7. Options: General Properties................................
In Chapters 1 and 4 we have seen simple examples of call and put options
in a one-step discrete-time setting. Here we shall establish some fundamental
properties of options, looking at them from a wider perspective and using con-
tinuous time. Nevertheless, many conclusions will also be valid in discrete time.
Chapter 8 will be devoted to pricing and hedging options.
7.1 Definitions...............................................
AEuropean call optionis a contract giving the holder the right to buy an asset,
called theunderlying,forapriceXfixed in advance, known as theexercise price
orstrike price,at a specified future timeT, called theexerciseorexpiry time.
AEuropean put optiongives the right to sell the underlying asset for the strike
priceXat the exercise timeT.
AnAmerican callorput optiongives the right to buy or, respectively, to
sell the underlying asset for the strike priceXat any time between now and
a specified future timeT, called theexpiry time. In other words, an American
option can be exercised at any time up to and including expiry.
The term ‘underlying asset’ has quite general scope. Apart from typical
assets such as stocks, commodities or foreign currency, there are options on
stock indices, interest rates, or even on the snow level at a ski resort. Some
underlying assets may be impossible to buy or sell. The option is then cleared
in cash in a fashion which resembles settling a bet. For example, the holder of
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