This notebook provides a complete implementation of a computational framework for pricing and hedging European options in financial markets characterized by Markov-modulated regime-switching dynamics. The underlying model extends the classical Black-Scholes paradigm by allowing key market parameters—volatility, drift, and interest rate—to evolve stochastically according to a finite-state continuous-time Markov chain.
The asset price
where
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Volterra-type Integral Equations: Solved numerically for the option price
$\phi(t, s, i)$ and hedge ratio$\psi(t, s, i) = \frac{\partial \phi}{\partial s}$ , offering improved efficiency over traditional PDE solvers. -
Monte Carlo Simulations: Assess hedging performance across thousands of regime paths, capturing the impact of transition frequency and volatility.
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High-Performance Computing: Leveraging
NumbaJIT compilation and vectorized operations, the framework achieves significant speed-ups suitable for large-scale risk analysis.
- Captures regime-dependent hedging performance and evaluates discrete vs continuous strategies.
- Highlights model risk from high-volatility transitions and tracking errors in practical hedging.
- Fully reproducible and extensible framework, enabling future research in incomplete markets and risk-aware derivatives trading.
📌 This implementation directly corresponds to the methods and experiments described in the paper: “Computational Methods for Optimal Hedging in Markov Modulated Markets” by Albin James Maliakal, Nagaraju Baydeti, Alen Peter Yimchunger, and Yongkong Kumzek Chang.