Optimal investment horizons

In stochastic finance, one traditionally considers the return as a competitive measure of an asset, i.e., the profit generated by that asset after some fixed time span Δt, say one week or one year. This measures how well (or how bad) the asset performs over that given period of time. It has been established that the distribution of returns exhibits “fat tails” indicating that large returns occur more frequently than what is expected from standard Gaussian stochastic processes [1-3]. Instead of estimating this “fat tail” distribution of returns, we propose here an alternative approach, which is outlined by addressing the following question: What is the smallest time interval needed for an asset to cross a fixed return level of say 10%? For a particular asset, we refer to this time as the investment horizon and the corresponding distribution as the investment horizon distribution. This latter distribution complements that of returns and provides new and possibly crucial information for portfolio design and risk-management, as well as for pricing of more exotic options. By considering historical financial data, exemplified by the Dow Jones Industrial Average, we obtain a novel set of probability distributions for the investment horizons which can be used to estimate the optimal investment horizon for a stock or a future contract. Received 20 February 2002 Published online 25 June 2002     

Entropy Based Student’s t-Process Dynamical Model

Volatility, which represents the magnitude of fluctuating asset prices or returns, is used in the problems of finance to design optimal asset allocations and to calculate the price of derivatives. Since volatility is unobservable, it is identified and estimated by latent variable models known as volatility fluctuation models. Almost all conventional volatility fluctuation models are linear time-series models and thus are difficult to capture nonlinear and/or non-Gaussian properties of volatility dynamics. In this study, we propose an entropy based Student’s t-process Dynamical model (ETPDM) as a volatility fluctuation model combined with both nonlinear dynamics and non-Gaussian noise. The ETPDM estimates its latent variables and intrinsic parameters by a robust particle filtering based on a generalized H-theorem for a relative entropy. To test the performance of the ETPDM, we implement numerical experiments for financial time-series and confirm the robustness for a small number of particles by comparing with the conventional particle filtering.     

Portfolio Optimization with a Mean–Absolute Deviation–Entropy Multi-Objective Model

Investors wish to obtain the best trade-off between the return and risk. In portfolio optimization, the mean-absolute deviation model has been used to achieve the target rate of return and minimize the risk. However, the maximization of entropy is not considered in the mean-absolute deviation model according to past studies. In fact, higher entropy values give higher portfolio diversifications, which can reduce portfolio risk. Therefore, this paper aims to propose a multi-objective optimization model, namely a mean-absolute deviation-entropy model for portfolio optimization by incorporating the maximization of entropy. In addition, the proposed model incorporates the optimal value of each objective function using a goal-programming approach. The objective functions of the proposed model are to maximize the mean return, minimize the absolute deviation and maximize the entropy of the portfolio. The proposed model is illustrated using returns of stocks of the Dow Jones Industrial Average that are listed in the New York Stock Exchange. This study will be of significant impact to investors because the results show that the proposed model outperforms the mean-absolute deviation model and the naive diversification strategy by giving higher a performance ratio. Furthermore, the proposed model generates higher portfolio mean returns than the MAD model and the naive diversification strategy. Investors will be able to generate a well-diversified portfolio in order to minimize unsystematic risk with the proposed model.     

PID feedback controller used as a tactical asset allocation technique: The G.A.M. model

The objective of this paper is to illustrate a tactical asset allocation technique utilizing the PID controller. The proportional-integral-derivative (PID) controller is widely applied in most industrial processes; it has been successfully used for over 50 years and it is used by more than 95% of the plants processes. It is a robust and easily understood algorithm that can provide excellent control performance in spite of the diverse dynamic characteristics of the process plant.In finance, the process plant, controlled by the PID controller, can be represented by financial market assets forming a portfolio. More specifically, in the present work, the plant is represented by a risk-adjusted return variable. Money and portfolio managers’ main target is to achieve a relevant risk-adjusted return in their managing activities. In literature and in the financial industry business, numerous kinds of return/risk ratios are commonly studied and used.The aim of this work is to perform a tactical asset allocation technique consisting in the optimization of risk adjusted return by means of asset allocation methodologies based on the PID model-free feedback control modeling procedure. The process plant does not need to be mathematically modeled: the PID control action lies in altering the portfolio asset weights, according to the PID algorithm and its parameters, Ziegler-and-Nichols-tuned, in order to approach the desired portfolio risk-adjusted return efficiently.     

