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Black-Scholes Equation

How much is the right to buy a stock in the future worth? Economics Nobel 1997. Connects calculus, PDEs, and probability in a single object.

Used in: Master's in Finance · Financial Engineering · Derivatives Desk · Market Risk

C=S⋅N(d1)−K e−rT N(d2)C = S \cdot N(d_1) - K\,\mathrm{e}^{-rT}\, N(d_2)

Black-Scholes Equation (1973) — price of a European call option. Where S is the current price of the asset, K the strike, r the risk-free rate, T the time to expiration, and N(·) the cumulative distribution function of the standard normal. Economics Nobel 1997.

Choose your door

Rigorous notation, full derivation, hypotheses

Black-Scholes Model (1973)

Hypotheses

  1. The price StS_t of the underlying asset follows a geometric Brownian motion under the risk-neutral measure Q\Q:
dSt=rSt dt+σSt dWtdS_t = r S_t \, dt + \sigma S_t \, dW_t
(1)
what this means · The asset price grows with expected rate rr (the risk-free rate) and fluctuates with volatility σ\sigma. WtW_t is a standard Brownian motion — infinitesimal random steps that accumulate into a continuous walk.
  1. Complete market, frictionless: no transaction costs, unlimited short positions, infinite divisibility.
  2. rr and σ\sigma constant along [0,T][0, T].
  3. No dividends (relaxable: replace rr with r−qr - q).
  4. Absence of arbitrage — fundamental principle.

The Black-Scholes PDE

By Itô's lemma applied to V(S,t)V(S, t) and construction of the replicating portfolio (long Δ\Delta shares, short 1 option), risk is eliminated and the portfolio must earn rr. Thus:

∂V∂t+12σ2S2∂2V∂S2+rS∂V∂S−rV=0\pdiff{V}{t} + \frac{1}{2}\sigma^2 S^2 \pdiff{^2 V}{S^2} + r S \pdiff{V}{S} - r V = 0
(2)
what this means · It is a parabolic PDE — analogous to the heat equation in physics! The term 12σ2S2VSS\frac{1}{2}\sigma^2 S^2 V_{SS} is the "heat" diffusing, rSVSr S V_S is a drift, and −rV-rV is discounting. All intuition from the physics of heat applies here.

with final condition V(S,T)=max⁡(S−K,0)V(S, T) = \max(S - K, 0) for a European call.

Closed-form solution

C(S,t)=S⋅N(d1)−K e−r(T−t)N(d2)C(S, t) = S \cdot N(d_1) - K\,\e^{-r(T-t)} N(d_2)
(3)
what this means · The call price is the risk-neutral expected value of the payoff. N(d2)N(d_2) is the probability the call finishes in-the-money. S⋅N(d1)S \cdot N(d_1) is the expected value of STS_T conditional on exercise.

where

d1=ln⁡(S/K)+(r+12σ2)(T−t)σT−t,d2=d1−σT−td_1 = \frac{\ln(S/K) + (r + \frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma\sqrt{T-t}
(4)
what this means · The d1d_1 measures how many standard deviations the log-price is above the log-strike, adjusted for drift. d2=d1−σT−td_2 = d_1 - \sigma\sqrt{T-t}.

and N(⋅)N(\cdot) is the cumulative distribution function of the standard normal:

N(x)=12π∫−∞xe−u2/2 duN(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} \e^{-u^2/2} \, du
(5)
what this means · The famous bell curve of Gauss, integrated from minus infinity to xx. It has no closed form — computed via rational approximations (Abramowitz & Stegun) or the complementary error function.

Put-call parity

C−P=S−Ke−r(T−t)C - P = S - K \e^{-r(T-t)}
(6)
what this means · Exact relationship between European calls and puts of the same strike and expiration. If this identity breaks in the market, there is pure arbitrage — in practice, violated by fractions of a penny for seconds.

The Greeks

Partial sensitivities of the option price. Every derivatives desk in the world monitors these numbers.

ΔC=∂C∂S=N(d1)\Delta_C = \pdiff{C}{S} = N(d_1)
(7)
what this means · Delta — how much the option price changes for each dollar move in the stock. For a deep-ITM call, Δ→1\Delta \to 1; deep-OTM, Δ→0\Delta \to 0.
Γ=∂2C∂S2=N′(d1)SσT−t\Gamma = \pdiff{^2 C}{S^2} = \frac{N'(d_1)}{S \sigma \sqrt{T-t}}
(8)
what this means · Gamma — convexity: how delta changes when price changes. Same for call and put. Largest near the strike and near expiration.
ν=∂C∂σ=ST−t⋅N′(d1)\nu = \pdiff{C}{\sigma} = S \sqrt{T-t} \cdot N'(d_1)
(9)
what this means · Vega — sensitivity to volatility. Same for call and put. On the trading floor, implied vol is what you operate, not price directly.
ΘC=−SσN′(d1)2T−t−rKe−r(T−t)N(d2)\Theta_C = -\frac{S \sigma N'(d_1)}{2\sqrt{T-t}} - r K \e^{-r(T-t)} N(d_2)
(10)
what this means · Theta — time decay: how much the option loses value per day, holding everything else constant. Θ<0\Theta < 0 for long options — option is an asset that "melts".

To continue

  • Black, F.; Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3): 637-654. The original paper.
  • Hull, J.C. Options, Futures, and Other Derivatives. 11th ed., Pearson. The bible of the field.
  • Wilmott, P. Paul Wilmott on Quantitative Finance. Wiley. A critical, practical voice.
  • Gatheral, J. The Volatility Surface. Wiley. After mastering BS, this book shows how the real market works.
  • Columbia Foundations FE — Black-Scholes Notes (official PDF). Free notes, careful derivation.
  • B3 — Options quote history.
  • Status Invest — PETR4 live quote.
  • Opções.Net.Br — PETR4 implied volatility.

Updated on 2026-05-11 · Author(s): Clube da Matemática

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