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.
Author(s): Dan Pirjol
Series: SpringerBriefs in Applied Sciences and Technology
Publisher: Springer
Year: 2022
Language: English
Pages: 137
City: Cham
513815_1_En_OFC
513815_1_En_BookFrontmatter_OnlinePDF
Preface
Contents
513815_1_En_1_Chapter_OnlinePDF
1 Introduction to Stochastic Exponential Growth
1.1 Discrete Time Proportionate Growth
1.2 Random Multiplicative Processes
1.2.1 Fluctuations
1.3 Exponential Stochastic Growth in Continuous Time
1.3.1 Fluctuations Around the Mean in Exponential Stochastic Growth
1.4 Stochastic Growth with Markovian Dependence
References
513815_1_En_2_Chapter_OnlinePDF
2 Stochastic Growth Processes with Exponential Growth Rates
2.1 Exponential Growth with Geometric Brownian Motion Rates
2.2 Exponential Growth with Geometric Random Walk Rates
2.3 Exponential Growth with Exponential Ornstein-Uhlenbeck Rates
2.4 Moment Explosion of the Bank Account in the BDT Model
2.5 Explosion Criterion
2.6 Growth Rates in the Model with Binomial Tree Growth
2.7 Appendix: Recursive Computation of Moments
2.7.1 Stochastic Growth with Geometric Brownian Motion Growth Rates
2.7.2 Stochastic Growth Process with Geometric Random Walk Growth Rates
References
513815_1_En_3_Chapter_OnlinePDF
3 Lattice Gas Analogy
3.1 Mapping to a One-Dimensional Lattice Gas
3.2 Temperature Definition
3.3 Thermodynamical Analogy
3.4 Numerical Simulations
References
513815_1_En_4_Chapter_OnlinePDF
4 One-Dimensional Lattice Gases with Linear Interaction
4.1 Preliminaries
4.2 Mean-Field Theory
4.3 Kac Potentials and the Lebowitz–Penrose Theory
4.4 Lattice Gas with Linear Attractive Interaction
4.4.1 Energy Spectrum
4.4.2 Exact Solution—Isothermal-Isobaric Ensemble
4.4.3 Van der Waals Approximation for the Lattice Gas with Linear Interaction
4.5 Grand Canonical Ensemble
4.5.1 Mean-Field Theory in the GC Ensemble
4.5.2 Lattice Gas with Linear Attractive Potentials in the Grand Canonical Ensemble
4.6 Discussion: The Physics of One-Dimensional Lattice Gases …
4.7 Appendix: Thermodynamical Tables for the Lattice Gas with Linear Potentials
References
513815_1_En_5_Chapter_OnlinePDF
5 One-Dimensional Lattice Gas with Exponential Attractive Potentials
5.1 The Kac-Helfand System in the Grand Canonical Ensemble
5.1.1 Solution: The Symmetric Case
5.1.2 Numerical Illustration
5.2 Bounds on the Thermodynamical Pressure
5.3 Van der Waals Limit and Comparison with the Literature
5.4 Explicit Result for the Bounds and Applications
5.5 Appendix: Proofs
References
513815_1_En_6_Chapter_OnlinePDF
6 Asymptotic Growth Rates for Exponential Stochastic Growth Processes
6.1 Stochastic Process Driven by a Geometric Brownian Motion
6.2 Reduction to Two Cases
6.3 The Solution of the Variational Problem
6.4 The Lyapunov Exponent lamda Subscript plus Baseline left parenthesis rho comma beta right parenthesisλ+(ρ,β)
6.5 Phase Transition in lamda Subscript plus Baseline left parenthesis rho comma beta right parenthesisλ+(ρ,β)
6.5.1 Mean-field Approximation for lamda Subscript plus Baseline left parenthesis rho comma beta right parenthesisλ+(ρ,β)
6.6 The Lyapunov Exponent lamda Subscript minus Baseline left parenthesis rho comma beta right parenthesisλ-(ρ,β)
6.7 Mathematical Aside: Fixed-time, Free-endpoint Variational Problem
6.8 Generalization: Sums of Geometric Brownian Motions
References
513815_1_En_7_Chapter_OnlinePDF
7 Applications
7.1 Stochastic Volatility Models with Log-normal Volatility
7.1.1 Time Discretization: Euler Scheme
7.1.2 Hull-White Model: Numerical Simulations
7.1.3 Implications for Monte Carlo Simulations
7.1.4 Asymptotic Growth Rates and Explosion Criteria
7.2 Applications to Interest Rates Models with Log-normal Rates
7.2.1 Bond Pricing in the Black-Derman-Toy Model
7.2.2 Asymptotic Bond Pricing in the BDT Model
7.3 Inverse Exponential Stochastic Growth Model
7.4 Appendix: Recursion for the Second Moment ...
7.5 Appendix: Code for Lyapunov Exponents
References
