Gamma

Gamma

The Curvature

Gamma is the exact change in delta per change in the underlying.

highlighted = computed this step

Gamma is the curvature

Gamma measures how fast delta itself changes: the change in delta over the change in the underlying, between the two first-step nodes. It is a discrete second difference of the option value.

Γ=Δu−ΔdSu−Sd\Gamma=\frac{\Delta_u-\Delta_d}{S_u-S_d}

Gamma at the root

Delta goes from 1 share at the up node to 1/9 share at the down node, over underlying $120.00 and $90.00. The exact gamma is 1/3375 per cent of the underlying.

Γ=Δu−ΔdSu−Sd=1/3375\Gamma=\frac{\Delta_u-\Delta_d}{S_u-S_d}=1/3375
Binomial treeStock and call values are recomputed at every node.todaydownupS $100.00V $10.20S $90.00V $1.43S $120.00V $20.00S $81.00V $0.00S $108.00V $3.00S $144.00V $39.00

Model note

Gamma here is a discrete curvature in the stated binomial model, not a continuous-time derivative. This is descriptive, not investment advice.

discrete curvature, stated model\text{discrete curvature, stated model}