New solvable stochastic volatility models for pricing volatility derivatives

In this paper we discuss a new approach to extend a class of solvable stochastic volatility models (SVM). Usually, classical SVM adopt a CEV process for instantaneous variance where the CEV parameter γ takes just few values: 0—the Ornstein–Uhlenbeck process, 1/2—the Heston (or square root) process, 1—GARCH, and 3/2—the 3/2 model. Some other models, e.g. with γ = 2 were discovered in Henry-Labordére (Analysis, geometry, and modeling in finance: advanced methods in option pricing. Chapman & Hall/CRC Financial Mathematics Series, London, 2009) by making connection between stochastic volatility and solvable diffusion processes in quantum mechanics. In particular, he used to build a bridge between solvable superpotentials (the Natanzon superpotentials, which allow reduction of a Schrödinger equation to a Gauss confluent hypergeometric equation) and existing SVM. Here we propose some new models with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma \in \mathbb{R}}$$\end{document} and demonstrate that using Lie’s symmetries they could be priced in closed form in terms of hypergeometric functions. Thus obtained new models could be useful for pricing volatility derivatives (variance and volatility swaps, moment swaps).