International Journal of Theoretical and Applied Finance

Physicists have recently begun doing research in finance, and even though this movement is less than five years old, interesting and useful contributions have already emerged. This article reviews these developments in four areas, including empirical statistical properties of prices, random-process models for price dynamics, agent-based modeling, and practical applications.

In recent years, a considerable number of physicists have started applying physics concepts and methods to understand economic phenomena. The term "Econophysics" is sometimes used to describe this work. Economic fluctuations can have many repercussions, and understanding fluctuations is a topic that many physicists have contributed to in recent years. Further, economic systems are examples of complex interacting systems for which a huge amount of data exist and it is possible that the experience gained by physicists in studying fluctuations in physical systems might yield new results in economics. Much recent work in econophysics is focused on understanding the peculiar statistical properties of price fluctuations in financial time series. In this talk, we discuss three recent results. The first result concerns the probability distribution of stock price fluctuations. This distribution decreases with increasing fluctuations with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. Further, this nonstable distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. The second result concerns the accurate quantification of volatility correlations in financial time series. While price fluctuations themselves have rapidly decaying correlations, the volatility estimated by using either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. We compare the statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks and a matrix with independent random elements, i.e., a random matrix. Contrary to first expectations, we find little or no deviation from the universal predictions of random matrix theory for all but a few of the largest eigenvalues of the cross-correlation matrix.

Analogous to the Richter scale for earthquakes, we introduce the Scale of Market Shocks (SMS), an "event" scale to quantify the size of shocks in financial markets. It is based on price volatilities and computed by integrating volatilities over time horizons ranging from 1 hour to 42 days. The SMS is computed using quality high frequency market data and can be constructed for any market.

We compute the SMS for the foreign exchange market. For two major FX rates, we study the relation between SMS peaks and major "world events". We measure also the correlation between the Scale of Market Shocks index and the size of the subsequent price movements and show a high correlation for short time intervals.

A time series model for the FX dynamics is presented which takes into account structural peculiarities of the market, namely its heterogeneity and an information flow from long to short time horizons. The model emerges from an analogy between FX dynamics and hydrodynamic turbulence. The heterogeneity of the market is modeled in the form of a multiplicative cascade of time scales ranging from several minutes to a few months, analogous to the Kolmogorov cascade in turbulence.

The model reproduces well the important empirical characteristics of FX rates for major currencies, as the heavy-tailed distribution of returns, their change in shape with the increasing time interval, and the persistence of volatility.

We consider the structure functions S^(q)(τ), i.e. the moments of order q of the increments X(t + τ)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S^(q)(τ)∝τ^ζ(q)). We demonstrate that the nonlinearity of the observed scaling exponent ζ(q) is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity of ζ(q) corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponent q_D > 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.

We perform a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major US stock markets. We consider (i) the trades and quotes (TAQ) database, for which we analyze 40 million records for 1000 US companies for the 2-year period 1994–95, and (ii) the Center for Research and Security Prices (CRSP) database, for which we analyze 35 million daily records for approximately 16,000 companies in the 35-year period 1962–96. We study the probability distribution of returns over varying time scales — from 5 min up to 4 years. For time scales from 5 min up to approximately 16 days, we find that the tails of the distributions can be well described by a power-law decay, characterized by an exponent α ≈ 3 — well outside the stable Lévy regime 0 < α < 2. For time scales greater than 16 days, we observe results consistent with a slow convergence to Gaussian behavior.

We analyze the Standard and Poor's 500 index data of the New York Stock Exchange for more than 32 years. It was suggested earlier that the high frequency data are well described by a truncated Lévy distribution and scaling with respect to the sampling time differences was found. The truncated character of the distribution implies that scaling must break down and that the distribution ultimately converges to a Gaussian. We show by comparing Lévy and Gaussian fits that the characteristic time of the break-down of scaling is of the order of few days. The analysis of the dependence of the kurtosis on the time differences shows that this is much shorter than the time needed for the convergence to the Gaussian being of the order of months.

