Session B33: Financial Markets and Other Applications of Statistical Physics

What physicists should learn about finance (if they want to)

There has been growing interest among physicists to Econophysics, i.e. analysis and modeling of financial and economic processes using the concepts of theoretical Physics. There has been also perception that the financial industry is a viable alternative for those physicists who are not able or are not willing to pursue career in their major field. However in our times, the Wall Street expects from applicants for quantitative positions not only the knowledge of the stochastic calculus and the methods of time series analysis but also of such concepts as option pricing, portfolio management, and risk measurement. Here I describe a synthetic course based on my book ``Quantitative Finance for Physicists'' (Elsevier, 2004) that outlines both worlds: Econophysics and Mathematical Finance. This course may be offered as elective for senior undergraduate or graduate Physics majors.   

Relation between bid-ask spread and volatility in financial markets

We establish empirically a linear relation between the bid-ask spread and the volatility per trade in stock markets. We give a theoretical argument explaining this relation and why it should hold on all electronic markets.   

An equation of state for the financial markets: connecting order flow to price formation.

Many of the peculiarities of price formation in the financial marketplace can be understood as the result of a few regularities in the placement and removal of trading orders. Based on a large data set from the London Stock Exchange we show that the distribution of prices where people place orders to buy or sell follows a surprisingly simple functional form that depends on the current best prices. In addition, whether or not an order is to buy or sell is described by a long-memory process, and the cancellation of orders can be described by a few simple rules. When these results are combined, simply by following the rules of the continuous double auction, the resulting simulation model produces good predictions for the distribution of price changes and transaction costs without any adjustment of parameters. We use the model to empirically derive equations of state relating order flow and the statistical properties of prices. In contrast to previous conjectures, our results demonstrate that these distributions are not universal, but rather depend on parameters of individual markets. They also show that factors other than supply and demand play an important role in price formation.   

An empirical analysis of waiting times for price changes and orders in a financial market

We discuss an empirical analysis of waiting time distribution for price changes and orders in a financial market and its Weibull approximation. It is widely assumed that trades in financial markets occur independently and the waiting time distribution is exponential. However, recent empirical results [Raberto et al 2002, Scalas et al 2005 etc] of high frequency financial data show that the distribution is non-exponential. Therefore, in order to understand market behavior quantitatively and systematically, it is important to check the validity of the exponential distribution hypothesis and which non-exponential distribution is appropriate. In this talk, we analyze the waiting times of Sony bank USD/JPY rate and orders. We show that the waiting time distribution for not only price changes, but also orders, is non-exponential by using non-double auction market data. We also measure exactly how much better the Weibull distribution is as an approximation by using the Weibull paper and divergence measurements. Moreover, the estimated value of the shape parameter in Weibull distribution is similar in both price changes and orders waiting time distributions.   

Dynamics of the return distribution in the Korean financial market

In this paper, we studied the dynamics of the log-return distribution of the Korean Composition Stock Price Index (KOSPI) from 1992 to 2004. Based on the microscopic spin model, we found that while the index during the late 1990s showed a power-law distribution, the distribution in the early 2000s was exponential. This change in distribution shape was caused by the duration and velocity, among other parameters, of the information that flowed into the market.   

Grouping in the stock markets of Japan and Korea

We investigated the temporally evolving network structures of the Japanese and Korean stock markets through the minimum spanning trees composed of listed stocks. We tested the validity of conventional grouping by industrial categories, and found a common trend of decrease for Japan and Korea. This phenomenon supports the increasing external effects on the markets due to the globalization of both countries. At last the Korean market are grouped with the MSCI Korea Index, a good reference for foreigners' trading, in the early 2000s. In the Japanese market, this tendency is strengthened more and more by burst of the bubble in 1990's.   

Dynamical Structures of High-Frequency Financial Data

We study the dynamical behavior for high-frequency data of the Korean stock price index (KOSPI) using the movement of returns in Korean financial markets. It is shown that the dynamical behavior for a binarized series of our models is not completely random. The conditional probability are numerically estimated from a return series of tick data in the KOSPI. Non-trivial probability structures can be constituted from binary time series of the autoregressive (AR), logit, and probit models for which the Akaike Information Criterion (IC) value shows a minimum value at the $15$th-order. From our result, the value of correct match ratio for the AR model is found to relatively have slightly larger than calculated findings of other models.   

Systematic Analysis of Group Identification in Stock Markets

We propose improved methods to identify stock groups using the correlation matrix of stock price changes. By filtering out the marketwide effect and the random noise, we construct the correlation matrix of stock groups in which nontrivial high correlations between stocks are found. Using the filtered correlation matrix, we successfully identify the multiple stock groups without any extra knowledge of the stocks by the optimization of the matrix representation and the percolation approach to the correlation-based network of stocks. These methods drastically reduce the ambiguities while finding stock groups using the eigenvectors of the correlation matrix.   

