Discrete Dupire equation
= Discrete Dupire equation
{title2=$\Delta_TC=\tfrac12\sigma^2\Delta_K^2C$}
For a <martingale> with increments in $\{-1,0,1\}$, <European call option> prices satisfy $C(T+1,K)-C(T,K)=\frac12\sigma^2(T,K)(C(T,K+1)-2C(T,K)+C(T,K-1))$. At positive-probability states, the <variance> and the upward/downward conditional transitions are determined by this identity; transitions at zero-probability states cannot be recovered.