6European Options

In this chapter, we will study in detail a specific form of a financial contract or conditional asset called a European option. Section 6.1 defines a European option and its characteristics. Section 6.2 introduces the concept of a complete financial market and its characterization in terms of a martingale. Section 6.3 resolves the problem of the valuation and hedging of European options in a general framework. Section 6.4 applies these results in a particular case of the Cox, Ross and Rubinstein model. Finally, the chapter ends with exercises in section 6.5 and practical work in section 6.6.

We still assume that Ω has a finite cardinal and that ℙ(ω) > 0 for any ω ∈ Ω. Let N be a time horizon and let (imagesn)0≤nN be a filtration such that images0 = {∅, Ω} and imagesN = images.

6.1. Definition

We start by writing the definition of a European option and giving the two chief examples: the put options and call options.

DEFINITION 6.1.– A European option with maturity N is a random XN ≥ 0 imagesN-measurable ...

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