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

Abstract

a 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.

Can the Heston Model Forecast Energy Generation? A Systematic Literature Review

“…The energy sector appears in the second position, since it also has volatile and unpredictable data (Dong et al, 2020). The main variables used were: option prices (European and American options, energy prices and simulated data); stock exchange indexes (S&P 500, CSI 300 and DJIA); personal income; and mean residence time (Balajewicz and Toivanen, 2017;Bi et al, 2016;Kyriakou et al, 2016;Li et al, 2016;Masoliver and Perelló, 2007;Richmond and Sabatelli, 2004). The application format of these variables were spot data (48%), variance swaps (28%) and series of returns (24%).…”

“…In Figure 8, "stochastic volatility" is the most quoted term, as the Heston model is a very successful closed-form model based on stochastic volatility (Zhu and Lian, 2012). Other trend topics are related to the justifications of the studies, since the Heston model is able to describe the dynamics of assets pricing and variance swaps (Balajewicz and Toivanen, 2017;Bi et al, 2016;Cao and Fang , 2017;Dong et al, 2020). In summary, the Heston model is a very well evaluated and widely used method in the financial area.…”

Variance swaps under the threshold Ornstein–Uhlenbeck model

“…Bi et al. investigate derivatives pricing with stochastic mean‐reverting levels in the principal component of the stochastic variance–covariance matrix. Zhu and Lian derive the closed‐form pricing formula for the discretely sampled variance swap under the Heston stochastic volatility (SV) model using a partial differential equation (PDE) approach.…”

Short time behavior of the ATM implied skew in the ADO-Heston model

“…( 9) is an SVM where the speed of mean-reversion of v t is stochastic, but fully correlated with v t . In the literature, there have been already some attempts to consider an extension of the Heston model by assuming the mean-reversion level θ to be stochastic; see [32,16]. In particular, in [32] it is shown that such a model is able to replicate a term structure of VIX (CBOE volatility index) options.…”