Calculate theoretical values and Greeks for European-style options using the Black-Scholes model
Option Parameters
Option Type
$
Current underlying price
$
Option exercise price
Time until option expiration
%
Annual risk-free interest rate
%
Annual dividend yield
%
Annual volatility of underlying
Convert Trading Days to Years
Time to Expiry (Years):1.00
Option Pricing Results
Option Price
$10.45
Call Option
Intrinsic Value
$0.00
Amount option is in-the-money
Time Value
$10.45
Premium above intrinsic value
Moneyness
At-the-Money
Ratio: 1.00
Option Greeks
Δ
Delta
0.528
Rate of change with respect to underlying price
Γ
Gamma
0.017
Rate of change of Delta with respect to underlying price
Θ
Theta
-0.048
Rate of change with respect to time (per day)
V
Vega
0.391
Rate of change with respect to volatility (per 1% change)
ρ
Rho
0.486
Rate of change with respect to interest rate (per 1% change)
Price Visualization
About the Black-Scholes Model
The Black-Scholes model, developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, is a mathematical model for pricing European-style options. It assumes that the price of the underlying asset follows a geometric Brownian motion with constant volatility.
Black-Scholes Formula
Call Price:
C = S₀e⁻ᵈᵗN(d₁) - Xe⁻ʳᵗN(d₂)
Put Price:
P = Xe⁻ʳᵗN(-d₂) - S₀e⁻ᵈᵗN(-d₁)
Where:
d₁ = [ln(S₀/X) + (r - d + σ²/2)t] / (σ√t)
d₂ = d₁ - σ√t
S₀Current stock price
XStrike price
tTime to expiration (in years)
rRisk-free interest rate
dDividend yield
σVolatility of the underlying
N(x)Cumulative normal distribution function
Key Assumptions
European-style options (no early exercise)
Efficient markets with no transaction costs or taxes
Stock prices follow a lognormal distribution
Volatility and risk-free rate are constant over the option's life
No arbitrage opportunities exist
Note: The Black-Scholes model has limitations and may not accurately price options in all market conditions, particularly during periods of high volatility or market stress.
Understanding the Greeks
ΔDelta
Measures the rate of change of the option price with respect to changes in the underlying asset's price. Delta ranges from 0 to 1 for calls and -1 to 0 for puts. It represents the equivalent exposure in the underlying asset.
ΓGamma
Measures the rate of change of Delta with respect to changes in the underlying price. High Gamma means the Delta is highly sensitive to price changes in the underlying asset.
ΘTheta
Measures the rate of change of the option price with respect to time (time decay). Theta is typically negative for both calls and puts, as options lose value as time passes.
VVega
Measures the rate of change of the option price with respect to changes in volatility. Higher implied volatility generally increases option prices.
ρRho
Measures the rate of change of the option price with respect to changes in the risk-free interest rate. Call options generally increase in value when interest rates rise, while put options decrease.