Is there a name for an option whose value is determined by a time difference?

I mean a derivative whose contract reads something like, "If stock $X$ goes below $Y$ at time $T_1$, and $T_1$ is before $T_0$, pay out $c\times(T_0-T_1)$".

The exposure is similar to that of an annuity, but different enough that I wondered whether it had a name of its own.

  • $\begingroup$ Is $T_1$ a stochastic time before $T_0$, or does the payoff depend on the stock going below $Y$ at any date up until $T_1$, with $T_1$ a fixed date? $\endgroup$ Sep 20 at 9:27
  • $\begingroup$ @Kermittfrog I should have been clearer about the parameters. The contract specifies $X$, $Y$, $T_0$ and $c$. If such a $T_1$ occurs, then there's a payout. This is the latter formulation you suggest. $\endgroup$
    – Volume Man
    Sep 20 at 15:55
  • $\begingroup$ Did you see such an option? Never heard of it, but would like to know what problem this would solve (what purpose the product has). $\endgroup$
    – AKdemy
    Sep 22 at 19:39

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