The Greeks – A Practical Approach

Demetri Papacostas and Francesco Tonin

Time to dive deep again, and put a more definitive structure around the way in which the premium changes as different market parameters move. We briefly discussed the first derivative of the option price with respect to movements in the forward.

$$\text{Recall: }{d}_{\text{1}}=\frac{\log \left(\frac{S}{K}\right)+\left(\left(r_d-r_f\right)+\frac{{\sigma }^2}{2}\right)\cdot t}{\sigma \sqrt{t}}$$

The change in the option premium divided by the corresponding change in the forward rate is called the delta (δ, lowercase, or Δ for capital) and indicated by N(d₁). It is also the slope of the option price when plotted as a function of the forward rate. The second derivative is called gamma (γ, Γ), and it is also the change in the slope of the option price when plotted against the forward rate, then the change in option price with respect to one interest rate is rho (ρ, P) and with respect to the other rate is phi (ϕ, Θ). Got that?

Don’t worry, we will review what is important gradually, starting with why these derivatives are called “the greeks”. It is just that Greek letters are used to represent these derivatives. Therefore, when speaking about the derivatives we will use lowercase, “g” for greeks. Having