Black-Scholes Model — Mathematics Knowledge System

Overview

The Black-Scholes model is the cornerstone of modern mathematical finance. It assumes the stock price follows geometric Brownian motion (GBM) and derives a PDE (the Black-Scholes equation) for the fair price of a European option. The key insight is that a continuously rebalanced portfolio of the stock and bond can perfectly replicate the option payoff, so by no-arbitrage the option price must equal the cost of the replicating portfolio. The resulting Black-Scholes formula gives an explicit closed-form price for European calls and puts.

Intuition

The central idea is hedging: for every option, there is a portfolio of stock and cash that exactly replicates the option's future payoff. Since the replicating portfolio and the option pay the same in all scenarios, they must cost the same today (no-arbitrage). Ito's formula translates the dynamics of the option price into a PDE, and solving that PDE gives the Black-Scholes formula. The formula depends on volatility sigma and time-to-maturity but NOT on the stock's expected return mu -- because mu cancels in the hedging argument.

Formal Definition

Definition

The stock price satisfies dS_t = mu S_t dt + sigma S_t dB_t (GBM). A European call option with strike K and maturity T has payoff (S_T - K)^+. The Black-Scholes price at time t is C(t, S) = S*N(d1) - K e^{-r(T-t)} N(d2), where d1 and d2 depend on S, K, r, sigma, T-t.

dS_t = \mu S_t\,dt + \sigma S_t\,dB_t \quad \text{(geometric Brownian motion)}

Stock price dynamics

\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS\frac{\partial C}{\partial S} - rC = 0

Black-Scholes PDE

d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma\sqrt{T-t}

d1 and d2

C(t,S) = S\,N(d_1) - K e^{-r(T-t)} N(d_2)

Black-Scholes call price

P(t,S) = K e^{-r(T-t)} N(-d_2) - S\,N(-d_1) \quad \text{(put by put-call parity)}

Black-Scholes put price

Notation

Notation	Meaning
$N(x)$	Standard normal CDF: N(x) = P(Z <= x) for Z ~ N(0,1)
$d_1, d_2$	Key arguments in the Black-Scholes formula; d2 = d1 - sigma*sqrt(T-t)
$r$	Risk-free interest rate (continuously compounded)
$\sigma$	Volatility of the stock price (annualized standard deviation of log returns)

Theorems

Theorem 1: Theorem 1 (Black-Scholes Formula)

\text{Under GBM with volatility } \sigma \text{ and risk-free rate } r, \text{ the price of a European call with strike } K \text{ and maturity } T \text{ is } C = S_0 N(d_1) - Ke^{-rT}N(d_2), \text{ where } d_{1,2} = \frac{\ln(S_0/K) + (r \pm \sigma^2/2)T}{\sigma\sqrt{T}}.

Theorem 2: Theorem 2 (Put-Call Parity)

\text{For European options on the same underlying with the same strike } K \text{ and maturity } T: C - P = S_0 - Ke^{-rT}. \text{ This is a model-independent no-arbitrage relation.}

Worked Examples

1
Compute d1 and d2. With S = K = 100, ln(S/K) = 0.
$d_1 = \frac{0 + (0.05 + 0.02)\cdot 1}{0.20 \cdot 1} = \frac{0.07}{0.20} = 0.35$
2
d2 = d1 - sigma * sqrt(T) = 0.35 - 0.20 = 0.15.
$d_2 = 0.35 - 0.20 = 0.15$
3
N(0.35) approx 0.6368, N(0.15) approx 0.5596. Discount factor: e^{-0.05} approx 0.9512.
$C = 100 \times 0.6368 - 100 \times 0.9512 \times 0.5596 \approx 63.68 - 53.23 = 10.45$

✓ Answer

C ≈ $10.45 (at-the-money European call with 20% vol and 5% risk-free rate for 1 year).

Practice Problems

Mediumfree response

What is the Black-Scholes price of a European call at maturity? Show that C(T, S) = (S - K)^+.

Hardfree response

State put-call parity and derive the Black-Scholes put price from it.

Common Mistakes

Common Mistake

Including mu (the drift) in the Black-Scholes formula

The Black-Scholes price does not depend on mu. The hedging argument and risk-neutral pricing replace mu with r. Including mu gives a wrong answer.

Common Mistake

Confusing d1 and d2 (swapping them)

d2 = d1 - sigma*sqrt(T-t). N(d1) is the Delta of the call (how much stock to hold). N(d2) is the risk-neutral probability the call expires in the money.

Quiz

Summary

GBM stock model: dS = mu S dt + sigma S dB. Option price C(t,S) satisfies the Black-Scholes PDE.
Black-Scholes formula: C = S N(d1) - K e^{-r(T-t)} N(d2) with d1 = [ln(S/K) + (r+sigma^2/2)(T-t)] / (sigma sqrt(T-t)).
Mu does not appear in the formula: it cancels via the hedging (Delta = C_S) argument.
N(d2) = risk-neutral probability of finishing in the money; N(d1) = option Delta.
Put-call parity: C - P = S - K e^{-rT} (model-independent).

References

BookBlack, F. and Scholes, M. -- The Pricing of Options and Corporate Liabilities (1973)
BookOksendal, B. -- Stochastic Differential Equations, Chapter 12
WebsiteWikipedia -- Black-Scholes model