Time changes of Markov processes: applications in finance

Vadim Linetsky (Northwestern)

Let X be a Markov process with the spectral representation of its transition semigroup available in closed form and T an independent time change process with the Laplace transform available in closed form. Then the spectral representation of the time changed process Y(t) = X(T(t)) is also available in closed form. This observation allows us to develop a model architecture to build analytically tractable financial models with rich features: state-dependent jumps, stochastic volatility, and state-dependent killing. In particular, we present a class of pure jump and jump-diffusion commodity models with mean-reverting jumps with analytical solutions for commodity futures options, a class of unified credit-equity models with analytical solutions for equity options and corporate bonds, and a class of correlated defaults models.