Credit risk models like Moody's KMV are now well established in the market and give bond managers reliable default probabilities for individual firms. Until now it has been hard to relate those probabilities to the actual credit spreads observed on the market for corporate bonds. Inspired by the existence of scaling laws in financial markets by Dacorogna et al. 2001 and DiMatteo et al. 2005 deviating from the Gaussian behavior, we develop a model that quantitatively links those default probabilities to credit spreads (market prices). The main input quantities to this study are merely industry yield data of different times to maturity and expected default frequencies (EDFs) of Moody's KMV. The empirical results of this paper clearly indicate that the model can be used to calculate approximate credit spreads (market prices) from EDFs, independent of the time to maturity and the industry sector under consideration. Moreover, the model is effective in an out-of-sample setting, it produces consistent results on the European bond market where data are scarce and can be adequately used to approximate credit spreads on the corporate level.

Results of fractal stability analysis of daily exchange rate fluctuations of more than 30 floating currencies for a 10-year period are presented. It is shown for the first time that small- and large-scale dynamical instabilities of national monetary systems correlate with deviations of the detrended fluctuation analysis (DFA) exponent from the value 1.5 predicted by the efficient market hypothesis. The observed dependence is used for classification of long-term stability of floating exchange rates as well as for revealing various forms of distortion of stable currency dynamics prior to large-scale crises. A normal range of DFA exponents consistent with crisis-free long-term exchange rate fluctuations is determined, and several typical scenarios of unstable currency dynamics with DFA exponents fluctuating beyond the normal range are identified. It is shown that monetary crashes are usually preceded by prolonged periods of abnormal (decreased or increased) DFA exponent, with the after-crash exponent tending to the value 1.5 indicating a more reliable exchange rate dynamics. Statistically significant regression relations (R=0.99, p<0.01) between duration and magnitude of currency crises and the degree of distortion of monofractal patterns of exchange rate dynamics are found. It is demonstrated that the parameters of these relations characterizing small- and large-scale crises are nearly equal, which implies a common instability mechanism underlying these events. The obtained dependences have been used as a basic ingredient of a forecasting technique which provided correct in-sample predictions of monetary crisis magnitude and duration over various time scales. The developed technique can be recommended for real-time monitoring of dynamical stability of floating exchange rate systems and creating advanced early-warning-system models for currency crisis prevention.

A diffusive model with a price dependent diffusion coefficient was recently proposed to explain the occurrence of non-Gaussian price return distributions observed empirically in real markets [J.L. McCauley and G.H. Gunaratne, Physica A 329, 178 (2003)]. Depending on the functional form of the diffusion coefficient, the exactly solved continuum limit of the model can produce either an exponential distribution, or a "fat-tailed" power-law distribution of returns. Real markets, however, are discrete, and, in this paper, the effects of discreteness on the model are explored. Discrete distributions from simulations and from numerically exact calculations are presented and compared to the corresponding distributions of the continuum model. A type of phase transition is discovered in discrete models that lead to fat-tailed distributions in the continuum limit, sheading light on the nature of such distributions. The transition is to a phase in which infinite price changes can occur in finite time.

Recent work based on a non-Gaussian statistical feedback model of stock returns is summarized. The model is outlined, as well as applications to option pricing and the pricing of credit. An extension of the original model which incorporates feedback over multiple time horizons is also briefly discussed.

It is well established that volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence the volatility cannot be characterized by a single correlation time. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. In this paper we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat-tails. We aim to find the most probable path that contributes to the action functional, that describes the dynamics of the entire system, by finding local minima. We obtain a second order differential equation for the functional return. This paper reviews our current progress and the remaining open questions.

There is intense interest in understanding the stochastic and dynamical properties of the global Foreign Exchange (FX) market, whose daily transactions exceed one trillion US dollars. This is a formidable task since the FX market is characterized by a web of fluctuating exchange rates, with subtle inter-dependencies which may change in time. In practice, traders talk of particular currencies being 'in play' during a particular period of time -- yet there is no established machinery for detecting such important information. Here we apply the construction of Minimum Spanning Trees (MSTs) to the FX market, and show that the MST can capture important features of the global FX dynamics. Moreover, we show that the MST can help identify momentarily dominant and dependent currencies.

