Module 3: Modelling with Randomness

Random Walks, Fat Tails and Financial Markets

This module adds randomness to your models. You start from the random walk and derive diffusion, the law behind the spread of molecules in a cell. You then see what happens when rare, very large steps dominate: fat tails, stable laws and Lévy flights. Finally you apply these ideas to financial prices and test the random walk model on a daily price series in the laboratory. Reusable code is in amlab.random_walks and amlab.finance.

Session Structure

Session 15: From random walks to diffusion

The Random Walk Mean Squared Displacement The Diffusion Equation Diffusion in Biology

Session 16: When the bell curve fails: fat tails and anomalous diffusion

The Central Limit Theorem Power Laws Stable Distributions Lévy Flights

Session 17: Modelling financial markets

Returns and Efficient Markets Stylized Facts of Returns Scaling of Returns

Session 18: Laboratory: financial time series

Loading Prices and Computing Returns Normality Tests Calibrated KS Test Tail Exponent Scaling of Returns in Data

Learning Goals

After this module you can:

Derive the mean and variance of a lattice random walk and the relation $\langle x^2 \rangle = 2Dt$.
Obtain the diffusion equation as the continuum limit of a random walk, with $D = \delta^2 / (2\tau)$.
State the central limit theorem, its assumptions, and the generalized version for infinite variance.
Estimate a power-law exponent by maximum likelihood and explain why a least-squares fit on a log-log histogram is biased.
Describe stable laws with index $0 < \alpha \le 2$ and simulate Lévy flights.
Compute log returns, test them for normality with a correctly calibrated Kolmogorov-Smirnov test, and estimate their tail index.
Compare real or synthetic return data with the random walk model and with the stylized facts of financial markets.

--- title: "Module 3: Modelling with Randomness" subtitle: "Random Walks, Fat Tails and Financial Markets" format: html --- This module adds randomness to your models. You start from the random walk and derive diffusion, the law behind the spread of molecules in a cell. You then see what happens when rare, very large steps dominate: fat tails, stable laws and Lévy flights. Finally you apply these ideas to financial prices and test the random walk model on a daily price series in the laboratory. Reusable code is in `amlab.random_walks` and `amlab.finance`. # Session Structure **Session 15: From random walks to diffusion** [The Random Walk](random-walk.qmd){.btn .btn-primary} [Mean Squared Displacement](mean-squared-displacement.qmd){.btn .btn-primary} [The Diffusion Equation](diffusion-equation.qmd){.btn .btn-primary} [Diffusion in Biology](diffusion-in-biology.qmd){.btn .btn-primary} **Session 16: When the bell curve fails: fat tails and anomalous diffusion** [The Central Limit Theorem](central-limit-theorem.qmd){.btn .btn-primary} [Power Laws](power-laws.qmd){.btn .btn-primary} [Stable Distributions](stable-distributions.qmd){.btn .btn-primary} [Lévy Flights](levy-flights.qmd){.btn .btn-primary} **Session 17: Modelling financial markets** [Returns and Efficient Markets](efficient-markets.qmd){.btn .btn-primary} [Stylized Facts of Returns](stylized-facts.qmd){.btn .btn-primary} [Scaling of Returns](scaling-of-returns.qmd){.btn .btn-primary} **Session 18: Laboratory: financial time series** [Loading Prices and Computing Returns](returns-data.qmd){.btn .btn-primary} [Normality Tests](normality-tests.qmd){.btn .btn-primary} [Calibrated KS Test](ks-bootstrap.qmd){.btn .btn-primary} [Tail Exponent](tail-exponent.qmd){.btn .btn-primary} [Scaling of Returns in Data](return-scaling-lab.qmd){.btn .btn-primary} --- # Learning Goals After this module you can: - Derive the mean and variance of a lattice random walk and the relation $\langle x^2 \rangle = 2Dt$. - Obtain the diffusion equation as the continuum limit of a random walk, with $D = \delta^2 / (2\tau)$. - State the central limit theorem, its assumptions, and the generalized version for infinite variance. - Estimate a power-law exponent by maximum likelihood and explain why a least-squares fit on a log-log histogram is biased. - Describe stable laws with index $0 < \alpha \le 2$ and simulate Lévy flights. - Compute log returns, test them for normality with a correctly calibrated Kolmogorov-Smirnov test, and estimate their tail index. - Compare real or synthetic return data with the random walk model and with the stylized facts of financial markets.