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We consider vector fixed point (FP) equations in large dimensional spaces involving random variables, and study their realization-wise solutions. We have an underlying directed random graph, that defines the connections between various components of the FP equations. Existence of an edge between nodes i, j implies the i th FP equation depends on the j th component. We consider a special case where any component of the FP equation depends upon an appropriate aggregate of that of the random neighbor components. We obtain finite dimensional limit FP equations (in a much smaller dimensional space), whose solutions approximate the solution of the random FP equations for almost all realizations, in the asymptotic limit (number of components increase). Our techniques are different from the traditional mean-field methods, which deal with stochastic FP equations in the space of distributions to describe the stationary distributions of the systems. In contrast our focus is on realization-wise FP solutions. We apply the results to study systemic risk in a large financial heterogeneous network with many small institutions and one big institution, and demonstrate some interesting phenomenon.
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Management of systemic risk in financial markets is traditionally associated with setting (higher) capital requirements for market participants. There are indications that while equity ratios have been increased massively since the financial crisis, systemic risk levels might not have lowered, but even increased. It has been shown that systemic risk is to a large extent related to the underlying network topology of financial exposures. A natural question arising is how much systemic risk can be eliminated by optimally rearranging these networks and without increasing capital requirements. Overlapping portfolios with minimized systemic risk which provide the same market functionality as empirical ones have been studied by [pichler2018]. Here we propose a similar method for direct exposure networks, and apply it to cross-sectional interbank loan networks, consisting of 10 quarterly observations of the Austrian interbank market. We show that the suggested framework rearranges the network topology, such that systemic risk is reduced by a factor of approximately 3.5, and leaves the relevant economic features of the optimized network and its agents unchanged. The presented optimization procedure is not intended to actually re-configure interbank markets, but to demonstrate the huge potential for systemic risk management through rearranging exposure networks, in contrast to increasing capital requirements that were shown to have only marginal effects on systemic risk [poledna2017]. Ways to actually incentivize a self-organized formation toward optimal network configurations were introduced in [thurner2013] and [poledna2016]. For regulatory policies concerning financial market stability the knowledge of minimal systemic risk for a given economic environment can serve as a benchmark for monitoring actual systemic risk in markets.
The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded cd processes, we show that this framework provides a systematic approach to the both issues of model ambiguity, and uncertainty about the time value of money. We also establish a link between risk measures for processes and BSDEs.
We propose the Hawkes flocking model that assesses systemic risk in high-frequency processes at the two perspectives -- endogeneity and interactivity. We examine the futures markets of WTI crude oil and gasoline for the past decade, and perform a comparative analysis with conditional value-at-risk as a benchmark measure. In terms of high-frequency structure, we derive the empirical findings. The endogenous systemic risk in WTI was significantly higher than that in gasoline, and the level at which gasoline affects WTI was constantly higher than in the opposite case. Moreover, although the relative influences degree was asymmetric, its difference has gradually reduced.
In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.
A meticulous assessment of the risk of impacts associated with extreme wind events is of great necessity for populations, civil authorities as well as the insurance industry. Using the concept of spatial risk measure and related set of axioms introduced by Koch (2017, 2019), we quantify the risk of losses due to extreme wind speeds. The insured cost due to wind events is proportional to the wind speed at a power ranging typically between 2 and 12. Hence we first perform a detailed study of the correlation structure of powers of the Brown-Resnick max-stable random fields and look at the influence of the power. Then, using the latter results, we thoroughly investigate spatial risk measures associated with variance and induced by powers of max-stable random fields. In addition, we show that spatial risk measures associated with several classical risk measures and induced by such cost fields satisfy (at least part of) the previously mentioned axioms under conditions which are generally satisfied for the risk of damaging extreme wind speeds. In particular, we specify the rates of spatial diversification in different cases, which is valuable for the insurance industry.
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