Delta-Gamma Price Estimate

Key idea. Gamma is the rate of change of delta, so a delta-gamma estimate replaces the constant delta with one that grows linearly in $S$.

With $dS=2$, $\Delta=0.5$ and $\Gamma=0.03$,

Hence

Alternative. Delta rises roughly linearly from $0.5$ at $S=13$ to $0.5+2(0.03)=0.56$ at $S=15$. The option’s gain is the integral of delta over the move. For a linear delta this equals the average delta, $0.53$, times $dS=2$, which gives $1.06$ again.

A dollar-by-dollar update instead uses $\Delta=0.5$ for the first dollar and $0.53$ for the second. That gives $2.3+0.5+0.53=3.33$. It is a left-endpoint Riemann sum and so slightly underestimates the gain.