Markovian Projection to a Displaced Volatility Heston Model

Markovian Projection is an optimal approximation of a complex underlying process with a simpler one, keeping essential properties of the initial process. The Heston process, as the Markovian Projection target, is an example.In this article, we generalize the results of Markovian Projection onto a Heston model to a wider class of approximating models, a Heston model with displaced volatility. As an important application, we derive an effective approximation for FX/EQ options for the Heston model, coupled with correlated Gaussian interest rates.The main technical result is an option evaluation for correlated Heston/Lognormal processes. Unlike the case of exactly solvable (affine) zero correlation or its uncorrelated displacement generalization,considered by Andreasen, non-trivial correlations destroy affine structure and exact solvability.Using the powerful technique of Markovian Projection onto a Heston model with displaced volatility, we produce an effective approximation and present its numerical confirmation