The first lesson left a gap. Four inputs can be looked up and the fifth has to be chosen, yet the market prints a firm price for every option all day long. Where does the market's volatility number come from? Nobody types it in. It is read out of the price.
Running the model in reverse
Forwards, the model takes five inputs and returns a price. Backwards, you take the price the market is showing, the four known inputs, and ask which volatility makes the model produce exactly that price. There is always one answer, because a higher volatility always means a higher price, so you can search for it. Try 30%: too high. Try 25%: too low. Try 27% and close in. The result is the implied volatility: the volatility the price implies, usually written IV.
A $100 stock with 30 days left. Hover along a line to pair each volatility with its price. Find $4.00 on the at-the-money line and read the vol underneath: that is the implied volatility of a $4.00 at-the-money option.
That chart is the whole idea. Every point on a line is a volatility and the price it produces. Pick the market price on the vertical axis, read across until you hit the line, then read down: that is the implied vol. Because each line only ever rises, there is exactly one crossing.
A worked example, with rates set to zero to keep it simple. A stock trades at $250 and the $250 call with 30 days left is offered at $14.00. At 40% volatility the model says the call is worth about $11.40. At 50% it says about $14.30. The volatility that lands on $14.00 is 49%, so that call's implied vol is 49. The market, by paying $14 for it, is saying that a one-standard-deviation yearly move of about 49% is the right assumption for this stock over the next month.
Why traders quote vol instead of price
A price of $14 tells you very little on its own. It depends on the strike, the stock price and the days left, so you cannot compare it with a $6 option on another stock or another expiration. 49 vol is comparable with all of them. Two options at 49 vol, on any strike, stock or date, are priced on the same view of how much their stock moves. The dollar price is what you pay. The implied vol is what you are paying for. That is why a desk quotes in vol and treats the dollar price as a consequence.
How steep the line is: vega
The slope of that price-against-vol line has a name, vega: the dollars the option gains per vol point. It gets its own lesson in Module 5. It decides how precisely a price pins down its implied vol. An at-the-money option with a vega of 30 cents turns a 15-cent change in price into half a vol point. A far out-of-the-money option with a vega of 2 cents turns the same 15 cents into more than seven points. So wing options with tiny prices have noisy implied vols: a one-tick move in the bid or ask can swing the IV column by points. When a far strike's IV looks odd, check its vega before believing it.
Dollars per vol point for three strikes. At low volatility the far strikes have almost no vega, so a few cents of price move their implied vol by points. Drag days down and the far strikes' vega shrinks further.
Every option has its own
Run the backwards calculation on every strike and both sides, and you get a full column of implied vols. They are not one number. Strikes below the stock usually show a higher vol than the at-the-money strike, strikes above often show a lower one, and the call and put at the same strike land close together but rarely identical, because each has its own bid and ask. That column is the IV column on the chain, and reading it is most of what this course teaches. If one volatility were enough to describe a stock, the column would be flat. It never is, and the next lesson explains why.
The figure is annualized percent, so a 45 on a 30-day option and a 45 on a 90-day one are already on the same footing. Vol is the unit. Price is the consequence.
The maths: how implied volatility is solved
There is no formula that turns a price straight into a volatility, so every system searches. Two methods cover almost all of it. Take the example above with a 4% rate this time: S = K = 250, 30 days, market price $14.00.
Bisection. Start with a bracket you know contains the answer, say 20% to 80%. Price at the middle, 50%: $14.68, too high, so the answer is in 20% to 50%. Try 35%: $10.40, too low, so 35% to 50%. Then 42.5% gives $12.54, 46.25% gives $13.61, 48.1% gives $14.14. Each step halves the bracket, so after 20 steps it is a millionth of the starting width. Slow, but it cannot fail, because price always rises with vol.
Newton's method. Use the slope to jump. The update is v_new = v_old + (market price - model price) / vega, with vega in dollars per unit of vol (per 1.00, so 100 times the chain's per-point figure). Start at 40%: the model says $11.83 and vega is 28.49, so the step is (14.00 - 11.83) / 28.49 = 0.0763, landing on 47.63%. At 47.63% the model says $13.999 and the next step is 0.00002. Two steps and the answer is 47.63% to four decimals. Near the money, where the line is almost straight, Newton is fast. Far from the money, where vega is tiny and the line bends, a step can overshoot, so real systems start with Newton and fall back to bisection when a step leaves the bracket.
Note that the rate matters: at 0% the same $14.00 implied 49%, at 4% it implies 47.6%, because part of a call's price is the interest on the strike you have not paid yet.
Your turn
Open TSLA. Read the implied vol of the at-the-money call, then of a put a few strikes below the stock, and compare them. Then read the at-the-money vega and work out how far the call's price would have to move to shift its IV by one point. Each strike has been priced on its own, so the two vols can agree or sit points apart, and what the gap means is where the next lesson starts.