Black-ScholesBlack-Scholes #

This app calculates the option price and Greeks up to the 3rd order according to the Black-Scholes model. Under the hood it is using my blackscholes open source Python library. To learn more about the implementation and formulas, check out the full documentation. If you are serious about studying option theory check out Paul Wilmott on Quantitative Finance and Nassim Taleb's Dynamic Hedging.

Disclaimer: This tool is for educational purposes only. Nothing on this page constitutes financial advice, and all calculations are provided "as is" without any guarantee of accuracy. While we strive for correctness, you should independently verify results before using them for financial decisions.

Current price of the underlying asset. 55 = $55
Price at which the option can be exercised. 50 = $50
Time until expiration in years (1 = 1 year)
Annual risk-free interest rate (0.05 = 5%)
Annual volatility of the underlying asset (0.15 = 15%)
Annual continuous dividend yield (e.g., 0.02 = 2%)

Type: Call S: 55.00 K: 50.00 T: 1.00 years r: 5.0% σ: 15.0% q: 2.0%
The theoretical value of the option according to the Black-Scholes model.
Price

7.2055

Rate of change in option price with respect to the forward price (1st derivative).
Delta

0.8025

Rate of change in delta with respect to the underlying asset price (2nd derivative). Measures the convexity of the option value.
Gamma

0.0313

Rate of change in option price with respect to volatility. Measures sensitivity to volatility changes.
Vega

14.4975

Rate of change in option price with respect to time (time decay). The value is annualized - divide by 365 for daily decay.
Theta

-2.0296

Rate of change in option price with respect to the dividend yield. Also known as psi.
Epsilon

-44.1366

Rate of change in option price with respect to the risk-free interest rate.
Rho

36.9311

Sensitivity of delta with respect to change in volatility (cross derivative). Shows how delta changes with volatility.
Vanna

-1.3362

Rate of change of delta over time (delta decay). Also known as the second order time decay.
Charm

0.0626

Second order sensitivity to volatility. Shows how vega changes with changes in volatility.
Vomma

66.9081

Rate of change in vega with respect to time. Shows how volatility sensitivity changes with time.
Veta

9.1523

Second order partial derivative with respect to strike price. Used in Breeden-Litzenberger formula for estimating risk-neutral probabilities.
Phi

0.0379

Rate of change in gamma with respect to the underlying price. Third derivative showing how gamma changes with price.
Speed

-0.0040

Rate of change of gamma with respect to volatility. Shows how gamma changes with volatility.
Zomma

-0.0642

Rate of change of gamma over time. Shows how gamma changes as time passes.
Color

-0.0111

Third derivative of option value with respect to volatility. Shows sensitivity of vomma to volatility changes.
Ultima

-1043.8701

First derivative of option price with respect to strike price. Shows sensitivity to strike price changes.
Dual_Delta

0.7386

Rate of change in dual delta with respect to the strike price (2nd derivative).
Dual_Gamma

0.0379

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