Prove Lemma 6.20. Lemma 6.20 Let (S, D) be a couple of price-dividend processes. The following are

Prove Lemma 6.20.

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Lemma 6.20 Let (S, D) be a couple of price-dividend processes. The following are equivalent: (i) the couple (S,D) admits an arbitrage opportunity; (ii) there exists t = {0,1,.... T 1} such that the couple (S1, P1+1) admits an arbitrage opportunity in the sense of Definition 4.17 in the period [t,t + 1]. According to the above lemma and recalling the event tree structure described at the beginning of the present chapter, there are no arbitrage opportunities in the multi-period economy if and only if there is no arbitrage opportunity (in the sense of Definition 4.17) in every elementary single-period sub-tree. This observation turns out to be useful for characterizing the absence of arbitrage in a multi-period setting, as shown in Theorem 6.23 below. As a preliminary, let us define the notion of risk neutral probability measure, extending Definition 4.21 to the present setting.