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

Vitaliy Golomoziy, Kamil Kladivko, Yuliya Mishura

TL;DR

This work addresses rigorous approximation of a Black-Scholes model with drift and volatility switching by a sequence of discrete-time, multiplicative markets. The authors establish weak convergence of the discrete-time switching Markov chains to the limiting continuous-time regime and, conditioning on the Markov path, prove that the discrete-time market prices converge to the geometric Brownian motion with Markov switching. The main result shows that the log-price process converges to $U_t=\int_0^t \sigma_{Y_s} dW_s + \int_0^t (\mu_{Y_s}-\tfrac12\sigma_{Y_s}^2) ds$, and hence the price process converges to $X_t$, under general assumptions on net profits and the switching generator. The analysis leverages Skorokhod convergence, conditional independence on regime paths, and uniformity lemmas for the Markov generator to justify discrete-to-continuous-time transitions with broad applicability for regime-switching financial models.

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.

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

TL;DR

This work addresses rigorous approximation of a Black-Scholes model with drift and volatility switching by a sequence of discrete-time, multiplicative markets. The authors establish weak convergence of the discrete-time switching Markov chains to the limiting continuous-time regime and, conditioning on the Markov path, prove that the discrete-time market prices converge to the geometric Brownian motion with Markov switching. The main result shows that the log-price process converges to , and hence the price process converges to , under general assumptions on net profits and the switching generator. The analysis leverages Skorokhod convergence, conditional independence on regime paths, and uniformity lemmas for the Markov generator to justify discrete-to-continuous-time transitions with broad applicability for regime-switching financial models.

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.
Paper Structure (6 sections, 9 theorems, 145 equations)

This paper contains 6 sections, 9 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete-time multiplicative approximation of the diffusion model with Markov switching
  4. Weak convergence of discrete-time Markov chains to the limit Markov process
  5. Weak convergence to a geometric Brownian motion with Markov switching in the multiplicative scheme of series
  6. Auxiliary lemmas

Key Result

Lemma 1

For every $\alpha > 0$ where $\lambda^* = \sup_i \lambda_i,\ \lambda_* = \inf_i \lambda_i$ are defined in (lambda_start_def) and $\Lambda = {\lambda^* \over \lambda_*}.$

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 9 more