Albert Ferreiro-Castilla

Pricing of contingent convertibles under smile conform models

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 Levy 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 Levy 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 Levy processes, known as β-processes, cf. Kuznetsov [12] and Kuznetsov et al. [14]. Indeed for this class of Levy 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 Kuznetsov et al. [13], 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 β-processes, known as β-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.

Efficient Pricing of Contingent Convertibles Under Smile Conform Models

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 Levy 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 Levy 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 Levy processes, known as beta-processes, cf. Kuzentsov and Kuzentsov et al.. Indeed for this class of Levy 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.

The β-Meixner model

We propose to approximate the Meixner model by a member of the @b-family introduced by Kuznetsov (2010) in [2]. The advantage of the approximation is the semi-explicit formulae for the running extrema under the @b-family processes which enables us to produce more efficient algorithms for pricing path dependent options through the Wiener-Hopf factors. We will explore the performance of the approximation both in an equity framework and in the credit risk setting, where we use the approximation to calibrate a surface of credit default swaps. The paper follows the approach of the study made by Schoutens and Damme (2010) in [1], where the aim was to approximate the variance gamma. We will contextualize the results by Schoutens and Damme (2010) in [1] and the ones here with respect to the approach taken by Jeannin and Pistorius (2010) in [15]. An asymptotic expression for the rate of convergence of the approximation is derived.

An Euler-Poisson scheme for Lévy driven stochastic differential equations

We describe an Euler scheme to approximate solutions of Levy driven Stochastic Differ- ential Equations (SDE) where the grid points are random and given by the arrival times of a Poisson process. This result extends a previous work of the authors in Ferreiro-Castilla et al. (11). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and proof that the approximation converges in mean square error with rate O(n −1/2 ). The only requirement of the methodology is to have exact samples from the resolvent of the Levy process driving the SDE; classic examples such as stable processes, subclasses of spectrally one sided Levy processes and new families such as meromorphic Levy processes (cf. Kuznetsov et al. (20)) are some examples for which the implementation of our algorithm is straightforward.