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

TL;DR

This paper derives precise small-time price estimates for basket options under a bi-variate SABR model using hyperbolic heat kernel techniques, revealing phase transitions in optimal paths.

Contribution

It extends small-time asymptotics for basket options to the bi-variate SABR model using hyperbolic heat kernel methods, connecting stochastic models with geometric analysis.

Findings

Derived sharp small-time asymptotics for basket call prices.

Identified phase transition in most-likely paths beyond a critical strike.

Numerical results support the theoretical asymptotic formulas.

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 $K^*$.