Efficient Pricing of Contingent Convertibles Under Smile Conform Models

Abstract

We look at the problem of pricing CoCo bonds 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 that is 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 process 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 beta-processes, cf. Kuzentsov and Kuzentsov et al.. 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, due to Kuzentsov et al., which capitalises on the randomised law of the infimum we show the efficient and effective numerical pricing of CoCos. We perform our analysis using a special class of beta-processes, known as beta-VG, which have similar characteristics to the classical 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.

Keywords: Wiener-Hopf factorization, Lévy processes, CoCo pricing, Monte-Carlo simulation

JEL Classification: C6, G13