Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phase-type jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phase-type case via martingale stopping and Wiener-Hopf factorisation. Also the first passage time problem is studied for a regime switching Lávy process with phase-type jumps. This is achieved by an embedding into a a semi-Markovian regime switching Brownian motion.

The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Lévy process and the solution of Gerber and Shiu [Astin

This research was supported by the Deutsche

Closed form of the ruin probability for a L'evy processes, possible killed at a constant rate, with arbitrary positive, and mixed exponentially negative jumps is given. Keywords: Ruin probability, closed form, L'evy process, mixed-exponential distributions. 1 Introduction 1.1 Let X = fX t g t0 be a real valued stochastic process defined on a stochastic basis(\Omega ; F ; F = (F t ) t0 ; P ) that satisfy the usual conditions. Assume that X is c`adl`ag, adapted, X 0 = 0, and for 0 s ! t the random variable X t \Gamma X s is independent of the oe-field F s with a distribution that only depends on the difference t \Gamma s. X is a process with stationary independent increments (PIIS), or a L'evy process. For q 2 R, /(q) denotes the characteristic exponent of X given by L'evy-Khinchine formula /(q) = 1 t log E(e iqX t ) = ibq \Gamma 1 2 oe 2 q 2 + Z R (e iqy \Gamma 1 \Gamma iqy1 fjyj!1g )\Pi(dy) where b and oe 0 are real constants, and \Pi is a positive measure on R \...

In this paper we overview the pricing of several so-called exotic options in the nowdays quite popular exponential Lévy models.

We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasi-variational inequality, that it is C 2 except at the stopping boundary and that it is C 1 everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each of which is a value function of an optimal stopping problem for a diffusion. This sequence, which converges to the value function of the American put option for jump diffusions, is constructed sequentially using a functional operator that maps a certain class of convex functions to smooth functions satisfying some quasi-variational inequalities. This sequence converges to the value function of the American put option uniformly and exponentially fast, therefore it provides a good approximation scheme. In fact, the value of the American put option is the fixed point of the functional operator we use. 1

We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasi-variational inequality, that it is C 2 except at the stopping boundary and that it is C 1 everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each of which is a value function of an optimal stopping problem for a diffusion. This sequence, which converges to the value function of the American put option for jump diffusions, is constructed sequentially using a functional operator that maps a certain class of convex functions to smooth functions satisfying some quasi-variational inequalities. This sequence converges to the value function of the American put option uniformly and exponentially fast, therefore it provides a good approximation scheme. In fact, the value of the American put option is the fixed point of the functional operator we use. 1

Elementary proofs of classical theorems on pricing perpetual call and put options in the standard Black-Scholes model are given. The method presented does not rely on stochastic calculus and is also applied to give prices and optimal stopping rules for perpetual call options when the stock is driven by a Levy process with no positive jumps, and for perpetual put options for stocks driven by a Levy process with no negative jumps 1 This work was partially written at the Laboratoire de Statistique et Probabilites de l'Universite Paul Sabatier, Toulouse, and beneted from helpful discussion with Walter Moreira. Elementary Proofs on Optimal Stopping Ernesto Mordecki y Facultad de Ciencias. Centro de Matematica. Igua 4225. CP 11400. Montevideo. Uruguay. December 29, 2000 1 Introduction 1.1 Consider a model of a nancial market with two assets, a savings account B = fB t g t0 and a stock S = fS t g t0 . The evolution of B is deterministic, with B t = B 0 e rt ; B 0 = 1; r >...

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Levy process. The main idea is to apply Girsanov's Theorem for Levy processes, in order to reduce the posed problem to the pricing of a one Levy driven stock in an auxiliary market, baptized as \dual market". In this way, we extend the results obtained by Gerber and Shiu (1996) for two dimensional Brownian motion. Also we examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov's Theorem, is called put{call duality. It includes as a particular case, the relation known as put{call symmetry. Necessary and sucient conditions for put{call symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Levy process.

These notes give a more detailed account of the material discussed in lectures. They are largely based on material found in books and research papers. In the