these embeddings.

On the Minimal Entropy Martingale Measure and Multinomial Lattices with Cumulants

In this article, we describe with relevant examples based on empirical data how to use the minimal entropy martingale measure (MEMM) to price European and American Options in multinomial lattices which take into account cumulants information. For trinomial lattices, we show that minimal entropy prices are close to results obtained using the Black and Scholes option pricing formula. For pentanomial lattices, minimal entropy prices are close to results obtained under the mean-correcting martingale measure using the discrete Fourier transform. The MEMM is very easy to compute and is therefore a good candidate for option pricing in multinomial lattices.

On the Join Dependency Relation in Multinomial Lattices

Aiming to understand equivalence relations that model concurrent computation, we investigate congruences of multinomial lattices \(\mathcal{L}(v)\) introduced by Bennett and Birkhoff (Algebra Univers. 32(1):115–144, 1994). Our investigation gives rise to an explicit description of the join dependency relation between two join irreducible elements and of its reflexive transitive closure. The explicit description emphasizes several properties and makes it possible to separate the equational theories of multinomial lattices by their dimensions. In their covering of non modular varieties Jipsen and Rose (Varieties of lattices, Lecture Notes in Mathematics, vol. 1533, Springer, Berlin, 1992) define a sequence of equations \(SD_{n}(\land)\), for n ≥ 0. Our main result sounds as follows: if\(v = (v_{1},\ldots,v_{n}) \in \mathbb{N}^{n}\)andvi > 0 for i = 1,...,n, then the multinomial lattice\(\mathcal{L}(v)\)satisfies\(SD_{n-1}(\land)\)and fails\(SD_{n-2}(\land)\)

Properties of Multinomial Lattices with Cumulants for Option Pricing and Hedging

In this paper, we analyze properties of multinomial lattices that model general stochastic dynamics of the underlying stock by taking into account any given cumulants (or moments). First, we provide a parameterization of multinomial lattices, and demonstrate that mean, variance, skewness, and kurtosis of the underlying may be matched using five branches. Then, we investigate the convergence of the multinomial lattice when the basic time period approaches zero, and prove that the limiting process of the multinomial lattice that matches annualized mean, variance, skewness and kurtosis is given by a compound Poisson process. Finally, we illustrate the effect of higher order moments in the underlying asset process on the price of derivative securities through numerical experiments using the multinomial lattice, and provide a comparison with jump-diffusion models.

Very High Order Lattice Methods for One Factor Models

Lattice methods are often used to value derivative instruments. Multinomial lattice methods can in principle converge to the true value of the derivative to very high order. In this paper we describe how very high order multinomial lattices can be constructed and implemented when the SDE followed by the underlying state variable can be solved. We illustrate with comparisons between methods with branching order 3, 7, 11, 15 and 19 applied to a geometric Brownian motion. Incorporating both a terminal correction and appropriate truncation methods we find for the heptanomial lattice convergence rates at its theoretical maximum for European style options. With 50 time steps per year our errors are of order 10E-11. With 100 time steps per year our errors are of order 10E-13, approaching the practical limit of the accuracy obtainable in our implementation. We discuss alternative methods of enabling the heptanomial lattice to achieve high convergence rates for payoff functions with a finite number of critical points. As an example we value a binary option to a high degree of accuracy. We also investigate applications to American and Bermudan options. Based on our comparisons, we conclude that the heptanomial lattice is the fastest and most accurate of the lattices of higher order, and recommend its use as standard in many one factor lattice implementations.

Accelerating American Option Pricing in Lattices

A ccelerating the pricing of American options in binomial and multinomial lattices in a Black-Scholes environment has been addressed in the literature (Kim and Byun [1994] (KB), Breen [1991]), but the method presented here (hereafter the diagonal method, or DM) is substantially more efficient and is applicable to a wider variety of cases. In particular, while the KB and Breen techniques yield an acceleration factor of three or four for the American put option, the DM results in acceleration factors in excess of an order of magnitude, while arriving at identical values as those obtained from the standard Cox, Ross, and Rubinstein [1979] (CRR) binomial method. The method is specific to CRR trees in that, starting at a fixed underlying price, an up move (u) followed by a down move (d) results in an underlying price equal to the original price, i.e., d = l/u. The DM also has potential for speeding the computation of American options on interest rate instruments and options involving multiple assets.

The order dimension of multinomial lattices

Using the notion of Ferrers dimension of incidence structures, the order dimension of multi-nomial lattices (i.e. lattices of ‘multi-permutations’) is determined. In particular, it is shown that the lattice of all permutations on ann-element set has dimensionn−1.