Value of A Call
This example leads to the following formal reasoning. Fix an underlying financial instrument. Let be a call option for this instrument, purchased at time, expiring at time, with exercise (strike) price ; and let be the price of the underlying instrument.
Assume the owner of the option, wants to make no loss, and does not want to actually possess the underlying instrument, . Then either (i) the person will exercise the option and purchase, and then immediately sell it; 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 call option at expiry, is
which is also written or .
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