An exotic option is described as follows:
Let $S_t$ be the underlying at $t$. The holder has the option to lock in the current price during the lifetime of the option, which he does for $S_{t}=50$. The strike price is $K=40$. At maturity, the option returns the payoff of a traditional European call or the intrinsic value at time $t$, whichever is greater. For example, if $S_T<50$ then the payoff is $10$. If $S_T>50$ then the payoff is the excess of the asset price over $50$.
I am quite confused, since by the description (if we ignore the last sentence) I understand the payoff to be simply the max of two calls: $\max(S_t-K,0)\textbf{1}_{S_T\le S_t}+\max(S_T-K,0)\textbf{1}_{S_T>S_s}$, where $\textbf{1}_{(\cdot)}$ is the indicator function.
After the last sentence, the payoff seems to be: $\max(\underbrace{S_t-K}_{=10},0) \textbf{1}_{S_T\le S_t} + \max(S_T-S_t,0) \textbf{1}_{S_T \gt S_t}$,
which is weird because e.g. for $S_T=55$ the payoff is $5$, i.e. less than $10$. If so, then the term "intrinsic value" refers to $S_T-S_t$ and not $S_t-K$.
Does this type of exotic option have a name? I tried searching "lock-in" option with no result so far. Is anyone familiar with what the text might be referring to and what the correct payoff is? Is there an error (contradiction) in the description?
