Related papers: The skewed multifractal random walk with applicati…
In this paper we briefly review the recently inrtroduced Multifractal Random Walk (MRW) that is able to reproduce most of recent empirical findings concerning financial time-series : no correlation between price variations, long-range…
We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). To our knowledge, it is the first multifractal processes with continuous dilation invariance properties and stationary increments. MRWs are…
We perform an extensive empirical analysis of scaling properties of equity returns, suggesting that financial data show time varying multifractal properties. This is obtained by comparing empirical observations of the weighted generalised…
We revisit the ``Smile Dynamics'' problem, which consists in relating the implied leverage (i.e. the correlation of the at-the-money volatility with the returns of the underlying) and the skew of the option smile. The ratio between these…
We study in details the skew of stock option smiles, which is induced by the so-called leverage effect on the underlying -- i.e. the correlation between past returns and future square returns. This naturally explains the anomalous…
This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in W ilmott magazine. We discuss how multiplicative cascades and related multifractal ideas might be relevant to model the main statistical features of financial…
We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with…
A multifractal random walk (MRW) is defined by a Brownian motion subordinated by a class of continuous multifractal random measures $M[0,t], 0\le t\le1$. In this paper we obtain an extension of this process, referred to as multifractal…
We derive a new, exact and transparent expansion for option smiles, which lends itself both to analytical approximation and, perhaps more importantly, to congenial numerical treatments. We show that the skew and the curvature of the smile…
We introduce a Multifractal Random Walk (MRW) defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional Brownian motion. Using the techniques of the Malliavin calculus, we study the existence…
The Multi Variate Mixture Dynamics model is a tractable, dynamical, arbitrage-free multivariate model characterized by transparency on the dependence structure, since closed form formulae for terminal correlations, average correlations and…
We define a large class of continuous time multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined…
We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential L\'evy models. This…
We give an explicit formula for the probability distribution based on a relativistic extension of Brownian motion. The distribution 1) is properly normalized and 2) obeys the tower law (semigroup property), so we can construct martingales…
We develop a powerful yet simple method that generates multifractal fields with fully controlled scaling properties. Adopting the Multifractal Random Walk (MRW) model of Bacry et al. (2001), synthetic multifractal fields are obtained from…
We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the…
Multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a nonsymmetric lattice of obstacles are investigated. Using the equivalence of the model to random walks on a locally…
Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and $L^1$ convergence of its structure function. This is an issue directly connected to the…
Multivariate probability density functions of returns are constructed in order to model the empirical behavior of returns in a financial time series. They describe the well-established deviations from the Gaussian random walk, such as an…
We study a Markov-Functional (MF) interest-rate model with Uncertain Volatility Displaced Diffusion (UVDD) digital mapping, which is consistent with the volatility-smile phenomenon observed in the option market. We first check the impact of…