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arXiv2011

De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process

Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal). We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario, whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin. This paper is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying dividends. For example in case of proportional reinsurance one can observe the interesting phenomenon that choice of the optimal barrier depends on the initial reserves. This is in contrast with the one-dimensional Cramér-Lundberg model where the optimal choice of the barrier is uniform for all initial reserves.

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arXiv2009

Improved and Developed Upper Bound of Price of Anarchy in Two Echelon Case

Price of anarchy, the performance ratio, which could characterize the loss of efficiency of the distributed supply chain management compared with the integrated supply chain management is discussed by utilizing newsvendor problem in single period which is well-known. In particular, some of remarkable distributed policies are handled, the performance ratios in each case which have been investigated in the previous works are analyzed theoretically and the tighter upper bound of price of anarchy and the lower bound are presented. Furthermore our approach is developed based on a generalized framework and a geometric interpretation of price of anarchy is appeared via the literature of convex optimization.

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arXiv2009

Heterogeneous Beliefs with Partial Observations

This paper examines a heterogeneous beliefs model in which there is a process that is only partially observed by the agents. The economy contains a risky asset producing dividends continuously in time. The dividends are observed by the agents. The dividends are assumed to be a known function of some other unobserved process. The agents use filtering to estimate the value of this unobserved process. The agents have different beliefs about the dynamics of the unobserved process and therefore form different estimates. We analyse this model and derive the state price density. We use this to derive the riskless rate. We also characterise the price of the risky asset in terms of the solution of a series of differential equations.

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arXiv2009

Heterogeneous Beliefs with Finite-Lived Agents

This paper will examine a model with many agents, each of whom has a different belief about the dynamics of a risky asset. The agents are Bayesian and so learn about the asset over time. All agents are assumed to have a finite (but random) lifetime. When an agent dies, he passes his wealth (but not his knowledge) onto his heir. As a result, the agents never become sure of the dynamics of the risky asset. We derive expressions for the stock price and riskless rate. We then use numerical examples to exhibit their behaviour.

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arXiv2009

A note on heterogeneous beliefs with CRRA utilities

This note will extend the research presented in Brown & Rogers (2009) to the case of CRRA agents. We consider the model outlined in that paper in which agents had diverse beliefs about the dividends produced by a risky asset. We now assume that the agents all have CRRA utility, with some integer coefficient of relative risk aversion. This is a generalisation of Brown & Rogers which considered logarithmic agents. We derive expressions for the state price density, riskless rate, stock price and wealths of the agents. This sheds light on the effects of risk aversion in an equilibrium with diverse beliefs.

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arXiv2009

Statistical Signatures in Times of Panic: Markets as a Self-Organizing System

We study properties of the cross-sectional distribution of returns. A significant anti-correlation between dispersion and cross-sectional kurtosis is found such that dispersion is high but kurtosis is low in panic times, and the opposite in normal times. The co-movement of stock returns also increases in panic times. We define a simple statistic $s$, the normalized sum of signs of returns on a given day, to capture the degree of correlation in the system. $s$ can be seen as the order parameter of the system because if $s= 0$ there is no correlation (a disordered state), whereas for $s \ne 0$ there is correlation among stocks (an ordered state). We make an analogy to non-equilibrium phase transitions and hypothesize that financial markets undergo self-organization when the external volatility perception rises above some critical value. Indeed, the distribution of $s$ is unimodal in normal times, shifting to bimodal in times of panic. This is consistent with a second order phase transition. Simulations of a joint stochastic process for stocks use a multi timescale process in the temporal direction and an equation for the order parameter $s$ for the dynamics of the cross-sectional correlation. Numerical results show good qualitative agreement with the stylized facts of real data, in both normal and panic times.

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arXiv2009

The International-Trade Network: Gravity Equations and Topological Properties

This paper begins to explore the determinants of the topological properties of the international - trade network (ITN). We fit bilateral-trade flows using a standard gravity equation to build a "residual" ITN where trade-link weights are depurated from geographical distance, size, border effects, trade agreements, and so on. We then compare the topological properties of the original and residual ITNs. We find that the residual ITN displays, unlike the original one, marked signatures of a complex system, and is characterized by a very different topological architecture. Whereas the original ITN is geographically clustered and organized around a few large-sized hubs, the residual ITN displays many small-sized but trade-oriented countries that, independently of their geographical position, either play the role of local hubs or attract large and rich countries in relatively complex trade-interaction patterns.

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arXiv2009

Optimal intervention in the foreign exchange market when interventions affect market dynamics

We address the problem of optimal Central Bank intervention in the exchange rate market when interventions create feedback in the rate dynamics. In particular, we extend the work done on optimal impulse control by Cadenillas and Zapatero to incorporate temporary market reactions, of random duration and level, to Bank interventions, and to establish results for more general rate processes. We obtain new explicit optimal impulse control strategies that account for these market reactions, and show that they cannot be obtained simply by adjusting the intervention cost in a model without market reactions.

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arXiv2010

Econophysics: Empirical facts and agent-based models

This article aims at reviewing recent empirical and theoretical developments usually grouped under the term Econophysics. Since its name was coined in 1995 by merging the words Economics and Physics, this new interdisciplinary field has grown in various directions: theoretical macroeconomics (wealth distributions), microstructure of financial markets (order book modelling), econometrics of financial bubbles and crashes, etc. In the first part of the review, we discuss on the emergence of Econophysics. Then we present empirical studies revealing statistical properties of financial time series. We begin the presentation with the widely acknowledged stylized facts which describe the returns of financial assets- fat tails, volatility clustering, autocorrelation, etc.- and recall that some of these properties are directly linked to the way time is taken into account. We continue with the statistical properties observed on order books in financial markets. For the sake of illustrating this review, (nearly) all the stated facts are reproduced using our own high-frequency financial database. Finally, contributions to the study of correlations of assets such as random matrix theory and graph theory are presented. In the second part of the review, we deal with models in Econophysics through the point of view of agent-based modelling. Amongst a large number of multi-agent-based models, we have identified three representative areas. First, using previous work originally presented in the fields of behavioural finance and market microstructure theory, econophysicists have developed agent-based models of order-driven markets that are extensively presented here. Second, kinetic theory models designed to explain some empirical facts on wealth distribution are reviewed. Third, we briefly summarize game theory models by reviewing the now classic minority game and related problems.

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arXiv2009

Utility Function and Optimum Consumption in the models with Habit Formation and Catching up with the Joneses

This paper analyzes popular time-nonseparable utility functions that describe "habit formation" consumer preferences comparing current consumption with the time averaged past consumption of the same individual and "catching up with the Joneses" (CuJ) models comparing individual consumption with a cross-sectional average consumption level. Few of these models give reasonable optimum consumption time series. We introduce theoretically justified utility specifications leading to a plausible consumption behavior to show that habit formation preferences must be described by a power CRRA utility function different from the exponential CARA used for CuJ.

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arXiv2021

Geometric Arbitrage Theory and Market Dynamics Reloaded

We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to: --Write arbitrage as curvature of a principal fibre bundle. --Parameterize arbitrage strategies by its holonomy. --Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization. --Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance. --Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where: -->Arbitrage is allowed but minimized. -->Arbitrage is not allowed. --Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition.

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arXiv2009

Diagnostics of Rational Expectation Financial Bubbles with Stochastic Mean-Reverting Termination Times

We propose two rational expectation models of transient financial bubbles with heterogeneous arbitrageurs and positive feedbacks leading to self-reinforcing transient stochastic faster-than-exponential price dynamics. As a result of the nonlinear feedbacks, the termination of a bubble is found to be characterized by a finite-time singularity in the bubble price formation process ending at some potential critical time $\tilde{t}_c$, which follows a mean-reversing stationary dynamics. Because of the heterogeneity of the rational agents' expectations, there is a synchronization problem for the optimal exit times determined by these arbitrageurs, which leads to the survival of the bubble almost all the way to its theoretical end time. The explicit exact analytical solutions of the two models provide nonlinear transformations which allow us to develop novel tests for the presence of bubbles in financial time series. Avoiding the difficult problem of parameter estimation of the stochastic differential equation describing the price dynamics, the derived operational procedures allow us to diagnose bubbles that are in the making and to forecast their termination time. The tests performed on three financial markets, the US S&P500 index from 1 February 1980 to 31 October 2008, the US NASDAQ composite index from 1 January 1980 to 31 July 2008 and the Hong Kong Hang Seng index from 1 December 1986 to 30 November 2008, suggest the feasibility of advance bubble warning.

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arXiv2009

Financial crises and the evaporation of trust

Trust lies at the crux of most economic transactions, with credit markets being a notable example. Drawing on insights from the literature on coordination games and network growth, we develop a simple model to clarify how trust breaks down in financial systems. We show how the arrival of bad news about a financial agent can lead others to lose confidence in it and how this, in turn, can spread across the entire system. Our results emphasize the role of hysteresis -- it takes considerable effort to regain trust once it has been broken. Although simple, the model provides a plausible account of the credit freeze that followed the global financial crisis of 2007/8, both in terms of the sequence of events and the measures taken (and being proposed) by the authorities.

