= Solution
{c}
At an entry date the holder chooses between entering the at-the-money call immediately and retaining the right to choose at the next date. The <Snell envelope> therefore gives, under <risk-neutral valuation>,
$$
B^{FSC}(T_i,s)=\max\left\{s\,c(T-T_i),
e^{-r(T_{i+1}-T_i)}
\mathbb E[B^{FSC}(T_{i+1},S_{T_{i+1}})\mid S_{T_i}=s]\right\},
$$
with terminal value zero if entry has never occurred.
<homogeneous function> makes $B^{FSC}(T_i,s)=s b_i$. The discounted stock <expected value> is $s$, so the recursion reduces to
$$
\boxed{b_{N+1}=0,\qquad
b_i=\max\{c(T-T_i),b_{i+1}\}
=\max_{j\geq i}c(T-T_j).}
$$
In the positive-rate case in the hint, put $\gamma=r/\sigma$ and $\beta=\sigma/2$. Then $c(u)$ is exactly its increasing function. The largest remaining maturity is $T-T_i$, so $b_i=c(T-T_i)$: immediate entry is optimal at every date while unentered.
At time zero this yields
$$
\boxed{B^{FSC}(0,S_0)=S_0c(T)
=C(0,S_0;T,S_0).}
$$
Thus the Bermudan choice of start date adds no value over the ordinary at-the-money <European call option> in this setting. The maximum formula remains the general recursion; using monotonicity to select immediate entry uses the rate assumption supplied with the hint. This is <immediate entry for a Bermudan forward-start call>.
Back to article page
