Lattice model (finance)

Résumé

In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem. Equity and commodity derivatives In general the approach is to divide

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https://en.wikipedia.org/wiki/Lattice_model_(finance)

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Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333-380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395-1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771-833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191-225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087-1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.

SPRINGER2022

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A relationship based upon analogy is explored between (i) a lattice-based model for percolation by cylinders that employs distinct site types with unequal occupation probabilities in order to capture heterogeneities in the particle dispersion, and (ii) the well-studied mean-field lattice gas model. The strength of the correlation that sites in the percolation model that have distinct occupation probabilities are adjacent to each other maps into the coupling constant for the associated lattice fluid problem. Pursuing the analogy further, the vapor-liquid coexistence curve for the lattice gas translates into a phase boundary for the percolation problem, in terms of a locus of correlation strengths that demarcate a region of phase separation into sites with different occupation probabilities and distinct percolation thresholds. The dependence of the percolation thresholds in the one- and two-phase regions as delineated by this analogy are calculated as functions of the degree of disparity between site occupation probabilities and the strength of the correlation between adjacent site types.

2018

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