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


Call Delta=P×N(d1)\begin{aligned} & Call \ Delta = P \times N(d_1) \end{aligned}
Put Delta=P×(N(d1)1)\begin{align*} & Put \ Delta = P \times (N(d_1) - 1) \end{align*}


Gamma=P×N(d1)F×σ×T\begin{align*} & Gamma = \frac{P \times N'(d_1)}{F \times \sigma \times \sqrt{T}} \end{align*}


Theta=P×F×N(d1)×σ2×T\begin{align*} & Theta = -P \times \frac{F \times N'(d_1) \times \sigma}{2 \times \sqrt{T}} \end{align*}


Vega=P×(F×T×N(d1))\begin{align*} & Vega = P \times (F \times \sqrt{T} \times N'(d_1)) \end{align*}
d1=ln(FK)+(σ2/2)TσT\begin{align*} & d1 = \frac{\ln(\frac{F}{K}) + (\sigma^2 / 2)T}{\sigma\sqrt{T}} \end{align*}
d2=ln(FK)(σ2/2)TσT=d1σT\begin{align*} & d2 = \frac{\ln(\frac{F}{K}) - (\sigma^2 / 2)T}{\sigma\sqrt{T}} = d1 - \sigma\sqrt{T} \end{align*}
N(x)=Cumulative Normal Distribution Function\begin{align*} & N(x) = \text{Cumulative Normal Distribution Function} \end{align*}
N(x)=ex222π\begin{align*} & N'(x) = \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}} \end{align*}

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