Ernesto Mordecki

Optimal stopping and perpetual options for Lévy processes

Abstract. Consider a model of a financial market with a stock driven by a Lévy process and constant interest rate. A closed formula for prices of perpetual American call options in terms of the overall supremum of the Lévy process, and a corresponding closed formula for perpetual American put options involving the infimum of the after-mentioned process are obtained. As a direct application of the previous results, a Black-Scholes type formula is given. Also as a consequence, simple explicit formulas for prices of call options are obtained for a Lévy process with positive mixed-exponential and arbitrary negative jumps. In the case of put options, similar simple formulas are obtained under the condition of negative mixed-exponential and arbitrary positive jumps. Risk-neutral valuation is discussed and a simple jump-diffusion model is chosen to illustrate the results.

Optimal stopping for a diffusion with jumps

Abstract. In this paper we give the closed form solution of some optimal stopping problems for processes derived from a diffusion with jumps. Within the possible applications, the results can be interpreted as pricing perpetual American Options under diffusion-jump information.

Symmetry and duality in Lévy markets

The aim of this paper is to introduce the notion of symmetry in a Lévy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, which is also reviewed in the paper, and that we call put–call duality. Symmetric Lévy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices. Put–call duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanovs theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanovs theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Lévy process, answering a question raised by Carr and Chesney (American put call symmetry, preprint, 1996). Some empirical evidence is shown, supporting that, in general, markets are not symmetric.

Optimal stopping of Hunt and Lévy processes

Infinite horizon (perpetual) optimal stopping problems for Hunt processes on R are studied via the representation theory of excessive functions. In particular, we focus on problems with one-sided structure, that is, there exists a point x* such that the stopping region is of the form . The main result states that if it is possible to find a Radon measure such that the excessive function induced by this measure via the spectral representation has some very intuitive properties then the constructed excessive function coincides with the value function of the problem. Corresponding results for two-sided problems are also indicated. Specializing to Lévy processes, we obtain, by applying the Wiener–Hopf factorization, a general representation of the value function in terms of the maximum of the Lévy process. To illustrate the results, an explicit expression for the Green kernel of Brownian motion with exponential jumps is computed and some optimal stopping problems for Poisson process with positive exponential jumps and negative drift are solved.

Ruin Probabilities for Levy Processes with Mixed-Exponential Negative Jumps

We give a closed form of the ruin probability for Levy processes, possible killed at a constant rate, with arbitrary, positive, and mixed exponentially negative jumps.

Adaptive Weak Approximation of Diffusions with Jumps

This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler-Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows and discrete dual backward functions. Combined with proper estimation of the statistical error, they lead to efficient and accurate computation of global error estimates, extending the results by A. Szepessy, R. Tempone, and G. E. Zouraris [Comm. Pure Appl. Math., 54 (2001), pp. 1169-1214]. Adaptive algorithms for either deterministic or trajectory-dependent time stepping are proposed. Numerical examples show the performance of the proposed error approximations and the adaptive schemes.

Cucker–Smale Flocking Under Hierarchical Leadership and Random Interactions

Consider a flock of birds that fly interacting between them. This flock is a hierarchical system in which each bird, at each time step, adjusts its own velocity according to its past velocity and a linear combination of the relative velocities of its superiors in the hierarchy. This linear combination has nonnegative random coefficients, including the case in which each of the birds can fail to see any of its superiors with certain probability. For this model with random interactions we prove that the flocking results, obtained for similar deterministic models, hold true.

SST anomaly variability in Southwestern Atlantic and El Niño/Southern oscillation

Abstract The purpose of this paper is to study the relation between Sea Surface Temperature anomalies in the Southwestern Atlantic Ocean and the presence of El Nino and La Nina events, in the period 1868–2000. SST anomalies were taken from 1856–2000 Kaplan data set, and El Nino and La Nina events from different authors. Significative differences were found in the SST anomalies in the winter period (from April to October) taken at 8 points in the Southwestern Atlantic, depending on the presence of El Nino or La Nina. On an average, the El Nino events (registered in December of each year) are characterized by negative SST anomalies at the Malvinas current and positive SST anomalies at the Brazil current in the winter of the corresponding year. On the contrary, La Nina events are characterized by anomalous cold Brazil current and warm Malvinas current. Decrease of SST anomalies in Malvinas current and for preservation of its trend until October is suggested as the precursor of El Nino. Using the monthly SST anomalies in Southwestern Atlantic, statistical indicators for ENSO events were computed. These indicators are used for reconstruction of the weak, moderate strength, and strong La Nina episodes for the 1868–2000 period. El Nino (except the strongest events), could be predicted three to six months in advance from of time-space SST anomaly variability in Southwestern Atlantic.

Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions

Consider the optimal stopping problem of a one-dimensional diffusion with positive discount. Based on Dynkins characterization of the value as the minimal excessive majorant of the reward and considering its Riesz representation, we give an explicit equation to find the optimal stopping threshold for problems with one-sided stopping regions, and an explicit formula for the value function of the problem. This representation also gives light on the validity of the smooth-fit (SF) principle. The results are illustrated by solving some classical problems, and also through the solution of: optimal stopping of the skew Brownian motion and optimal stopping of the sticky Brownian motion, including cases in which the SF principle fails.

Rice Formula for processes with jumps and applications

We extend Rice Formula to a process which is the sum of two independent processes: a smooth process and a pure jump process with finitely many jumps. Formulas for the mean number of both continuous and discontinuous crossings through a fixed level on a compact time interval are obtained. We present examples in which we compute explicitly the mean number of crossings and compare which kind of crossings dominates for high levels. In one of the examples the leading term of the tail of the distribution function of the maximum of the process over a compact time interval as the level goes to infinity is obtained. We end giving a generalization, to the non-stationary case, of Borovkov-Last’s Rice Formula for Piecewise Deterministic Markov Processes.