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