ABSTRACT
Allowing correlation to be local, i.e., state-dependent, in multi-asset models allows better hedging by incorporating correlation moves in the delta. When options on a basket, be it a stock index, a cross FX rate, or an interest rate spread, are liquidly traded, one may want to calibrate a local correlation to these option prices. In this chapter we discuss calibration methods for local correlation models. In particular, we introduce a new general technique that produces a whole family of local correlation models. With this new family at hand, one can pick a model that not only calibrates to the smile of basket options but also has extra desirable properties, like fitting a view on correlation skew, mimicking historical correlation, or matching prices of exotic options. The models are described by nonlinear SDEs and built using the particle method, which was introduced in Section 10.2. We also show how this technique generalizes at no extra cost to (i) models that combine stochastic interest rates, stochastic dividend yield, local stochastic volatility, and local correlation; and (ii) single-asset path-dependent volatility models. Our numerical tests in the FX context show the wide variety of admissible correlations and give insight on lower bounds/upper bounds on general multiasset option prices given the smile of a basket and the smiles of its constituents. Parts of this research have been published in [123].
