Abstract not found

We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

As is well known, the classic Black-Scholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that time-changed Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

The de nition and properties of Levy-driven CARMA (continuous-time ARMA) processes are reviewed. Gaussian CARMA processes are special cases in which the driving Levy process is Brownian motion. The use of more general Levy processes permits the speci cation of CARMA processes with a wide variety ofmarginal distributions which may be asymmetric and heavier tailed than Gaussian. Non-negative CARMA processes are of special interest, partly because of the introduction by Barndor-Nielsen and Shephard (2001) of non-negativeLevy-driven Ornstein-Uhlenbeck processes as models for stochastic volatility. Replacing the Ornstein-Uhlenbeck process byaLevy-driven CARMA process with non-negative kernel permits the modelling of non-negative, heavy-tailed processes with a considerably larger range of autocovariance functions than is possible in the Ornstein-Uhlenbeck framework. We also de ne a class of zero-mean fractionally integrated Levy-driven CARMA processes, obtained by convoluting the CARMA kernel with a kernel corresponding to Riemann-Liouville fractional integration, and derive explicit expressions for the kernel and autocovariance functions of these processes. They are long-memory in the sense that their kernel and autocovariance functions decay asymptotically at hyperbolic rates depending on the order of fractional integration. In order to introduce long-memory into non-negative Levy-driven CARMA processes we replace the fractional integration kernel with a closely related absolutely integrable kernel. This gives a class of stationary non-negative continuous-time Levy-driven processes whose autocovariance functions at lag h also converge to zero at asymptotically hyperbolic rates.

This paper examines long-term dependence in times between trades on financial markets. The autocorrelation functions of several intertrade duration series show a slow, hyperbolic rate of decay typical for long memory processes. For example, a shock to times between trades of the Alcatel stock on the Paris Stock Exchange (SBF Paris Bourse) may persist in the transactions time for a long period of 1000 or 2000 ticks. With an average duration of 52 seconds between transactions this may amount to sixteen or thirty two hours in calendar time. This paper introduces a fractionally integrated autoregressive conditional duration (FIACD) model for intertrade duration series. It also examines transformed duration processes representing times between consecutive returns to states of null, positive or negative returns. This approach captures the relationship between the duration persistence and return dynamics. The times elapsed between returns to various states feature very similar auto...

A new class of graphical models capturing the dependence structure of events that occur in time is proposed. The graphs represent so–called local independencies, meaning that the intensities of certain types of events are independent of some (but not necessarilly all) events in the past. This dynamic concept of independence is asymmetric, similar to Granger non–causality, so that the corresponding local independence graphs differ considerably from classical graphical models. Hence a new notion of graph separation, called δ–separation, is introduced and implications for the underlying model as well as for likelihood inference are explored. Benefits regarding facilitation of reasoning about and understanding of dynamic dependencies as well as computational simplifications are discussed.

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in scalar diffusion models. We study this link using two notions of temporal dependence: β − mixing and ρ − mixing. We show that β − mixing and ρ − mixing with exponential decay are essentially equivalent concepts for scalar diffusions. For stationary diffusions that fail to be ρ −mixing, we show that they are still β −mixing except that the decay rates are slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. Some have spectral densities that diverge at frequency zero in a manner similar to that of stochastic processes with long memory. Finally we show how nonlinear, state-dependent, Poisson sampling alters the unconditional distribution as well as the temporal dependence

The notions of self-similarity, scaling, fractional processes and long range dependence have been repeatedly used to describe properties of financial time series: stock prices, foreign exchange rates, market indices and commodity prices. We discuss the relevance of these concepts in the context of financial modelling, their relation with the basic principles of financial theory and possible economic explanations for their presence in financial time series.

Abstract. We consider the classical Merton problem of finding the optimal consumption rate and the optimal portfolio in a Black-Scholes market driven by fractional Brownian motion B H with Hurst parameter H> 1/2. The integrals with respect to B H are in the Skorohod sense, not pathwise which is known to lead to arbitrage. We explicitly find the optimal consumption rate and the optimal portfolio in such a market for an agent with logarithmic utility functions. A true self-financing portfolio is found to lead to a consumption term that is always favorable to the investor. We also present a numerical implementation by Monte Carlo simulations. 1.

We use a general representation of continuous Gaussian processes as the limit of a sequence of processes in the associated reproducing kernel Hilbert space, to Gaussian processes represented as Volterra type stochastic integrals with respect to Brownian motion, including the fractional Brownian motion. As special cases of this representation we obtain for example, the Karhunen-Love decomposition for standard Brownian motion and a wavelet representation for fractional Brownian motion. We also show how the representation can be used to estimate parameters. In particular we derive an estimator for the mean-reverting parameter in an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. 1.