Applications of Semi-Markov Chains in Bond and Energy Markets

In this paper, we consider applications of semi-Markov chains (SMCs) in bond and energy markets. The research work was conducted for semi-Markov-modulated exponential-affine bond prices, and for pricing and hedging European-style options under a semi-Markov-switching diffusion model. In the former case, we will focus on the Hull–White model and generalize the results to three states and more for the SMC. In the latter case, based on real data (daily Alberta electricity prices from 1 January 2020 to 31 December 2025, Alberta Electric System Operator (AESO)), it is observed that occupation time does not follow an exponential distribution for two out of four volatility states, which violates the Markov process conditions. Thus, we calculated call and put option prices for different maturities and strikes under the semi-Markov condition by applying the finite-difference method. We also calculate the option prices using the Markov-diffusion model, and the classical Black–Scholes (BS) call option price equation for comparison. The contributions of the paper are twofold: the application of SMCs with any number of states to bond pricing, and the applications of SMCs in energy markets with real market data.