Payoff of A Put
These examples lead to the following formal reasoning. Fix an underlying financial instrument. Let be a put option for this instrument, purchased at time, expiring at time, with exercise (strike) price of ; and let be the price of the underlying instrument.
Assume the owner of the option, wants to not take a loss, and does not want to actually possess the underlying instrument, . Then either (i) the person will purchase at expiry, and then immediately exercise the selling option; or (ii) the person will not exercise the option (which subsequently becomes worthless).
In (i), the pay-off would be ; in (ii) the pay-off would be . So if (i) or (ii) occurs; if then (ii) occurs.
Hence the pay-off, i.e. the value of the put option at expiry, is
which is alternatively written or .
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