Black-Scholes Formula

The Nobel Prize Formula

On May 1, 1973 — exactly five days after the CBOE opened — the Journal of Political Economy published "The Pricing of Options and Corporate Liabilities" by Fischer Black and Myron Scholes. The paper had been rejected by two journals before publication. In 1997, Scholes and Robert Merton shared the Nobel Prize in Economics for this work (Black had died in 1995). The committee called it "a major contribution to economic sciences."

The formula gave options a theoretical price for the first time. Before 1973, traders priced options by feel. After 1973, there was a number — and within years, almost every options trader on earth was using it. The CBOE even distributed hand-held calculators pre-programmed with Black-Scholes to traders on its floor.

The Setup: Geometric Brownian Motion

Black and Scholes modeled stock price movements as geometric Brownian motion (GBM). The key intuition: the percentage change in a stock price over a small time interval is normally distributed with mean μ dt and standard deviation σ √dt, where μ is the expected return and σ is the volatility.

In math: dS = μS dt + σS dW_t, where dW_t is an increment of a Wiener process (continuous-time random walk). This implies that log returns — log(S_t/S₀) — are normally distributed, making the stock price itself log-normally distributed. Log-normal makes sense: prices can't go below zero, and a 10% gain followed by a 10% loss doesn't return you to the starting price.

Under this model, the stock price at time T starting from S today is:

S_T = S · exp((μ − σ²/2)T + σ√T · Z)

where Z ~ N(0,1). The σ²/2 term is a Jensen's inequality correction — because log is concave, the expected log return is slightly less than μ.

The Key Insight: Risk-Neutral Pricing

Black and Scholes made a stunning discovery: the option price does not depend on the expected return μ of the stock. This seems wrong at first — surely a stock that's expected to rise is worth more as a call option?

The logic is subtle. If you hold a call and continuously delta-hedge it (buy and sell the underlying to stay delta-neutral), you can eliminate all directional risk from the position. The resulting hedged portfolio must earn the risk-free rate — otherwise there's an arbitrage. But this means the drift μ cancels out of the pricing equation entirely. You can price the option as if the stock grows at the risk-free rate r, regardless of its true expected return. This is risk-neutral pricing.

Mathematically, this means replacing μ with r in the GBM, which gives:

S_T = S · exp((r − σ²/2)T + σ√T · Z), where Z ~ N(0,1)

The option price is then the expected payoff under this risk-neutral measure, discounted at the risk-free rate.

The Black-Scholes Formula

Taking the expected value of max(S_T − K, 0) under the log-normal distribution and discounting gives:

C = S · N(d₁) − K · e^−rT · N(d₂)
P = K · e^−rT · N(−d₂) − S · N(−d₁)

where:

• d₁ = [ln(S/K) + (r + σ²/2)·T] / (σ·√T)
• d₂ = d₁ − σ·√T = [ln(S/K) + (r − σ²/2)·T] / (σ·√T)
• N(·) is the standard normal CDF — the probability that a standard normal variable is below a given value

Interpreting Each Term

The formula has a clean interpretation. For a call:

• N(d₂) is the risk-neutral probability that the call expires in-the-money (S_T > K). It's approximately the probability you'll end up with the stock.
• N(d₁) is a delta-adjusted probability — the expected fraction of the stock price you effectively "own" through the option. It's always slightly larger than N(d₂) because of the log-normal skew.
• S · N(d₁) is the present value of receiving the stock conditional on the call being exercised.
• K · e^−rT · N(d₂) is the present value of paying the strike K conditional on exercise.

The call price is what you get (the stock) minus what you pay (the strike), each probability-weighted and discounted.

What d₁ and d₂ Measure

Think of d₁ and d₂ as standardized measures of how far in-the-money the option is:

• ln(S/K) measures how far the stock is from the strike in log-space. Positive means S > K (ITM call).
• (r + σ²/2)·T adjusts for the drift of the log-normal process.
• σ·√T normalizes by the uncertainty over the life of the option.

As T → 0 at expiration, the formula collapses to the intrinsic value: C → max(S − K, 0). As σ → 0 (certain world), the formula gives the discounted payoff with certainty. Both limits make sense.

The Assumptions (and Their Violations)

Black-Scholes makes five key assumptions, all of which are violated to some degree in practice:

1. Constant volatility: Real volatility changes over time and across strikes. This is why traders use the "volatility smile" — different implied vols for different strikes — a direct contradiction of the model.
2. Log-normal returns: Real stock returns have "fat tails" — extreme moves happen far more often than the normal distribution predicts. The 1987 Black Monday crash (Dow −22.6% in one day) was a 27-sigma event under Black-Scholes — theoretically impossible.
3. No jumps: Stocks can gap overnight or on news. GBM assumes continuous paths.
4. Continuous trading at zero cost: Delta hedging requires continuous rebalancing. In practice, trading is discrete and has transaction costs.
5. Constant risk-free rate: Interest rates move, especially over longer option maturities.

Despite these violations, Black-Scholes remains the lingua franca of options markets. Traders don't use it to believe its prices are correct — they use it as a common language to quote implied volatility, which we'll study in Lesson 8.