We apply a method to filter relevant information from the correlation coefficient matrix by extracting a network of relevant interactions. This method succeeds to generate networks with the same hierarchical structure of the Minimum Spanning Tree but containing a larger amount of links resulting in a richer network topology allowing loops and cliques. In Tumminello et al.,1 we have shown that this method, applied to a financial portfolio of 100 stocks in the USA equity markets, is pretty efficient in filtering relevant information about the clustering of the system and its hierarchical structure both on the whole system and within each cluster. In particular, we have found that triangular loops and 4 element cliques have important and significant relations with the market structure and properties. Here we apply this filtering procedure to the analysis of correlation in two different kind of interest rate time series (16 Eurodollars and 34 US interest rates).

Many asset classes, such as interest rates, exchange rates, commodities, and equities, often exhibit a strong relationship between asset prices and asset volatilities. This paper examines an analytical model that takes into account this level dependence of volatility. We demonstrate how prices of European options under stochastic volatility can be calculated analytically via inverse Laplace transformations. We also examine a Hull-White stochastic volatility expansion. While a success of this expansion in approximate computation of option prices has already been established empirically, the question of convergence has been left unanswered. We demonstrate, in this paper, that this expansion diverges essentially for all possible stochastic volatility processes. In contrast to a majority of volatility expansion models reported in the literature, we construct expansions that explicitly show the contribution of all of the variance moments. Such complete expansions are very useful in analyzing properties of option prices, as we demonstrate by examining why empirical volatility surfaces plotted as a function of the rescaled strike can sometimes exhibit striking time invariance.

It is known since Bachelier 1900 that price changes are nearly uncorrelated, leading to a random-walk like behaviour of prices. We provide evidence for a very subtle compensation mechanism that underlies the 'random' nature of price changes. This compensation drives the market close to a critical point, which may explain the sensitivity of financial markets to small perturbations, and its propensity to enter bubbles and crashes. We argue that the resulting unpredictability of price changes is very far from the neo-classical view that markets are informationally efficient.

A new theory for pricing of options is presented. It is based on the assumption that successive movements depend on the value of the return. The solution to the Fokker-Planck equation is shown to be an asymmetric exponential distribution, similar to those observed in intra-day currency markets. The "volatility smile", used by traders to correct the Black-Scholes pricing is shown to be a heuristic mechanism to implement options pricing formulae derived from our theory.

We briefly review the main stylized facts observed in financial markets and show how a multifractal process naturally captures those effects. In particular we generalize the construction of the multifractal random walk (MRW) due to Bacry, Delour and Muzy to take into account the asymmetric character of the financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. Explicit scaling exponents are computes and are shown to behave differently for even and odd moments. We illustrate the usefulness of this "skewed" MRW by computing the resulting shape of the volatility smiles generated by such a process. A large variety of smile surfaces can be reproduced.

The cornerstone of Boltzmann-Gibbs (BG) statistical mechanics is the Boltzmann-Gibbs-Jaynes-Shannon entropy S_BG≡ -k &sh; dx f(x) ln f(x), where k is a positive constant and f(x) a probability density function. This theory has exibited, along more than one century, great success in the treatment of systems where short spatio/temporal correlations dominate. There are, however, anomalous natural and artificial systems that violate the basic requirements for its applicability. Different physical entropies, other than the standard one, appear to be necessary in order to satisfactorily deal with such anomalies. One of such entropies is S_q ≡ k (1-&sh; dx [f(x)]^q)=(1-q) (with S₁ = S_BG), where the entropic index q is a real parameter. It has been proposed as the basis for a generalization, referred to as nonextensive statistical mechanics, of the BG theory. S_q shares with S_BG four remarkable properties, namely concavity (8q > 0), Lesche-stability (8q > 0), finiteness of the entropy production per unit time (q 2 <), and additivity (for at least a compact support of q including q = 1). The simultaneous validity of these properties suggests that S_q is appropriate for bridging, at a macroscopic level, with classical thermodynamics itself. In the same natural way that exponential probability functions arise in the standard context, power-law tailed distributions, even with exponents out of the Levy range, arise in the nonextensive framework. In this review, we intend to show that many processes of interest in economy, for which fat-tailed probability functions are empirically observed, can be described in terms of the statistical mechanisms that underly the nonextensive theory.

