A q-binomial extension of the CRR asset pricing model

TL;DR

This paper introduces a $q$-binomial extension of the CRR asset pricing model, enabling more flexible modeling of default probabilities and option pricing with convergence to Black-Scholes formulas.

Contribution

It develops a novel $q$-binomial framework for the CRR model, incorporating time-dependent switching and tilt parameters, and analyzes its convergence to continuous-time models.

Findings

Model includes tilt and stretch parameters for increment control.

Convergence to Black-Scholes formula with rate O(N^{-1/2}).

Application to default with logistic failure rates.

Abstract

We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on $q$-binomial (or Kemp) random walks, with application to default with logistic failure rates. This model allows us to consider time-dependent switching probabilities varying according to a trend parameter on a non-self-similar binomial tree. In particular, it includes tilt and stretch parameters that control increment sizes. Option pricing formulas are written using $q$-binomial coefficients, and we study the convergence of this model to a Black-Scholes type formula in continuous time. A convergence rate of order $O(N^{-1/2})$ is obtained.