One of the key features of wavelet analysis is its ability to decompose non-stationary signals according to time and scale. In this work, we use discrete wavelets to analyze the influence of detrending techniques on the time-scale information structure of daily financial data. We examine the use of log returns, a linear trend and the Hodrick–Prescott (HP) filter. Quantitative measurements of information distortion are given using the mean-squared error (MSE) and correlation of the wavelet coefficients between the detrended and original data. We find that log returns and linear detrending are most distortional. We also conclude that the HP-filter is most effective, depending on appropriate selection of the filter parameter, λ, which is for the given data set.

We present compelling empirical evidence for a new interpretation of the Forward Rate Curve (FRC) term structure. We find that the average FRC follows a square-root law, with a prefactor related to the spot volatility, suggesting a Value-at-Risk like pricing. We find a striking correlation between the instantaneous FRC and the past spot trend over a certain time horizon. This confirms the idea of an anticipated trend mechanism proposed earlier and provides a natural explanation for the observed shape of the FRC volatility. We find that the one-factor Gaussian Heath–Jarrow–Morton model calibrated to the empirical volatility function fails to adequately describe these features.

We show that results from the theory of random matrices are potentially of great interest when trying to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series. We find a remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). Finally, we give a specific example to show how this idea can be sucessfully implemented for improving risk management.

We address the question of how to precisely identify correlated behavior between different firms in the economy by applying methods of random matrix theory (RMT). Specifically, we use methods of random matrix theory to analyze the cross-correlation matrix of price changes of the largest 1000 US stocks for the 2-year period 1994–1995. We find that the statistics of most of the eigenvalues in the spectrum of agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. To prove that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations. We demonstrate that shares universal properties with the Gaussian orthogonal ensemble of random matrices. In addition, we quantify the number of significant participants, that is companies, of the eigenvectors using the inverse participation ratio, and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum — a situation reminiscent of results in localization theory.

We select n stocks traded in the New York Stock Exchange and form a statistical ensemble of daily stock returns for each of the k trading days of our database from the stock price time series. We analyze each ensemble of stock returns by extracting its first four central moments. We observe that these moments are fluctuating in time and are stochastic processes themselves. We characterize the statistical properties of central moments by investigating their probability density function and temporal correlation properties.

We have performed a Zipf-like analysis of financial data. We emphasize new results: (i) the existence of long-range correlations for the sign of daily fluctuations of selected size; (ii) the existence of patterns and correlations in the world major trading places. The Zipf analysis allows also to measure the "domino effect" between trading places in the world.

On experimental data the historical volatility is usually calculated by averaging the local variance (or its generalizations) over a finite time window. Already in the case of a constant volatility in the Gaussian model the resulting historical volatility is non-Gaussian distributed. We will calculate historical volatility distributions in the Gaussian and GARCH(1,1) model for different time window sizes and compare them with those obtained from the S&P500 data [1].

Selected data from Polish and USA stock and bond markets as well as foreign-exchange data have been analysed by the use of recurrence plots and the Hurst method. It has been found that there exist significant correlations in some of analysed data chains. Values of recurrence ratios and ratios of determinism calculated from recurrence diagrams decrease significantly if one shuffles the data. The corresponding values of Hurst exponents are in the range 0.56–0.74 and they also decrease after shuffling. The lowest values of the Hurst exponent have been found for single shares at Polish stock market while the highest values are related to foreign-exchange data. The mean length of the cycle calculated from the behaviour of the Hurst exponent for Dow Jones index and S&P500 index is about 5 years while the Warsaw Stock Index WIG possesses the corresponding cycle of order of 11 months. The performed analysis shows that in the economical dynamics the main role is played by stochastic behaviour but traces of deterministic origin can be also seen.

By developing a simple model for market making, price formation can be studied in a nonequilibrium setting. Commonly used trading strategies, such as value investing based on fundamentals, or technical trading based on past price history, generate characteristic price dynamics, including excess volatility, clustered volatility, and long tails. The long term evolution of capital in the market can be modeled in terms of reinvestment of profits, which when combined with the price formation rule leads to generalized Lotka–Volterra equations, originally developed to model predator-prey systems in ecology. Sensible estimates of the characteristic time for substantial capital reallocations are years to decades, suggesting that the progression toward market efficiency may be rather slow.

