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

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  • Martin Forde
  • Hongzhong Zhang

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 $K^*$.

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  • Martin Forde & Hongzhong Zhang, 2016. "Small-time asymptotics for basket options -- the bi-variate SABR model and the hyperbolic heat kernel on $\mathbb{H}^3$," Papers 1603.02896, arXiv.org, revised Apr 2016.
    • Handle: RePEc:arx:papers:1603.02896
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    1. John Armstrong & Martin Forde & Matthew Lorig & Hongzhong Zhang, 2013. "Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion," Papers 1312.2281, arXiv.org, revised Sep 2016.
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