Continuously monitored barrier options under Markov processes

In this paper we present an algorithm for pricing barrier options in one-dimensional Markov models. The approach rests on the construction of an approximating continuous-time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, including a local Levy process and a local volatility jump-diffusion. We also provide a convergence proof and error estimates for this algorithm.Comment: 35 pages, 5 figures, to appear in Mathematical Financ

Numerical methods for Lévy processes

We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model

Essays on American Options

This thesis deals with the pricing of American equity options exposed to correlated interest rate and equity risks. The first article, American options on high dividend securities: a numerical investigation by F. Rotondi, investigates the Monte Carlo-based algorithm proposed by Longstaff and Schwartz (2001) to price American options. I show how this algorithm might deliver biased results when valuing American options that start out of the money, especially if the dividend yield of the underlying is high. I propose two workarounds to correct for this bias and I numerically show their strength. The second article, American options and stochastic interest rates by A. Battauz and F. Rotondi introduces a novel lattice-based approach to evaluate American option within the Vasicek model, namely a market model with mean-reverting stochastic interest rates. Interestingly, interest rates are not assumed to be necessarily positive and non standard optimal exercise policy of American call and put options arise when interest rates are just mildly negative. The third article, Barrier options under correlated equity and interest rate risks by F. Rotondi deals with derivatives with barrier features within a market model with both equity and interest rate risk. Exploiting latticebased algorithm, I price European and American knock-in and knock-out contracts with both a discrete and a continuous monitoring. Then, I calibrate the model to current European data and I document how models that assume either a constant interest rate, or strictly positive stochastic interest rates or uncorrelated interest rates deliver sizeable pricing errors

Randomisation and recursion methods for mixed-exponential Levy models, with financial applications

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Levy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Levy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Levy processes, which includes Brownian motion with drift, Kou's double-exponential model and hyper-exponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Levy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient

Advanced Monte Carlo methods for barrier and related exotic options

International audienceIn this work, we present advanced Monte Carlo techniques applied to the pricing of barrier options and other related exotic contracts. It covers in particular the Brownian bridge approaches, the barrier shifting techniques (BAST) and their extensions as well. We leverage the link between discrete and continuous monitoring to design efficient schemes, which can be applied to the Black-Scholes model but also to stochastic volatility or Merton's jump models. This is supported by theoretical results and numerical experiments