3.3.1 Black–Scholes–Merton from Zero

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

This page builds the why of Black–Scholes–Merton with no prior derivatives knowledge needed. The objective is one idea: an option can be exactly replicated by a continuously-rebalanced portfolio of the underlying stock and the risk-free bond, and therefore its price is forced by the no-arbitrage principle - it cannot be anything else.

Start with the dumbest question: why does an option have a price at all? A stock has a price because it pays future cash flows. An option is just a contract - a bet written as $\max (S_T-K,0)$ for a call. Its value comes from a different mechanism: you can manufacture that payoff by trading. If the manufacturing cost is knowable, then the option must trade at that cost, or someone can lock in a risk-free profit by selling the option and manufacturing the payoff cheaply (or vice-versa).

Three steps:

1. The stock’s expected drift is irrelevant. The stock moves as $dS=\mu Sdt+\sigma SdW$. The probability of ending in the money depends on $\mu$. But because we replicate by holding a delta-share position, the option’s price cancels $\mu$ entirely. You never need to forecast the direction - you only need the size of the wiggles $\sigma$.

2. Volatility, not drift, is the pricing parameter. If two stocks have the same volatility but different expected returns, options on them must have the same price. Only $\sigma$ matters. This feels absurd at first and is exactly right: the hedge removes the return, leaving only the uncertainty that volatility quantifies.

3. Risk-neutral expectation is a bookkeeping trick, not a belief. The “risk-neutral measure” $\mathbb{Q}$ does not say the market is risk-neutral. It is the measure under which discounted prices are martingales, and its only job is to make the drift equal $r$ so that pricing becomes “discount the expected payoff.” It is a change of measure (Girsanov), not a change of opinion.

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2. Mathematical Ground Truth & Derivations

The replication story in three pictures.

One-step binomial (the discrete seed). Suppose the stock is $S_0$ and, over one period, moves to either $S_{0}u$ (up) or $S_{0}d$ (down). Buy $\Delta$ shares and finance with the bond. Choose $\Delta$ so the portfolio’s payoff matches the option in both states:

$$\Delta =\frac{f_u-f_d}{S_{0}u-S_{0}d},$$

where $f_u,f_d$ are the option payoffs in the up/down states. The option price is the (discounted) cost of this replicating portfolio. Letting the number of steps $\to \infty$ and the step size $\to 0$ (Cox–Ross–Rubinstein) converges exactly to the BSM closed form.

Continuous limit. In continuous time the underlying obeys geometric Brownian motion

$$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t.$$

By Itô’s lemma the option $V(t,S)$ moves as

$$dV=(\frac{\partial V}{\partial t}+\mu S\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2})dt+\sigma S\frac{\partial V}{\partial S}dW.$$

Hold $\Pi =V-\Delta S$. Pick $\Delta =\partial V/\partial S$ to kill the $dW$ term - the portfolio becomes riskless, so it must earn $r$ (else arbitrage):

$$\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}=r(V-S\frac{\partial V}{\partial S})⟹\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}-rV=0.$$

This is the BSM PDE (Hull eq. 15.16; Shreve II 4.5.14; Björk Thm 7.7). Notice: $\mu$ is gone. That is the entire point of the model.

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3. Computational Implementation - the binomial tree converges to BSM

This is the single most convincing way to see the theory: build the CRR tree (Haug §4.1, hull Ch 13), price a European put, and watch it converge to the closed form. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. The “it’s just expected discounted payoff” trap. Under the physical measure $\mathbb{P}$, ${\mathbb{E}}^{\mathbb{P}}[e^{-rT}\max (S_T-K,0)]$ is not the correct price - it depends on $\mu$. Only under $\mathbb{Q}$ (drift $=r$) does it price correctly. The beginner error is to discount under the wrong measure.
2. Replication is not prediction. The hedge removes drift, but only if you can trade continuously. If you cannot (costs, gaps, jumps), $\mu$-independence breaks down - that is the seed of every failure mode in 05 · Failure Modes.
3. The $\frac{1}{2}{\sigma }^2$ correction surprises beginners. $\ln S_T$ has drift $\mu -\frac{1}{2}{\sigma }^2$, not $\mu$. Itô’s lemma inserts the half-variance term because $d{W}^2=dt$ has nonzero quadratic variation. Missing it misprices everything.

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5. References

• Björk, Arbitrage Theory in Continuous Time
• Hull, Options, Futures, and Other Derivatives
• Haug, The Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Base: Stochastic Calculus & Itô · No-Arbitrage & Binomial Trees
• Continue: 02 · PDE & Derivation · Index Hub