Correlation Estimation in Hybrid Systems

A simple method is proposed to estimate the instantaneous correlations between state variables in a hybrid system from the empirical correlations between observable market quantities such as spot rate, stock price and implied volatility. The new algorithm is extremely fast since only low-dimension linear systems are involved. If the resulting matrix from the linear systems is not positive semidefinite, the shrinking method, which requires only bisection-style iterations, is recommended to convert the matrix to positive semidefinite. The square of short-term at-the-money implied volatility is suggested as the proxy for the unobservable stochastic variance. When the implied volatility is not available, a simple trick is provided to fill in the missing correlations. Numerical study shows that the estimates are reasonably accurate, when using more than 1,000 data points. In addition, the algorithm is robust to misspecified interest rate model parameters and the short-sampling-period assumption. G2++ and Heston are used for illustration but the method can be extended to other affine term structure, local volatility and jump diffusion models, with or without stochastic interest rate.