Optimal investment of variance-swaps in jump-diffusion market with regime-switching

We consider a general jump-diffusion market with regime-switching where the jump risk is modeled as a Markov-modulated Poisson random measure. In this incomplete market, we price the variance-swaps using a combination of the Esscher transform and change of measure on time-inhomogeneous Markov chains. We study the dynamic optimal investment problem of the variance-swaps and characterize the optimal feedback strategy. Moreover, a closed-form solution to the HJB PDE associated with the stochastic control problem is established and the verification theorem is proved. The numerical analysis based on a two-state Markov chain uncovers some robust features of the optimal investment strategy.