Conjuncted: Speculatively Accelerated Capital – Trading Outside the Pit.

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High Frequency Traders (HFTs hereafter) may anticipate the trades of a mutual fund, for instance, if the mutual fund splits large orders into a series of smaller ones and the initial trades reveal information about the mutual funds’ future trading intentions. HFTs might also forecast order flow if traditional asset managers with similar trading demands do not all trade at the same time, allowing the possibility that the initiation of a trade by one mutual fund could forecast similar future trades by other mutual funds. If an HFT were able to forecast a traditional asset managers’ order flow by either these or some other means, then the HFT could potentially trade ahead of them and profit from the traditional asset manager’s subsequent price impact.

There are two main empirical implications of HFTs engaging in such a trading strategy. The first implication is that HFT trading should lead non-HFT trading – if an HFT buys a stock, non-HFTs should subsequently come into the market and buy those same stocks. Second, since the HFT’s objective would be to profit from non-HFTs’ subsequent price impact, it should be the case that the prices of the stocks they buy rise and those of the stocks they sell fall. These two patterns, together, are consistent with HFTs trading stocks in order to profit from non-HFTs’ future buying and selling pressure. 

While HFTs may in aggregate anticipate non-HFT order flow, it is also possible that among HFTs, some firms’ trades are strongly correlated with future non-HFT order flow, while other firms’ trades have little or no correlation with non-HFT order flow. This may be the case if certain HFTs focus more on strategies that anticipate order flow or if some HFTs are more skilled than other firms. If certain HFTs are better at forecasting order flow or if they focus more on such a strategy, then these HFTs’ trades should be consistently more strongly correlated with future non-HFT trades than are trades from other HFTs. Additionally, if these HFTs are more skilled, then one might expect these HFTs’ trades to be more strongly correlated with future returns. 

Another implication of the anticipatory trading hypothesis is that the correlation between HFT trades and future non-HFT trades should be stronger at times when non-HFTs are impatient. The reason is anticipating buying and selling pressure requires forecasting future trades based on patterns in past trades and orders. To make anticipating their order flow difficult, non-HFTs typically use execution algorithms to disguise their trading intentions. But there is a trade-off between disguising order flow and trading a large position quickly. When non-HFTs are impatient and focused on trading a position quickly, they may not hide their order flow as well, making it easier for HFTs to anticipate their trades. At such times, the correlation between HFT trades and future non-HFT trades should be stronger. 

Entangled State Vectors

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Let R1, R2 be von Neumann algebras on H such that R1 ⊆ R′2. Recall that a state ω of R12 is called a normal product state just in case ω is normal, and there are states ω1 of R1 and ω2 of R2 such that

ω(AB) = ω1(A)ω2(B) —– (1)

∀ A ∈ R1, B ∈ R2. Werner, in dealing with the case of B(Cn) ⊗ B(Cn), defined a density operator D to be classically correlated — the term separable is now more commonly used — just in case D can be approximated in trace norm by convex combinations of density operators of form D1 ⊗ D2. Although Werner’s definition of nonseparable states directly generalizes the traditional notion of pure entangled states, he showed that a nonseparable mixed state need not violate a Bell inequality; thus, Bell correlation is in general a sufficient, though not necessary condition for a state’s being non-separable. On the other hand, it has since been shown that nonseparable states often possess more subtle forms of nonlocality, which may be indicated by measurements more general than the single ideal measurements which can indicate Bell correlation.

In terms of the linear functional representation of states, Werner’s separable states are those in the norm closed convex hull of the product states of B(Cn) ⊗ B(Cn). However, in case of the more general setup — i.e., R1 ⊆ R′2, where R1, R2 are arbitrary von Neumann algebras on H — the choice of topology on the normal state space of R12 will yield in general different definitions of separability. Moreover, it has been argued that norm convergence of a sequence of states can never be verified in the laboratory, and as a result, the appropriate notion of physical approximation is given by the (weaker) weak-∗ topology. And the weak-∗ and norm topologies do not generally coincide even on the normal state space.

For the next proposition, then, we will suppose that the separable states of R12 are those normal states in the weak-∗ closed convex hull of the normal product states. Note that β(ω) = 1 if ω is a product state, and since β is a convex function on the state space, β(ω) = 1 if ω is a convex combination of product states. Furthermore, since β is lower semicontinuous in the weak-∗ topology, β(ω) = 1 for any separable state. Conversely, any Bell correlated state must be nonseparable.

