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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“All that we call ideal in Greek or any other art, because to us it is false and visionary, was, to the makers of it, true and existent.”
—John Ruskin (1819–1900)