The Mathematics of Arbitrage

2.5 Change of Num ́eraire 27 Proof.As observed in the proof of Theorem 2.3.2 and Proposition 2.3.1, every Q∈Me(S) can be written ...

28 2 Models of Financial Markets on Finite Probability Spaces S=(S^1 ,S^2 ,...,Sd)= ( Ŝ^1 Ŝ^0 ,..., Ŝd Ŝ^0 ) . Before we pro ...

2.5 Change of Num ́eraire 29 V. It suffices to show that, for a one-dimensional predictable processL,the quantitiesLt∆Vtare inK( ...

30 2 Models of Financial Markets on Finite Probability Spaces The converse inclusion follows by symmetry. In the financial marke ...

2.6 Kramkov’s Optional Decomposition Theorem 31 2.6 Kramkov’s Optional Decomposition Theorem We now present a dynamic version of ...

32 2 Models of Financial Markets on Finite Probability Spaces Proof of Theorem 2.6.1.First assume thatT= 1, i.e., we have a one- ...

3 Utility Maximisation on Finite Probability Spaces In addition to the modelSof a financial market, we now consider a function U ...

34 3 Utility Maximisation on Finite Probability Spaces We note that it is natural from an economic point of view to require that ...

3.1 The Complete Case 35 under the constraint EQ[XT]= ∑N n=1 qnξn ≤ x. (3.3) To verify that (3.2) and (3.3) are indeed equivalen ...

36 3 Utility Maximisation on Finite Probability Spaces form (3.5) of the Lagrangian and fixingy>0, the optimisation problem o ...

3.1 The Complete Case 37 We now apply these general facts about the Legendre transform to calcu- lateΨ(y). Using definition (3.9 ...

38 3 Utility Maximisation on Finite Probability Spaces ∑N n=1 qnξ̂n=x, (3.16) and ∑N n=1 pnU(̂ξn)=L(ξ̂ 1 ,...,ξ̂N,̂y(x)). (3.17) ...

3.1 The Complete Case 39 (ii) The optimiserX̂T(x)in (3.19) exists, is unique and satisfies X̂T(x)=I ( y dQ dP ) , or, equivalent ...

40 3 Utility Maximisation on Finite Probability Spaces proportionality relation fails to hold. Then one immediately deduces from ...

3.2 The Incomplete Case 41 now is, that the optimally investing economic agent is indifferent of first order towards a marginal ...

42 3 Utility Maximisation on Finite Probability Spaces u(x)= sup H∈H E[U(x+(H·S)T)] = sup XT∈C(x) E[U(XT)]. The Lagrangian now i ...

3.2 The Incomplete Case 43 where (q 1 ,...,qN) denotes the probability vector ofQ∈Ma(S). The min- imisation ofΨwill be done in t ...

44 3 Utility Maximisation on Finite Probability Spaces (ii) The optimisersX̂T(x)andQ̂(y)in (3.27) and (3.28) exist, are unique, ...

3.3 The Binomial and the Trinomial Model 45 H∈H}, the constant function 1, and (f^1 ,...,fk) linearly spanL∞(Ω,F,P). Define thek ...

46 3 Utility Maximisation on Finite Probability Spaces We assume w.l.o.g. thatu ̃≥−d ̃; we do so — mainly for notational conve- ...