Derivation of the Black-Scholes Formula

Motivation: why this matters in quant finance

The Black-Scholes formula is the canonical result in derivatives pricing. It prices a European call by combining three ideas: a stock following geometric Brownian motion, continuous rebalancing through Itô's lemma, and no-arbitrage valuation under a risk-neutral measure.

The final call price is familiar:

C_0=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2),

where

d_1=\frac{\ln(S_0/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt T}, \qquad d_2=d_1-\sigma\sqrt T.

The derivation matters more than the formula. It explains why the physical drift $\mu$ vanishes, why volatility remains, why $\Phi(d_1)$ is the delta-like stock weight, and why $\Phi(d_2)$ is the risk-neutral exercise probability.

The informal idea

A European call payoff is nonlinear: $(S_T-K)^+$ . Black-Scholes prices it by showing that the option can be replicated by dynamically trading the stock and a risk-free account. If a trading strategy replicates the payoff in every state, no-arbitrage says the strategy and the option must have the same price.

There are two equivalent routes:

Delta-hedging PDE: Build a self-financing portfolio that cancels the local Brownian risk and earns the risk-free rate.
Risk-neutral expectation: Move to $\mathbb{Q}$ , under which the discounted stock is a martingale, and compute $e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)^+]$ .

The expectation route gives the closed form most directly. The PDE route explains replication.

Model assumptions

Assume a non-dividend stock satisfies

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t^{\mathbb{P}}, \qquad S_0>0,

with constant volatility

\sigma>0

, constant risk-free rate

r

, frictionless trading, continuous rebalancing, and no arbitrage. By change of measure, under the risk-neutral measure

\mathbb{Q}

the stock follows

dS_t=rS_t\,dt+\sigma S_t\,dW_t^{\mathbb{Q}}.

Therefore

S_T=S_0\exp\left((r-\tfrac12\sigma^2)T+\sigma\sqrt T Z\right), \qquad Z\sim\mathcal{N}(0,1).

Risk-neutral valuation

The call price is

C_0=e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)^+].

Split the expectation:

C_0=e^{-rT}\left(\mathbb{E}^{\mathbb{Q}}[S_T\mathbf{1}_{S_T>K}]-K\mathbb{Q}(S_T>K)\right).

The event $S_T>K$ is equivalent to

Z>\frac{\ln(K/S_0)-(r-\tfrac12\sigma^2)T}{\sigma\sqrt T}=-d_2.

\mathbb{Q}(S_T>K)=\Phi(d_2).

For the stock-weighted term, completing the square in the normal density gives

e^{-rT}\mathbb{E}^{\mathbb{Q}}[S_T\mathbf{1}_{S_T>K}]=S_0\Phi(d_1),

where $d_1=d_2+\sigma\sqrt T$ . Combining terms gives

C_0=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2).

For a European put,

P_0=Ke^{-rT}\Phi(-d_2)-S_0\Phi(-d_1),

which also follows from put-call parity.

PDE derivation sketch

Let $V(S,t)$ be the option value. Applying Itô's lemma under $\mathbb{P}$ gives

dV=\left(V_t+\mu S V_S+\tfrac12\sigma^2S^2V_{SS}\right)dt+\sigma S V_S\,dW_t.

Hold one option and short $\Delta=V_S$ shares of stock. The random term cancels in the hedged portfolio. Since the portfolio is locally riskless, it must earn $r$ . Rearranging gives the Black-Scholes PDE:

V_t+rS V_S+\tfrac12\sigma^2S^2V_{SS}-rV=0,

with terminal condition $V(S,T)=(S-K)^+$ for a call. The closed-form formula above is the solution to this PDE.

The disappearance of $\mu$ is not an algebra accident. It is the replication argument saying that an option's price depends on hedgeable risk, not on the expected return demanded by stock investors.

Worked example

Let $S_0=100$ , $K=100$ , $r=0.05$ , $\sigma=0.20$ , and $T=1$ . Then

d_1=\frac{0+(0.05+0.02)}{0.20}=0.35, \qquad d_2=0.15.

Using $\Phi(0.35)\approx0.6368$ and $\Phi(0.15)\approx0.5596$ ,

C_0=100(0.6368)-100e^{-0.05}(0.5596)\approx10.45.

This is the benchmark value used to validate Monte Carlo and finite-difference pricers.

Common confusions and pitfalls

"The formula assumes the stock earns the risk-free rate in reality." No. The stock earns drift

\mu

under

\mathbb{P}

. Pricing uses

r

under

\mathbb{Q}

because the option payoff is replicated.

" $\Phi(d_2)$ is the real-world exercise probability." It is the risk-neutral exercise probability. Real-world exercise probability would use

\mu

instead of

r

"Delta hedging is free." The ideal derivation assumes continuous trading, no transaction costs, and constant volatility. Real hedging has discrete rebalancing error, costs, and model risk.

"Black-Scholes is true because the formula is elegant." It is true under its assumptions. Volatility smiles, jumps, dividends, rates, and market frictions are precisely the ways real markets depart from those assumptions.

Where this goes next

Black-Scholes PDE: The PDE route in more detail.
Delta: The hedge ratio $\Phi(d_1)$ for a call.
Implied Volatility: Inverts Black-Scholes to read volatility from market prices.
Barrier Options: Shows how closed forms change for path-dependent payoffs.
Risk-Neutral Valuation: Generalises the expectation formula beyond vanilla calls.