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Valuation in Incomplete Markets

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5 1.2 Existence of Equivalent Martingale Measure The central and most important principle in any market model, is the no-arbitrage condition. Now we will define the mathematical part of this economic principle. Definition (1.2.1) Let i ) )be a set of self-financing strategies. A strategy i Μ )is called an arbitrage opportunity or arbitrage strategy with respect to )if ⊥  001PV Μ , and the terminal wealth of Μ satisfies ⊥  01PV T Μ τ and ⊥  00PV T Μ ! !. So an arbitrage opportunity is a self-financing strategy with zero initial value, which produces a non-negative final value with probability one and has a positive probability of a positive final value. Arbitrage opportunities are always defined with respect to a certain class of trading strategies. Definition (1.2.2) We say that a security market M is arbitrage-free if there are no arbitrage opportunities in the class )of trading strategies. For example we can use this case. We observe a realization ,St Ζof the price process St. We want to know which sample point Ζ : we have. Information about : is captured in the filtration ⊥  t FF . In this setting we can switch to the unique sequence partitions ⊥  t P corresponding to the filtration ⊥  t F . So at time t we know the set tt A Pwith t A Ζ . Now recall the structure of the subsequent partitions. A set t A P is the disjoint union of sets 12 1 , ,..., kt AA A P . Since Suis u F -measurable St is constant on A and 1St is constant on the k A , 1,2,..., K . So we can think of A as the time 0 state in a single-period model and each k A corresponds to a state time 1 in the single-period model. We can therefore think of a multi-period market model as a collection of consecutive single-period markets. This is the effect of a “global” no-arbitrage condition on the single-period markets.

Anteprima della Tesi di Luca Cassani

Anteprima della tesi: Valuation in Incomplete Markets, Pagina 7

Tesi di Laurea

Facoltà: Scienze economiche statistiche e sociali

Autore: Luca Cassani Contatta »

Composta da 129 pagine.

 

Questa tesi ha raggiunto 760 click dal 05/05/2005.

 

Consultata integralmente 6 volte.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.