Question 1

What is the Black–Scholes formula?

Accepted Answer

For a European option on an asset with price S, strike K, time to expiry T, risk-free rate r, volatility σ and a continuous dividend yield q: call = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2) and put = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1), where d1 = [ln(S/K) + (r − q + σ²/2)·T] ÷ (σ·√T), d2 = d1 − σ·√T, and N is the standard normal cumulative distribution. Black and Scholes published it in 1973 for stocks without dividends; Merton added the dividend yield the same year. Rates, volatility and yield are entered as percents and compound continuously.

Question 2

Example

Accepted Answer

A stock at $42.00, a $40.00 strike, six months to expiry (T = 0.5), a 10% risk-free rate and 20% volatility, no dividends: d1 = 0.7693, d2 = 0.6278, call = $4.76, put = $0.81. These are the inputs in the form. With a 3% dividend yield the call falls to $4.28 and the put rises to $0.96. The rates here are example inputs, not market data.

Question 3

What do the Greeks mean?

Accepted Answer

Delta is the change in the option price for a $1 move in the stock (0.7791 for the example call). Gamma is how much delta changes for that $1 move (0.0500). Theta is the change as time passes — shown per year and per calendar day (the example call loses $0.0125 a day). Vega is the change for a 1 percentage-point rise in volatility ($0.0881), and rho the change for a 1 percentage-point rise in the interest rate ($0.1398 for the call). Vega per volatility point and theta per calendar day follow the Options Industry Council's conventions.

Question 4

What is implied volatility and how is it found?

Accepted Answer

It is the volatility that makes the model price equal to a market price. The model price rises steadily with volatility, so the calculator halves a range from 0.01% to 500% until it matches. For the call in the example priced at $4.76 it finds 20.01%. It works only when the price is between the option's lowest and highest possible values — a call must be worth more than max(S·e^(−qT) − K·e^(−rT), 0) and less than S·e^(−qT); a put more than max(K·e^(−rT) − S·e^(−qT), 0) and less than K·e^(−rT) — otherwise the calculator says so.

Question 5

What is put–call parity?

Accepted Answer

For European options on the same stock, strike and date, call − put = S·e^(−qT) − K·e^(−rT). The calculator shows both sides; they match to rounding. It is a check on the prices, and it means a European call and put with the same strike and date share the same implied volatility.

Question 6

How is time to expiry counted?

Accepted Answer

Choose days or years. Days are divided by 365 (calendar days), so 183 days is 0.5014 years; the theta per day also divides by 365. Traders sometimes use 252 trading days a year instead, which changes the numbers slightly — enter years directly if you prefer another basis.

Question 7

What does the model assume and leave out?

Accepted Answer

Constant volatility, interest rate and dividend yield; stock prices that follow a lognormal random walk; no transaction costs or taxes; European exercise at expiry only. American options — which most U.S. listed stock options are — can be exercised early, so their value can be higher, and early exercise is not covered. For known discrete dividends a common approach is to subtract their present value from the stock price instead of using a yield. The results are educational, not investment advice.

Black–Scholes calculator