Existence of a calibrated Regime Switching Local Volatility model
Abstract
By Gyongy's theorem, a local and stochastic volatility model is calibrated to
the market prices of all call options with positive maturities and strikes if
its local volatility function is equal to the ratio of the Dupire local
volatility function over the root conditional mean square of the stochastic
volatility factor given the spot value. This leads to a SDE nonlinear in the
sense of McKean. Particle methods based on a kernel approximation of the
conditional expectation, as presented by Guyon and Henry-Labord\`ere (2011),
provide an efficient calibration procedure even if some calibration errors may
appear when the range of the stochastic volatility factor is very large. But so
far, no existence result is available for the SDE nonlinear in the sense of
McKean. In the particular case where the local volatility function is equal to
the inverse of the root conditional mean square of the stochastic volatility
factor multiplied by the spot value given this value and the interest rate is
zero, the solution to the SDE is a fake Brownian motion. When the stochastic
volatility factor is a constant (over time) random variable taking finitely
many values and the range of its square is not too large, we prove existence to
the associated Fokker-Planck equation. Thanks to Figalli (2008), we then deduce
existence of a new class of fake Brownian motions. We then extend these results
to the special case of the LSV model called Regime Switching Local Volatility,
where the stochastic volatility factor is a jump process taking finitely many
values and with jump intensities depending on the spot level. Under the same
condition on the range of its square, we prove existence to the associated
Fokker-Planck PDE. We then deduce existence of the calibrated model by
extending the results in Figalli (2008).