Principal component models with stochastic mean‐reverting levels. Pricing and covariance surface improvements

In this work, we create a family of simple stochastic covariance models, which display stochastic mean‐reverting levels of covariance as an additional level of stochastic behavior beyond well‐known stochastic volatility and correlation. The one‐dimensional version of our model is inspired by Heston model, while the multidimensional model generalizes the principal component stochastic volatility model. Their main contribution is that they capture stochastic mean‐reversion levels on the volatility and on the eigenvalues of the instantaneous covariance matrix of the vector of stock prices, with direct implications on the correlations as well. Our focus is on the multidimensional model; we investigate its properties and derive a closed‐form expression for the characteristic function. This allows us to study the pricing of financial derivatives, such as correlation and spread options. Those prices are compared with simulated Monte Carlo prices for correctness. A sensitivity analysis is performed on the parameters of the stochastic mean‐reverting level of volatilities to study their impact on the price. Finally, implied volatility curves and correlation surfaces are built to reveal the additional flexibility gained within the new model. Copyright © 2016 John Wiley & Sons, Ltd.