 Binomial call-price recursion

Let a binomial options pricing model multiply the stock price by $u$ or $d$ in each period, with $0<d<R<u$ and bank account gross return $R$. The unique one-step risk-neutral probability is $q=(R-d)/(u-d)$. If $C_N(s,K)$ is the time-zero price of an $N$-period call option starting at $s$ with strike $K$, thenThis is conditional expectation after the first move, with one period of discounting. The terminal payoff's positive homogeneity gives $C_N(as,K)=a{C}_N(s,K/a)$ for $a>0$, henceThe coefficients are positive because the no-arbitrage inequalities put $q$ strictly between zero and one. This strike-rescaling recursion is useful when the shorter-maturity call curve is available. The physical up probability does not enter the recursion.