Black-Scholes explained without the stochastic calculus

Black-Scholes is the standard model for pricing European options. You will not be asked to derive it in a trading interview, but you are expected to know what it says and where it breaks.

The formula

For a call on a stock that pays no dividends:

C = S × N(d1) − K × e^(−rT) × N(d2)

where

d1 = [ln(S / K) + (r + σ² / 2) × T] / (σ × √T) and d2 = d1 − σ × √T

S is the stock price, K the strike, T the time to expiry in years, r the interest rate, σ the volatility and N the standard normal cumulative distribution.

The put follows from put-call parity: P = C − S + K × e^(−rT).

What the pieces mean

• N(d2) is the risk-neutral probability that the option finishes in the money. The second term is the present value of paying the strike, weighted by that probability.
• N(d1) is the delta: the number of shares that replicates the option over a short period.
• The call is therefore worth what you expect to receive in stock minus what you expect to pay in cash.

The idea behind it

An option can be replicated by holding delta shares and borrowing cash, and adjusting the holding as the stock moves. If the replication is exact, the option must cost what the replicating portfolio costs. Nobody's view of where the stock is going enters the price. Only volatility does.

Assumptions

1. The stock follows a lognormal random walk with constant volatility.
2. You can trade continuously with no costs.
3. Interest rates are constant.
4. No jumps.

Where it fails

• Volatility is not constant. Out-of-the-money puts on equity indices trade at higher implied volatility than at-the-money options. This is the skew, and it exists because the model's tails are too thin.
• Prices jump. Earnings and news create gaps that continuous hedging cannot follow.
• Hedging costs money and can only be done at intervals.

Traders keep the formula anyway and use it as a quoting convention. They put the "wrong" volatility into it to get the right price, a different volatility for each strike and expiry.

Interview questions

• What is implied volatility? The volatility that makes the model price equal the market price.
• Why does the expected return of the stock not appear? Because the option is priced by replication, and the hedge removes exposure to direction.
• What is the price of an at-the-money call roughly? About 0.4 × S × σ × √T when rates are low.
• What does the volatility smile tell you? That the market assigns more probability to large moves than a lognormal distribution does.