Virtual Volatility, an Elementary New Concept with Surprising Stock Market Consequences

Textbook investors start by predicting the future price distribution, PDF, of a candidate stock (or portfolio) at horizon T, e.g. a year hence. A (log)normal PDF with center (=drift =expected return) $\mu $T and width (=volatility) $\sigma \surd $T is often assumed on Central Limit Theorem grounds, i.e. by a random walk of daily (log)price increments $\Delta $s. The standard deviation, stdev, of historical (\textit{ex post}) $\Delta $s `s is usually a fair predictor of the coming year's (\textit{ex ante}) stdev($\Delta $s) = $\sigma _{daily}$, but the historical mean E($\Delta $s)\textit{ at best }roughly limits the true, to be predicted, drift by $\mu _{true}$T$\sim \quad \mu _{hist}$T $\pm $ $\sigma _{hist}\surd $T. Textbooks take a PDF with $\sigma \quad \sim $ $\sigma _{daily }$ and $\mu $ as somehow known, as if accurate predictions of $\mu $ were possible. It is elementary and presumably new to argue that an average of PDF's over a range of $\mu $ values should be taken, e.g. an average over forecasts by different analysts. We estimate that this leads to a PDF with a `virtual' volatility $\sigma \quad \sim $ 1.3$\sigma _{daily.}$ It is indeed clear that uncertainty in the value of the expected gain parameter increases the risk of investment in that security by most measures, e. g. Sharpe's ratio $\mu $T/$\sigma \surd $T will be 30{\%} smaller because of this effect. It is significant and surprising that there are investments which\textit{ benefit }from this 30{\%} virtual increase in the volatility   

Volatility, Persistence, and Survival in Financial Markets

We study the temporal fluctuations in time-dependent stock prices (both individual and composite) as a stochastic phenomenon using general techniques and methods of nonequilibrium statistical mechanics. In particular, we analyze stock price fluctuations as a non-Markovian stochastic process using the first-passage statistical concepts of persistence and survival. We report the results of empirical measurements of the normalized $q$-order correlation functions $f_q(t)$, survival probability $S(t)$, and persistence probability $P(t)$ for several stock market dynamical sets. We analyze both minute-to-minute and higher frequency stock market recordings. We find that the fluctuating stock price is multifractal and the choice of the sampling time has no effect on the qualitative multifractal behavior displayed by the $1/q$-dependence of the generalized Hurst exponent $H_q$. The probability $S(t)$ of the stock price remaining above the average up to time $t$ is very sensitive to the total measurement time $t_m$ and the sampling time. The probability $P(t)$ of the stock not returning to the initial value within an interval $t$ has a universal power-law behavior, $P(t)\sim t^{-\theta}$, with a persistence exponent $\theta$ close to $0.5$ that agrees with the prediction $\theta=1-H_2$. The empirical financial stocks also present an interesting feature found in turbulent fluids, the extended self-similarity. This work is partially supported by the NSF and U.S. ONR.   

Comparing Extremal and Hysteretic Optimization on the Satisfiability Problem

We apply physically inspired optimization methods to the classical combinatorial Satisfiablity problem. Treating the usual boolean variables as Ising spins and each clause as a p-spin interaction we can use the pre-existing physical intuition about spin glasses and magnetic systems to find the optimal solution for this problem (the ground state energy). We compare the performance of Extremal Optimization\footnote{PRL 23:5211, 2001} ($\tau EO$) and Hysteretic Optimization\footnote{PRL 89:150201, 2002} ($HO$) and determine the parameter values that provide the best results. Comparisons with previously published results on well known benchmarks\footnote{DIMACS 35:393, 1997} are also made.   

What is the most interesting team sport?

What is the most interesting team sport? We answer this question via an extensive statistical survey of game scores, consisting of more than 1/4 million games in over a century. We propose the likelihood of upsets as a measure of competitiveness. We demonstrate the utility of this measure via a comparative analysis of several popular team sports including soccer, baseball, hockey, basketball, and football. We also develop a mathematical model, in which the stronger team is favored to win a game. This model allows to us conveniently estimate the likelihood of upsets from the more easily-accessible standings data.   

Diagnosis of weaknesses in modern error correction codes: a physics approach

One of the main obstacles to the wider use of the modern error-correction codes is that, due to the complex behavior of their decoding algorithms, no systematic method which would allow characterization of the bit-error-rate (BER) is known. This is especially true at the weak noise where many systems operate and where coding performance is difficult to estimate because of the diminishingly small number of errors. We show how the instanton method of physics allows one to solve the problem of BER analysis in the weak noise range by recasting it as a computationally tractable minimization problem. The material is based on Phys. Rev. Lett. 95 (22) 228701 (2005).