About the Black-Scholes Calculator
Estimate the fair value of a European call and put option with the Black-Scholes model, and their Greeks: delta, gamma, vega, theta and rho. Enter the share price, strike, days to expiry, volatility, interest rate and dividend yield.
The page opens with Example 15.6 from John Hull's Options, Futures, and Other Derivatives, so you can see it reproduce the textbook's call value of 4.76 and put value of 0.81 before entering your own numbers.
How it works
The model works out d1 = (ln(S ÷ K) + (r − q + σ² ÷ 2) × T) ÷ (σ√T) and d2 = d1 − σ√T, where S is the share price, K the strike, T the time in years, r the interest rate, q the dividend yield and σ the volatility. The call is worth S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), and the put K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1), where N is the normal distribution.
The Greeks show how the value changes: delta per 1 of share price, gamma as the change in delta, vega per 1 percentage point of volatility, theta per calendar day, and rho per 1 percentage point of interest. With Hull's other example (S = 49, K = 50, 20 weeks, 20% volatility, 5% interest) the call's delta is 0.522 and its vega 0.121 per point.
The normal distribution is computed to about fifteen decimal places, and 200 random options are checked against Python's own implementation.
Assumptions and limits
- European exercise: the option can only be exercised at expiry. American options on dividend-paying shares can be worth more.
- Volatility, interest rate and dividend yield are constant until expiry.
- Days are calendar days divided by 365; theta is per calendar day.
Worked examples
Hull's Example 15.6
Share 42, strike 40, six months, 20% volatility, 10% interest: call 4.76, put 0.81.
Hull's Greeks example
Share 49, strike 50, 20 weeks, 20% volatility, 5% interest: call 2.40, delta 0.522, theta −0.0118 a day.
Higher volatility
Raising volatility raises both the call and the put, because a wider range of outcomes makes both more valuable.
How to use the Black-Scholes Calculator
- Enter the option. Share price, strike and days to expiry.
- Add the market inputs. Volatility, interest rate and dividend yield.
- Read value and Greeks. For the call and the put.
Frequently asked questions
What is the Black-Scholes formula?
It values a European option from the share price, strike, time to expiry, volatility, interest rate and dividend yield. The call value is S·N(d1) − K·e^(−rT)·N(d2), adjusted for dividends, where d1 and d2 combine the inputs and N is the normal distribution.
Which volatility should I enter?
For a fair value, use the volatility you expect over the option's life. To compare with market prices, use the implied volatility shown on the option chain; entering it should give a value close to the option's market price.
What does delta mean?
Delta is how much the option's value changes when the share price rises by 1. A call delta of 0.52 means the call gains about 0.52 for each 1 rise; it is also used as a rough chance that the option ends in the money.
Why is my broker's price different?
Market prices reflect supply, demand and the volatility traders expect, and many listed stock options are American-style. Black-Scholes gives a theoretical value from your inputs, so differences usually come from the volatility used.
What is theta?
Theta is the change in the option's value as one day passes with everything else unchanged. It is usually negative for options you buy, because time value melts away as expiry approaches, and faster near the end.
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