Greeks vs. Time to Maturity

Here we are inspecting what happens with the \(\Delta-S\) curves when we let time to maturity change.

• Delta:

  • At maturity, \(\Delta\) has a digital shape around the strike (0 or 1); the further we are from expiry, the flatter the \(\Delta\!-\!S\) curve becomes (see photo for what do I mean by "flatter").

• Gamma:

  • Given what we said above, its peak (for ATM) rises as maturity approaches while gamma for OTM/ITM decreases. As time to maturity increases the opposite happens. This is clear given the insights from delta.

• Theta:

  • Time decay \(|\Theta|\) grows larger as we near maturity/expiry. This holds especially for ATM options.

• Vega:

  • \(\mathrm{Vega}\) weakens as expiry approaches but grows with longer \(T\). We note that both \(\mathrm{Gamma}\) and \(\mathrm{Vega}\) peak ATM, though \(\mathrm{Gamma}\) strengthens near maturity while \(\mathrm{Vega}\) weakens.

• Rho:

  • \(\rho\) increases (respectively decreases for puts) with time to maturity. Pretty intuitive.