  The calibration of a local volatility models to a given set of option prices
is a classical problem of mathematical finance. It was considered in multiple
papers where various solutions were proposed. In this paper an extension of the
approach proposed in LiptonSepp2011 is developed by i) replacing a piecewise
constant local variance construction with a piecewise linear one, and ii)
allowing non-zero interest rates and dividend yields. Our approach remains
analytically tractable; it combines the Laplace transform in time with an
analytical solution of the resulting spatial equations in terms of Kummer's
degenerate hypergeometric functions.
