The Black-Scholes formula stands as one of the most significant achievements in quantitative finance, providing a theoretical framework for pricing European options. Aspiring quant researchers, especially those interviewing at top-tier firms like Citadel, are frequently tested on their understanding of this formula—not just its application, but its derivation and underlying assumptions. In this comprehensive guide, we will walk through the step-by-step derivation of the Black-Scholes formula, clarify every mathematical and financial concept involved, and discuss the essential assumptions that underpin the model. This knowledge is crucial for anyone preparing for a Citadel Quant Research Intern interview or seeking a deeper grasp of financial mathematics.

Citadel Quant Research Intern Interview Question: Deriving the Black-Scholes Formula

Table of Contents

• Introduction to the Black-Scholes Formula
• Assumptions of the Black-Scholes Model
• Mathematical Preliminaries
• Stochastic Processes and the SDE for Stock Prices
• Portfolio Construction and Hedging
• Derivation of the Black-Scholes Partial Differential Equation (PDE)
• Risk-Neutral Valuation
• Solving the Black-Scholes PDE
• The Final Black-Scholes Formula
• Implications and Limitations of the Model

Introduction to the Black-Scholes Formula

The Black-Scholes formula, developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s, revolutionized the field of financial engineering. It provides an analytic solution for pricing European-style options—contracts that can only be exercised at expiration. The formula is a direct consequence of modeling the evolution of asset prices using stochastic calculus and constructing a riskless hedged portfolio.

Understanding the Black-Scholes formula is not only crucial for passing quant interviews at elite firms like Citadel, but also foundational for anyone pursuing a career in quantitative finance or derivatives trading.

Assumptions of the Black-Scholes Model

Before diving into the mathematical derivation, it is critical to understand the assumptions underlying the Black-Scholes model. These assumptions simplify the real financial markets and make the mathematics tractable.

• Frictionless Markets: No transaction costs or taxes; assets are infinitely divisible.
• Continuous Trading: Trading can occur continuously in time.
• Short Selling Allowed: Investors can short-sell securities with full use of proceeds.
• Constant Risk-Free Rate: The risk-free interest rate \( r \) is constant and known.
• Log-Normal Asset Prices: The price of the underlying asset follows a geometric Brownian motion with constant drift (\( \mu \)) and volatility (\( \sigma \)).
• No Dividends: The underlying asset does not pay dividends during the option's life.
• European Option: The option can only be exercised at maturity.

These idealized conditions are not fully met in the real world, but they are useful for building a tractable model.

Mathematical Preliminaries

The Black-Scholes model is built upon several mathematical concepts. Understanding these is essential for following the derivation.

Brownian Motion and Geometric Brownian Motion

• Brownian Motion (\( W_t \)): A stochastic process with continuous paths and stationary, independent increments that are normally distributed. Formally, \( W_0 = 0 \), and for all \( t \), \( W_t \sim N(0, t) \).
• Geometric Brownian Motion: If \( S_t \) denotes the asset price at time \( t \), then it follows the SDE:
  $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$

Itô's Lemma

Itô's Lemma is a fundamental result in stochastic calculus. It provides the differential of a function of a stochastic process. For a function \( f(t, S_t) \):

We will use Itô's Lemma to derive the dynamics of an option price on a stock following geometric Brownian motion.

Stochastic Processes and the SDE for Stock Prices

The Black-Scholes model assumes the following stochastic differential equation (SDE) for the asset price:

• \( S_t \): Asset price at time \( t \)
• \( \mu \): Expected return (drift)
• \( \sigma \): Volatility of returns
• \( dW_t \): Increment of a Wiener process (Brownian motion)

This SDE indicates that the asset price evolves with a deterministic trend (\( \mu S_t dt \)), and a stochastic component (\( \sigma S_t dW_t \)), both proportional to the current price.

Solution to the SDE

The explicit solution to this SDE is:

This demonstrates that the log-returns of the asset are normally distributed, hence the asset price is log-normally distributed.

Portfolio Construction and Hedging

A central idea in the Black-Scholes derivation is constructing a riskless hedged portfolio by holding a position in the option (whose price we want to determine) and a position in the underlying stock. By dynamically adjusting the portfolio to eliminate risk, we can argue that the portfolio should earn the risk-free rate.

The Replicating Portfolio

Let \( V(S, t) \) denote the price of an option as a function of the asset price \( S \) and time \( t \). Construct a portfolio \( \Pi \) as follows:

• Long 1 option (value \( V \))
• Short \( \Delta \) shares of the underlying (value \( -\Delta S \))

Here, \( \Delta \) is chosen to eliminate risk.

Derivation of the Black-Scholes Partial Differential Equation (PDE)

By applying Itô's Lemma to \( V(S, t) \), we have:

Recall that under the SDE:

and

because \( (dW)^2 = dt \) in stochastic calculus.

Therefore,

Portfolio Dynamics

The change in the hedged portfolio is:

Substitute the expressions for \( dV \) and \( dS \):

Group the \( dt \) and \( dW \) terms:

Eliminating Risk

To create a riskless portfolio, we set the coefficient of \( dW \) to zero:

Thus, our portfolio will be riskless over an infinitesimal interval.

Portfolio Growth at Risk-Free Rate

Since the portfolio is riskless, it should grow at the risk-free rate \( r \):

Recall:

Plugging in the above expressions:

Rearrange terms:

This is the Black-Scholes Partial Differential Equation (PDE).

Risk-Neutral Valuation

A key insight is that in the Black-Scholes world, the expected return (\( \mu \)) of the underlying does not appear in the PDE. Instead, the only relevant rate is the risk-free rate (\( r \)). This is a consequence of risk-neutral valuation.

Risk-Neutral Measure (\( \mathbb{Q} \))

Under the risk-neutral measure, all assets are expected to grow at the risk-free rate. The SDE for the underlying under \( \mathbb{Q} \) is:

where \( dW_t^{\mathbb{Q}} \) is a Brownian motion under the risk-neutral measure.

The value of a derivative (such as a European option) is the discounted expected payoff under the risk-neutral measure:

• \( \Phi(S_T) \): Option payoff at maturity
• \( T \): Expiry time

For a European call option,

where \( K \) is the strike price.

Solving the Black-Scholes PDE

Let’s derive an explicit formula for a European call option. We need to solve the Black-Scholes PDE with the terminal condition:

Transformation to the Heat Equation

We can transform the Black-Scholes PDE into the classic heat/diffusion equation. Define:

• \( \tau = T - t \) (time to maturity)
• \( x = \ln\left( \frac{S}{K} \right) \) (log-moneyness)

After substitution and some algebra, the Black-Scholes PDE becomes a heat equation, which has a known solution. For brevity, we’ll state the final result:

Solution: The Black-Scholes Formula

The price of a European call option is:

import math
from scipy.stats import norm

def black_scholes_call(S, K, T, r, sigma):
    """
    S: Spot price
    K: Strike price
    T: Time to maturity (in years)
    r: Risk-free rate
    sigma: Volatility
    """
    d1 = (math.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * math.sqrt(T)
    call = S * norm.cdf(d1) - K * math.exp(-r * T) * norm.cdf(d2)
    return call