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

Appendix to: Efficient European and American Option Pricing Under a Jump-diffusion Process

When the underlying asset of an option displays oscillations, spikes or heavy-tailed distributions, modeling it with the lognormal diffusion process is inadequate. In order to overcome these real world difficulties, Merton proposed a jump-diffusion model, where the dynamics of the price of the underlying are subject to variations due to a Brownian process and also to possible jumps, driven by a compound Poisson process. There have been a lot of attempts to obtain a discretization of the Merton model with tree methods in order to price American or more complex options, e. g. Amin, the O(n3) procedure by Hilliard and Schwartz and the O(n2:5) procedure by Dai et al. Here, starting from the implementation of the seven-nodes procedure by Hilliard and Schwartz, we prove theoretically that it is possible to reduce the complexity of this method to O(n2 ln n) in the American put case. Our method is based on a suitable truncation of the lattice structure; the proofs provide closed formulas for the truncation limitations

Efficient derivatives evaluation under a jump-diffusion process

Merton (1976) provides a jump-diffusion model, where the dynamics of the price of the underlying are subject to variations due to a Brownian process and also to possible jumps. Under the Black and Scholes (1973) assumptions on the Brownian component and considering a compound Poisson process for the jump part, the model admits a series solution for the European option pricing. Amin (1993) proposes a procedure for derivative pricing by discretising the distribution of the underlying allowing the jumps to have a random amplitude which must be a multiple of the Brownian move. Hilliard and Schwartz (HS, 2005) considering the independency of the two factors develop a multinomial lattice: one variable mimicking the diusion process and the second one the log-normal jumps in the compound Poisson process. HS procedure provides more accurate results than Amin and the weak convergence of the discrete price is ensured in the special case of deterministic jump amplitude and numerically justified otherwise. HS bivariate tree can be applied to the evaluation of American options. The time complexity of the HS backward procedure is O(n3). Dai et al. (2010) build on the HS procedure reducing complexity to O(n2:5) by dissolving the intermediate nodes on the tree introduced by the jumps in the nearest diffusion node, therefore providing a one-dimensional tree. Here, also starting from the HS technique we introduce a procedure which further reduces the complexity both in the European and in the American case. We prove this theoretically when the log-normal distribution is discretised by a variable with tree possible states (up, down and no jump). In the general case, our procedure is numerically justified as in the Hilliard and Schwartz paper

Fair evaluation of life insurance policies with periodic rebalancing between asset portfolios and interest rate guarantee

We consider the problem of evaluating at fair rates an innovativelife insurance policy with a rebalancing scheme between different as-set portfolios and an embedded interest rate guarantee. The premiumsare invested in two portfolios of assets characterized by different levelsof risk and sums are transferred from one fund to the other at someprefixed dates, depending on the performance of the funds. The dy-namics of each fund is approximated by means of binomial lattices but,since the remixing feature makes the evolution of the riskier fund path-dependent, we propose a model based on \u201crepresentative\u201d values to keepthe computational cost of the evaluation problem at a reasonable level.The usual backward induction coupled with linear interpolation allowsus to determine the policy value at inception