Pricing of pseudo-swaps based on pseudo-statistics

The main problem in pricing variance, volatility, and correlation swaps is how to determine the evolution of the stochastic processes for the underlying assets and their volatilities. Thus, sometimes it is simpler to consider pricing of swaps by so-called pseudo-statistics, namely, the pseudo-variance, -covariance, -volatility, and -correlation. The main motivation of this paper is to consider the pricing of swaps based on pseudo-statistics, instead of stochastic models, and to compare this approach with the most popular stochastic volatility model in the Cox-Ingresoll-Ross (CIR) model. Within this paper, we will demonstrate how to value different types of swaps (variance, volatility, covariance, and correlation swaps) using pseudo-statistics (pseudo-variance, pseudo-volatility, pseudo-correlation, and pseudo-covariance). As a result, we will arrive at a method for pricing swaps that does not rely on any stochastic models for a stochastic stock price or stochastic volatility, and instead relies on data/statistics. A data/statistics-based approach to swap pricing is very different from stochastic volatility models such as the Cox-Ingresoll-Ross (CIR) model, which, in comparison, follows a stochastic differential equation. Although there are many other stochastic models that provide an approach to calculating the price of swaps, we will use the CIR model for comparison within this paper, due to the popularity of the CIR model. Therefore, in this paper, we will compare the CIR model approach to pricing swaps to the pseudo-statistic approach to pricing swaps, in order to compare a stochastic model to the data/statistics-based approach to swap pricing that is developed within this paper.