Immediate entry for a Bermudan forward-start call ($B_i(S)=S{\max }_{j\ge i}c(T-T_j)$)

Let $c(u)$ be the Black-Scholes formula value divided by stock price for an at-the-money call of duration $u$. On a discrete set of possible entry dates, homogeneous function and the discounted stock martingale reduce the Snell envelope to $b_i=\max (c(T-T_i),b_{i+1})$, with zero terminal value. Thus the displayed maximum is exact. When $r>0$ and volatility is positive, $c$ increases with maturity, so immediate entry is optimal and the initial value equals the ordinary at-the-money European call option. The maximum recursion remains useful if a different parameter regime loses that monotonicity.