We discuss the stationary states of a model economy in which N heterogeneous adaptive consumers purchase commodity bundles repeatedly from P sellers. The system undergoes a transition from an inefficient to an efficient state as the number of consumers increases. In the latter phase, however, price fluctuations may be much larger than in the inefficient regime. Results from dynamical mean-field theory obtained in the 'thermodynamic limit' compare fairly well with computer simulations.

Searching for a closed-form exact solution for American put options under the Black-Scholes framework has been a long standing problem in the past; many researchers believe that it is impossible to find such a solution. In this paper, a closed-form exact solution, in the form of a Taylor's series expansion, of the well-known Black-Scholes equation is presented for the first time. As a result of this analytic solution, the optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration.

We consider a simple pure exchange economy with two assets, one riskless, yielding a constant return on investment, and one risky, paying a stochastic dividend. Trading takes place in discrete time and in each trading period the price of the risky asset is fixed through the market clearing condition. Individual demands are expressed as fractions of traders wealth and depend on traders forecasts about future price movement. Under these assumptions, we derive the stochastic dynamical system that describes the evolution of price and wealth. We study the cases in which one or two agents operate in the market, identifying the possible equilibria and discussing their stability conditions. The main novelty of this paper rests in the abstraction from the precise characterization of agents' beliefs and preferences. In this respect our results generalize several previous contributions in the field. In particular, we show that, irrespectively of agents' behavior, the system can only possess isolated generic equilibria where a single agent dominates the market and continuous manifolds of non-generic equilibria where heterogeneous agents hold finite shares of the aggregate wealth. Moreover, we show that all possible equilibria belong to a one dimensional "Equilibria Market Line". Finally we discuss the role of different parameters for the stability of equilibria and the selection principle governing market dynamics.

In this paper, empirical analyses and computational experiments are presented on high-frequency data for a double-auction (book) market. Main objective of the paper is to generalize the order waiting time process in order to properly model such empirical evidences. The empirical study is performed on the best bid and best ask data of 7 U.S. financial markets, for 30-stock time series. In particular, statistical properties of trading waiting times have been analyzed and quality of fits is evaluated by suitable statistical tests, i.e., comparing empirical distributions with theoretical models. Starting from the statistical studies on real data, attention has been focused on the reproducibility of such results in an artificial market. The computational experiments have been performed within the Genoa Artificial Stock Market. In the market model, heterogeneous agents trade one risky asset in exchange for cash. Agents have zero intelligence and issue random limit or market orders depending on their budget constraints. The price is cleared by means of a limit order book. The order generation is modelled with a renewal process. Based on empirical trading estimation, the distribution of waiting times between two consecutive orders is modelled by a mixture of exponential processes. Results show that the empirical waiting-time distribution can be considered as a generalization of a Poisson process. Moreover, the renewal process can approximate real data and implementation on the artificial stocks market can reproduce the trading activity in a realistic way.

We consider a simple binary market model containing N competitive agents. The novel feature of our model is that it incorporates the tendency shown by traders to look for patterns in past price movements over multiple time scales, i.e. multiple memory-lengths. In the regime where these memory-lengths are all small, the average winnings per agent exceed those obtained for either (1) a pure population where all agents have equal memory-length, or (2) a mixed population comprising sub-populations of equal-memory agents with each sub-population having a different memory-length. Agents who consistently play strategies of a given memory-length, are found to win more on average -- switching between strategies with different memory lengths incurs an effective penalty, while switching between strategies of equal memory does not. Agents employing short-memory strategies can outperform agents using long-memory strategies, even in the regime where an equal-memory system would have favored the use of long-memory strategies. Using the many-body 'Crowd-Anticrowd' theory, we obtain analytic expressions which are in good agreement with the observed numerical results. In the context of financial markets, our results suggest that multiple-memory agents have a better chance of identifying price patterns of unknown length and hence will typically have higher winnings.