We now introduce some notation that will aid us in the proof of our result. For a state ω of the von Neumann algebra R and an operator A ∈ R, define the state ωA on R by

ωA(X) ≡ ω(A∗XA)/ω(A∗A) —– (2)

if ω(A∗A) ≠ 0, and let ωA = ω otherwise. Suppose now that ω(A∗A) ≠ 0 and ω is a convex combination of states:

ω = ∑i=1nλiωi —– (3)

Then, letting λAi ≡ ω(A∗A)−1ωi(A∗A)λi, ωA is again a convex combination

ωA = ∑i=1n λAiωAi —– (4)

Moreover, it is not difficult to see that the map ω → ωA is weak-∗ continuous at any point ρ such that ρ(A∗A) ≠ 0. Indeed, let O1 = N(ρA : X1,…,Xn, δ) be a weak-∗ neighborhood of ρA. Then, taking O2 = N(ρ : AA,AX1A,…,AXnA, δ) and ω ∈ O2, we have

|ρ(AXiA) − ω(AXiA)| < δ —– (5)

for i = 1,…,n, and

|ρ(AA) − ω(AA)| < δ —– (6)

By choosing δ < ρ(AA) ≠ 0, we also have ω(AA) ≠ 0, and thus

A(Xi) − ωA(Xi)| < O(δ) ≤ δ —– (7)

for an appropriate choice of δ. That is, ωA ∈ O1 forall ω ∈ O2 and ω → ωA is weak-∗ continuous at ρ.

Specializing to the case where R1 ⊆ R′2, and R12 = {R1 ∪ R2}”, it is clear from the above that for any normal product state ω of R12 and for A ∈ R1, ωA is again a normal product state. The same is true if ω is a convex combination of normal product states, or the weak-∗ limit of such combinations. Summarizing the results of this discussion in the following lemma:

Lemma: For any separable state ω of R12 and any A ∈ R1, ωA is again separable.

Proposition: Let R1,R2 be nonabelian von Neumann algebras such that R1 ⊆ R′2. If x is cyclic for R1, then ωx is nonseparable across R12.

Proof: There is a normal state ρ of R12 such that β(ρ) = √2. But since all normal states are in the (norm) closed convex hull of vector states, and since β is norm continuous and convex, there is a vector v ∈ S such that β(v) > 1. By the continuity of β (on S), there is an open neighborhood O of v in S such that β(y) > 1 ∀ y∈O. Since x is cyclic for R1,there is an A ∈ R1 such that Ax ∈ O. Thus, β(Ax) > 1 which entails that ωAx = (ωx)A is a nonseparable state for R12. This, by the preceding lemma, entails that ωx is nonseparable.

Note that if R1 has at least one cyclic vector x ∈ S, then R1 has a dense set of cyclic vectors in S. Since each of the corresponding vector states is nonseparable across R12, Proposition shows that if R1 has a cyclic vector, then the (open) set of vectors inducing nonseparable states across R12 is dense in S. On the other hand, since the existence of a cyclic vector for R1 is not invariant under isomorphisms of R12, Proposition does not entail that if R1 has a cyclic vector, then there is a norm dense set of nonseparable states in the entire normal state space of R12. Indeed, if we let R1 = B(C2) ⊗ I, R2 = I ⊗ B(C2), then any entangled state vector is cyclic for R1; but, the set of nonseparable states of B(C2) ⊗ B(C2) is not norm dense. However, if in addition to R1 or R2 having a cyclic vector, R12 has a separating vector (as is often the case in quantum field theory), then all normal states of R12 are vector states, and it follows that the nonseparable states will be norm dense in the entire normal state space of R12.

What Drives Investment? Or How Responsible is Kelly’s Optimum Investment Fraction?

