Stochastic Exponential Growth and Lattice Gases: Statistical Mechanics of Stochastic Compounding Processes

ISBN-13: 9783031111426 / Angielski / Miękka / 2022

Dan Pirjol

Opis

The book discusses a class of discrete time stochastic growth processes for which the growth rate is proportional to the exponential of a Gaussian Markov process. These growth processes appear naturally in problems of mathematical finance as discrete time approximations of stochastic volatility models and stochastic interest rates models such as the Black-Derman-Toy and Black-Karasinski models. These processes can be mapped to interacting one-dimensional lattice gases with long-range interactions. The book gives a detailed discussion of these statistical mechanics models, including new results not available in the literature, and their implication for the stochastic growth models. The statistical mechanics analogy is used to understand observed non-analytic dependence of the Lyapunov exponents of the stochastic growth processes considered, which is related to phase transitions in the lattice gas system. The theoretical results are applied to simulations of financial models and are illustrated with Mathematica code. The book includes a general introduction to exponential stochastic growth with examples from biology, population dynamics and finance. The presentation does not assume knowledge of mathematical finance. The new results on lattice gases can be read independently of the rest of the book. The book should be useful to practitioners and academics studying the simulation and application of stochastic growth models.

Spis treści

1.  Introduction to stochastic growth processes

A.    Growth processes in economics, biology and ecology

B.    Stochastic growth with multiplicative noise

C.    Stochastic growth with Markovian dependence

2.  Stochastic growth processes with exponential growth rates

D.    Exponential growth model driven by a geometric Brownian motion

E.     Exponential growth model with binomial tree growth rates

F.     Exp-Ornstein-Uhlenbeck model

3.  Lyapunov exponents of the exponential stochastic growth processes

G.   Numerical illustration: the bank account in the Black-Derman-Toy model

H.    Lyapunov exponents and their analyticity

I.      Lattice gas analogy

J.      Recursion relation for the moments

4.  One-dimensional lattice gas models with linear attractive interaction

K.    Lattice gases with mutual exclusion

L.     Lattice gas with universal interaction; mean-field theory

M.  Kac potentials and the Lebowitz-Penrose theory

N.   One-dimensional lattice gas with linear attractive interaction: exact results

5.  Lattice gas with exponential attractive interactions

O.   Connection to the Black-Karasinski model

P.    Exact result for the Lyapunov exponents

Q.   Limiting case: Kac-Helfand lattice gas and the van der Waals theory

6.  Applications

R.    Monte Carlo simulation of stochastic volatility models

S.     Asymptotic bond pricing in the Black-Derman-Toy model

 

O autorze

Dan Pirjol is working at the interface of mathematical physics, probability theory and mathematical finance. His main research interests are in applied probability, mathematical finance and statistical physics. After doing research in theoretical high energy physics he worked on financial engineering and model risk management for Merrill Lynch, Markit and JP Morgan, most recently in the Model Risk group. Since 2019 he joined the faculty of the School of Business at Stvens Institute of Technology.