Stochastic Correlation and Volatility Mean-reversion – Empirical Motivation and Derivatives Pricing via Perturbation Theory

Abstract

The dependence structure is crucial when modelling several assets simultaneously. We show for a real-data example that the correlation structure between assets is not constant over time but rather changes stochastically, and we propose a multidimensional asset model which fits the patterns found in the empirical data. The model is applied to price multi-asset derivatives by means of perturbation theory. It turns out that the leading term of the approximation corresponds to the Black–Scholes derivative price with correction terms adjusting for stochastic volatility and stochastic correlation effects. The practicability of the presented method is illustrated by some numerical implementations. Furthermore, we propose a calibration methodology for the considered model.

Key Words:

• Multivariate asset price model
• stochastic correlation
• perturbation theory
• derivatives pricing

Notes

1 In the transformation from the risk-neutral to the historical measure, we set the market price of volatility risk to zero. As we examine the scale of mean-reversion only, this simplification does not disrupt our results.

2 Applying a 10-points median filter to a time series means replacing by the median of the 10-dimensional vector

3 The remaining parameters are set to , , , , , , , , for , , , , .