We explore various extensions of Challet and Zhang's Minority Game in an attempt to gain insight into the dynamics underlying financial markets. First we consider a heterogeneous population where individual traders employ differing "time horizons" when making predictions based on historical data. The resulting average winnings per trader is a highly non-linear function of the population's composition. Second, we introduce a threshold confidence level among traders below which they will not trade. This can give rise to large fluctuations in the "volume" of market participants and the resulting market "price".

We review recent exact analytical results on Minority Game — a binary exactly solvable El Farol's bar problem. Inductive agents minimize the available information, not their losses, thus the stationary state differs from a Nash equilibrium. On the other hand, the same learning dynamics leads to a Nash equilibrium when agents take into account their impact on the market.

New continuous and stochastic extensions of the Minority Game, devised as a fundamental model for a market of competitive agents, are introduced and studied in the context of statistical physics. The new formulation reproduces the key features of the original model, without the need for some of its special assumptions and, most importantly, it demonstrates the crucial role of stochastic decision-making. Furthermore, this formulation provides the exact but novel nonlinear equations for the dynamics of the system.

We investigate the Minority Game, a toy model for agents buying and selling a commodity. The dynamics of the model are decomposed into two processes, firstly, an agent's choice of active strategy, secondly, the interaction between agents while they play. The latter is suitably described by the Hamming distance between strategies. Here, we argue that the first process is described by the history distribution.

The Minority Game was introduced to study the competition between agents with bounded rationality. As the amount of information available decreases, agents manage to arbitrage away all the information, and collective gain is then reduced. This crowd effect arises from the fact that only a minority can profit at each moment, while all agents make their choices using the same input. The properties of the model change drastically if agents make choices based on their individual histories, keeping all remaining rules unaltered. This variation reduces the intrinsic frustration of the model, and improves the tendency towards cooperation and self organization. Finally, we study the stable mixing of individual and cooperative behavior.

We propose a model with heterogeneous interacting traders which can explain the observed cross-correlation between stock return volatility and trading volume. Transaction costs are introduced which, by responding to price movements, create a feedback mechanism on future trading and generates volatility clustering.

It is believed that trading agents often imitate the behaviour of those around them. In its excessive form this imitation can help lead to large increases or decreases in asset-prices over a small time, often described as bubbles and crashes. In this paper we examine a model in which rational agents repeatedly trade one asset whose price is influenced by supply and demand together with a stochastic noise term. Each agent is able to observe and remember the actions of her nearest neighbours. Furthermore the agents receive private information about the asset-price. We find that profit-maximization implies that agents should to some extent imitate the behaviour of the people around them allowing the use of the Ising Spin Model to investigate agent-agent interactions.

Using Chowdhury's simplification of the Cont–Bouchaud percolation algorithm, and including inertia (trading momentum) as well as a nonlinear restoring force (pricing pressure), we find spontaneous oscillations, which for suitable parameters have at first an exponentially increasing period. They seem to mimic the approximate log-periodic oscillations found by Johansen and Sornette in the Nikkei index after the burst of the Japanese bubble a decade ago.

We present a modelling environment for the simulation of interactions between agents of different strategies similar to that reported by Lux (Nature 397 1999). Our freely available software allows market parameters to be adjusted via a simple interface. The incorporation of different population size distributions extends previous work.

I describe some recent advances in the pricing and hedging of derivative instruments. The theme is the reduction of model risk using non-probabilistic models. We see how the nonlinearity in the models leads to natural and optimal choices for static hedges for the reduction of market risk.

We study long-term growth-optimal strategies on a simple market with transaction costs. We show that several problems of this sort can be solved in closed form, and explicit the non-analytic dependance of optimal strategies and expected frictional losses of the friction parameter.

We study investment strategy in different models of financial markets, where the investors cannot attain a perfect knowledge about available assets. The investor spends a certain effort to get information; this allows him to better choose the investment strategy, and puts a selective pressure upon assets. The best strategy is then a compromise between diversification and effort to get information.

