  We present a method for the arbitrage-free interpolation of plain-vanilla
option prices and implied volatilities, which is based on a system of integral
equations that relates terminal density and option prices. Using a
discretization of the terminal density, we write these integral equations as a
system of linear equations. We show that the kernel matrix of this system is in
general ill-conditioned, so that it can not be solved for the discretized
density using a naive approach. Instead, we construct a sparse model for the
kernel matrix using the singular value decomposition (SVD), which allows us not
only to systematically improve the condition number of the kernel matrix, but
also determines the computational effort and accuracy of our method. In order
to allow for the treatment of realistic inputs that may contain arbitrage, we
reformulate the system of linear equations as an optimization problem, in which
the SVD-transformed density minimizes the error between the input prices and
the arbitrage-free prices generated by our method. To further stabilize the
method in the presence of noisy input prices or arbitrage, we apply an
$L_1$-regularization to the SVD-transformed density. Our approach, which is
inspired by recent progress in theoretical physics, offers a flexible and
efficient framework for the arbitrage-free interpolation of plain-vanilla
option prices and implied volatilities, without the need to explicitly specify
a stochastic process, expansion basis functions or any other kind of model. We
demonstrate the capabilities of our method on a number of artificial and
realistic test cases.
