Journal of Computational Finance

Risk.net

Calibration of local correlation models to basket smiles

Julien Guyon

  • Allowing correlation to be local in multi-asset models allows better hedging by incorporating correlation moves in the delta.
  • We explain how to build all the local correlation models that are calibrated to a basket smile
  • One can now pick a calibrated model so as to fit a view on correlation skew, reproduce historical correlation, or match exotic option prices.
  • The range of prices of an exotic option in calibrated models gives precious insight on model risk.
  • Our original approach combines the particle method and a simple affine transform

Allowing correlation to be local, ie, 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-foreign exchange rate or an interest rate spread- are liquidly traded, one may want to calibrate a local correlation to these option prices. Only two particular solutions have been suggested so far in the literature. Both impose a particular dependency of the correlation matrix on the asset values that one has no reason to undergo. They may also fail to be admissible, ie, positive semi-definite. We explain how, by combining the particle method presented in "The smile calibration problem solved" by Guyon and Henry-Labordère (2011) with a simple affine transform, we can build all the calibrated local correlation models. The two existing models appear as special cases (if admissible). For the first time, one can now choose a calibrated local correlation in order to fit a view on the correlation skew, or reproduce historical correlation, or match some exotic option prices, thus improving the pricing, hedging and risk-management of multi-asset derivatives. This technique is generalized to calibrate models that combine stochastic interest rates,stochastic dividend yield, local stochastic volatility and local correlation. Numerical results show the wide variety of calibrated local correlations and give insight into a difficult (still unsolved) problem: finding lower bounds/upper bounds on general multi-asset option prices given the whole surfaces of implied volatilities of a basket and its constituents.

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