Downside Risk analysis applied to the Hedge Funds universe

Hedge Funds are considered as one of the portfolio management sectors which shows a fastest growing for the past decade. An optimal Hedge Fund management requires an appropriate risk metrics. The classic CAPM theory and its Ratio Sharpe fail to capture some crucial aspects due to the strong non-Gaussian character of Hedge Funds statistics. A possible way out to this problem while keeping the CAPM simplicity is the so-called Downside Risk analysis. One important benefit lies in distinguishing between good and bad returns, that is: returns greater or lower than investor's goal. We revisit most popular Downside Risk indicators and provide new analytical results on them. We compute these measures by taking the Credit Suisse/Tremont Investable Hedge Fund Index Data and with the Gaussian case as a benchmark. In this way, an unusual transversal lecture of the existing Downside Risk measures is provided.     

Noisy covariance matrices and portfolio optimization

According to recent findings [#!bouchaud!#,#!stanley!#], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [#!bouchaud!#], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [#!bouchaud!#,#!stanley!#] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions. Received 31 December 2001     

Stochastic relaxational dynamics applied to finance: Towards non-equilibrium option pricing theory

Non-equilibrium phenomena occur not only in the physical world, but also in finance. In this work, stochastic relaxational dynamics (together with path integrals) is applied to option pricing theory. Equilibrium in financial markets is defined as the absence of arbitrage, i.e. profits “for nothing”. A recently proposed model (by Ilinski et al.) considers fluctuations around this equilibrium state by introducing a relaxational dynamics with random noise for intermediate deviations called “virtual” arbitrage returns. In this work, the model is incorporated within a martingale pricing method for derivatives on securities (e.g. stocks) in incomplete markets using a mapping to option pricing theory with stochastic interest rates. The arbitrage return is considered as a component of a fictitious short-term interest rate in a virtual world. The influence of intermediate arbitrage returns on the price of derivatives in the real world can be recovered by performing an average over the (non-observable) arbitrage return at the time of pricing. Using a famous result by Merton and with some help from the path integral method, exact pricing formulas for European call and put options under the influence of virtual arbitrage returns (or intermediate deviations from economic equilibrium) are derived where only the final integration over initial arbitrage returns needs to be performed numerically. This result, which has not been given previously and is at variance with results stated by Ilinski et al., is complemented by a discussion of the hedging strategy associated to a derivative, which replicates the final payoff but turns out to be not self-financing in the real world, but self-financing when summed over the derivative's remaining life time. Numerical examples are given which underline the fact that an additional positive risk premium (with respect to the Black-Scholes values) is found reflecting extra hedging costs due to intermediate deviations from economic equilibrium. Received 16 June 1999 and Received in final form 26 September 1999     

The Information Conveyed in a SPAC′s Offering

The popularity of SPACs (Special Purpose Acquisition Companies) has grown dramatically in recent years as a substitute for the traditional IPO (Initial Public Offer). We modeled the average annual return for SPAC investors and found that this financial tool produced an annual return of 17.3%. We then constructed an information model that examined a SPAC′s excess returns during the 60 days after a potential merger or acquisition had been announced. We found that the announcement had a major impact on the SPAC’s share price over the 60 days, delivering on average 0.69% daily excess returns over the IPO portfolio and 31.6% cumulative excess returns for the entire period. Relative to IPOs, the cumulative excess returns of SPACs rose dramatically in the next few days after the potential merger or acquisition announcement until the 26th day. They then declined but rose again until the 48th day after the announcement. Finally, the SPAC’s structure reduced the investors’ risk. Thus, if investors buy a SPAC stock immediately after a potential merger or acquisition has been announced and hold it for 48 days, they can reap substantial short-term returns.     

Characteristics of the volatility in the Korea composite stock price index

We empirically analyze the time series of the Korea Composite Stock Price Index (KOSPI) from March of 1992 to February of 2007 using methods from the hydrodynamic turbulence. To this end, we focus on characteristics of the return and volatility, which are respectively the price change and a measure of the financial market fluctuation over a time interval. With these, we show that the non-Gaussian probability distribution of the return can be modeled by the convolution of the conditional probability distribution of the return given the volatility and the distribution of the volatility per se. From this model, we suggest that the non-Gaussian characteristic of the return results from the fluctuation of the volatility. That is, a large return is partly, if not entirely, due to the market fluctuation in a long time scale influencing the fluctuation in a short time scale via net information flow. We further show that the volatility has a multi-fractal property, which resembles the multifractality of the energy dissipation in the turbulence.     

