Mathematical finance: continuous time by Ewald C.-O.

By Ewald C.-O.

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This is reasonable at least, since at real world markets, trading only happens in discrete 1,1 time steps. We denote this market with MBS . 1 we ∗ defined a new measure P on (Ω, F) as 42 P∗ (A) := EP (ZT · 1A ) = Ω ZT · 1A 1 2 where Zt := eθWt − 2 θ t and θ := r−b . Since ZT > 0 it is clear that P and σ ∗ P are equivalent measures. 17) is a Brownian motion with respect to the measure P∗ . Clearly W0∗ = W0 = 0 P- and hence also P∗ -almost sure. Also Wt∗ −Ws∗ = Wt −Ws −θ(t− s) is independent of Fs .

2 ). In this way, martingale measures should be interpreted as pricing systems. They should be considered as ( linear ) functionals on the space of contingent claims with values in the space of fair prices. The interesting thing is, that is some cases if |P(Mm,n )| > 1 not all of them give the same prices. It might be, that one equivalent martingale measure prices a contingent claim more expensive than another one, but 40 still both prices are fair. Nevertheless the question arises, which of the equivalent martingale measures one should use to price contingent claims.

Let X k be a sequence of stochastic processes. We say X k converges ucp ( uniformly on compacts in probability ) if 55 P ∀t ∈ I : sup0≤s≤t |Xsk − Xs | /0 :⇔ Xk ucp /X The following proposition shows that we can get any lcrl process as an ucp limit of elementary processes. 1. Let X be an lcrl adapted process. Then there exists ucp /X . a sequence X k of elementary processes, such that X k Proof. As before we define an increasing sequence of stopping times as σn := inf{t : |Xt | ≥ n}. We have limn→∞ σn = ∞ and hence =0 lim P( sup |Xsσn − Xs | ≥ ǫ) ≤ n→∞ 0≤s≤t lim P( sup |Xsσn − Xs | ≥ ǫ) n→∞ 0≤s≤σn =0 + ≤ lim P( sup |Xsσn − Xs | ≥ ǫ) n→∞ σn ≤s≤t lim P(σn ≤ t) = 0 n→∞ ucp / X and all X n are bounded processes.

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