Black–Scholes: what it says and what breaks it

Black–Scholes is the continuous limit of the binomial argument: hedge continuously and the option’s risk disappears, leaving a price that depends on volatility but not on expected return. Every desk knows it is wrong in specific ways, and the interview is almost always about those ways rather than the formula.

Equation 2.1

The formula

With the two arguments given by:

C = S_0 \Phi(d_1) - Ke^{-rT}\Phi(d_2)

$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$: And $d_2 = d_1 - \sigma\sqrt{T}$ .
$\Phi$: The standard normal cumulative distribution.
$\sigma$: Volatility — the only input you cannot observe.

Figure 2.2 · Delta and exercise probability are different weights. At the money with zero rates, 20% volatility and one year to expiry, d₁ = 0.1 and d₂ = −0.1. The call delta is about 0.5398, while the risk-neutral probability of finishing in the money is about 0.4602. The two terms in the formula weight different things.

Proposition 2.3

What the two terms mean

Read the formula as "what you get minus what you pay, each weighted by the chance you get to do it". $\Phi(d_2)$ is the risk-neutral probability the option finishes in the money, so the second term is the strike paid in exactly those states. $\Phi(d_1)$ is the delta, and the first term is the stock received, weighted by that hedge ratio.

Holds when

$\Phi(d_2)$ is the probability of exercise; $\Phi(d_1)$ is the delta. They are not the same number and confusing them is a common slip.
Both are risk-neutral probabilities, not real-world ones.

Where the drift went. The expected return of the stock does not appear. This is the continuous version of the binomial result: because the option can be hedged, the only thing that matters is how far the stock moves, not which way it tends to go. Volatility is in the formula because it measures movement; drift is absent because hedging removes the direction.

Assumption	What breaks it in practice
Continuous hedging at no cost	Spreads and commissions; you hedge discretely and pay for it
Constant volatility	The smile — every strike implies a different one
Lognormal returns, no jumps	Gaps, earnings, crashes; real tails are far fatter
Constant known rate	Minor for equities, material for long-dated and rates products
No dividends	Handled by an adjustment, not a real limitation
Frictionless shorting	Borrow costs on hard-to-borrow names

Table 2.4 · The assumptions, and what violates each. The first three are the ones worth talking about. The others are patched routinely and nobody considers them interesting.

Black–Scholes: what it says and what breaks it

After this lesson you should be able to

The formula

What the two terms mean

The rest of this lesson is in Premium