Multivariate Lévy processes with dependent jump intensity

In this work we propose a new and general approach to build dependence in multivariate Lévy processes. We fully characterize a multivariate Lévy process whose margins are able to approximate any Lévy type. Dependence is generated by one or more common sources of jump intensity separately in jumps of any sign and size and a parsimonious method to determine the intensities of these common factors is proposed. Such a new approach allows the calibration of any smooth transition between independence and a large amount of linear dependence and provides greater flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. The model is analytically tractable and a straightforward multivariate simulation procedure is available. An empirical analysis shows an accurate multivariate fit of stock returns in terms of linear and nonlinear dependence. A numerical illustration of multi-asset option pricing emphasizes the importance of the proposed new approach for modeling dependence.     

Forward-looking portfolio selection with multivariate non-Gaussian models

In this study, we suggest a portfolio selection framework based on time series of stock log-returns, option-implied information, and multivariate non-Gaussian processes. We empirically assess a multivariate extension of the normal tempered stable (NTS) model and of the generalized hyperbolic (GH) one by implementing an estimation method that simultaneously calibrates the multivariate time series of log-returns and, for each margin, the univariate observed one-month implied volatility smile. To extract option-implied information, the connection between the historical measure P and the risk-neutral measure Q, needed to price options, is provided by the multivariate Esscher transform. The method is applied to fit a 50-dimensional series of stock returns, to evaluate widely known portfolio risk measures and to perform a forward-looking portfolio selection analysis. The proposed models are able to produce asymmetries, heavy tails, both linear and non-linear dependence and, to calibrate them, there is no need for liquid multivariate derivative quotes.     

Estimating non-linear serial and cross-interdependence between financial assets

This paper proposes an approach based on copula families to determine shape and magnitude of non-linear serial and cross-interdependence between returns and volatilities of financial assets. It is evident the predominance of the student’s t copula in returns relationships. Association in tails is generally larger than the absolute. There is a fast decrease in association along time, but even after 5 days, there is still dependence between returns. For volatilities, Joe copula predominates in estimated bivariate relationships fit. Clayton copula rotated 180° (survival), Gumbel, BB6 and BB8 copulas also fit some relationships. The magnitude of lagged associations is larger for risks than returns. Persistence in the dependences is very high, and decreases very little after the first lag. The tail dependence has larger values than the absolute in most relationships. We present a practical application of the proposed approach, based on optimal investment allocation and risk prediction.     

Non-linear dependence and conditional heteroscedasticity in stock returns evidence from the norwegian thinly traded equity market

Abstract

The paper investigates the presence of non-linear dependencies in stock returns for the Norwegian equity market as it is very difficult to interpret the unconditional distribution of stock returns and its economic implications if the i.i.d. assumption is violated. Standard tests of non-linear dependence give strong evidence for the presence of non-linearity in raw returns. Modelling non-linear dependence must distinguish between models that are non-linear in mean and hence depart from the Martingale hypothesis, and models that are non-linear in variance and hence depart from independence but not from the Martingale hypothesis. Therefore, three non-linear models of asset returns are formulated applying ARMA-GARCH specifications for the conditional mean and variance equations. The paper goes on to answer which model has the necessary characteristics that are sufficient to account for most of the non-linear dependence. In the Norwegian equity market most of the non-linear dependence seems to be conditional heteroscedasticity. However, the most thinly traded assets still report significant non-linear dependence for all non-linear specifications. These results imply that the independence hypothesis can be rejected for all assets, portfolios and indices. Moreover, for thinly traded assets the Martingale hypothesis can also be rejected. The economic implications from the unconditional distributions of thinly traded assets are therefore very difficult to interpret and are unfamiliar territory for those who are accustomed to thinking analytically, intuitively and linearly.     

Non-linear Dependence in Stock Returns: Does Trading Frequency Matter?

