Skip to main content

TL;DR

The foundational option pricing model. For a European call: \( C = S_0 N(d_1) - Ke^{-rT}N(d_2) \) where \( d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \) and \( d_2 = d_1 - \sigma\sqrt{T} \).

By Valenke Exam Prep Team·Last updated 2026-06-03

Black-Scholes Model

The foundational option pricing model. For a European call: \( C = S_0 N(d_1) - Ke^{-rT}N(d_2) \) where \( d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \) and \( d_2 = d_1 - \sigma\sqrt{T} \).

Why it matters for interviews

The most important formula in quantitative finance. Every quant interview expects fluency with Black-Scholes: derivation, assumptions, Greeks, and limitations. It is the benchmark against which all other models are compared.

Definition and Mathematical Foundation

The foundational option pricing model. For a European call: \( C = S_0 N(d_1) - Ke^{-rT}N(d_2) \) where \( d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \) and \( d_2 = d_1 - \sigma\sqrt{T} \).

Application in Quantitative Finance

The most important formula in quantitative finance. Every quant interview expects fluency with Black-Scholes: derivation, assumptions, Greeks, and limitations. It is the benchmark against which all other models are compared.

Related Concepts

Brownian Motion (Wiener Process)

Related Terms

Ready to practice for the Quant Trading Interview?

Adaptive practice powered by Item Response Theory targets your weak areas. Start with 3 free sessions.

Start free practice →

Frequently Asked Questions

What are the assumptions of Black-Scholes?▼
Constant volatility, log-normal stock prices (GBM), no dividends (basic version), no transaction costs, continuous trading, constant risk-free rate, and European exercise only.
How is the Black-Scholes PDE derived?▼
Construct a delta-hedged portfolio (long option, short \( \Delta \) shares). Apply Ito's lemma to the option price. The stochastic term cancels, leaving a deterministic PDE: \( \frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = rV \).
Why does the Black-Scholes model produce a volatility smile?▼
It does not -- Black-Scholes assumes constant volatility. The volatility smile is an empirical phenomenon where implied volatility varies with strike price, indicating that the model's assumptions (especially log-normality) are violated.