We introduce a faithful representation of the heavy tail multivariate distribution of asset returns, as parsimonious as the Gaussian framework. Using calculation techniques of functional integration and Feynman diagrams borrowed from particle physics, we characterize precisely, through its cumulants of high order, the distribution of wealth variations of a portfolio composed of an arbitrary mixture of assets. This approach makes quantitative and rigorous the well-known fact that minimizing the variance, i.e. the relatively "small" risks, often increases larger risks as measured by higher normalized cumulants and the Value-at-Risk.

The standard model defined in the Capital Adequacy Directive issued by the EEC in 1993 imposes nonlinear constraints on certain parts of the trading portfolios of financial institutions. It is shown that an institution that complies with the rules of the standard model but wants to optimize its portfolio according to some internal criteria, such as minimizing the variance or the VaR, faces a computational problem equivalent to finding the ground states of a long range spin glass. This problem is known to be NP-complete, and to have exponentially many solutions which are extremely sensitive to any changes of the input parameters.

A general class of truncated Lévy processes is introduced, and possible ways of fitting parameters of the constructed family of truncated Lévy processes to data are discussed. For a market of a riskless bond and a stock whose log-price follows a truncated Lévy process, TLP-analogs of the Black–Scholes equation, the Black–Scholes formula, the Dynkin derivative and the Leland's model are obtained, a locally risk-minimizing portfolio is constructed, and an optimal exercise price for a perpetual American put is computed.

In recent years there has been much attention paid to pricing and hedging models that are more general than the Black and Scholes' one, whose hypotheses often aren't satisfied by true market data. Sato and Takayasu proposed a market model that can produce price fluctuations with infinite variance from a deterministic behaviour of many market's dealers as it emerged from their simulations. We check the possibility to apply this model to an Italian stock, the "Assicurazioni Generali" and determine the density function of the multiplicative and additive noise appearing in the proposed Langevin equation.

We find also the Lévy distribution for the changes. This distribution is consistent with the histograms and the Kernel estimates. These results are used for hedging following the approach of Sornette and Bouchaud.

We develop a recombining tree method for pricing of options by using a general two-factor stochastic-variance (SV) diffusion model for asset price dynamics. We show that it is possible to construct a riskless hedge by including additional short-term options in the hedging portfolio. This procedure gives us Partial Differential Equation (PDE) that can be solved by using standard numerical techniques giving us a unique option's price. We show that the option's price does not depend on the long run volatility forecast but only on the parameters of the model, which are related to the volatility of variance. We show one particular transformation of PDE to the Finite Difference Equation (FDE) that leads to the three-dimensional lattice method similar to the standard binomial-tree method. Our tree grows in the price-variance space and similarly to the binomial-tree, the coefficients of the FDE can be interpreted as risk neutral probabilities for jumping between the tree nodes. By investigating an error accumulation in the tree we found the stability criterion and the method that can be applied in order to achieve a stabile procedure. Our option pricing method can be used for European and American options and various payoffs. Since the procedure converges quickly and can be easily implemented, we believe that it could be useful for practitioners. The SV model we used here was shown earlier to be a diffusion limit of various GARCH-type models giving the possibility of using parameters obtained in the discrete-time GARCH framework as an input for our option pricing method.

We study weak convergence of a sequence of (approximating) asset price models Sⁿ to a limiting model S: both Sⁿ and S are multi-dimensional asset price processes with some physical probability measures Pⁿ resp. P, and a natural notion of process convergence is the weak convergence of the induced path probability measures, denoted by (Sⁿ|Pⁿ) resp. (S|P), on the abstract topological space of possible asset price trajectories.

For the purpose of no-arbitrage pricing of options or more general derivatives on the model assets, there are two different aspects of this convergence: (i) convergence under the given physical probability measures, (Sⁿ|Pⁿ) → (S|P) and (ii) convergence under suitably chosen equivalent martingale measures (EMM) relevant for pricing derivatives, (Sⁿ|Qⁿ) → (S|Q). A simple example is the convergence of a sequence of discrete-time binomial models to the Black–Scholes model (geometric Brownian motion), where the model markets are complete and hence the choice of Qⁿ resp. Q is unique. This example and the general case of complete limit markets (S|P) have been studied in [1].

