Smiling Hyperbolas

We propose using hyperbolic splines for arbitrage free interpolation of implied volatilities in the strike domain. Hyperbolic splines allow for perfect fit to input data and have carry computational cost since there is no root finding or calibration: spline parameters are expressed directly in terms of elementary mathematical functions. We demonstrate that hyperbolic splines work just as well in the extrapolation region providing a tool for fixing wings produced by arbitrage prone methods. Finally we present a family of global hyperbolic splines that have time-dependent extensions with an intuitive interpretation in terms of local diffusions coupled with a jump to default.