Going Deep Without Drowning: A Gentle Dive into Option Pricing
Where High Finance Meets High School Math—and Somehow Makes Sense
Options theory was developed on the foundations of stochastic calculus.
While building the models we use today required some deep and abstract mathematical concepts, it is still possible to deeply understand how option prices originate and move by exploring the core mechanisms using ideas that are accessible to anyone with a high school-level education.
Understanding a model or even building one does not allow us to predict the future, with all the consequences that come with that.
What it does allow is a clearer understanding of how options react. Options are derivative instruments, meaning their value is derived from another underlying asset. But beyond simply reflecting that asset's value, options add extra layers of parameters. It's like moving from a two-dimensional world to a multidimensional one.
Understanding how option prices move and what they depend on is essential in order to use them effectively, whether for hedging, for speculation, or to strengthen an existing strategy.
The most well-known model for pricing options is the Black-Scholes model, and despite all the criticism, it remains the standard reference.
Its main characteristic is that it assumes volatility is constant over the life of the option. Of course, this isn't true in reality: the potential movement of prices changes over time. For example:
At time zero, we expect the price to move between 10 and 15 by time 2;
Anyway, by time 1, the potential range might have widened to something like 5 to 15, even though there is less time left until expiration.
To address this limitation, more sophisticated models introduce variable volatility. This, in turn, requires building models that can estimate volatility itself, and there are many approaches for doing so:
Stochastic volatility models
GARCH-family models
Local volatility models
…and so on.
Each of these frameworks attempts to more accurately capture how volatility evolves over time, addressing the limitations of the constant-volatility assumption in Black-Scholes.
That said, if one model truly outperformed all others, we would expect it to dominate the field. But that's not the case. This suggests that our ability to forecast volatility is inherently limited, and as a result, the benefit of using more complex models over Black-Scholes is also limited.
Unless you're part of a top-tier research team or working in a highly specialized institutional context, the best approach for a retail investor is often to deeply understand Black-Scholes and stick with it.
Our goal is to measure the price of an option and understand how it changes.
The start is to use a simplification of the Black-Scholes model, the binomial model, and to move on step by step.
The binomial model breaks down the life of the option into discrete time steps and assumes that at each step, the price of the underlying asset can move to one of two possible values: up or down. This simple framework helps us visualize how option pricing works and sets the foundation for more complex models like Black-Scholes.
We start with two basic assumptions:
No arbitrage opportunities exist: you can't make a risk-free profit by buying and selling the asset. This is a B&S assumption, too.
Arbitrage is essentially making a profit without taking any risk. A simple example is: I buy something for 5 and immediately sell it for 10. That’s a risk-free profit of 5.
In the real world, arbitrage is something merchants have always done. Think of a small store in a tourist area that sells bottled water for 2 euros, even though you could buy the same bottle in a supermarket for 1 euro. The tourist shop makes a profit because it offers a “service-convenience”. People pay more because the shop is right there when they need it. That profit isn’t "pure" arbitrage in the financial sense because the shop has a strategic advantage: location. That location isn’t easily replicable by others, so the price gap persists.
In financial markets, such logistical advantages don’t exist anymore. Everyone can access the same platforms and buy or sell in different markets simultaneously. That’s why, in theory, all arbitrage opportunities get absorbed quickly. If one asset is priced differently in two places, traders will immediately buy where it’s cheaper and sell where it’s more expensive. This flow of trades pushes prices toward equilibrium and eliminates the profit gap.
This assumption—that arbitrage does not exist or gets absorbed quickly—is a foundational principle in pricing models like Black-Scholes. If it didn’t hold, models would break down because risk-free profits would distort asset prices and option values.
Volatility is known, constant, and modeled discretely in this framework: the underlying asset can only jump to certain values at each step. As we reduce the size of the steps (and increase the number of them), the binomial model converges to the continuous Black-Scholes model.
Here is a visualization of this concept:
This is the binomial tree. Each branch represents a potential up or down move, with a fixed daily change of 10%:
Next time, we will see how calls are priced in this simplified market.