Option pricing is one of the foundational problems in derivatives, quantitative finance, and trading. While the payoff of an option looks simple, its 

 depends on uncertainty, time, volatility, interest rates, and market behavior. Over the years, multiple pricing frameworks have evolved—each suited for different products, assumptions, and market realities.

, from classical textbook models to modern quant-driven approaches, and explains 

Analytical models provide a direct mathematical formula to compute the option price.

The most famous option pricing model, introduced in 1973.

2. Tree‑Based Models (Discrete Time Models)

Tree models simulate the evolution of the underlying price step‑by‑step over discrete time intervals.

Monte Carlo methods price options by simulating thousands or millions of possible future price paths.

4. Partial Differential Equation (PDE) / Finite Difference Methods

Most option pricing models reduce to a PDE. Finite difference methods solve this PDE numerically on a grid.

These models relax the assumption of constant volatility.

Volatility itself follows a random process.

Instead of randomness, volatility is a deterministic function of price and time.

Markets do not move smoothly—jumps happen.

In practice, traders rarely focus on price directly.

* American options via Longstaff–Schwartz

 for the risk you want to measure. Every trading desk, fund, and pricing engine uses a blend of these methods depending on speed, accuracy, and product complexity.