= Convex-payoff delta monotonicity in a binomial market
{title2=$\Delta_{r+1}(us)\geq\Delta_r(s)\geq\Delta_{r+1}(ds)$}
A <convex> terminal payoff has a convex pricing extension in the <binomial market>, because <risk-neutral pricing> is a positive weighted sum of rescaled copies of the payoff. Let the stock factors be $u,d$, the riskless gross factor be $R$, and $q=(R-d)/(u-d)$. The <replicating portfolio in a binomial market> satisfies
$$
\Delta_r(s)=\frac{qu}{R}\Delta_{r+1}(us)+\frac{(1-q)d}{R}\Delta_{r+1}(ds).
$$
The weights sum to one. Convexity orders the secant slopes on the adjacent intervals $[d^2s,uds]$ and $[uds,u^2s]$, proving the displayed inequalities. The inequalities need not be strict: an affine payoff has constant stock holding.
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