= Immediate entry for a Bermudan forward-start call
{title2=$B_i(S)=S\max_{j\geq i}c(T-T_j)$}
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.
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