Stochastic Finance, 4th Edition

5.3European contingent claims

A key topic of mathematical finance is the analysis of derivative securities or contingent claims, i.e., of certain assets whose payoff depends on the behavior of the primary assets S0, S1, . . . , Sd and, in some cases, also on other factors.

Definition 5.19. A nonnegative random variable C on (Ω,FT , P) is called a European contingent claim. A European contingent claim C is called a derivative of the underlying assets S0, S1, . . . , Sd if C is measurable with respect to the σ-algebra generated by the price process (St)t=0,...,T.

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A European contingent claim has the interpretation of an asset which yields at time T the amount C(ω), depending on the scenario ω of the market evolution. T is called the expiration ...