On the Weak Error for Local Stochastic Volatility Models

Local stochastic volatility refers to a popular model class in applied mathematical finance that allows for “calibration-on-the-fly,” typically via a particle method, derived from a formal McKean–Vlasov equation. Well-posedness of this limit is a well-known problem in the field; the general case is largely open, despite recent progress in Markovian situations. Our take is to start with a well-defined Euler approximation to the formal McKean–Vlasov equation, followed by a newly established half-step-scheme, allowing for good approximations of conditional expectations. In a sense, we do Euler first and particle second, in contrast to previous works that start with the particle approximation. We show weak order one for the Euler discretization, plus error terms that account for the said approximation. The case of particle approximation is discussed in detail, and the error rate is given independent of all parameters used.