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The Notion of Stochastic Time

The notion of stochastic time has gained traction amongst both practitioners and academics over the past decade thanks mostly due to introduction of financial derivatives such as Timer option, etc. In these, "maturity" or "time" in a Black-Scholes world - if such a world is indeed Black-Scholes - is random. The model explicitly assumes time to be stochastic (the maturity of such options can happen anytime) and is mathematically modeled in that way. However, the notion of stochastic time has been with us for a long time now. For example, even though while pricing a barrier option (knock-in or knock-out) we do not explicitly assume - and, mathematically model - time to be stochastic (random), in essence time is stochastic in the case of barrier options. That is why many in the good old days of options trading would call such options "Sudden Birth and Death" options. In the life insurance industry such "sudden birth and death" options are also very common; an example of a sudden-death option in insurance is the Guaranteed Minimum Death Benefit (GMDB).

Reference: The Handbook of Exotic Options by Israel Nelken, Irwin Publishing, 1996

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