The Mathematics of Arbitrage

212 10 Counter-Example ∫ {σ<∞} Lσ∧τ= ∫∞ 0 P[σ∈dt]E[Lτ∧t] =P[σ<∞]. The first line follows from the independence ofσand the ...

10.3 Incomplete Markets 213 Proof.The proof is broken up in different lemmata. LetW′be the martin- gale component in the Doob-Me ...

214 10 Counter-Example We takeu 1 small enough so thatR[νu 1 <∞]> 1 −ε 1. From the definition ofνu 1 it follows that〈N, N〉 ...

10.3 Incomplete Markets 215 1 n ∑n k=1 ( logXk−ER [ logXk ∣ ∣ ∣Fνuk− 1 ]) → 0. On the set{νuk− 1 < ∞}we have thatER [ logXk ∣ ...

216 10 Counter-Example (4) two processes β^1 and β^2 that are Brownian motions with respect to ( ̃Ft)t≥ 0 and such that〈β^1 ,β^2 ...

11 The No-Arbitrage Property under a Change of Num ́eraire (1995) Abstract.For a price process that has an equivalent risk neutr ...

218 11 Change of Num ́eraire arbitrage stands for the existence of an equivalent risk neutral (i.e. for a local martingale) meas ...

11.2 Basic Theorems 219 11.2 Basic Theorems Before proving the main results of the paper we need to recall some definitions and ...

220 11 Change of Num ́eraire The proof of the fundamental theorem is quite complicated and we cannot repeat it here. The basic i ...

11.2 Basic Theorems 221 Proof.The necessity is clear. IfQis an equivalent local martingale mea- sure, then the Radon-Nikod ́ym d ...

222 11 Change of Num ́eraire Me(P)= { Q ∣ ∣ ∣ ∣ Qis equivalent toP and the processSis aQ-local martingale } M(P)= { Q ∣ ∣ ∣ ∣ Qi ...

11.3 Duality Relation 223 sup Q∈Me(P) EQ[f]= sup Q∈M(P) EQ[f] =inf{x|∃h∈Cx+h≥f} =inf{x|∃h∈Cx+h=f} =inf{x|(f−x)∈C} =inf{x|∃h∈Kx+h ...

224 11 Change of Num ́eraire Theorem 11.3.4.Suppose that the locally bounded martingaleSsatisfies the (NFLVR) property with resp ...

11.4 Hedging and Change of Num ́eraire 225 11.4 Hedging and Change of Num ́eraire Before we give a martingale characterisation o ...

226 11 Change of Num ́eraire Theorem 11.4.2.LetW be a semi-martingale, taking values inRd.LetV be a strictly positive semi-marti ...

11.4 Hedging and Change of Num ́eraire 227 thatY∞−1=(L·X)∞is non-negative and strictly positive on a non-negligible set. This sh ...

228 11 Change of Num ́eraire Remark 11.4.5.We conjecture that the assumption thatV−^1 is locally bounded can be removed.† Proof. ...

11.4 Hedging and Change of Num ́eraire 229 (1) (H·S)∞is maximal inK (2)there isQ∈Me(P)such thatEQ[(H·S)∞]=0 (3)there isQ∈Me(P)su ...

230 11 Change of Num ́eraire (1)fcan be hedged, (2)there isQinMe(P)such that EQ[f]=sup{ER[f]|R∈Me(P)}<∞. Proof.(1)implies(2): ...

12 The Existence of Absolutely Continuous Local Martingale Measures (1995) Abstract.We investigate the existence of an absolutel ...