How to Calculate European Option Greeks by Hand and in Python

Differentiate the Black–Scholes price with respect to spot, volatility, calendar time, and the rate. Evaluate the resulting formulas at the same inputs used for the price. State the units for each Greek. A mathematical vega of $37.52$ corresponds to about $0.3752$ in option value for a one-percentage-point volatility increase.

Let $\tau=T-t$, let $N$ be the standard normal CDF and $\phi$ its density, and assume zero dividends. Use

From $C=SN(d_1)-Ke^{-r\tau}N(d_2)$, the identity $S\phi(d_1)=Ke^{-r\tau}\phi(d_2)$ cancels the chain-rule terms in the spot derivative. Hence $\Delta_C=N(d_1)$. Put–call parity $C-P=S-Ke^{-r\tau}$ gives $\Delta_P=N(d_1)-1$, and differentiating either delta yields $\Gamma=\phi(d_1)/(S\sigma\sqrt{\tau})$ for both. Differentiating with respect to volatility gives $\nu=S\phi(d_1)\sqrt{\tau}$ for both.

Define $a=-S\phi(d_1)\sigma/(2\sqrt{\tau})$. Calendar theta is $\Theta_C=a-rKe^{-r\tau}N(d_2)$ for the call and $\Theta_P=a+rKe^{-r\tau}N(-d_2)$ for the put. Finally, $\rho_C=K\tau e^{-r\tau}N(d_2)$ and $\rho_P=-K\tau e^{-r\tau}N(-d_2)$. These expressions assume positive spot, strike, time to maturity, and volatility.

To see why call delta simplifies, differentiate each term in $C=SN(d_1)-Ke^{-r\tau}N(d_2)$. The derivative of the leading $S$ produces $N(d_1)$. The two CDF derivatives are proportional to the same $\partial d_1/\partial S=\partial d_2/\partial S$. Their coefficients cancel by the density identity above. Gamma then comes from differentiating the remaining $N(d_1)$. For the put you can use $C-P=S-Ke^{-r\tau}$ and differentiate the parity equation. It immediately tells you that put delta is call delta minus one and that their gammas and vegas agree.

Set $S=K=100$, $\tau=1$, $r=5\%$, and $\sigma=20\%$. The call and put deltas are approximately $0.6368$ and $-0.3632$. Both gammas are approximately $0.01876$ and both vegas are approximately $37.52$ for a unit change in decimal volatility. The call theta is about $-6.41$ per year, or $-0.0176$ per calendar day using a 365-day convention. The put theta is about $-1.66$ per year in this example. Multiply per-share Greeks by the contract multiplier to obtain exposures for one contract.

Change one input at a time in the options pricer if you want a visual check of the signs of delta and vega.