The Mathematics of Arbitrage

4.4 The Black-Scholes Model 67 Keeping in mind that this was achieved during an interval of total length 2 dt(which corresponds ...

68 4 Bachelier and Black-Scholes Why was the calculation ofEQ[f 2 ] so easy? Simply because the factor Ke−rTappearing inf 2 is a ...

4.4 The Black-Scholes Model 69 EQ[f 1 ]=S 0 Φ ⎛ ⎝ ln (S 0 K ) + ( r+σ 2 2 ) T σ √ T ⎞ ⎠, which is the first term appearing in th ...

5 The Kreps-Yan Theorem Let us turn back to the no-arbitrage theory developed in Chap. 2 to raise again the question: what can w ...

72 5 The Kreps-Yan Theorem We first will make a technical assumption, namely that the processSis locally bounded, i.e., that the ...

5.1 A General Framework 73 Throughout this chapter we callHadmissibleif, in addition, the stopped processSτnand the functionsh 1 ...

74 5 The Kreps-Yan Theorem Lemma 5.1.3.LetQbe a probability measure onFwhich is absolutely con- tinuous w.r. toP. A locally boun ...

5.1 A General Framework 75 Proof.The proof proceeds by induction onnand yields additionally that the stopping timesσ 1 andσ 2 ca ...

76 5 The Kreps-Yan Theorem such that (εn)∞n=1are independent random variables taking the values +1 or −1 with probabilities P[εn ...

5.2 No Free Lunch 77 Definition 5.2.1.(compare [K 81])Ssatisfies the condition ofno free lunch (NFL) if the closureCofCsimple, t ...

78 5 The Kreps-Yan Theorem Hence,g 0 +g 1 wouldbeanelementofGwhose support hasP-measure strictly bigger thanP[{g 0 > 0 }], a ...

5.2 No Free Lunch 79 (i) Is it possible, in general, to replace the net (gα)α∈Iabove by a sequence (gn)∞n=0? (ii) Can we choose ...

80 5 The Kreps-Yan Theorem prove versions of the theorem, where the closure with respect to the weak- star topology is replaced ...

5.2 No Free Lunch 81 reasoning. However, both arguments amount to the same, and the difference is only superficial. Fix 1≤p≤∞,an ...

82 5 The Kreps-Yan Theorem (iii) For allf∈Cwe haveEQ[f]= ∑∞ n=1βnEQn[f]≤0. The final assertion is obvious.
We now turn to the ...

5.2 No Free Lunch 83 (iv) C∩ball∞is closed with respect to the topology of convergence in measure. Proof.(ii)⇒(i): This is the c ...

6 The Dalang-Morton-Willinger Theorem 6.1 Statement of the Theorem In Chap. 2 we only dealt with finite probability spaces. This ...

86 6 The Dalang-Morton-Willinger Theorem The proof of the theorem is not a trivial extension of the case of finite Ω (we remark, ...

6.2 The Predictable Range 87 F 0 -measurably parameterised subspace ofRdwhereXtakes its values. This idea was used in [S 92]. Th ...

88 6 The Dalang-Morton-Willinger Theorem φ 1 =|ff| (^1) A 1 is still inE 1 and it has maximal support among all elements of E 1. ...