Methodol Comput Appl Probab DOI 10.1007/s11009-015-9462-7 Option Pricing Under Jump-Diffusion Processes with Regime Switching Nikita Ratanov 1 Received: 11 May 2015 / Revised: 18 August 2015 / Accepted: 1 September 2015 © Springer Science+Business Media New York 2015 Abstract We study an incomplete market model, based on jump-diffusion processes with parameters that are switched at random times. The set of equivalent martingale measures is determined. An analogue of the fundamental equation for the option price is derived. In the case of the two-state hidden Markov process we obtain explicit formulae for the option prices. Furthermore, we numerically compare the results corresponding to different equivalent martingale measures. Keywords Jump-telegraph process · Jump-diffusion process · Martingales · Relative entropy · Financial modelling · Option pricing · Esscher transform Mathematics Subject Classification (2010) 91B28 · 60G44 · 60J75 · 60K99 1 Introduction Consider the jump-diffusion process with time-dependent deterministic driving parameters that are simultaneously switched at random times, X(t) = T c (t) + J h (t) + W σ (t), t ≥ 0. Here T c is path-by-path integral of the alternating at random times velocity regimes c i (t), J h we denote the associated pure jump part, i.e. the stochastic integral w.r.t. counting pro- cess N = N(t) applied to the alternating functions h i (t), and W σ denotes the Wiener part, defined by the stochastic integral (w.r.t. Brownian motion B) of the process, which is formed by deterministic functions σ i (t), i ∈ D, D := {1,...,d },d ≥ 2, alternating simultaneously with T c and J h .  Nikita Ratanov nratanov@urosario.edu.co 1 Universidad del Rosario, Cl. 12c, No.4-69, Bogot´ a, Colombia