Transaction fees and optimal rebalancing in the growth-optimal portfolio

The growth-optimal portfolio optimization strategy pioneered by Kelly is based on constant portfolio rebalancing which makes it sensitive to transaction fees. We examine the effect of fees on an example of a risky asset with a binary return distribution and show that the fees may give rise to an optimal period of portfolio rebalancing. The optimal period is found analytically in the case of lognormal returns. This result is consequently generalized and numerically verified for broad return distributions and returns generated by a GARCH process. Finally we study the case when investment is rebalanced only partially and show that this strategy can improve the investment long-term growth rate more than optimization of the rebalancing period.     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000     

Wavelet multiscale analysis for Hedge Funds: Scaling and strategies

The wide acceptance of Hedge Funds by Institutional Investors and Pension Funds has led to an explosive growth in assets under management. These investors are drawn to Hedge Funds due to the seemingly low correlation with traditional investments and the attractive returns. The correlations and market risk (the Beta in the Capital Asset Pricing Model) of Hedge Funds are generally calculated using monthly returns data, which may produce misleading results as Hedge Funds often hold illiquid exchange-traded securities or difficult to price over-the-counter securities. In this paper, the Maximum Overlap Discrete Wavelet Transform (MODWT) is applied to measure the scaling properties of Hedge Fund correlation and market risk with respect to the S&P 500. It is found that the level of correlation and market risk varies greatly according to the strategy studied and the time scale examined. Finally, the effects of scaling properties on the risk profile of a portfolio made up of Hedge Funds is studied using correlation matrices calculated over different time horizons.     

Risk minimization through portfolio replication

We use a replica approach to deal with portfolio optimization problems. A given risk measure is minimized using empirical estimates of asset values correlations. We study the phase transition which happens when the time series is too short with respect to the size of the portfolio. We also study the noise sensitivity of portfolio allocation when this transition is approached. We consider explicitely the cases where the absolute deviation and the conditional value-at-risk are chosen as a risk measure. We show how the replica method can study a wide range of risk measures, and deal with various types of time series correlations, including realistic ones with volatility clustering.     

Stylized facts in internal rates of return on stock index and its derivative transactions

Universal features in stock markets and their derivative markets are studied by means of probability distributions in internal rates of return on buy and sell transaction pairs. Unlike the stylized facts in normalized log returns, the probability distributions for such single asset encounters incorporate the time factor by means of the internal rate of return, defined as the continuous compound interest. Resulting stylized facts are shown in the probability distributions derived from the daily series of TOPIX, S & P 500 and FTSE 100 index close values. The application of the above analysis to minute-tick data of NIKKEI 225 and its futures market, respectively, reveals an interesting difference in the behavior of the two probability distributions, in case a threshold on the minimal duration of the long position is imposed. It is therefore suggested that the probability distributions of the internal rates of return could be used for causality mining between the underlying and derivative stock markets. The highly specific discrete spectrum, which results from noise trader strategies as opposed to the smooth distributions observed for fundamentalist strategies in single encounter transactions may be useful in deducing the type of investment strategy from trading revenues of small portfolio investors.     

基于上证指数高频数据的中国资本市场微观特性研究

以上证指数高频数据为研究对象,基于上涨、平缓和下跌三个市场状态分析我国金融市场的微观特性.通过分析上证指数在不同时间间隔下的概率分布、自相关性和多分形三个特性,发现上证指数对数增量序列存在厚尾、列维非高斯分布特征,且随着时间间隔的增大,收益序列愈收敛于正态分布,其中,下降趋势收敛于正态分布的速度更快,拟合于列维分布的效果更好.最为突出的是,在自相关函数分析中,上证指数的收益率无长期记忆性,而波动率则具有较强的记忆性.同时,波动率的自相关性存在明显的周期性特征,即T=240 min,且在下降趋势时其相关性最高.在以时间增量刻画的多重分形结构中,对于不同的时间序列、时间间隔,由于受投资期限和流动性的影响,三种股市状态的收益率波动存在着短期和长期性的差异.上证指数的总体宏观行为与国际成熟股市较为一致,但在微观特性上仍存在显著差异,其所特有的周期性是投资者的惯性反冲所致,而自相关性函数较之成熟股市衰减较慢,则表明投资者的投资行为更多地受历史信息的影响.     

