By Walter Schachermayer
A classical subject in Mathematical Finance is the idea of portfolio optimization. Robert Merton's paintings from the early seventies had huge, immense impression on educational study in addition to at the paradigms guiding practitioners.
One of the ramifications of this subject is the research of (small) proportional transaction charges, equivalent to a Tobin tax. The lecture notes current a few amazing contemporary result of the asymptotic dependence of the appropriate amounts while transaction expenses are inclined to zero.
An beautiful function of the glory of transaction bills is that it permits the 1st time to reconcile the no arbitrage paradigm with using non-semimartingale versions, akin to fractional Brownian movement. This ends up in the culminating theorem of the current lectures which approximately reads as follows: for a fractional Brownian movement inventory rate version we consistently discover a shadow expense procedure for given transaction charges. This strategy is a semimartingale and will consequently be handled utilizing the standard equipment of mathematical finance.
Keywords: Portfolio optimization, transaction bills, shadow rate, semimartingale, fractional Brownian movement
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1 The frictionless case Proof. 9) with initial value V0 = 1. 3). 8), we again obtain ∫ T ∫ T π2 σ2 E [log(VT )] = E πt σt dWt + πt µt − t t dt 2 0 0 ∫ T 2 2 π σ =E πt µt − t t dt . 2 0 It is obvious that, for fixed 0 ≤ t ≤ T and ω ∈ Ω, the function π → π µt (ω) − attains its unique maximum at πˆ t (ω) = ∫ E [log(VT )] ≤ E =E µt (ω) σt2 (ω) T 0 ∫ 0 π 2 σt2 (ω) , 2 T so that πˆ t µt − µ2t 2σt2 π∈R πˆ t2 σt2 dt 2 dt = E log(VˆT ) . More generally, for stopping times 0 ≤ ≤ τ ≤ T, we obtain ∫ τ π2 σ2 Vτ E log =E πt µt − t t dt V 2 ∫ τ 2 πˆ σ 2 Vˆτ ≤E πˆ t µt − t t dt = E log 2 Vˆ .
Then it is economically obvious (and easily checked) that it is optimal not to trade at all (even under transaction costs λ = 0). 2 are given by ϕˆt ≡ (X, 0), and Qˆ = P, as well as yˆ = U (x). For the optimal shadow price process S˜ we may take S˜ = S. But this choice is not unique. In fact, we may take any P-martingale S˜ = (S˜t )Tt=0 taking values in the bid–ask spread ([(1 − λ)St , St ])Tt=0 . 9) and denote by Sˆ˜ the process E ZˆT1 | Ft Sˆ˜t = , E ZˆT0 | Ft t = 0, . . , T, ˆ which is a martingale under Q(y).
2. 1. 8) and prepares the spirit for the local duality arguments developed below. 34 3 Growth-optimal portfolio in the Black–Scholes model Let Vˆt be the wealth process for the logarithmic utility and Vt any competing Vt process. It follows from Itô’s lemma that V ˆ t is a positive local martingale and therefore a supermartingale. Hence, by Jensen’s inequality, we obtain E log(Vt ) − log(Vˆt ) ≤ log E Vt Vˆ t ≤ 0. 10 below) we want to develop the heuristics to find the shadow price process (S˜t )t ≥0 for the utility maximization problem of optimizing the expected growth of a portfolio.
Asymptotic Theory of Transaction Costs by Walter Schachermayer