Small-time asymptotics for basket options -- the bi-variate SABR model and the hyperbolic heat kernel on ${\mathbb{H}}^3$

Abstract

We compute a sharp small-time estimate for the price of a basket call under a bi-variate SABR model with both $\beta$ parameters equal to $1$ and three correlation parameters, which extends the work of Bayer,Friz&Laurence [BFL14] for the multivariate Black-Scholes flat vol model. The result follows from the heat kernel on hyperbolic space for $n=3$ combined with the Bellaiche [Bel81] heat kernel expansion and Laplace's method, and we give numerical results which corroborate our asymptotic formulae. Similar to the Black-Scholes case, we find that there is a phase transition from one "most-likely" path to two most-likely paths beyond some critical $K^{\ast }$.

24 pages, 10 figures

References in corpus (1)

• Exponential functionals of Brownian motion, II: Some related diffusion processes