European and American options: The semi-Markov case

In this paper, we assume that the log return of the underlying asset follows a semi-Markov process, then from the knowledge of the kernel we derive an explicit expression for the value of the option and for the bare risk in the case of the European call (put) option and, by means of a recursive system, we derive the value and the bare risk in the case of the American option. The prices and risks we obtained depend explicitly on the waiting-time distributions of the asset and they are duration dependent. The link with models based on Markov Chains and Continuous Time Random Walks is debated.     

Investment strategies and hidden variables

The present study shows how the information on `hidden' market variables effects optimal investment strategies. We take the point of view of two investors, one who has access to the hidden variables and one who only knows the quotes of a given asset. Following Kelly's theory on investment strategies, the Shannon information and the doubling investment rate are quantified for both investors. Thanks to his privileged knowledge, the first investor can follow a better investment strategy. Nevertheless, the second investor can extract some of the hidden information looking at the past history of the asset variable. Unfortunately, due to the complexity of his strategy, this investor will have computational difficulties when he tries to apply it. He will than follow a simplified strategy, based only on the average sign of the last l quotes of the asset. This results have been tested with some Monte Carlo simulations.     

Financial Network Analysis on the Performance of Companies Using Integrated Entropy–DEMATEL–TOPSIS Model

In this paper, we propose a multi-criteria decision making (MCDM) model by integrating the entropy–DEMATEL with TOPSIS model to analyze the causal relationship of financial ratios towards the financial performance of the companies. The proposed model is illustrated using the financial data of the companies of Dow Jones Industrial Average (DJIA). The financial network analysis using entropy–DEMATEL shows that the financial ratios such as debt to equity ratio (DER) and return on equity (ROE) are classified into the cause criteria group, whereas current ratio (CR), earnings per share (EPS), return on asset (ROA) and debt to assets ratio (DAR) are categorized into the effect criteria group. The top three most influential financial ratios are ROE, CR and DER. The significance of this paper is to determine the causal relationship of financial network towards the financial performance of the companies with the proposed entropy–DEMATEL–TOPSIS model. The ranking identification of the companies in this study is beneficial to the investors to select the companies with good performance in portfolio investment. The proposed model has been applied and validated in the portfolio investment using a mean-variance model based on the selection of companies with good performance. The results show that the proposed model is able to generate higher mean return than the benchmark DJIA index at minimum risk. However, short sale is not allowed for the applicability of the proposed model in portfolio investment.     

Volatility return intervals analysis of the Japanese market

We investigate scaling and memory effects in return intervals between price volatilities above a certain threshold q for the Japanese stock market using daily and intraday data sets. We find that the distribution of return intervals can be approximated by a scaling function that depends only on the ratio between the return interval τ and its mean 〈τ〉. We also find memory effects such that a large (or small) return interval follows a large (or small) interval by investigating the conditional distribution and mean return interval. The results are similar to previous studies of other markets and indicate that similar statistical features appear in different financial markets. We also compare our results between the period before and after the big crash at the end of 1989. We find that scaling and memory effects of the return intervals show similar features although the statistical properties of the returns are different.     

Estimating risk of foreign exchange portfolio: Using VaR and CVaR based on GARCH-EVT-Copula model

This paper introduces GARCH-EVT-Copula model and applies it to study the risk of foreign exchange portfolio. Multivariate Copulas, including Gaussian, t and Clayton ones, were used to describe a portfolio risk structure, and to extend the analysis from a bivariate to an n-dimensional asset allocation problem. We apply this methodology to study the returns of a portfolio of four major foreign currencies in China, including USD, EUR, JPY and HKD. Our results suggest that the optimal investment allocations are similar across different Copulas and confidence levels. In addition, we find that the optimal investment concentrates on the USD investment. Generally speaking, t Copula and Clayton Copula better portray the correlation structure of multiple assets than Normal Copula.