Journal of Statistical Mechanics: Theory and Experiment, 2008
We study the pricing problem for a European call option when the volatility of the underlying ass... more 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.
Financial markets provide an ideal frame for studying decision making in crowded environments. Bo... more Financial markets provide an ideal frame for studying decision making in crowded environments. Both the amount and accuracy of the data allows to apply tools and concepts coming from physics that studies collective and emergent phenomena or self-organised and highly heteregeneous systems. We analyse the activity of 29 930 non-expert individuals that represent a small portion of the whole market trading volume. The very heterogeneous activity of individuals obeys a Zipf's law, while synchronization network properties unveil a community structure. We thus correlate individual activity with the most eminent macroscopic signal in financial markets, that is volatility, and quantify how individuals are clearly polarized by volatility. The assortativity by attributes of our synchronization networks also indicates that individuals look at the volatility rather than imitate directly each other thus providing an interesting interpretation of herding phenomena in human activity. The results can also improve agent-based models since they provide direct estimation of the agent's parameters.
Volatility measures the amplitude of price fluctuations. Despite it is one of the most important ... more Volatility measures the amplitude of price fluctuations. Despite it is one of the most important quantities in finance, volatility is not directly observable. Here we apply a maximum likelihood method which assumes that price and volatility follow a two-dimensional diffusion process where volatility is the stochastic diffusion coefficient of the log-price dynamics. We apply this method to the simplest versions of the expOU, the OU and the Heston stochastic volatility models and we study their performance in terms of the log-price probability, the volatility probability, and its Mean First-Passage Time. The approach has some predictive power on the future returns amplitude by only knowing current volatility. The assumed models do not consider long-range volatility auto-correlation and the asymmetric return-volatility cross-correlation but the method still arises very naturally these two important stylized facts. We apply the method to different market indexes and with a good performance in all cases.
The Feller process is an one-dimensional diffusion process with linear drift and state-dependent ... more The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single neuron firing to volatility of financial assets.
We study financial distributions within the framework of the continuous time random walk (CTRW). ... more We study financial distributions within the framework of the continuous time random walk (CTRW). We review earlier approaches and present new results related to overnight effects as well as the generalization of the formalism which embodies a non-Markovian formulation of the CTRW aimed to account for correlated increments of the return.
Options are financial instruments designed to protect investors from the stock market randomness.... more Options are financial instruments designed to protect investors from the stock market randomness. In 1973, Fisher Black, Myron Scholes and Robert Merton proposed a very popular option pricing method using stochastic differential equations within the Itô interpretation. Herein, we derive the Black-Scholes equation for the option price using the Stratonovich calculus along with a comprehensive review, aimed to physicists, of the classical option pricing method based on the Itô calculus. We show, as can be expected, that the Black-Scholes equation is independent of the interpretation chosen. We nonetheless point out the many subtleties underlying Black-Scholes option pricing method.
We study the activity, i.e., the number of transactions per unit time, of financial markets. Usin... more We study the activity, i.e., the number of transactions per unit time, of financial markets. Using the diffusion entropy technique we show that the autocorrelation of the activity is caused by the presence of peaks whose time distances are distributed following an asymptotic power law which ultimately recovers the Poissonian behavior. We discuss these results in comparison with ARCH models, stochastic volatility models and multi-agent models showing that ARCH and stochastic volatility models better describe the observed experimental evidences.
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Papers by Josep Perello