This paper presents an agent-based model of a power exchange. Supply of electric power is provided by competing generating companies, whereas demand is assumed to be inelastic with respect to price and is constant over time. The transmission network topology is assumed to be a fully connected graph and no transmission constraints are taken into account. The price formation process follows a common scheme for real power exchanges: a clearing house mechanism with uniform price, i.e., with price set equal across all matched buyer-seller pairs. A single class of generating companies is considered, characterized by linear cost function for each technology. Generating companies compete for the sale of electricity through repeated rounds of the uniform auction and determine their supply functions according to production costs. However, an individual reinforcement learning algorithm characterizes generating companies behaviors in order to attain the expected maximum possible profit in each auction round. The paper investigates how the market competitive equilibrium is affected by market microstructure and production costs.

A new agent-based Cellular Automaton (CA) computational algorithm for option pricing is proposed. CAs have been extensively used in modeling complex dynamical systems but not in modeling option prices. Compared with traditional tools, which rely on guessing volatilities to calculate option prices, the CA model is directly addressing market mechanisms and simulates price fluctuation from aggregation of actions made by interacting individual market makers in a large population. This paper explores whether CA models can provide reasonable good answers to pricing European options. The Black-Scholes model and the Binomial Tree model are used for comparison. Comparison reveals that CA models perform reasonably well in pricing options, reproducing overall characteristics of random walk based model, while at the same time providing plausible results for the 'fat-tail' phenomenon observed in many markets. We also show that the binomial tree model can be obtained from a CA rule. Thus, CA models are suitable tools to generalize the standard theories of option pricing.

Standard macroeconomics, based on a reductionist approach centered on the representative agent, is badly equipped to explain the empirical evidence where heterogeneity and industrial dynamics are the rule. In this paper we show that a simple agent-based model of heterogeneous financially fragile agents is able to replicate a large number of scaling type stylized facts with a remarkable degree of statistical precision.

Investment strategies in multiplicative Markovian market models with transaction costs are defined using growth optimal criteria. The optimal strategy is shown to consist in holding the amount of capital invested in stocks within an interval around an ideal optimal investment. The size of the holding interval is determined by the intensity of the transaction costs and the time horizon. The inclusion of financial derivatives in the models is also considered. All the results presented in this contributions were previously derived in collaboration with E. Aurell.

The rate of convergence of numerical methods for integration of some convex functionals of ordinary stochastic differential equations (SDEs) is discussed. In particular, we answer how rates of p-th mean convergence carry over to rates of weak convergence for non-smooth and convex functionals of SDEs. Qualitative behavior of some numerical approximations such as nonnegativity of balanced implicit Milstein methods (BMMs) is investigated as well. Nonstandard integration techniques such as partial- and linear-implicit ones seem to be the most promissing. As a main result we obtain some justification for the choice of approximation schemes of discounted price functionals and their ingredients of random interest rates and volatility processes involved in dynamic asset pricing.

The rapid growth of the hedge fund industry presents significant business opportunity for the institutional investors particularly in the form of portfolio diversification. To facilitate this, there is a need to develop a new set of risk analytics for investments consisting of hedge funds, with the ultimate aim to create transparency in risk measurement without compromising the proprietary investment strategies of hedge funds. As well documented in the literature, use of dynamic options like strategies by most of the hedge funds make their returns highly non-normal with fat tails and high kurtosis, thus rendering Value at Risk (VaR) and other mean-variance analysis methods unsuitable for hedge fund risk quantification. This paper looks at some unique concerns for hedge fund risk management and will particularly concentrate on two approaches from physical world to model the non-linearities and dynamic correlations in hedge fund portfolio returns: Self Organizing Criticality (SOC) and Random Matrix Theory (RMT).Random Matrix Theory analyzes correlation matrix between different hedge fund styles and filters random noise from genuine correlations arising from interactions within the system. As seen in the results of portfolio risk analysis, it leads to a better portfolio risk forecastability and thus to optimum allocation of resources to different hedge fund styles. The results also prove the efficacy of self-organized criticality and implied portfolio correlation as a tool for risk management and style selection for portfolios of hedge funds, being particularly effective during non-linear market crashes.

