Small-time asymptotics for basket options -- the bi-variate SABR model and the hyperbolic heat kernel on $\mathbb{H}^3$. (arXiv:1603.02896v1 [q-fin.PR])

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, and we show that the [BFL14] result has to be adjusted for strikes above a certain critical value $K^*$ where the convolution density admits a higher order saddlepoint. 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 $K^*$.