Seeing Black-Scholes
Yesterday's ₹30 premium was a number I gave you. Today you see where a number like that comes from, without a single equation until the very end.
Mission link: "explain theta to a friend without notes" and "pick a strike with a reason". You cannot do either until you can see what an option's price is.
The one idea
An option's price is the average of what it pays across every plausible future, weighted by how likely each future is, discounted to today.
That's it. Black and Scholes didn't invent a mysterious formula. They wrote down a specific way of drawing "every plausible future" (a fan of random paths whose spread grows with the square root of time), and then computed that average exactly instead of by simulation. The formula is a shortcut for the picture below. Learn the picture; the formula follows.
The picture
Reliance is ₹2,500. You're looking at the ₹2,600 call, 30 days out. Don't touch the sliders yet. Just read the two charts.
The first chart is forty guesses at the next 30 days. None of them is a forecast. Together they say: "Reliance will probably be somewhere in this band." The second chart is what happens if you run two thousand of those guesses and count where they end up. Most land near ₹2,500. A few land far away in either direction.
The shaded green part is the only part the call cares about. Every ending above ₹2,600 pays (ending − 2,600). Every ending below pays nothing. Average those payoffs across all 2,000 futures and you get the "average payoff" box. The model price next to it is the same thing computed exactly. They should be within a rupee or two of each other. The gap is simulation noise.
Why the model price is the fair premium. If a seller charges less than the average payoff, then over many trades they lose. If they charge more, buyers walk to another seller. Competition pins the premium near the average. Yesterday's ₹30 was roughly this number for a ₹2,600 call.
Now predict, then test
For each move below, write down (or say out loud) what happens to the model price and to the "chance in the money" before you drag the slider. Then drag it and check. Prediction before observation is what turns this from a demo into a thing you know.
- Days to expiry from 30 to 5. Watch the fan narrow. What happens to the shaded area?
- Days to expiry from 30 to 90. The fan widens. Why does a wider fan make the call worth more, even though the middle of the fan hasn't moved?
- Volatility from 25% to 50%. Same question, different cause.
- Strike from 2,600 to 2,500. The dashed line moves to the middle of the pile. Chance in the money should be close to half.
- Strike from 2,600 to 2,900. Almost nothing lands there. What does a nearly-worthless option cost?
- Switch to Put with strike 2,400. The shading flips to the left tail. Same machine, other side.
Set everything back to 2,500 / 2,600 / 30 days / 25% before the quiz.
Three things you should now be able to see
Theta is the fan shrinking. Every day that passes, the fan of futures narrows, fewer endings land above the strike, and the shaded area shrinks. The premium falls. Nothing about the stock changed. Only the amount of uncertainty left. That is all theta is.
Vega is the fan widening. Higher volatility means the model draws wilder paths. More of them cross the strike. A long option gains; a short option loses. Sellers sell when the fan is drawn too wide, buyers buy when it's drawn too narrow.
Out-of-the-money options are cheap because few futures reach them. The ₹2,900 call is a lottery ticket. The chance is real but small, and the price is the chance times the prize.
The formula, now that you can read it
For a call: C = S · N(d₁) − K · e−rT · N(d₂)
| Piece | In the picture | In words |
|---|---|---|
| N(d₂) | Share of the pile to the right of the strike | Chance the call finishes in the money |
| K · e−rT · N(d₂) | The strike, times that chance, in today's money | What you expect to pay for the shares |
| S · N(d₁) | The average value of the shares in the futures where you exercise, in today's money | What you expect to receive |
| C | Green area, discounted | Receive minus pay, on average |
N(d₁) is slightly bigger than N(d₂) because when you do end up exercising, the stock is not merely above the strike, it's on average well above, and that tilts the receive side up. If that sentence doesn't land yet, it doesn't matter. The picture is the understanding; the formula is bookkeeping.
What the model assumes, and where it lies. It assumes the fan is symmetric in log terms, no jumps, constant volatility, and that you can hedge continuously with no costs. Real markets crash more than they spike, so real put prices sit above the model's, which is why the "IV" you see on Dhan differs by strike. Traders don't use Black-Scholes to find the price. They use the market price to find the IV, then compare IVs. The model is a ruler, not an oracle.
Retrieve it
Sliders back to 2,500 / 2,600 / 30 / 25%. Answer from the picture in your head, not the one on screen.
Before the next lesson
Open Dhan's NIFTY chain. Pick the at-the-money call for the nearest Tuesday expiry, note its premium and IV. Put spot, strike, days and that IV into Zerodha's calculator. The model price should be close to the market premium. It's close because the IV was backed out of the market price, which is the point of the last aside. Then do the same for a strike 500 points out and notice how the IV differs.