Implied volatility is the volatility figure that makes a pricing model return the price an option is actually trading at. Delta, gamma and theta are three of the Greeks; vega and rho are the other two. Together they state how that model price moves when one input moves and the rest stay still.
Definition and Formula
The Black-Scholes model takes six inputs. Four are ordinary numbers: the underlying price, the strike, the time to expiry and an interest rate. Two more complete the list: a dividend yield and a volatility figure. It returns one number, a theoretical price. Five of the six inputs can be read off directly. Volatility cannot.
Implied volatility is that missing input, worked backwards. The market price of the option is taken as given, and the volatility figure that would produce it is searched for. There is no direct way to solve for it. So the search is done by narrowing a range until the model price and the market price agree.
The Greeks are the rates of change of the model price. Delta states the change for a one-rupee move in the underlying. Gamma states how much delta itself moves. Vega states the change for a one-point move in implied volatility. Rho does the same for the interest rate.
Theta is stated per day on this site. The yearly figure is divided by 365, which changes the number by a factor of several hundred. A theta quoted without its unit is not a measurement at all.
Each of these numbers belongs to the model, not to the market. The model assumes exercise only at expiry, volatility that stays constant, and returns that follow a particular distribution. None of those hold exactly, which is why the output is called theoretical.
Formula: T = days ÷ 365 · d1 = [ln(spot ÷ strike) + (r − q + σ² ÷ 2) × T] ÷ (σ × √T) · d2 = d1 − σ × √T. Call = spot × e^(−q×T) × N(d1) − strike × e^(−r×T) × N(d2)
A Worked Example
| Underlying | 24,280.00 |
|---|---|
| Strike | 24,300.00 |
| Days to expiry | 7 |
| Implied volatility | 12.5% |
| Risk-free rate | 6.5% |
| Theoretical price | 172.83 |
|---|---|
| Delta | 0.5132 |
| Gamma | 0.000949 |
| Theta (per day) | -14.1585 |
| Vega | 13.4067 |
| Rho | 2.3566 |
Each figure in the result column is computed by the same code that runs the Option Greeks calculator on this site. It uses the five inputs above.
What This Number Tells You
Implied volatility states what volatility figure the current price of the option corresponds to under this model. It is a restatement of the price in another unit.
Each Greek states a rate of change with everything else held still. That condition is useful for reading one input at a time. But it is one the market never actually meets.
And What It Does Not Tell You
Implied volatility is not a forecast of how much the underlying will move. It is the model input consistent with today’s price, and no more.
The Greeks are not a statement about what will happen. Holding each other input still is an assumption made for the arithmetic, not a description of a trading session.
A theoretical price differing from the traded price does not mean either one is wrong. It means the market is not the model, which is true of any option at any time.
Frequently Asked Questions
Why is implied volatility searched for instead of calculated?
Because there is no closed formula running backwards from price to volatility. The range is narrowed until the model price matches the price supplied. If no figure in the range fits, this site says so instead of printing a number.
Is theta per day or per year?
Per day on this site. The yearly figure is divided by 365 to get there. The two differ by a factor of several hundred, so a theta without its unit says nothing.
Why can the model price differ from the traded price?
Because the model’s assumptions are never met exactly: exercise only at expiry, constant volatility, and a particular distribution of returns. The difference is expected, not an error.
What happens at zero days to expiry?
The term σ × √T becomes zero and d1 has to be divided by it, so gamma and theta come out as infinity. That is not an answer, which is why zero days is not accepted.