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

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