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Option Pricing via Probability Models

Every option pricing method computes the same thing:

The pricing expectation
V0=e−rTEQ[payoff(ST)]V_0 = e^{-rT}E^{\mathbb{Q}}\left[\text{payoff}(S_T)\right]

Every option price in one line. The work is entirely in specifying the measure and the payoff.

The discounted expected payoff under the risk-neutral measure. The methods differ only in how they evaluate that expectation.

Closed form

Black-Scholes solves the integral analytically for a European option under geometric Brownian motion.

Strengths: instant, exact, differentiable, so Greeks come out in closed form too. This matters enormously in production, where a desk reprices thousands of options many times a second.

Limits: European exercise only, one underlying, constant volatility, no path dependence. Closed forms exist for a handful of other cases and run out quickly.

Binomial trees

Discretise time and price into a recombining lattice and work backwards.

Strengths: handles early exercise naturally, since you compare exercise against continuation at every node. Also handles discrete dividends and barriers cleanly.

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