Lecture 2: Linear Algebra

Quick Overview

Peter Kempthorne introduces the first linear algebra lecture, outlining a curriculum covering basic vector algebra, special matrices, eigenvectors, decompositions, and the Perron-Frobenius theorem, emphasizing the utility of linear algebra in quantitative finance for simplifying computations.

Key Points: The lecture covers basic fundamentals including vector algebra, special matrices, eigenvectors and eigenvalue, decompositions, and the Perron-Frobenius theorem, noting linear algebra makes computations in quantitative finance easier. A portfolio holding S&P 500 stocks is represented by a vector q, and its value is calculated using the dot product, q dot p, where p is the vector of closing prices. Short selling involves borrowing stock, selling it, and buying it back later, which credits the brokerage account with positive cash flow, potentially allowing investment exceeding 100% of wealth. A zero-cost portfolio is one where the components are not all zero, but the net value of the portfolio is zero, and an arbitrage portfolio has zero cost but a positive profit and loss. The stationary distribution (pi star) for a Markov chain satisfies pi star equals A pi star, provided the chain is acyclic, and this relationship is fundamentally tied to eigenvalues, where one eigenvalue is 1. The matrix equation for eigenvalues and eigenvectors, Av = lambda v, is solved by finding roots of the determinant of (A minus lambda i) equals 0. In multi-asset models, no arbitrage exists if a pricing measure Q exists where the initial price of asset j equals a discount factor times the expected terminal price of that asset across states.

Context: This transcript is from Lecture 2 of a course on Linear Algebra, presented by Peter Kempthorne, specifically tailoring the mathematical concepts to applications in quantitative finance. The lecture begins by defining basic vector notation, such as using bold letters for vectors, and immediately connects these concepts to financial examples like stock prices in the S&P 500 index and portfolio valuation using the dot product.