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Black Scholes Model Derivation Essentials

The derivation matters because its central trick, delta hedging, is what options market makers do all day. The mathematics is the formalisation of a practice.

The setup

Assume the underlying follows geometric Brownian motion:

dS=μS dt+σS dWdS = \mu S\,dt + \sigma S\,dW

A deterministic drift plus a random shock proportional to the price, which keeps SS positive and makes returns rather than prices normally distributed.

Ito's lemma

For an option value V(S,t)V(S,t), ordinary calculus is insufficient because SS is not differentiable. Ito's lemma adds a second-order term:

dV=(∂V∂t+μS∂V∂S+12σ2S2∂2V∂S2)dt+σS∂V∂S dWdV = \left(\frac{\partial V}{\partial t} + \mu S\frac{\partial V}{\partial S} + \tfrac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S\frac{\partial V}{\partial S}\,dW

That extra 12σ2S2∂2V∂S2\frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2} term is the whole difference from ordinary calculus, and it exists because (dW)2=dt(dW)^2 = dt rather than vanishing. It is also where gamma enters, so this term is not a technicality: it is the mathematical statement that convexity has value.

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