Hyperbolic normal stochastic volatility model

TL;DR

This paper introduces a new continuous-time stochastic volatility model based on hyperbolic functions, offering closed-form simulation and heavy-tailed distribution properties, improving upon existing models like SABR.

Contribution

It proposes a novel hyperbolic stochastic volatility model with closed-form simulation and links to Johnson's $S_U$ distribution, enhancing heavy-tailed modeling in finance.

Findings

Model has a closed-form Monte-Carlo simulation scheme.

Transition probability follows Johnson's $S_U$ distribution.

Empirically similar but analytically superior to normal SABR.

Abstract

For option pricing models and heavy-tailed distributions, this study proposes a continuous-time stochastic volatility model based on an arithmetic Brownian motion: a one-parameter extension of the normal stochastic alpha-beta-rho (SABR) model. Using two generalized Bougerol's identities in the literature, the study shows that our model has a closed-form Monte-Carlo simulation scheme and that the transition probability for one special case follows Johnson's $S_U$ distribution---a popular heavy-tailed distribution originally proposed without stochastic process. It is argued that the $S_U$ distribution serves as an analytically superior alternative to the normal SABR model because the two distributions are empirically similar.