Optimal hedging of Derivatives with transaction costs

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free ("Black-Scholes") pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a Black-Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.Comment: Revised version, expanded introduction and references 17 pages, submitted to International Journal of Theoretical and Applied Finance (IJTAF

Models of Passive and Reactive Tracer Motion: an Application of Ito Calculus

By means of Ito calculus it is possible to find, in a straight-forward way, the analytical solution to some equations related to the passive tracer transport problem in a velocity field that obeys the multidimensional Burgers equation and to a simple model of reactive tracer motion.Comment: revised version 7 pages, Latex, to appear as a letter to J. of Physics

On the canonically invariant calculation of Maslov indices

After a short review of various ways to calculate the Maslov index appearing in semiclassical Gutzwiller type trace formulae, we discuss a coordinate-independent and canonically invariant formulation recently proposed by A Sugita (2000, 2001). We give explicit formulae for its ingredients and test them numerically for periodic orbits in several Hamiltonian systems with mixed dynamics. We demonstrate how the Maslov indices and their ingredients can be useful in the classification of periodic orbits in complicated bifurcation scenarios, for instance in a novel sequence of seven orbits born out of a tangent bifurcation in the H\'enon-Heiles system.Comment: LaTeX, 13 figures, 3 tables, submitted to J. Phys.

Passive scalar turbulence in high dimensions

Exploiting a Lagrangian strategy we present a numerical study for both perturbative and nonperturbative regions of the Kraichnan advection model. The major result is the numerical assessment of the first-order $1/d$-expansion by M. Chertkov, G. Falkovich, I. Kolokolov and V. Lebedev ({\it Phys. Rev. E}, {\bf 52}, 4924 (1995)) for the fourth-order scalar structure function in the limit of high dimensions $d$'s. %Two values of the velocity scaling exponent $\xi$ have been considered: %$\xi=0.8$ and $\xi=0.6$. In the first case, the perturbative regime %takes place at $d\sim 30$, while in the second at $d\sim 25$, %in agreement with the fact that the relevant small parameter %of the theory is $\propto 1/(d (2-\xi))$. In addition to the perturbative results, the behavior of the anomaly for the sixth-order structure functions {\it vs} the velocity scaling exponent, $\xi$, is investigated and the resulting behavior discussed.Comment: 4 pages, Latex, 4 figure

Shell Model for Time-correlated Random Advection of Passive Scalars

We study a minimal shell model for the advection of a passive scalar by a Gaussian time correlated velocity field. The anomalous scaling properties of the white noise limit are studied analytically. The effect of the time correlations are investigated using perturbation theory around the white noise limit and non-perturbatively by numerical integration. The time correlation of the velocity field is seen to enhance the intermittency of the passive scalar.Comment: Replaced with final version + updated figure

Manifestation of anisotropy persistence in the hierarchies of MHD scaling exponents

The first example of a turbulent system where the failure of the hypothesis of small-scale isotropy restoration is detectable both in the `flattening' of the inertial-range scaling exponent hierarchy, and in the behavior of odd-order dimensionless ratios, e.g., skewness and hyperskewness, is presented. Specifically, within the kinematic approximation in magnetohydrodynamical turbulence, we show that for compressible flows, the isotropic contribution to the scaling of magnetic correlation functions and the first anisotropic ones may become practically indistinguishable. Moreover, skewness factor now diverges as the P\'eclet number goes to infinity, a further indication of small-scale anisotropy.Comment: 4 pages Latex, 1 figur

Eddy diffusivity of quasi-neutrally-buoyant inertial particles

We investigate the large-scale transport properties of quasi-neutrally-buoyant inertial particles carried by incompressible zero-mean periodic or steady ergodic flows. We show howto compute large-scale indicators such as the inertial-particle terminal velocity and eddy diffusivity from first principles in a perturbative expansion around the limit of added-mass factor close to unity. Physically, this limit corresponds to the case where the mass density of the particles is constant and close in value to the mass density of the fluid, which is also constant. Our approach differs from the usual over-damped expansion inasmuch as we do not assume a separation of time scales between thermalization and small-scale convection effects. For a general flow in the class of incompressible zero-mean periodic velocity fields, we derive closed-form cell equations for the auxiliary quantities determining the terminal velocity and effective diffusivity. In the special case of parallel flows these equations admit explicit analytic solution. We use parallel flows to show that our approach sheds light onto the behavior of terminal velocity and effective diffusivity for Stokes numbers of the order of unity.Peer reviewe

Generally covariant state-dependent diffusion

Statistical invariance of Wiener increments under SO(n) rotations provides a notion of gauge transformation of state-dependent Brownian motion. We show that the stochastic dynamics of non gauge-invariant systems is not unambiguously defined. They typically do not relax to equilibrium steady states even in the absence of extenal forces. Assuming both coordinate covariance and gauge invariance, we derive a second-order Langevin equation with state-dependent diffusion matrix and vanishing environmental forces. It differs from previous proposals but nevertheless entails the Einstein relation, a Maxwellian conditional steady state for the velocities, and the equipartition theorem. The over-damping limit leads to a stochastic differential equation in state space that cannot be interpreted as a pure differential (Ito, Stratonovich or else). At odds with the latter interpretations, the corresponding Fokker-Planck equation admits an equilibrium steady state; a detailed comparison with other theories of state-dependent diffusion is carried out. We propose this as a theory of diffusion in a heat bath with varying temperature. Besides equilibrium, a crucial experimental signature is the non-uniform steady spatial distribution.Comment: 24 page

Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of $\xi$, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.Comment: Latex (11pt) 43 pages, 22 figures (Feynman diagrams). The reader interested in the technical details of the calculations presented in the paper may want to visit: http://www.math.helsinki.fi/mathphys/paolo_files/passive_scalar/passcal.htm