In contrast we have several choices for Qⁿ resp. Q when all the model markets are incomplete. A natural choice is the minimum entropy EMM [2, 3], defined as the (unique) EMM R minimizing the relative entropy H(R|P) to the physical measure P, among all EMMs. We prove the following: given that the approximating models converge under the physical measures, (Sⁿ|Pⁿ) → (S|P) — with some mild assumptions on P and on the minimum entropy EMMs Rⁿ for Sⁿ resp. R for S — entropy number convergence implies weak convergence of the minimum entropy option price processes: H(Rⁿ|Pⁿ) → H(R|P) implies (Sⁿ|Rⁿ) → (S|R). Several rigorous examples illustrate the result; cf. also [4].

We extend the neural-network approach for the valution of financial derivatives developed by Hutchinson et al. [1] to the case of fat-tailed distributions of the underlying asset returns. We use a two-layer perceptron with three inputs, four hidden neurons, and one output. The input parameters of the network are: the simulated price of the underlying asset F divided by the strike price E, the time-to-maturity T, and the ratio |F-E|/T. The latter takes into account the volatility smile, whereas the price F is generated by the method of Gorenflo et al. [2] based on fractional calculus. The output parameter is the call price C over E. The learning-set option price C is computed by means of a formula given by Bouchaud and Potters [3, 4]. Option prices obtained by means of this learning scheme are compared with LIFFE option prices on German treasury bond (BUND) futures.

A diffusion equation for the price evolution of the Italian share "Olivetti" is found by investigating a series of its data. The coefficients of this equation are found by using the maximum likelihood method based on martingale theory. We evaluate pricing and hedging strategy by the Sornette and Bouchaud approach.

This paper develops a corporate bond valuation model that incorporates a default barrier with dynamics depending on stochastic interest rates and variance of the corporate bond function. Since the volatility of the firm value affects the level of leverage over time through the variance of the corporate bond function, more realistic default scenarios can be put into the valuation model. When the firm value touches the barrier, bondholders receive an exogenously specified number of riskless bonds. We derive a closed-form solution of the corporate bond price as a function of firm value and a short-term interest rate, with time-dependent model parameters governing the dynamics of the firm value and interest rate. The numerical results show that the dynamics of the barrier has material impact on the term structures of credit spreads. This model provides new insight for future research on risky corporate bonds analysis and modelling credit risk.

In value-at-risk (VaR) methodology of option risk measurement, the determination of market values of the current option positions under various market scenarios is critical. Under the full revaluation and factor sensitivity approach which are accepted by regulators, accurate revaluation and precise factor sensitivity calculation of options in response to significant moves in market variables are important for measuring option risks in terms of VaR figures. This paper provides a method for pricing equity options in the constant elasticity variance (CEV) model environment using the Lie-algebraic technique when the model parameters are time-dependent. Analytical solutions for option values incorporating time-dependent model parameters are obtained in various CEV processes. The numerical results, which are obtained by employing a very efficient computing algorithm similar to the one proposed by Schroder [11], indicate that the option values are sensitive to the time-dependent volatility term structures. It is also possible to generate further results using various functional forms for interest rate and dividend term structures. From the analytical option pricing formulae, one can achieve more accuracy to compute factor sensitivities using more realistic term-structures in volatility, interest rate and dividend yield. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black–Scholes model, more precise risk management in equity options can be achieved by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.

Bagging different predictors for an exchange index can provide higher returns than by only using a single strategy.

A model for a stock market bubble is based upon the assumption that market agents incorporate trend and deviation of the actual price from the fundamental value into their expectations. Possible price dynamics are analyzed, and necessary conditions for large price deviations are obtained.

We analyse the Hopf bifurcations in the Kaldor–Kalecki business cycle model represented by a delay differential equation system.