Many studies in Economics and other disciplines have been reporting distributions following power-law behavior (i.e distributions of incomes (Pareto’s law), city sizes (Zipf’s law), frequencies of words in long sequences of text etc.)[1, 6, 7]. This widespread observed regularity has been explained in many ways: generalized Lotka-Volterra (GLV) equations, self-organized criticality and highly o...

Abstract The critical price $$S^{*}\left( t\right) $$ S ? t of an American put option is the underlying stock level that triggers its immediate optimal exercise. We provide a new perspecti...

We study drawdowns and rallies of Brownian motion. A rally is defined as the difference of the present value of the Brownian motion and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. This paper determines the probability that a drawdown of a units precedes a rally of b units. We apply this result to examine stock market c...

We solve explicitly the Almgren–Chriss optimal liquidation problem where stock price process follows a geometric Brownian motion. Our technique is to work in terms of cash and use functional analysis tools. show that this framework extends readily case stochastic drift for portfolio.

This paper has four goals: (a) relate ladder height distributions to option values; (b) show how Laguerre expansions may be used in the computation of densities, distribution functions and option prices; (c) derive some new results on the integral of geometric Brownian motion over a finite interval; (d) apply the preceding results to the determination of the distribution of the integral of geom...

There are several models for the evolutionary process forming a species tree. We examine the Birth-and-Death model (BDM), the Proportional-to-Distinguishable Arrangements (PDA) model, the Kirkpatrick and Slatkin (KS) model, the BetaSplitting (BS) model and a model where birth rates evolve according to a Geometric Brownian Motion Process (GBM). For testing and calibrating the models, we evaluate...

We consider the problem of optimal portfolio selection for a multidimensional geometric Brownian motion model. We look for portfolios that maximise the probability of outperforming a stochastic benchmark. More specifically, we seek to maximise the decay rate of the shortfall probability and (or) to minimise the decay rate of the outperformance probability in the long run. A simple heuristic ena...

Carbon finance play an essential role in the promotion of carbon peaking and neutrality, one support for development is structured deposit launched by banks. This article first examines pricing rationality a using risk neutrality pricing, GARCH model, Cholesky decomposition, BS Model, Monte Carlo simulation, geometric Brownian motion, Heston model Merton jump-diffusion etc., parameters used ass...