Martingales
The martingale methodology plays an important role in the modern theory of finance 2, 7, 8 . Martingale is a stochastic process X t that satisfies the following condition E X t 1 X t , X t - 1 , X t The equivalent definition is given by E X t 1 - X t X t , X t - 1 , 0 Both these definitions are easily generalized for the continuum presentation where the time interval, dt, between two sequent moments t 1 and t approaches zero dt 0 . The notion of martingale is rooted in the gambling theory. It...
Multifractals
Let us turn to the generic notion of multifractals see, e.g., 5 . Consider the map filled with a set of boxes that are used in the box-counting fractal dimension. What matters for the fractal concept is whether the given box belongs to fractal. The basic idea behind the notion of multifractals is that every box is assigned a measure m that characterizes some probability density e.g., intensity of color between the white and black limits . The so-called multiplicative process or cascade defines...
Discrete Systems The Logistic Map
The logistic map is a simple discrete model that was originally used to describe the dynamics of biological populations see, e.g., 5 and references therein . Let us consider a variable number of individuals in a population, N. Its value at the k-th time interval is described with the following equation Parameter A characterizes the population growth that is determined by such factors as food supply, climate, etc. Obviously, the population grows only if A gt 1. If there are no restrictive...
Stochastic Integral
Now that the stochastic differential has been introduced, let us discuss how to perform its integration. First, the Riemann-Stieltjes integral should be defined. Consider a deterministic function f t on the interval t 2 0, T . In order to calculate the Riemann integral of f t over the interval 0, T , this interval is divided into n sub-intervals t0 0 lt t1 lt lt tn T and the following sum should be computed where ti 2 ti-1, ti . The Riemann integral is the limit of Sn f t dt lim Sn, max ti -...
fxi ti x2 t2xk tkxki tki xk2 tk2
f xi, ti x2, t2 xk, tk xk i, tk i 4.i.3 This means that evolution of the system is determined with the initial condition i.e., with the value xk i at time tk i . It follows for the Markov process that f xi, ti x2, t2 x3, t3 f xi, ti x2, t2 f x2, t2 x3, t3 4.i.4 Using the definition of the conditional probability density, one can introduce the general equation f xi, ti x2, t2 x3, t3 f x2, t2 x3, t3 dx2 4.i.5 f xi, ti x2, t2 x3, t3 f xi, ti x2, t2 , 4.i.6 Then the substitution of equation 4.i.6...
Market Efficiency
Asset prices generally obey the Law of One Price, which says that prices of equivalent assets in competitive markets must be the same 6 . This implies that if a security replicates a package of other securities, the price of this security and the price of the package it replicates must be equal. It is expected also that the asset price must be the same worldwide, provided that it is expressed in the same currency and that the transportation and transaction costs can be neglected. Violation of...
Introduction
This book is written for those physicists who want to work on Wall Street but have not bothered to read anything about Finance. This is a crash course that the author, a physicist himself, needed when he landed a financial data analyst job and became fascinated with the huge data sets at his disposal. More broadly, this book addresses the reader with some background in science or engineering college-level math helps who is willing to learn the basic concepts and quantitative methods used in...