arXiv subjects

Ernesto Mordecki

Publications and source records attributed to Ernesto Mordecki.

34 records · Page 2Linked to original sources

arXiv2015

Modelling the Uruguayan debt through gaussians models

We model bond's price curves corresponding to the sovereign uruguayan debt nominated in USD, as an alternative to the official bond prices publication released by the Central Bank of Uruguay (CBU). Four different gaussian models are fitted, based on historical data issued by the CBU, corresponding to some of the more frequently traded bonds. The main difficulty we approach is the absence of liquidity in the bond market. Nevertheless the adjustment is relatively good, giving the possibility of non-arbitrage pricing of the whole family of non traded instruments, and also the possibility of pricing derivative securities.

q-fin.PR↗

arXiv2015

Optimal stopping for Levy processes with polynomial rewards

Explicit solution of an infinite horizon optimal stopping problem for a Levy processes with a polynomial reward function is given, in terms of the overall supremum of the process, when the solution of the problem is one-sided. The results are obtained via the generalization of known results about the averaging function associated with the problem. This averaging function can be directly computed in case of polynomial rewards. To illustrate this result, examples for general quadratic and cubic polynomials are discussed in case the process is Brownian motion, and the optimal stopping problem for a quartic polynomial and a Kou's process is solved.

math.PR↗

arXiv2015

On the number of open knight's tours

We review the state of the art in the problem of counting the number open knight tours, since the publication in internet of a computation of this quantity.

math.CO↗

arXiv2015

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 dominate 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 Rice Formula for Piecewise Deterministic Markov Processes.

math.PR↗

arXiv2014

Computing Greeks for Lévy Models: The Fourier Transform Approach

The computation of Greeks for exponential Lévy models are usually approached by Malliavin Calculus and other methods, as the Likelihood Ratio and the finite difference method. In this paper we obtain exact formulas for Greeks of European options based on the Lewis formula for the option value. Therefore, it is possible to obtain accurate approximations using Fast Fourier Transform. We will present an exhaustive development of Greeks for Call options. The error is shown for all Greeks in the Black-Scholes model, where Greeks can be exactly computed. Other models used in the literature are compared, such as the Merton and Variance Gamma models. The presented formulas can reach desired accuracy because our approach generates error only by approximation of the integral.

q-fin.PR↗

arXiv2014

A finite exact algorithm to solve a dice game

We provide an algorithm to find the value and an optimal strategy of the solitaire variant of the Ten Thousand dice game in the framework of Markov Control Processes. Once an optimal critical threshold is found, the set of non-stopping states of the game becomes finite, and the solution is found by a backwards algorithm that gives the values for each one of these states of the game. The algorithm is finite and exact.The idea to find the critical threshold comes from the continuous pasting condition used in optimal stopping problems for continuous-time processes with jumps.

math.OC↗

arXiv2014

Robustness of Cucker-Smale flocking model

Consider a system of autonomous interacting agents moving in space, adjusting each own velocity as a weighted mean of the relative velocities of the other agents. In order to test the robustness of the model, we assume that each pair of agents, at each time step, can fail to connect with certain probability, the failure rate. This is a modification of the (deterministic) Flocking model introduced by Cucker and Smale in Emergent behavior in flocks, IEEE Trans. on Autom. Control, 2007, 52 (May) pp. 852-862. We prove that, if this random failures are independent in time and space, and have linear or sub-linear distance dependent rate of decay, the characteristic behavior of flocking exhibited by the original deterministic model, also holds true under random failures, for all failure rates.

math.PR↗

arXiv2013

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 Dynkin's 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 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 smooth fit principle fails.

math.PR↗

arXiv2011

Is a Brownian skew?

We study the asymptotic behavior of the maximum likelihood estimator corresponding to the observation of a trajectory of a Skew Brownian motion, through a uniform time discretization. We characterize the speed of convergence and the limiting distribution when the step size goes to zero, which in this case are non-classical, under the null hypothesis of the Skew Brownian motion being an usual Brownian motion. This allows to design a test on the skewness parameter. We show that numerical simulations that can be easily performed to estimate the skewness parameter, and provide an application in Biology.

math.PR↗

arXiv2009

Optimal minimax strategy in a dice game

Each of two players, by turns, rolls a dice several times accumulating the successive scores until he decides to stop, or he rolls an ace. When stopping, the accumulated turn score is added to the player account and the dice is given to his opponent. If he rolls an ace, the dice is given to the opponent without adding any point. In this paper we formulate this game in the framework of competitive Markov decision processes (also known as stochastic games), show that the game has a value, provide an algorithm to compute the optimal minimax strategy, and present results of this algorithm in three different variants of the game.

math.PR↗

arXiv2009

Cucker-Smale Flocking Under Hierarchical Leadership and Random Interactions

Consider a flock of birds that fly interacting between them. The interactions are modelled through a hierarchical system in which each bird, at each time step, adjusts its own velocity according to his past velocity and a weighted mean of the relative velocities of its superiors in the hierarchy. We consider the additional fact, that each of the birds can fail to see any of its superiors with certain probability, that can depend on the distances between them. For this model with random interactions we prove that the flocking phenomena, obtained for similar deterministic models, holds true.

math.PR↗

arXiv2008

Skewness Premium with Lévy Processes

We study the skewness premium (SK) introduced by Bates (1991) in a general context using Lévy Processes. Under a symmetry condition Fajardo and Mordecki (2006) obtain that SK is given by the Bate's $x%$ rule. In this paper we study SK under the absence of that symmetry condition. More exactly, we derive sufficient conditions for SK to be positive, in terms of the characteristic triplet of the Lévy Process under the risk neutral measure.

math.PR↗

arXiv2008

Symmetry and Time Changed Brownian Motions

In this paper we examine which Brownian Subordination with drift exhibits the symmetry property introduced by Fajardo and Mordecki (2006). We obtain that when the subordination results in a Lévy process, a necessary and sufficient condition for the symmetry to hold is that drift must be equal to -1/2.

math.PR↗

arXiv2007

Flocking in noisy environments

We consider a perturbed version of the dynamics of a flock introduced by Cucker and Smale ("Emergent behaviour in flocks") and prove, under similar conditions, that nearly-alignment (a concept that is precised in the text) is achieved with a certain probability, bounded from below.

math.DS↗

arXiv2006

Counting Knight's Tours through the Randomized Warnsdorff Rule

We give an estimate of the number of geometrically distinct open tours $\G$ for a knight on a chessboard. We use a randomization of Warnsdorff rule to implement importance sampling in a backtracking scheme, correcting the observed bias of the original rule, according to the proposed principle that ``most solutions follow Warnsdorff rule most of the time''. After some experiments in order to test this principle, and to calibrate a parameter, interpreted as a distance of a general solution from a Warnsdorff solution, we conjecture that $\G=1.22\times 10^{15}$.

math.PR↗

arXiv2006

Optimal stopping of Hunt and Lévy processes

The optimal stopping problem for a Hunt processes on $\R$ is considered via the representation theory of excessive functions. In particular, we focus on infinite horizon (or perpetual) problems with one-sided structure, that is, there exists a point $x^*$ such that the stopping region is of the form $[x^*,+\infty)$. Corresponding results for two-sided problems are also indicated. The main result is a spectral representation of the value function in terms of the Green kernel of the process. Specializing in 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.

math.PR↗