شبکه های پیچیده: به حداقل رساندن ریسک در بازارهای مالی از طریق اسپین شیشه ای کشورهای قابل دسترس

Recurrent international financial crises inflict significant damage to societies and stress the need for mechanisms or strategies to control risk and tamper market uncertainties. Unfortunately, the complex network of market interactions often confounds rational approaches to optimize financial risks. Here we show that investors can overcome this complexity and globally minimize risk in portfolio models for any given expected return, provided the margin requirement remains below a critical, empirically measurable value. In practice, for markets with centrally regulated margin requirements, a rational stabilization strategy would be keeping margins small enough. This result follows from ground states of the random field spin glass Ising model that can be calculated exactly through convex optimization when relative spin coupling is limited by the norm of the network’s Laplacian matrix. In that regime, this novel approach is robust to noise in empirical data and may be also broadly relevant to complex networks with frustrated interactions that are studied throughout scientific fields.

Large and abnormal fluctuations in financial markets can spread to other parts of the global economy with untoward and often incalculable effects. Therefore, a key priority is to minimize risks and contain their propagation in spite of the tendencies of current financial markets to the contrary [1] and [2]. Important examples of such market places include exchanges where stocks, commodities, futures and other financial products can be bought and sold short by using leverage on margin accounts held by investors. A central financial decision problem in these markets is, for a given expected return rPrP, to distribute the available capital among multiple assets, which comprise a portfolio PP of size nn, so as to minimize the overall risk. In portfolio selection models this goal can be mathematically formulated as finding the global minimum of a risk function [3], [4], [5] and [6], View the MathML sourceR=1/2∑i,k=1nCikpipk−∑i=1npiri−γ∑i=1npisi, where pipi is the positive or negative amount of capital invested in asset ii, and View the MathML sourcesi=sign(pi)∈{−1,1} are binary spin variables; riri is the expected return of asset ii such that View the MathML sourcerP=∑i=1nripi;Cik is the covariance between assets ii and kk; and γγ is the margin account requirement which sets the fraction of capital that the investor must deposit in a margin account before buying or selling short assets. With the inverse C−1C−1 of the covariance matrix CC the minimum risk distribution p=(p1,…,pn)p=(p1,…,pn) becomes View the MathML sourcepi=∑k=1nCik−1rk+γ∑k=1nCik−1sk. It is known that finding the absolute risk minimum is computationally equivalent to the ground state problem of the random field Ising model [3] and [5]. This is evident after inserting pp into the risk function while neglecting fixed terms that do not depend on spin variables which gives View the MathML sourceR=−1/2∑i,k=1nJiksisk−∑i=1nhisi, and where we introduced an interaction term View the MathML sourceJik=γCik−1 and a random local field h=(h1,…,hn)h=(h1,…,hn) with View the MathML sourcehi=∑k=1nCik−1rk. Covariance among assets can be both positive and negative (see, for example, inset in Fig. 1(A)), and globally minimizing risk means finding a ground state of the random field Ising model with random spin glass interactions, which in general belongs to the class of NP-complete decision problems [7] and [8] and for which efficient algorithms are not known. This computational intractability arises from the non-convexity of the cost function RR; non-convex problems are much harder to solve computationally than convex optimization problems for which efficient algorithms do exist [9]. In the context of financial markets, the non-convexity of the spin glass model prevents equilibration into an optimum ground state and is viewed as an inherent source of risk [10] and [3]. Full-size image (88 K) Fig. 1. (A) Portfolio risk can be globally and rationally minimized if the relative margin requirement satisfies γ/γc<1γ/γc<1. In contrast, for γ/γc>1γ/γc>1 the estimated risk undergoes large fluctuations above the optimum. Red data points (“TAP”) give the risk from solutions of the TAP equation for n=16n=16 with randomly selected assets from the S&P500 price data, and with a random field h=(h1,…,hn)h=(h1,…,hn) with |hi|≤1|hi|≤1. Blue data points (“Local field”) depict the risk obtained by taking the sign of local field hh. Error bars represent standard deviations after 128 random trials. Inset shows the distribution of price correlations between all pairs in the m=395m=395 assets taken from the S&P500 index. (B) Estimated critical margin requirement as a function of portfolio size n≤mn≤m and for three different choices of price samples, t={2511,193,78}t={2511,193,78}, where stock prices were selected every {1, 13, 32} days, respectively. Error bars represent standard deviations from 128 random selections in the S&P500 price data. Black solid and dashed graphs represent the function View the MathML source(1−n/t)2. (C) The inverse partition ratio View the MathML sourceNs=∑l=1m(ulk)4 for each normalized eigenvector ukuk of the m×mm×m correlation matrix View the MathML sourceC̃ ranked by its increasing eigenvalues [11]. Red dots represent the unfiltered correlation matrix which, up to a rank of k=372k=372, follow a semicircle distribution; blue dots represent the filtered correlation matrix after setting all eigenvalues with lower rank to zero, i.e. those in size smaller than View the MathML sourceλmax. Inset shows the resulting histogram of pairwise price correlations after filtering. (D) Estimated critical margin requirement γcγc from the S&P500 correlation matrix View the MathML sourceC̃ before (red) and after (blue) eigenvalue filtering. Black solid line represents the graph (2n)−1(2n)−1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Risk minimization in portfolio selection models with short selling is equivalent to the ground state problem of the random field Ising model with spin glass interactions. Because calculating its ground state is computationally hard in general, globally minimizing the risk has been regarded as unfeasible with a computationally efficient and rational approach [10] and [3]. Our result shows that, under realistic conditions, finding the ground state and thus efficient risk minimization is rationally possible. As a direct consequence in financial markets, this may provide an instrument for curbing volatility if financial products are traded below the critical margin requirement, and if investors and traders rationally optimize their portfolios. The second condition is both desirable and realistic in today’s highly computerized markets, although it may have been less realistic in the past when computers were not widespread and therefore complex financial decisions were to a lesser degree rational. But the first condition seems to be in conflict with interests of traders and lenders who, in individual contracts, seek to reduce default risk by increasing margins. From a collective market perspective, however, higher margin requirements may have a destabilizing effect through higher transaction costs, which can drive traders from the market place; this may lead to a lower overall liquidity thus making the market more susceptible to volatility [17] and [18]. Hence, in financial markets where minimum margin requirements are regulated a reduction of risk by lowering margins is conceivable. Historically, the possibility of such a regulatory approach is indirectly supported by the fact that both the 1987 and the 1929 financial market crashes were accompanied by an increase in margin requirements which exacerbated liquidity problems and which might have contributed to rapid downfall [19] and [20]. Of practical relevance may be the observation that for portfolio sizes above n≈10n≈10 our estimates on the critical margin requirement from the recent American stock market fall below one (Fig. 1(B)), thus potentially imposing realistic upper limits on margin requirements. The efficient access to an optimum is not restricted to portfolio risk models; in general, an efficient computation of a ground state is possible in any spin glass Ising model with a random field if the relative coupling strength between spins falls below the critical value. Further applications may follow in frustrated systems that routinely occur in artificial [21] and [22] and in biological [23] and [24] networks and where the goal is to find a ground state.