Deriving Greeks

Deriving Greeks

The Greeks of OLP and the Greeks of specific options positions are also calculated using the Black-76 model. Options' Greeks are used for calculating Execution Price of options, determining the Bid/Ask implied volatility, and monitoring OLP risk.

P : Size and directionality of the requested option position by the trader (Long : + / Short : -)

F : Futures price of the underlying asset adjusted for the expiration date

σ : log-normal with constant volatility

r : Risk-free interest rate

K : Strike price

T : Time to maturity

Delta

$$\begin{aligned} & Call \ Delta = P \times N(d_1) \end{aligned}$$

$$\begin{align*} & Put \ Delta = P \times (N(d_1) - 1) \end{align*}$$

Gamma

$$\begin{align*} & Gamma = \frac{P \times N'(d_1)}{F \times \sigma \times \sqrt{T}} \end{align*}$$

Theta

$$\begin{align*} & Theta = -P \times \frac{F \times N'(d_1) \times \sigma}{2 \times \sqrt{T}} \end{align*}$$

Vega

$$\begin{align*} & Vega = P \times (F \times \sqrt{T} \times N'(d_1)) \end{align*}$$

$$Where,$$

$$\begin{align*} & d1 = \frac{\ln(\frac{F}{K}) + (\sigma^2 / 2)T}{\sigma\sqrt{T}} \end{align*}$$

$$\begin{align*} & d2 = \frac{\ln(\frac{F}{K}) - (\sigma^2 / 2)T}{\sigma\sqrt{T}} = d1 - \sigma\sqrt{T} \end{align*}$$

$$\begin{align*} & N(x) = \text{Cumulative Normal Distribution Function} \end{align*}$$

$$\begin{align*} & N'(x) = \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}} \end{align*}$$