Discrete-time weak approximation of a Black-Scholes model with drift and volatility Markov switching

TL;DR

This paper develops a discrete-time approximation scheme for a Black-Scholes model with Markov switching in drift and volatility, proving weak convergence to the continuous model under broad conditions.

Contribution

It introduces a novel multiplicative scheme of discrete-time markets with Markov switching and proves their weak convergence to the continuous-time Black-Scholes model with switching.

Findings

Discrete-time Markov chains weakly converge to the continuous-time process.

Discrete market models converge to the Black-Scholes model with switching.

Convergence holds under broad assumptions on profits and generator.

Abstract

We consider a continuous-time financial market with an asset whose price is modeled by a linear stochastic differential equation with drift and volatility switching driven by a uniformly ergodic jump Markov process with a countable state space (in fact, this is a Black-Scholes model with Markov switching). We construct a multiplicative scheme of series of discrete-time markets with discrete-time Markov switching. First, we establish that the discrete-time switching Markov chains weakly converge to the limit continuous-time Markov process. Second, having this in hand, we apply conditioning on Markov chains and prove that the discrete-time market models themselves weakly converge to the Black-Scholes model with Markov switching. The convergence is proved under very general assumptions both on the discrete-time net profits and on a generator of a continuous-time Markov switching process.