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

Paper: arXiv 2501.06895

Authors: Vitaliy Golomoziy, Kamil Kladivko, Yuliya Mishura

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework for the discrete-time weak approximation of a Black-Scholes model under Markov regime switching with a countable state space. It is strictly theoretical with high mathematical complexity, featuring formal proofs of convergence in the Skorokhod topology, but completely lacks empirical validation or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Construct discrete-time market <br>weakly converging to Black-Scholes <br>with Markov switching"] --> B["Inputs: Continuous-time Black-Scholes SDE <br>& Countable state Markov switching generator"]
    
    B --> C["Construct multiplicative scheme <br>of discrete-time markets with Markov chains"]
    
    C --> D["Step 1: Prove weak convergence of <br>discrete Markov chains to continuous process"]
    
    D --> E["Step 2: Apply conditioning on Markov chains <br>under general net profit assumptions"]
    
    E --> F["Key Outcome: Prove discrete-time market <br>weakly converges to Black-Scholes model <br>with Markov switching"]