Anatoliy Swishchuk

In this paper, we consider the problem of pricing variance, volatility, covariance and correlation swaps for financial markets with semi-Markov volatilities. The paper’s motivation derives from the fact that in many financial markets, the inter-arrival times between book events are not independent or exponentially distributed but instead have an arbitrary distribution, which means they can be accurately modelled using a semi-Markov process. Through the results of the paper, we hope to answer the following question: Is it possible to calculate averaged swap prices for financial markets with semi-Markov volatilities? This question has not been considered in the existing literature, which makes the paper’s results novel and significant, especially when one considers the increasing popularity of derivative securities such as swaps, futures and options written on the volatility index VIX. Within this paper, we model financial markets featuring semi-Markov volatilities and price-averaged variance, volatility, covariance and correlation swaps for these markets. Formulas used for the numerical evaluation of averaged variance, volatility, covariance and correlation swaps with semi-Markov volatilities are presented as well. The formulas that are detailed within the paper are innovative because they provide a new, simplified method to price averaged swaps, which has not been presented in the existing literature. A numerical example involving the pricing of averaged variance, volatility, covariance and correlation swaps in a market with a two-state semi-Markov process is presented, providing a detailed overview of how the model developed in the paper can be used with real-life data. The novelty of the paper lies in the closed-form formulas provided for the pricing of variance, volatility, covariance and correlation swaps with semi-Markov volatilities, as they can be directly applied by derivative practitioners and others in the financial industry to price variance, volatility, covariance and correlation swaps.

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.