Lesson 8 of 20 · 8 min

The formula, once, with every term in words

Two terms, one for stock and one for cash, each weighted by a probability-like number.

Here is the formula. It appears exactly once in this curriculum, and every symbol in it is explained in words underneath. If you have read the previous six lessons, nothing in it should be new — it is the replication argument written down.

C = S · N(d₁) − K · e^(−rT) · N(d₂)

C: The call's price. What the model says the contract is worth right now.
S: The underlying's current price. Input one.
K: The strike. Input two.
T: Time to expiry as a fraction of a year. Input three.
r: The risk-free rate. Input five.
e^(−rT): The discount factor: what a dollar at expiry is worth today. Over 30 days at 4% it is 0.99672.
N(·): The cumulative normal function. It converts a distance measured in standard deviations into a number between 0 and 1.
d₁: How far the underlying sits above the strike, measured in standard deviations, adjusted for drift and volatility. N(d₁) is the option's delta.
d₂: The same distance one volatility-unit lower. N(d₂) is the model's probability that the option finishes in the money.

Read it as a sentence and it says: the option is worth the stock you would hold, times how much of it you hold, minus the cash you would owe, times the chance you actually pay it. The first term is the stock leg of the replicating package. The second is the financing leg.

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

ln(S/K): How far the price is from the strike, in proportional rather than dollar terms.
(r + σ²/2)T: An adjustment for financing and for the fact that proportional returns compound.
σ√T: Expected wandering over the remaining life: volatility scaled by the square root of time.

Everything divides by σ√T because the model works in units of expected movement, not dollars. Being five dollars from the strike is a lot with one day left and very little with a year left, and dividing by σ√T is what converts dollars into that comparison.

The put formula is the same object rearranged, and it does not need separate memorising. Put-call parity from lesson 4 gives it directly: the put equals the call minus the underlying plus the discounted strike. Anyone who can price a call can price the corresponding put with one subtraction, which is a good reason to learn the call version properly rather than learning two formulas badly.

Two habits make the formula useful rather than decorative. First, always compute σ√T before anything else, because it is the yardstick everything else is measured against and it tells you immediately whether the contract has much life in it. Second, look at the two legs separately. When the stock leg and the cash leg are nearly equal, as they are at the money, the price is a small difference between two large numbers — which is exactly why at-the-money premiums are so sensitive to small changes in the inputs.

Key terms

Cumulative normal function N(·): The probability that a standard normal variable falls below a given value. Turns a distance in standard deviations into a number between 0 and 1.
d₁ and d₂: Standardised distances from the strike. N(d₁) is delta; N(d₂) is the model's chance of finishing in the money.
Risk-neutral probability: The probability used inside the model, consistent with no-arbitrage pricing. Related to, but not the same as, a real-world probability.

Worked example

The base case, computed term by term

Underlying 100, strike 100, 30 days (T = 0.08219), rate 4%, volatility 25%.

σ√T: 0.0717
d₁: 0.0817
d₂: 0.0100
N(d₁), N(d₂): 0.5326, 0.5040
Stock leg: S × N(d₁): $53.26
Cash leg: K × 0.99672 × N(d₂): $50.23
Call price: $3.02

The $3.02 that every other lesson in this track quotes is these seven lines. Nothing is hidden inside the formula.

Check yourself

In the base case, N(d₁) is 0.5326 and N(d₂) is 0.5040. What does each one mean in words?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.