We look at the problem of pricing contingent convertible bonds (CoCos) where the underlying risky asset dynamics are given by a smile conform model, more precisely, an exponential Lévy process incorporating jumps and heavy tails. A core mathematical quantity, needed in closed form in order to produce an exact analytical expression for the price of a CoCo, is the law of the infimum of the underlying equity price process at a fixed time. With the exception of Brownian motion with drift, no such closed analytical form is available within the class of Lévy processes that are suitable for financial modeling. Very recently, however,there has been some remarkable progress made with the theory of a large family of Lévy processes, known as β-processes. Indeed, for this class of Lévy processes, the law of the infimum at an independent and exponentially distributed random time can be written down in terms of the roots and poles of its characteristic exponent, all of which are easily found within regularly spaced intervals along one of the axes of the complex plane. Combining these results together with a recently suggested Monte Carlo technique, which capitalizes on the randomized law of the infimum, we show the efficient and effective numerical pricing of CoCos. We perform our analysis using a special class of β-processes, known as β-VG, which have similar characteristics to the classic variance-Gamma model. The theory is put to work by performing two case studies. After calibrating our model to market data, we price and analyze one of the Lloyds CoCos as well as the first Rabo CoCo.