A hybrid approach for the implementation of the Heston model

We propose a hybrid tree-finite difference method in order to approximate the Heston model. We prove the convergence by embedding the procedure in a bivariate Markov chain and we study the convergence of European and American option prices. We finally provide numerical experiments that give accurate option prices in the Heston model, showing the reliability and the efficiency of the algorithm

Pricing and Hedging GLWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models

Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Forsyth and Vetzal the Black and Scholes framework seems to be inappropriate for such long maturity products. They propose to use a regime switching model. Alternatively, we propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GLWB variables annuities: a hybrid tree-finite difference method and a hybrid Monte Carlo method, an ADI finite difference scheme, and a standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GLWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal and optimal withdrawal (including lapsation) strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions

Efficient pricing of swing options in Lévy-driven models

International audienceWe consider the problem of pricing swing options with multiple exercise rights in Lévy-driven models. We propose an efficient Wiener-Hopf factorisation method that solves multiple parabolic partial integro-differential equations associated with the pricing problem. We compare the proposed method with a finite difference algorithm. Both proposed deterministic methods are related to the dynamic programming principle and lead to the solution of a multiple optimal stopping problem. Numerical examples illustrate the efficiency and the precision of the proposed methods

Tree methods

International audienceTree methods are among the most popular numerical methods to price financial derivatives. Mathematically speaking, they are easy to understand and do not require severe implementation skills to obtain algorithms to price financial derivatives. Tree methods basically consist in approximating the diffusion process modeling the underlying asset price by a discrete random walk. In this contribution, we provide a survey of tree methods for equity options, which focus on multiplicative binomial Cox-Ross-Rubinstein model

Fast binomial procedures for pricing Parisian/ParAsian options

The discrete procedures for pricing Parisian/ParAsian options depend, in general, by three dimensions: time, space, time spent over the barrier. Here we present some combinatorial and lattice procedures which reduce the computational complexity to second order. In the European case the reduction was already given by Lyuu-Wu \cite{WU} and Li-Zhao \cite{LZ}, in this paper we present a more efficient procedure in the Parisian case and a different approach (again of order 2) in the ParAsian case. In the American case we present new procedures which decrease the complexity of the pricing problem for the Parisian/ParAsian knock-in options. The reduction of complexity for Parisian/ParAsian knock-out options is still an open problem.Les méthodes à temps discret pour le pricing des options parisienne et parAsian dépendent généralement de trois paramètres : Le temps, l'espace et le temps écoulé proche de la barrière. Dans ce travail, nous présentons des procédures combinatoires et de treillis qui permettent de réduire d'ordre 2 la complexité du calcul. Ces simplifications ont déjà été utilisées par Lyuu-Wu et Li-Zhao dans le cas des options européennes. Dans cet article, une technique plus efficace est employée pour les options parisienne et parAsian. Nous introduisons aussi de nouvelles méthodes rapides pour les options américaines applicables aux parisiennes et parAsians knock-in. La généralisation de ce type de procédures aux options parisienne/parAsian knock-out reste un problème ouvert

Pricing Ratchet equity-indexed annuities with early surrender risk in a CIR++ model

International audienceIn connection with a problem posed by Kijima and Wong \cite{kw}, we propose a lattice algorithm for pricing simple Ratchet equity-indexed annuities (EIAs) with early surrender risk and global minimum contract value when the asset value depends on the CIR++ stochastic interest rates. In addition we present an asymptotic expansion technique which permits to obtain a first order approximation formula for the price of simple Ratchet EIAs without early surrender risk and without global minimum contract value. Numerical comparisons show the reliability of the proposed methods.Suite au problème posé par Kijima et Wong, nous proposons un algorithme de treillis pour le pricing du Ratchet sur action avec risque de rachat anticipé. Il inclut aussi une valeur minimale globale lorsque l'actif dépend du processus CIR++ du taux d'intérêt. Par ailleurs, nous présentons une technique de développement asymptotique permettant une approximation de premier ordre pour le prix du Ratchet EIAs, sans risque de rachat anticipé ni valeur minimale. Des expériences numériques montrent une bonne qualité des résultats obtenus par la méthode proposée