The Mathematics of Arbitrage

61TheStoryinaNutshell or, written more explicitly, C 1 (g)=Π 1 (g), (1.2) C 1 (b)=Π 1 (b). (1.3) We are confident that the reade ...

1.5 Martingale Measures 7 1.4 Variationsof the Example Although the preceding “toy example” is extremely simple and, of course, ...

81TheStoryinaNutshell EQ[C 1 ]=^13. Clearly we suspect that this numerical match is not just a coincidence. At this stage it is, ...

1.6 The Fundamental Theorem of Asset Pricing 9 (i) There exists a probability measureQequivalent toPunder whichSis a sigma-marti ...

2 Models of Financial Markets on Finite Probability Spaces 2.1 Descriptionof the Model In this section we shall develop the theo ...

12 2 Models of Financial Markets on Finite Probability Spaces Readers who are not so enthusiastic about this mainly formal and e ...

2.1 Description of the Model 13 The interpretation goes as follows. By changing the portfolio fromĤt− 1 toĤtthere is no input/ ...

14 2 Models of Financial Markets on Finite Probability Spaces Ft-measurable. In economic terms the above argument is rather obvi ...

2.1 Description of the Model 15 VT=V 0 + ∑T t=1 (Ht,∆St)=V 0 +(H·S)T, where (H·S)T = ∑T t=1(Ht,∆St) is the notation for a stocha ...

16 2 Models of Financial Markets on Finite Probability Spaces 2.2 No-Arbitrage and the Fundamental Theorem of Asset Pricing of A ...

2.2 No-Arbitrage and the FTAP 17 Recall thatL^0 (Ω,F,P) denotes the space of allF-measurable real-valued functions andL^0 +(Ω,F, ...

18 2 Models of Financial Markets on Finite Probability Spaces Indeed, (2.5) holds true iff for eachFt− 1 -measurable setAwe have ...

2.2 No-Arbitrage and the FTAP 19 (Q,f)≤α,forf∈K, (Q,h)≥β,forh∈P. SinceKis a linear space, we haveα≥0 and may replaceαby 0. Hence ...

20 2 Models of Financial Markets on Finite Probability Spaces a=EQ[f],anda+(H·S)t=EQ[f|Ft],for 0 ≤t≤T. (2.7) Proof.As regards th ...

2.2 No-Arbitrage and the FTAP 21 After these general observations we pass to the concrete setting of the coneC⊆L∞(Ω,F,P) of cont ...

22 2 Models of Financial Markets on Finite Probability Spaces In this casea=EQ[f], the stochastic integralH·Sis unique and we ha ...

2.4 Pricing by No-Arbitrage 23 Proof.Obviously (i) implies (ii), since there are less strategies in each single period market th ...

24 2 Models of Financial Markets on Finite Probability Spaces For givenf∈L∞(Ω,F,P), we calla∈Ranarbitrage-free price,ifin additi ...

2.4 Pricing by No-Arbitrage 25 view ofQ∗∼Pimplies thatf−π(f)≡g;inotherwordsfis attainable at priceπ(f). This in turn implies tha ...

26 2 Models of Financial Markets on Finite Probability Spaces Remark 2.4.5.Before we prove the theorem let us remark that the “s ...