Black-Scholes Option Pricing Calculator

Calculate theoretical prices of European call and put options

Option Parameters

Results

Call Option Price
-
The theoretical price to buy the asset at the strike price. Higher when the stock price is above strike (in the money).
Put Option Price
-
The theoretical price to sell the asset at the strike price. Higher when the stock price is below strike (in the money).
Delta (Δ)
-
Sensitivity of option price to changes in the underlying asset price. Calls: 0 to 1, Puts: -1 to 0.

Understanding the Black-Scholes Model

Variables
Formula
Interpreting Results
Stock Price (S)
Current market price of the underlying asset. As S increases, call options become more valuable and put options less valuable.
Strike Price (K)
Price at which the option holder can buy/sell the asset. Options with lower strike prices are more valuable for calls, and vice versa for puts.
Time to Expiry (T)
Time remaining until option expiration (in years). More time means higher chance to become profitable, increasing value.
Risk-Free Rate (r)
Theoretical return of a risk-free investment (like Treasury bonds). Higher rates increase call values and decrease put values.
Volatility (σ)
Annualized standard deviation of stock returns. Higher volatility increases both call and put values due to greater potential movement.
Dividend Yield (q)
Expected dividend payments during option life. Higher dividends decrease call values (stock price drops) and increase put values.

The Black-Scholes formula calculates the theoretical price of European options:

C = S₀e⁻ᵞᵀN(d₁) - Ke⁻ʳᵀN(d₂)
P = Ke⁻ʳᵀN(-d₂) - S₀e⁻ᵞᵀN(-d₁)

where:
d₁ = [ln(S₀/K) + (r - q + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T

N(x) = cumulative standard normal distribution

Where C is the call price, P is the put price, and other variables are as defined above.

Understanding Your Results

Option Price: The theoretical fair value of the option based on the inputs. Compare this to market prices to identify potential opportunities.

In-the-Money (ITM): For calls, when stock price > strike price. For puts, when stock price < strike price. ITM options have intrinsic value.

Out-of-the-Money (OTM): Opposite of ITM. These options have only time value.

Delta (Δ): Measures how much the option price changes for a $1 change in the stock price. Near 0.5 for at-the-money options.

Practical Implications

• Higher volatility increases option prices (more chance to be profitable)

• More time until expiration increases option prices

• Calls benefit from higher underlying prices, puts from lower prices

• Dividends reduce call values and increase put values