Small-Time Asymptotics for Basket Options - the Bivariate SABR Model and the Hyperbolic Heat Kernel on ℍ3

We compute a sharp small-time estimate for the price of a basket call under a bi-variate SABR model with both β 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∗. 1