I learned very early the difference between knowing the name of something and knowing something.
Richard Feynman24 April 2024
The prime factorization of 999999 allows us to compute repeating decimals for some common fractions. I work through this idea.
1Simulating Geometric Brownian Motion
13 April 2024
I work through a simple Python implementation of geometric Brownian motion and check it against the theoretical model.
204 January 2024
In probability theory, Bienaymé's identity is a formula for the variance of random variables which are themselves sums of random variables. I provide a little intuition for the identity and then prove it.
317 December 2023
I derive some basic properties of the lognormal distribution.
409 December 2023
A useful view of a covariance matrix is that it is a natural generalization of variance to higher dimensions. I explore this idea.
529 October 2023
A mean–variance optimizer will hedge correlated assets. I explain why and then work through a simple example.
608 October 2023
In finance, the "Greeks" refer to the partial derivatives of an option pricing model with respect to its inputs. They are important for understanding how an option's price may change. I discuss the Black–Scholes Greeks in detail.
710 September 2023
The VIX is a benchmark for market-implied volatility. It is computed from a weighted average of variance swaps. I first derive the fair strike for a variance swap and then discuss the VIX's approximation of this formula.
819 August 2023
I work through a well-known approximation of the Black–Scholes price of at-the-money (ATM) options.
9Proof the Binomial Model Converges to Black–Scholes
03 June 2023
The binomial options-pricing model converges to Black–Scholes as the number of steps in fixed physical time goes to infinity. I present Chi-Cheng Hsia's 1983 proof of this result.
10Binomial Options-Pricing Model
03 June 2023
I present a simple yet useful model for pricing European-style options, called the binomial options-pricing model. It provides good intuition into pricing options without any advanced mathematics.
1113 May 2023
I describe the process of using ChatGPT-3.5 to write a program that uses OpenAI's API. The program generates LLM fortunes a la the Unix command 'fortune'.
12Problem Solving with Dimensional Analysis
11 February 2023
Dimensional analysis is the technique of analyzing relationships through their base quantities. I demonstrate the power of this approach by approximating a Gaussian integral without calculus.
13Estimating Square Roots in Your Head
01 February 2023
I explore an ancient algorithm, sometimes called Heron's method, for estimating square roots without a calculator.
1426 January 2023
In the options-pricing literature, the Carr–Madan formula equates a derivative's nonlinear payoff function with a portfolio of options. I describe and prove this relationship.
1507 December 2022
The binomial options-pricing model is a numerical method for valuing options. I explore this model over a single time period and focus on two key ideas, the no-arbitrage condition and risk-neutral pricing.
1617 September 2022
Principal component analyis (PCA) is a simple, fast, and elegant linear method for data analysis. I explore PCA in detail, first with pictures and intuition, then with linear algebra and detailed derivations, and finally with code.
17Matrices as Functions, Matrices as Data
28 August 2022
I discuss two views of matrices: matrices as linear functions and matrices as data. The second view is particularly useful in understanding dimension reduction methods.
18Scaling Factors for Hidden Markov Models
13 August 2022
Inference for hidden Markov models (HMMs) is numerically unstable. A standard approach to resolving this instability is to use scaling factors. I discuss this idea in detail.
1909 August 2022
Weighted least squares (WLS) is a generalization of ordinary least squares in which each observation is assigned a weight, which scales the squared residual error. I discuss WLS and then derive its estimator in detail.
2029 June 2022
The Sharpe ratio measures a financial strategy's performance as the ratio of its reward to its variability. I discuss this metric in detail, particularly its relationship to the information ratio and -statistics.
21How Dangerous Is Biking in New York?
18 June 2022
I estimate my probability of serious injury or death from bike commuting to work in New York, using public data from city's Department of Transportation.
2204 June 2022
I discuss moving or rolling averages, which are algorithms to compute means over different subsets of sequential data.
2324 May 2022
A common heuristic for time-aggregating volatility is the square root of time rule. I discuss the big idea for this rule and then provide the mathematical assumptions underpinning it.
2417 May 2022
Many phenomena can be modeled as exponential decay. I discuss this model in detail, focusing on natural exponential decay (base ) and various useful properties.
2512 April 2022
I discuss multi-factor modeling, which generalizes many early financial models into a common prediction and risk framework.
2627 March 2022
During my PhD, I went hiking alone in a remote region of Iceland. Over the years, I've come to view this trip as analogous to the PhD process. Graduate school was hard, but on the warm days, the views were spectacular.
2720 March 2022
Conjugate gradient descent (CGD) is an iterative algorithm for minimizing quadratic functions. CGD uses a kind of orthogonality (conjugacy) to efficiently search for the minimum. I present CGD by building it up from gradient descent.
28The Capital Asset Pricing Model
06 March 2022
In finance, the capital asset pricing model (CAPM) was the first theory to measure systematic risk. The CAPM argues that there is a single type of risk, market risk. I derive the CAPM from the mean–variance framework of modern portfolio theory.
2903 March 2022
I discuss generalized least squares (GLS), which extends ordinary least squares by assuming heteroscedastic errors. I prove some basic properties of GLS, particularly that it is the best linear unbiased estimator, and work through a complete example.
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