This paper proposes a model of endogenous fluctuations in investment. A monopolistic producer has an incentive to invest when the aggregate demand is high. The investment at the firm level is also known to exhibit a threshold behavior called an (S,s) policy. These two facts lead us to consider that the fluctuation in aggregate investment is generated by the global coupling of the non-linear oscillators. From this perspective, we characterize the probability distribution of the investment clustering in a partial equilibrium of product markets, and show that its variance can be large enough to match the observed investment fluctuations. We then implement this mechanism in a dynamic general equilibrium model to explore an investment-driven business cycle. By calibrating the model with the SIC 4-digit level industry data, we numerically show that the model replicates the basic structure of the business cycles.

We study the serial correlation of high-frequency intraday returns on the Italian stock index futures (FIB30) in the period 2000-2002. We adopt three different methods of analysis: the spectral density via Fast Fourier Transform, Detrended Fluctuation Analysis (DFA) and the Variance Ratio test. We find that intraday autocorrelation is mostly negative for time scales lower than 20 minutes, but we support the efficiency of the Italian futures market.

The correlation between stock price changes is useful information. Through the correlation matrix, we construct a portfolio with its minimum spanning tree. We make the minimum spanning tree of the Korean stock market, a representative emerging market, which is different from that of the mature market. It is due to the emerging market's less abundant liquidity than the mature market. And we find the distribution of the correlation coefficient is different for several periods. As the market is developing, many changes from inside and outside the market occurs, and several parameters of the stock market network are changed. The Korean stock market is under an evolution.

There has been growing interest in Econophysics, i.e. analysis and modeling of financial time series using the theoretical Physics concepts (scaling, fractals, chaos). Besides the scientific stimuli, this interest is backed by perception that the financial industry is a viable alternative for those physicists who are not able or are not willing to pursue an academic career. However, the times when any Ph.D. in Physics had a chance to find a job on the Wall Street are gone (if they ever existed). Indeed, not every physicist wields the stochastic calculus, non-normal statistical distributions, and the methods of time series analysis. Moreover, now that many universities offer courses in mathematical finance, the applicants for quantitative positions in finance are expected to know such concepts as option pricing, portfolio management, and risk measurement. Here I describe a synthetic course based on my book [1] that outlines both worlds: Econophysics and Mathematical Finance. The course may be offered as elective for senior undergraduate or graduate Physics majors.

In this paper, in order to de-noise a chaotic signal, we compare the time-frequency deconvolution method with the wavelets method. We apply our results on different dynamical systems and show the capability of wavelets' method to reconstruct the attractor of a chaotic time series. Then, we de-noise different data sets in order to re-built their attractor using the wavelets method. The applications concern temperatures and wind fluctuations, electricity spot prices and financial data sets.

This paper studies the time-varying behavior of scaling laws and long-memory. This is motivated by the earlier finding that in the FX markets a single scaling factor might not always be sufficient across all relevant timescales: a different region may exist for intradaily time-scales and for larger time-scales. In specific, this paper investigates (i) if different scaling regions appear in stock market as well, (ii) if the scaling factor systematically differs from the Brownian, (iii) if the scaling factor is constant in time, and (iv) if the behavior can be explained by the heterogenuity of the players in the market and/or by intraday volatility periodicity. Wavelet method is used because it delivers a multiresolution decomposition and has excellent local adaptiviness properties. As a consequence, a wavelet-based OLS method allows for consistent estimation of long-memory. Thus issues (i)-(iv) shed light on the magnitude and behavior of a long-memory parameter, as well. The data are the 5-minute volatility series of Nokia Oyj at the Helsinki Stock Exchange around the burst of the IT-bubble. Period one represents the era of "irrational exuberance" and another the time after it. The results show that different scaling regions (i.e. multiscaling) may appear in the stock markets and not only in the FX markets, the scaling factor and the long-memory parameter are systematically different from the Brownian and they do not have to be constant in time, and that the behavior can be explained for a significant part by an intraday volatility periodicity called the New York effect. This effect was magnified by the frenzy trading of short-term speculators in the bubble period. The found stronger long-memory is also attributable to irrational exuberance.