What a couple of days.

I explored mathematics, the essence of derivatives (i.e. options, futures, etc.), and some pretty awesome intersections between the two. I’m not understanding everything yet, but I’m trying my best, and finding it fascinating.

I first dove into what a derivative is in financial markets, and how it can become so expensive that the total market value exceeds the combined GDP of the world. I found European options to be the most straightforward to understand. They’re effectively a promise: you may buy or sell an asset, whether it’s a stock, a tangible item, or anything else, for a specific price at a specific time. You’re not obligated to act, but you have the right to do so, and only at that time.

An American option works differently. It can be bought or sold at any time up to expiration, which makes the pricing more complex.

This ties into my attempt to understand the Black-Scholes model, still far from mastering it, but a bit closer now. I then tried to grasp Itô’s Lemma and Itô calculus. I failed, sort of, but I now have a rough idea. Itô’s Lemma is effectively a chain rule of standard differential calculus, but adapted to stochastic processes. It’s used to derive a stochastic differential equation by taking randomness into account.

Roughly speaking, you take the original function, derive it normally, then add terms that account for randomness. These additional parameters help you compute the rate of change in a probabilistic context. This is used in the derivation of the Black-Scholes equation. I recognized this because the variables in Itô’s Lemma matched those in the simplified form of the Black-Scholes equation:

\(
dS = \mu S,dt + \sigma S,dW_t
\)

As well as:

\(
\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} – rV = 0
\)

I also realized that the “Greeks”, risk measures like delta, gamma, etc., are derived from the Black-Scholes formula. That made a lot of sense. I learned a bit about delta, which shows how much the option price changes relative to the stock price. For example, if a stock goes up by $1, and delta is 0.5, the option price increases by $0.50.

Next, I explored Brownian motion, a stochastic process that models random behavior. Think of it like standing at the base of an infinite staircase, flipping a coin. Heads: move up. Tails: move down. Do this 10 times. Then repeat that experiment multiple times. You’ve just modeled uncertainty, with no memory of past or future steps.

Geometric Brownian Motion, which is derived using Itô’s Lemma, modifies this by factoring in the current value of the asset. What this really means? I don’t know yet. But I can explain it in one sentence, which is more than I could do two days ago.

That’s what I’ve learned over the past 48 hours. Stick around, I’ll keep building my understanding, one concept at a time.