On the calibration of distortion risk measures to bid-ask prices

We investigate the calibration of a non-linear pricing model to quoted bid-ask prices and show the existence of a solution in a broad class of distortion risk measures, following the frameworks of Cherny and Madan [Int. J. Theor. Appl. Financ., 2010, 13(8), 1149-1177] and Bannör and Scherer [Eur. Actuarial J., 2013, 3(1), 97-132]. We present an approximation of distortion risk measures by a piecewise linear approximation of concave distortions. This is used to construct a tractable non-parametric calibration procedure to bid-ask prices based on piecewise linear concave distortion functions. To analyze the specific structure of distortion functions, we calibrate quoted bid-ask prices non-parametrically and w.r.t. parametric families and obtain a jump-linear structure. Hence, we suggest considering the parametric family of -expectation convex combinations as a possible family of distortion functions. This family allows fast and efficient calibration and has an appealing economic interpretation.