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A reasonable way to describe assets price variations (on a given time-scale) is to assume them to be multiplicative random walks with log-normal step. This comes from the assumption that growth rates of prices are more significant than their absolute variations. So, we describe the price of a financial assets as a time-dependent multiplicative random process. We introduce a set of N Gaussian random variables xi(t) depending on a time parameter t. By this set, we define N independent multiplicative Gaussian random walks, whose assigned discrete time evolution is given by

pi(t+1) = exi(t)pi(t) —– (1)

for i = 1,…,N, where each xi(t) is not correlated in time. To optimize an investment, one can choose different risk-return strategies. Here, by optimization we will mean the maximization of the typical capital growth rate of a portfolio. A capital W(t), invested into different financial assets who behave as multiplicative random walks, grows almost certainly at an exponential rate ⟨ln W (t+1)/W (t)⟩, where one must average over the distribution of the single multiplicative step. We assume that an investment is diversified according to the Kelly’s optimum investment fraction, in order to maximize the typical capital growth rate over N assets with identical average return α = ⟨exi(t)⟩ − 1 and squared volatility ∆ = ⟨e2xi(t)⟩ − ⟨exi(t)⟩2. It should be noted that Kelly capital growth criterion, which maximizes the expected log of final wealth, provides the strategy that maximizes long run wealth growth asymptotically for repeated investments over time. However, one drawback is found in its very risky behavior due to the log’s essentially zero risk aversion; consequently it tends to suggest large concentrated investments or bets that can lead to high volatility in the short-term. Many investors, hedge funds, and sports bettors use the criterion and its seminal application is to a long sequence of favorable investment situations. On each asset, the investor will allocate a fraction fi of his capital, according to the return expected from that asset. The time evolution of the total capital is ruled by the following multiplicative process

W(t+1) = [1 + ∑i=1Nfi(exi(t) -1)] W(t) —– (2)

First, we consider the case of an unlimited investment, i.e. we put no restriction tothe value of ∑i=1Nfi. The typical growth rate

Vtyp = ⟨ln[1+  ∑i=1Nfi(exi -1)]⟩ —– (3)

of the investor’s capital can be calculated through the following 2nd-order expansion in exi -1, if we assume that fluctuations of prices are small and uncorrelated, that seems to be quite reasonable

Vtyp ≅ ∑i=1Nfi(⟨exi⟩ – 1) – fi2/2(⟨e2xi⟩ – 2⟨exi⟩ + 1 —– (4)

By solving d/df(Vtyp = 0), it easy to show that the optimal value for fi is fiopt (α, Δ) = α / (α2 + Δ) ∀ i. We assume that the investor has little ignorance about the real value of α, that we represent by a Gaussian fluctuation around the real value of α. In the investor’s mind, each asset is different, because of this fluctuation αi = α + εi. The εi are drawn from the same distribution, with ⟨εi⟩ = 0 as errors are normally distributed around the real value. We suppose that the investor makes an effort E to investigate and get information about the statistical parameters of the N assets upon which he will spread his capital. So, his ignorance (i.e. the width of the distribution of the εi) about the real value of αi will be a decreasing function of the effort “per asset” E ; more, we suppose that an even infinite effort will not make N this ignorance vanish. In order to plug these assumptions in the model, we write the width of the distribution of ε as

⟨ε2i⟩ = D0 + (N/E)γ —– (5)

with γ > 0. As one can see, the greater is E, the more exact is the perception, and better is the investment. D0 is the asymptotic ignorance. All the invested fraction fopt (αi, Δ) will be different, according to the investor’s perception. Assuming that the εi are small, we expand all fi(α + εi) in equation 4 up to the 2nd order in εi, and after averaging over the distribution of εi, we obtain the mean value of the typical capital growth rate for an investor who provides a given effort E:

Vtyp = N[A − (D0 + (N/E)γ )B] —– (6)

where

A = (α (3Δ – α2))/(α2 + Δ)3 B = -(α2 – Δ)2/2(α2 + Δ)3 —– (7)

We are now able to find the optimal number of assets to be included in the portfolio (i.e., for which the investment is more advantageous, taken into account the effort provided to get information), by solving d/dNVtyp = 0, it is easy to see that the number of optimal assets is given by

Nopt(E) = E {[(A – D0]/(1 + γ)B}1/γ —– (8)

that is an increasing function of the effort E. If the investor has no limit in the total capital fraction invested in the portfolio (so that it can be greater than 1, i.e. the investor can invest more money than he has, borrowing it from an external source), the capital can take negative values, if the assets included in the portfolio encounter a simultaneous negative step. So, if the total investment fraction is greater than 1, we should take into account also the cost of refunding loss to the bank, to predict the typical growth rate of the capital.