Hyperbolic normal stochastic volatility model

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 SU distribution—a popular heavy‐tailed distribution originally proposed without stochastic process. It is argued that the SU distribution serves as an analytically superior alternative to the normal SABR model because the two distributions are empirically similar.

A Black-Scholes user's guide to the Bachelier model

“…When the models have the same ATM volatility, the volatility smiles are very close to each other. See Choi et al (2019) for further numerical evidence supporting the equivalence between the two models.…”

“…[µ] t = Z t + µ t denotes BM with drift µ, and ρ * = 1 − ρ 2 . Choi et al (2019) shows that the terminal price F T is distributed as…”

“…Figure5: The Bachelier volatility smile generated by the NSVh(Choi et al, 2019) (left column) and SABR models with β = 0 (right column). From the base parameters, F 0 = 100, σn(F 0 ) = 20, ν = 0.2, ρ = 0.1, we vary ν (top row) and ρ (bottom row) to illustrate that the vol-of-vol, ν, and the correlation, ρ, control the convexity and slope of the volatility smile.…”

The Equivalent Constant-Elasticity-of-Variance (CEV) Volatility of the Stochastic-Alpha-Beta-Rho (SABR) Model

“…In particular, when the interest rate hovered near 0 after the global financial crisis in 2008, the normal SABR (i.e., β = 0) was a popular model choice to allow a negative interest rate (Antonov, 2015). In the normal SABR model, the option price can be expressed by integration (Henry-Labordère, 2005;Korn and Tang, 2013), and an exact simulation scheme is available as a closed-form formula (Choi et al, 2019).…”

“…Further, Lorig et al (2015) obtain the implied BS volatility up to the third order in time, which is valid near the money. The exact solution of the SABR model, however, requires full-scale methods such as the finite difference method (Park, 2014), continuous time Markov chain (Cui et al, 2018), (numerical) multidimensional integration (Henry-Labordère, 2005;Islah, 2009;Antonov et al, 2013;Korn and Tang, 2013), and exact Monte-Carlo simulation (Cai et al, 2017;Choi et al, 2019).…”