Option pricing with Correlation Risk

In this paper we develop a novel market model where asset variances–covariances evolve stochastically. In addition shocks on asset return dynamics are assumed to be linearly correlated with shocks driving the variance–covariance matrix. Analytical tractability is preserved since the model is linear-affine and the conditional characteristic function can be determined explicitly. Quite remarkably, the model provides prices for vanilla options consistent with observed smile and skew effects, while making it possible to detect and quantify the correlation risk in multiple-asset derivatives like basket options. In particular, it can reproduce and quantify the asymmetric conditional correlations observed on historical data for equity markets. As an illustrative example, we provide explicit pricing formulas for rainbow “Best-of” options. The model solves a long standing problem in the area of asset and derivatives valuation, in fact its definition has two interesting properties that for a long time have appeared to be incompatible. On one hand it is a model linear-affine in the factors, hence it is analytically and econometrically tractable, on the other hand the stochastic factors model risks on first and second moments of each factor and also on their instantaneous covariations. In addition the parametric restrictions which are required to preserve the positive definiteness of the variance covariance matrix are explicitly given. Beyond its importance for the academic research, the model is currently implemented by a number of leading financial institutions as a solution to properly account for “correlations risks”. In fact the recent crises (notice that the paper has been written before its start) has revealed that the assumption of constant correlations is one of the leading sources of model risk in trading and managing the risk of equity and credit financial derivatives.