# Black-Scholes Option Pricing Calculator

This calculator finds an estimate for the fair value of a European put or call option using the Black-Scholes options pricing model.

Input the spot price, the strike price, the expiration date, the volatility, the risk-free interest rate, and the dividend yield if it exists, then press `Calculate`.

35.71
0.91
Call option Delta
0.926
Put option Delta
-0.074
Option Gamma
0.0053

Call Option Theta
-0.165
Put Option Theta
-0.111
Call Option Rho
0.112
Put Option Rho
-0.011
Option Vega
0.092

## Black-Scholes Assumptions

The Black-Scholes model holds the following assumptions:

• The option is European and can only be exercised at expiration
• The returns of the underlying asset are normally distributed
• Markets are random, in that market movements cannot be predicted
• No dividends are paid out during the life of the option
• No transaction costs for buying the option
• Risk-free interest rate and volatility of the underlying asset are known and constant

The original model does not account for dividends paid during the option’s lifetime. Still, there are versions of the model that account for dividends by determining the ex-dividend date value of the underlying asset. The model has also been adapted for the effect of options that can be exercised before expiration.

## The Option Greeks Explained

Option Greeks are a set of quantities representing an option’s price sensitivity to its underlying parameters.

### Delta (Δ)

Delta is a measure of the sensitivity of an option’s price changes to changes in the value of the underlying security. In other words, it identifies how much the option’s price may change if the underlying price changes by `\$1`.

For example, if an option has a Delta of `.40`, the price would increase by `\$0.40` if the underlying increases by `\$1.00`.

Delta estimates the probability of expiring in the money. For example, an option with a Delta of `.40` has a `40%` chance of expiring in the money. The lower the Delta, the lower the probability that the option will expire in the money.

Delta does not have a constant rate of change and grows as an option moves further in the money and shrinks as an option moves further out of the money. The greek Gamma determines the rate of change.

### Gamma (Γ)

Gamma represents Delta’s rate of change relative to the price of the underlying security.

For example, if an option has a Delta of `.40` and a Gamma of `0.05`, the price will change by `\$0.40` with the first `\$1.00` move in the underlying. The option price change with the next dollar move is Delta + Gamma to find the new Delta: `\$0.45`.

### Theta (ϴ)

Theta represents the rate of time decay of an option. It estimates how much value an option loses with each passing day. If an option has a Theta of `0.05`, the option price drops by `\$0.05` every day.

### Vega (V)

Vega represents an option’s sensitivity to volatility. It estimates how much the option price changes with each percentage point change in implied volatility. If an option has a vega of `0.04` and the implied volatility decreases one percentage point, the option price will decrease by `\$0.04`.

External factors like financial news or political conditions can cause a spike in implied volatility.

The further an option’s expiration date is, the higher Vega will be. In other words, options with more time to expiry may react more to a change in volatility.

### Rho (ρ)

Rho represents how sensitive the price of an option is relative to interest rates. Interest rates change slowly and thus have a small impact on options trading compared to the other Greeks.

For further reading on deriving the Options Greeks from the Black-Scholes options pricing formula, go to the article:

If you want to calculate the Implied Volatility for a priced option, go to the calculator:

For further reading on how to write the Black-Scholes options pricing formula in Python and R, go to the articles: