Strategy correlations and timing of adaptation in Minority Games

We study the role of strategy correlations and timing of adaptation for the dynamics of Minority Games, both simulationally and analytically. Using the exact generating functional approach a la De Dominicis we compute the phase diagram and the behaviour of batch and on-line games with correlated strategies, complementing exisiting replica studies of their statics. It is shown that the timing of adaptation can be relevant; while conventional games with uncorrelated strategies are nearly insensitive to the choice of on-line versus batch learning, we find qualitative differences when anti-correlations are present in the strategy assignments. The available standard approximations for the volatility in terms of persistent order parameters in the stationary ergodic states become unreliable in batch games under such circumstances. We then comment on the role of oscillations and the relation to the breakdown of ergodicity. Finally, it is discussed how the generating functional formalism can be used to study mixed populations of so-called `producers' and `speculators' in the context of the batch minority games.Comment: 15 pages, 13 figures, EPJ styl

Market response to external events and interventions in spherical minority games

We solve the dynamics of large spherical Minority Games (MG) in the presence of non-negligible time dependent external contributions to the overall market bid. The latter represent the actions of market regulators, or other major natural or political events that impact on the market. In contrast to non-spherical MGs, the spherical formulation allows one to derive closed dynamical order parameter equations in explicit form and work out the market's response to such events fully analytically. We focus on a comparison between the response to stationary versus oscillating market interventions, and reveal profound and partially unexpected differences in terms of transition lines and the volatility.Comment: 14 pages LaTeX, 5 (composite) postscript figures, submitted to Journal of Physics

Pricing exotic options in the incomplete market: an imprecise probability method

This paper considers a novel exotic option pricing method for incomplete markets. Nonparametric Predictive Inference (NPI) is applied to the option pricing procedure based on the binomial tree model allowing the method to evaluate exotic options with limited information and few assumptions. As the implementation of the NPI method is greatly simplified by the monotonicity of the option payoff in the tree, we categorize exotic options by their payoff monotonicity and study a typical type of exotic option in each category, the barrier option and the look-back option. By comparison with the classic binomial tree model, we investigate the performance of our method either with different moneyness or varying maturity. All outcomes show that our model offers a feasible approach to price the exotic options with limited information, which makes it can be utilized for both complete and incomplete markets

Predictive inference for system reliability after common-cause component failures

This paper presents nonparametric predictive inference for system reliability following common-cause failures of components. It is assumed that a single failure event may lead to simultaneous failure of multiple components. Data consist of frequencies of such events involving particular numbers of components. These data are used to predict the number of components that will fail at the next failure event. The effect of failure of one or more components on the system reliability is taken into account through the system׳s survival signature. The predictive performance of the approach, in which uncertainty is quantified using lower and upper probabilities, is analysed with the use of ROC curves. While this approach is presented for a basic scenario of a system consisting of only a single type of components and without consideration of failure behaviour over time, it provides many opportunities for more general modelling and inference, these are briefly discussed together with the related research challenges

Non‐parametric predictive inference for the validation of credit rating systems

Credit rating or credit scoring systems are important tools for estimating the obligor's creditworthiness and for providing an indication of the obligor's future status. The discriminatory power of a credit rating or credit scoring system refers to its ex ante ability to distinguish between two or more classes of borrowers. One of the most popular tools for the validation of the power of credit rating or credit scoring models to distinguish between two (or more) classes of borrowers is the receiver operating characteristic (ROC) curve (hypersurface) and its widely used overall summary, the area (hypervolume) under the curve (hypersurface). As the end goal of building such models is to predict and quantify uncertainty about future loans, prediction methods are especially valuable in this context. For this, non‐parametric predictive inference is a promising candidate for such inference as it is a frequentist statistical method that is explicitly aimed at using few modelling assumptions, enabled through the use of lower and upper probabilities to quantify uncertainty. The aim of the paper is to introduce non‐parametric predictive inference for ROC analysis within a banking context, for which novel results on ROC hypersurfaces for more than three groups are presented. Examples are provided to illustrate the method

The survival signature for quantifying system reliability: an introductory overview from practical perspective

The structure function describes the functioning of a system dependent on the states of its components, and is central to theory of system reliability. The survival signature is a summary of the structure function which is sufficient to derive the system’s reliability function. Since its introduction in 2012, the survival signature has received much attention in the literature, with developments on theory, computation and generalizations. This paper presents an introductory overview of the survival signature, including some recent developments. We discuss challenges for practical use of survival signatures for large systems

Dynamical Probability Distribution Function of the SK Model at High Temperatures

The microscopic probability distribution function of the Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as a function of time by a high-temperature expansion. The resulting formula to the third order of the inverse temperature shows that an assumption made by Coolen, Laughton and Sherrington in their recent theory of dynamics is violated. Deviations of their theory from exact results are estimated quantitatively. Our formula also yields explicit expressions of the time dependence of various macroscopic physical quantities when the temperature is suddenly changed within the high-temperature region.Comment: LaTeX, 6 pages, Figures upon request (here revised), To be published in J. Phys. Soc. Jpn. 65 (1996) No.

Nonparametric Predictive Inference for European Option Pricing based on the Binomial Tree Model

In finance, option pricing is one of the main topics. A basic model for option pricing is the Binomial Tree Model, proposed by Cox, Ross, and Rubinstein in 1979 (CRR). This model assumes that the underlying asset price follows a binomial distribution with a constant upward probability, the so-called risk-neutral probability. In this paper, we propose a novel method based on the binomial tree. Rather than using the risk-neutral probability, we apply Nonparametric Predictive Inference (NPI) to infer imprecise probabilities of movements, reflecting more uncertainty while learning from data. To study its performance, we price the same European options utilizing both the NPI method and the CRR model and compare the results in two different scenarios, firstly where the CRR assumptions are right, and secondly where the CRR model assumptions deviate from the real market. It turns out that our NPI method, as expected, cannot perform better than the CRR in the first scenario, but can do better in the second scenario

Dynamics of a spherical minority game

We present an exact dynamical solution of a spherical version of the batch minority game (MG) with random external information. The control parameters in this model are the ratio of the number of possible values for the public information over the number of agents, and the radius of the spherical constraint on the microscopic degrees of freedom. We find a phase diagram with three phases: two without anomalous response (an oscillating versus a frozen state), and a further frozen phase with divergent integrated response. In contrast to standard MG versions, we can also calculate the volatility exactly. Our study reveals similarities between the spherical and the conventional MG, but also intriguing differences. Numerical simulations confirm our analytical results.Comment: 16 pages, 3 figures; submitted to J. Phys.