A pricing model is a recipe. Give it a short list of facts about an option and it hands back a fair price. Every desk runs one, the Terminal's calculators run one, and every number on the chain was made by one. So the first thing to learn is not the maths inside the model but what goes in, and how the price answers when each input moves.
There are five inputs: the stock price, the strike, the time left to expiration, the interest rate together with any dividends, and the volatility. The first four you can look up. The fifth you cannot. Volatility, in plain words how much the stock is expected to move, is a guess about the future, and it is the reason this course exists. The other four are bookkeeping.
Stock price and strike
These two set where the option sits. A call gains when the stock rises and loses when it falls. A put does the opposite. Move the strike instead and the picture flips: a higher strike makes a call cheaper, because it is the right to buy at a worse price, and makes a put dearer, because it is the right to sell at a better one. Take a stock at $200. A $190 call is worth more than a $210 call, and a $210 put is worth more than a $190 put.
What matters to the model is the ratio between the two. A $100 call on a $100 stock and a $50 call on a $50 stock are the same option at half the size.
Three strikes on the same $100 stock with 30 days left. Switch to puts and the order reverses: the highest strike becomes the most valuable. Hover to read the value at any stock price.
Each curve rises slowly far from its strike and fastest near it. Until expiry there is always some chance the stock ends on the other side, so the model blends worthless and valuable instead of jumping at the strike.
Time
More time means more chances for the stock to travel, so more time makes both calls and puts worth more, and as the days run down both lose value. A 90-day option is not three times the price of a 30-day option, though. At the money it is closer to the square root of three, a little over 1.7 times, for a reason the next lesson explains. The chart shows the shape: steep near expiry, flatter further out.
An at-the-money option on a $100 stock at three volatilities, with expiry at the left. The curves climb steeply over the first weeks and flatten after: value grows with the square root of time, not in a straight line.
Volatility
Higher volatility raises the price of every option, call or put. A stock that might move 20% has more ways to finish deep in the money than one that might move 10%, and an option owner only feels the good side of that, because the loss is capped at what they paid. Volatility goes into the model as a yearly percentage: 25 means the model assumes moves of about 25% over a year are typical. It has to be chosen, and the next lessons are about how.
Every line rises with volatility, for calls and puts alike. At the money the line is nearly straight; drag days up and it steepens, because a longer option has more room for volatility to matter.
At the money the line is almost straight: each extra vol point adds about the same number of cents. That straightness is what makes it possible to run the model backwards later in the module. Out of the money the line starts flat, because at low volatility a far strike is nearly unreachable, then bends upward.
Interest rate and dividends
These are the small inputs. A call lets you control the stock without paying for it yet, so the cash you keep earns interest. That makes calls a little more valuable when rates are higher, and puts a little less. Dividends go the other way: a share bought through a call misses any dividend paid before you exercise, so an expected dividend trims the call and adds to the put. On a stock yielding under 1% a year, the effect on a one-month option is cents. On a long-dated option, or a high-yield stock, it is real money. These two inputs are the cost of carry from Module 2, seen from the option's side.
A $100 strike with a year and with a month left. Drag the rate from 0% to 8%: the one-year call lifts visibly while the one-month line barely moves. Add a dividend and the call sinks; switch to puts and they gain.
| If this input rises | Call price | Put price |
|---|---|---|
| Stock price | Up | Down |
| Strike | Down | Up |
| Time to expiration | Up | Up |
| Interest rate | Up a little | Down a little |
| Expected dividends | Down a little | Up a little |
| Volatility | Up | Up |
The maths: the Black-Scholes formula, symbol by symbol
The standard model prices a European call as C = S e^(-qT) N(d1) - K e^(-rT) N(d2) and a put as P = K e^(-rT) N(-d2) - S e^(-qT) N(-d1), where d1 = [ln(S/K) + (r - q + v²/2) T] / (v √T) and d2 = d1 - v √T.
Sis the stock price today andKthe strike. They only ever appear as the ratioln(S/K)inside d1 and d2, which is the moneyness point above.Tis the time to expiry in years: 30 days is 30/365 = 0.0822.vis the volatility as a decimal, 30% is 0.30. It appears asv √T, the size of a typical move over the option's life.ris the interest rate andqthe dividend yield, both yearly and continuous.e^(-rT)turns a dollar paid at expiry into today's money;e^(-qT)takes out the dividends the option holder misses.N(x)is the chance that a standard bell-curve draw lands below x.N(d2)is the model's chance the call finishes in the money;N(d1), timese^(-qT), is the call's delta.
Read the call as what you expect to receive, the stock weighted by how likely you are to own it, minus what you expect to pay, the discounted strike weighted by the chance you pay it.
A worked example: S = 100, K = 100, 30 days, v = 0.30, r = 0.04, q = 0. Then v √T = 0.30 × 0.2867 = 0.0860. d1 = (0 + (0.04 + 0.045) × 0.0822) / 0.0860 = 0.081 and d2 = 0.081 - 0.086 = -0.005. N(d1) = 0.532 and N(d2) = 0.498. The call is 100 × 0.532 - 100 × 0.9967 × 0.498 = $3.59. The put, by parity, is the call minus S - K e^(-rT), or $3.59 - $0.33 = $3.26. The 33 cents between them is the interest on $100 for a month: the rate input at work.
Your turn
Open AAPL. Read the call and put prices the model has to explain off the chain, then find the dividend yield on Price Context. With the stock price in the quote bar and the strike and expiration you chose, that is four of the five inputs in hand.