Calculators
Black-Scholes & Greeks
Price a European option and see all its greeks update live, with a chart of value and delta versus the underlying price.
≈ how much the price moves per $1 move in the underlying
how fast Delta itself changes as the underlying moves
price change per 1-point (1%) rise in volatility
value lost each day as time ticks by (time decay)
price change per 1% rise in the interest rate
Option value vs. spot price
Black-Scholes estimates a fair price for a European option, the smooth curve, while the dashed line shows its intrinsic value, what it would be worth if it expired right now. The gap between them is time value: extra worth from the chance the price moves in your favour before expiry. One big caveat: the model assumes volatility stays constant and prices move in a smooth, bell-shaped (lognormal) way, so real-market prices routinely drift from what it predicts.
Learn how it works
Five worked examples. Read a couple before you dive in, try to answer first, then reveal the solution.
At-the-money call: price and every Greek explained
Inputs: spot S = $100, strike K = $100, time to expiry T = 1 year, risk-free rate r = 5%, volatility σ = 20%. Type: CALL. Find the fair price and the main Greeks.
Show solution
Price ≈ $10.45.
- Delta ≈ 0.64, the option moves about 64 cents for every $1 the stock moves. Loosely, it also implies a ~64% chance of finishing in the money.
- Gamma ≈ 0.0188, delta itself changes by about 0.019 per $1 stock move, so the option speeds up as the stock rises.
- Vega ≈ 0.375, each 1 percentage-point rise in volatility (20% → 21%) adds about 37.5 cents.
- Theta ≈ −$0.018 per day, with nothing else changing, time decay bleeds about 1.8 cents a day (≈ −$6.41 over a whole year).
- Rho ≈ 0.532, a 1 percentage-point rise in rates (5% → 6%) adds about 53 cents.
Because the stock and strike are equal, none of the $10.45 is intrinsic value, it is all time value, the price of one year of uncertainty.
Out-of-the-money call is cheaper with a smaller delta
Same as before but the strike is raised to K = $110 (S = $100, T = 1, r = 5%, σ = 20%, CALL). How do the price and delta compare with the at-the-money call?
Show solution
Price ≈ $6.04, delta ≈ 0.45.
The stock has to climb $10 just to reach the strike, so this call is worth less than the $100-strike call ($10.45). Its delta is lower too (0.45 vs 0.64): it moves only about 45 cents per $1 stock move and has a smaller (~45%) chance of finishing in the money. Cheaper option, but the stock has more work to do before it pays off.
In-the-money put
Spot S = $100, strike K = $110, T = 1, r = 5%, σ = 20%. Type: PUT. A put whose strike is above the stock is already in the money. Find the price and delta.
Show solution
Price ≈ $10.68, delta ≈ −0.55.
Because you could sell the stock at $110 while it trades at $100, the put already holds $10 of intrinsic value (110 − 100). The remaining ≈ $0.68 is time value. The delta is negative (−0.55): a put gains when the stock falls, so a $1 drop in the stock adds about 55 cents to the put.
Raise volatility to 40% and the call nearly doubles
Take the at-the-money call (S = $100, K = $100, T = 1, r = 5%, CALL) and lift volatility from σ = 20% to σ = 40%. What happens to the price?
Show solution
Price jumps from ≈ $10.45 to ≈ $18.02, a rise of about 72%, close to double.
This is vega at work: options love volatility, because a wilder stock has a better shot at a big favorable move while the loss stays capped at the premium paid. Doubling σ nearly doubles what buyers will pay. (It is not exactly proportional because vega itself shifts as volatility changes.)
Time decay: a shorter expiry is worth less
Take the at-the-money call (S = $100, K = $100, r = 5%, σ = 20%, CALL) but shorten the time to expiry from T = 1 year to T = 0.25 year (3 months). What happens to the price?
Show solution
Price falls from ≈ $10.45 to ≈ $4.61.
With only a quarter of the time, there is far less room for a big move, so the option is worth less than half as much. Every cent of it is time value (the strike equals the stock), and that is exactly what theta grinds away day by day. Less time on the clock → less optionality → lower price.
What you'll learn
How an option's price and its greeks respond to spot, strike, time, rate and volatility, the core intuition behind options trading.