Some money is better than others. This is foundational to how money and banking works. I explore this idea and its many consequences, without any prerequisites or jargon.

1

I explain the Black–Scholes formula using only basic probability theory and calculus, with a focus on the big picture and intuition over technical details.

2

I work through a simple Python implementation of geometric Brownian motion and check it against the theoretical model.

3

A mean–variance optimizer will hedge correlated assets. I explain why and then work through a simple example.

4

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.

5

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.

6

I work through a well-known approximation of the Black–Scholes price of at-the-money (ATM) options.

7

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.

8

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.

9

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.

10

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.

11

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 $t$-statistics.

12

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.

13

I discuss multi-factor modeling, which generalizes many early financial models into a common prediction and risk framework.

14

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.

15

I discuss prices, returns, cumulative returns, and log returns, with a special focus on some nice mathematical properties of log returns.

16

Some important financial ideas are encoded in the geometry of the efficient frontier, such as the tangency portfolio and the Sharpe ratio. The goal of this post is to re-derive these ideas geometrically, showing that they arise from the mean–variance analysis framework.

17

Drawdown measures the decline of a time series variable from a historical peak. I explore visualizing and computing drawdown-based metrics.

18

The casual investor knows that diversification matters. This intuition is grounded in the mathematics of modern portfolio theory. I define diversification and formalize how diversification helps maximize risk-adjusted returns.

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