Mathematics of the Financial Markets by Alain Ruttiens

Author:Alain Ruttiens
Language: eng
Format: epub, pdf
ISBN: 9781118513484
Publisher: Wiley
Published: 2013-03-31T16:00:00+00:00

and, obviously, the outcomes of these four situations are not at all comparable.

Regarding the European or American feature of an option contract, the difference will be considered later on (cf. Section 5.1). Meanwhile, let us agree that, later in this book, if it is not specified as American, the option will be supposed to be a European one.

The strike price of the option contract is contractually fixed at any price level, whatever it is, the current spot price, or the current forward price (corresponding to the option maturity), or any other price level. Of course, the choice of the strike will affect the option premium: the right to buy something @ $100 may not cost the same premium as the right to buy it @ $120 or @ $80.

In particular, if the strike corresponds to the current forward underlying price, the option is said to be at the money (ATM). Because many practitioners tend to use ATM when talking about a strike equal to the current spot underlying price, it is worth specifying ATMF (at the money forward) or ATMS (at the money spot) accordingly.1

Similarly, if an option strike price is more attractive (i.e., granting the right to buy cheaper, or to sell at a higher price) than the corresponding current forward price, the option is said to be in the money (ITM) and otherwise out of the money (OTM). We also use the acronyms DOTM and DITM for “deep” OTM or ITM, namely if the forward is very far from the strike price.

It is also important to notice that the meaning of option price does not refer to is exercise price, but refers to the option contract value, that is, the premium paid at the option contract inception, and later on, its revaluation. Given the crucial uncertainty for an option to be exercised or not, the option contract valuation is the core topic of this chapter.

As a first step in valuing an option contract, we must define its intrinsic value:

At option maturity T, the contract will be exercised or not, depending on the current spot underlying price level vis-à-vis the option strike: for a call option, the intrinsic value IV is 0 in case of no exercise, or the positive difference between the spot ST at T and the strike K:

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