Jaume Masoliver

Continuous-time random-walk model for financial distributions

We apply the formalism of the continuous-time random walk to the study of financial data. The entire distribution of prices can be obtained once two auxiliary densities are known. These are the probability densities for the pausing time between successive jumps and the corresponding probability density for the magnitude of a jump. We have applied the formalism to data on the U.S. dollar-deutsche mark future exchange, finding good agreement between theory and the observed data.

Some two and three-dimensional persistent random walks

Abstract We formulate non-Markovian versions of the persistent random walk in two and three dimensions and in continuous time. These models can be regarded as being generalizations of the original Pearson random walk and of the freely jointed chain which has been of some importance in polymer physics. Solutions for the probability density of the displacement of the random walker can be furnished for a restricted (essentially Markovian) set of these models. It is shown that in two dimensions the solution to one of these models is equivalent to the solution to an inhomogeneous telegraphers equation. It does not appear to be possible, starting from a similar model in three dumensions, to find any form of the telegraphers equation that follows from our solution of the equations in the Fourier-Laplace transform domains.

A continuous-time generalization of the persistent random walk

We develop the formalism for a continuous-time generalization of the persistent random walk, by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject. This generalization leads to evolution equations, in the time domain, that differ and are of higher order than the telegraphers equation which is found in the case of the Markovian persistent random walk.

The continuous time random walk formalism in financial markets

We adapt continuous time random walk (CTRW) formalism to describe asset price evolution and discuss some of the problems that can be treated using this approach. We basically focus on two aspects: (i) the derivation of the price distribution from high-frequency data, and (ii) the inverse problem, obtaining information on the market microstructure as reflected by high-frequency data knowing only the daily volatility. We apply the formalism to financial data to show that the CTRW offers alternative tools to deal with several complex issues of financial markets.

Properties of resonant activation phenomena

The phenomenon of resonant activation of a Brownian particle over a fluctuating barrier is revisited. We discuss the important distinctions between barriers that can fluctuate among ‘‘up’’ and ‘‘down’’ configurations, and barriers that are always ‘‘up’’ but that can fluctuate among different heights. A resonance as a function of the barrier fluctuation rate is found in both cases, but the nature and physical description of these resonances is quite distinct. The nature of the resonances, the physical basis for the resonant behavior, and the importance of boundary conditions are discussed in some detail. We obtain analytic expressions for the escape time over the barrier that explicitly capture the minima as a function of the barrier fluctuation rate, and show that our analytic results are in excellent agreement with numerical results. @S1063-651X~98!11304-1#

A CORRELATED STOCHASTIC VOLATILITY MODEL MEASURING LEVERAGE AND OTHER STYLIZED FACTS

We analyze a stochastic volatility market model in which volatility is correlated with return and is represented by an Ornstein-Uhlenbeck process. In the framework of this model we exactly calculate the leverage effect and other stylized facts, such as mean reversion, leptokurtosis and negative skewness. We also obtain a close analytical expression for the characteristic function and study the heavy tails of the probability distribution.

Option Pricing Under Stochastic Volatility: The Exponential Ornstein-Uhlenbeck Model

We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbeck subordinated process describing the randomness of the log-volatility. We derive an approximate option price that is valid when (i) the fluctuations of the volatility are larger than its normal level, (ii) the volatility presents a slow driving force toward its normal level and, finally, (iii) the market price of risk is a linear function of the log-volatility. We study the resulting European call price and its implied volatility for a range of parameters consistent with daily Dow Jones Index data.

First-passage and risk evaluation under stochastic volatility

We solve the first-passage problem for the Heston random diffusion model. We obtain exact analytical expressions for the survival and the hitting probabilities to a given level of return. We study several asymptotic behaviors and obtain approximate forms of these probabilities which prove, among other interesting properties, the nonexistence of a mean-first-passage time. One significant result is the evidence of extreme deviations-which implies a high risk of default-when certain dimensionless parameter, related to the strength of the volatility fluctuations, increases. We confront the model with empirical daily data and we observe that it is able to capture a very broad domain of the hitting probability. We believe that this may provide an effective tool for risk control which can be readily applicable to real markets both for portfolio management and trading strategies.

The continuous-time random walk description of photon motion in an isotropic medium

Lattice random walk models are useful alternatives to diffusion theory for describing photon migration in turbid media. Results obtained from random walk models were, in the past, based on approximating the propagator as a Gaussian function. This is equivalent to assuming that a large number of steps have been taken. We show that a particular, physically plausible, continuous-time random walk (CTRW) model can be solved exactly. At sufficiently long times the solution reduces to that for a random walk in discrete time. We provide exact solutions to problems related to cw reflection measurements and indicate some generalizations to time-gated measurements made on a uniform medium.

Finite-velocity diffusion

The phenomenon of diffusion modelled as Brownian motion or in terms of the diffusion equation is deficient in that it permits an infinite speed of signal propagation. The simplest diffusion-like equation with a bounded speed of signal propagation is the so-called telegraphers equation which implicitly contains a type of momentum. We discuss some properties, solutions and problems related to this equation. These are generally more complicated in form than corresponding ones for the diffusion equation, exhibiting wave-like properties at short times and diffusion-like properties in the long-time limit. Resumen. El fenomeno de la difusion, modelado tanto como movimiento Browniano como mediante la ecuacion de difusion, es deficiente en el sentido que permite una velocidad de propagacion infinita. La denominada ecuacion del telegrafista es una de las ecuaciones mas simples con una velocidad de propagacion acotada y que contiene de forma implicita un cierto tipo momento. En este trabajo se discuten algunas propiedades, soluciones y cuestiones relacionadas con esta ecuacion. Estas propiedades y soluciones son generalmente mas complicadas que las correspondientes a la ecuacion de difusion ordinaria, presentando, a tiempos cortos, propiedades de tipo ondulatorio y propiedades difusivas en el limite asintotico.