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arXiv2013

On the Existence of Consistent Price Systems

We formulate a sufficient condition for the existence of a consistent price system (CPS), which is weaker than the conditional full support condition (CFS) introduced by Guasoni, Rasonyi, and Schachermayer [Ann. Appl. Probab., 18(2008), pp. 491-520] . We use the new condition to show the existence of CPSs for certain processes that fail to have the CFS property. In particular this condition gives sufficient conditions, under which a continuous function of a process with CFS admits a CPS, while the CFS property might be lost.

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arXiv2009

Credit derivatives: instruments of hedging and factors of instability. The example of ?Credit Default Swaps? on French reference entities

Through a long-period analysis of the inter-temporal relations between the French markets for credit default swaps (CDS), shares and bonds between 2001 and 2008, this article shows how a financial innovation like CDS could heighten financial instability. After describing the operating principles of credit derivatives in general and CDS in particular, we construct two difference VAR models on the series: the share return rates, the variation in bond spreads and the variation in CDS spreads for thirteen French companies, with the aim of bringing to light the relations between these three markets. According to these models, there is indeed an interdependence between the French share, CDS and bond markets, with a strong influence of the share market on the other two. This interdependence increases during periods of tension on the markets (2001-2002, and since the summer of 2007).

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arXiv2011

Causal Links Between US Economic Sectors

In this paper, we perform a comparative segmentation and clustering analysis of the time series for the ten Dow Jones US economic sector indices between 14 February 2000 and 31 August 2008. From the temporal distributions of clustered segments, we find that the US economy took one and a half years to recover from the mid-1998-to-mid-2003 financial crisis, but only two months to completely enter the present financial crisis. We also find the oil & gas and basic materials sectors leading the recovery from the previous financial crisis, while the consumer goods and utilities sectors led the descent into the present financial crisis. On a macroscopic level, we find sectors going earlier into a crisis emerge later from it, whereas sectors going later into the crisis emerge earlier. On the mesoscopic level, we find leading sectors experiencing stronger and longer volatility shocks, while trailing sectors experience weaker and shorter volatility shocks. In our shock-by-shock causal-link analysis, we also find shorter delays between corresponding shocks in more closely related economic sectors. In addition, our analysis reveals evidences for complex sectorial structures, as well as nonlinear amplification in the propagating volatility shocks. From a perspective relevant to public policy, our study suggests an endogeneous sectorial dynamics during the mid-2003 economic recovery, in contrast to strong exogeneous driving by Federal Reserve interest rate cuts during the mid-2007 onset. Most interestingly, we find for the sequence of closely spaced interest rate cuts instituted in 2007/2008, the first few cuts effectively lowered market volatilities, while the next few cuts counter-effectively increased market volatilities. Subsequent cuts evoked little response from the market.

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arXiv2009

Finitely additive probabilities and the Fundamental Theorem of Asset Pricing

This work aims at a deeper understanding of the mathematical implications of the economically-sound condition of absence of arbitrages of the first kind in a financial market. In the spirit of the Fundamental Theorem of Asset Pricing (FTAP), it is shown here that absence of arbitrages of the first kind in the market is equivalent to the existence of a finitely additive probability, weakly equivalent to the original and only locally countably additive, under which the discounted wealth processes become "local martingales". The aforementioned result is then used to obtain an independent proof of the FTAP of Delbaen and Schachermayer. Finally, an elementary and short treatment of the previous discussion is presented for the case of continuous-path semimartingale asset-price processes.

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arXiv2010

What Causes Business Cycles? Analysis of the Japanese Industrial Production Data

We explore what causes business cycles by analyzing the Japanese industrial production data. The methods are spectral analysis and factor analysis. Using the random matrix theory, we show that two largest eigenvalues are significant. Taking advantage of the information revealed by disaggregated data, we identify the first dominant factor as the aggregate demand, and the second factor as inventory adjustment. They cannot be reasonably interpreted as technological shocks. We also demonstrate that in terms of two dominant factors, shipments lead production by four months. Furthermore, out-of-sample test demonstrates that the model holds up even under the 2008-09 recession. Because a fall of output during 2008-09 was caused by an exogenous drop in exports, it provides another justification for identifying the first dominant factor as the aggregate demand. All the findings suggest that the major cause of business cycles is real demand shocks.

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