The UK has a quote-driven pure dealer market structure that is very different from order driven markets such as the NYSE and Japanese markets. This paper investigates non-linear dependence in stock returns for an exhaustive sample of UK stocks for a 21 year period. The results are analysed on the basis of trading frequency. It is found that non-linear dependence is highly significant in all cases for both individual stocks and stock portfolios formed on the basis of trading frequency. The non-linear dependence is primarily over a one day interval, although statistically significant non-linear dependence exists consistently even up to five trading days. Most of the non-linear dependence is in the form of ARCH-type conditional heteroskedasticity. However, statistically significant non-linearity in addition to an EGARCH(1,1) dependence also appears to be present. This additional non-linearity is greater for individual stocks than for portfolios and greater for smaller, less-liquid portfolios. Non-linear dependence does not appear to be caused by non-stationarity in underlying economic fundamentals or by non-linearity in the conditional mean. However, low dimensional chaos is not generally supported. The limited evidence on chaotic behaviour is stronger for portfolios with long price adjustment delays across component stocks. The main results are consistent with US studies on stock indices, suggesting that the process generating non-linear dependence is not dependent on market microstructure characteristics.     

The αVG model for multivariate asset pricing: calibration and extension

Luciano and Semeraro proposed a class of multivariate asset pricing models where the asset log-returns are modeled by a multivariate Brownian motion time-changed by a multivariate subordinator which consists of the weighted sum of a common and an idiosyncratic subordinator. In the original setting, Luciano and Semeraro imposed some constraints on the subordinator parameters such that the multivariate subordinator is of the same subordinator sub-class as its components, leading to asset log-returns of a particular Lévy type. This restriction leads to marginal characteristic functions which are independent on the common subordinator setting. In this paper, we propose to extend the original model by relaxing the constraints on the subordinator parameters, leading to marginal characteristic functions which become a function of the whole parameter set. Under this generalized version, the volatility of the log-returns depends on both the common and idiosyncratic subordinator settings, and not only on the idiosyncratic one, which makes the generalized model more in line with the empirical evidence of the presence of both an idiosyncratic and a common component in the business clock. For the numerical study, we compare the calibration fit of both univariate option surfaces and market implied correlations for a period extending from the 2nd of June 2008 until the 30th of October 2009 under the two model settings and assess the calibration risk arising from different calibration procedures by pricing traditional multivariate exotic options. In particular we show that the decoupling calibration procedure fails to accurately replicate the market dependence structure under the original model for highly correlated asset returns and we propose an alternative methodology which rests on a joint calibration of the univariate and the dependence structure and which leads to an accurate fit of the market reality under both the generalized and original models.     

High-dimensional Hawkes processes for limit order books: modelling,empirical analysis and numerical calibration

High-dimensional Hawkes processes with exponential kernels are used to describe limit order books in order-driven financial markets. The dependencies between orders of various types are carefully studied and modelled, based on a thorough empirical analysis. The observation of inhibition effects is particularly interesting, and leads us to the use of non-linear Hawkes processes. Specific attention is devoted to the calibration problem, in order to account for the high dimensionality of the problem and the very poor convexity properties of the MLE. Our analyses show a good agreement between the statistical properties of order book data and those of the model.     

Nonlinear analysis for forecasting currencies: are they useful to the portfolio manager?

The importance of a time-varying specification for both the return and the risk of financial assets is well known. The purpose of this study is to investigate if some of the most recently developed econometric models, combined with technical indicators often used by practitioners, can significantly predict future returns. While most studies have focused on either univariate series or in-sample analyses of a given econometric specification, this study considers a multivariate framework where a US based investor daily reallocates a portfolio of three currencies (Deutschmark, Swiss Franc and Japanese Yen). Series of three years out-of-sample forecasts are analysed in terms of risk and return and it is shown that some of the tested speciications can indeed signiicantly predict future daily returns and correlations over this three-year period.     

On the persistence and volatility in European,American and Asian stocks bull and bear markets

In this paper we examine the statistical properties of several stock market indices in Europe, the US and Asia by means of determining the degree of dependence in both the level and the volatility of the processes. In the latter case, we use the squared returns as a proxy for the volatility. We also investigate the cyclical pattern observed in the data and in particular, if the degree of dependence changes depending on whether there is a bull or a bear period. We use fractional integration and GARCH specifications. The results indicate that the indices are all nonstationary I(1) processes with the squared returns displaying a degree of long memory behaviour. With respect to the bull and bear periods, we do not observe a systematic pattern in terms of the degree of persistence though for some of the indices (FTSE, Dax, Hang Seng and STI) there is a higher degree of dependence in both the level and the volatility during the bull periods.     

A multivariate jump-driven financial asset model

We discuss a Lévy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behaviour of a series of stocks or indexes and to study a multi-firm, value-based default model. Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a stochastic time-change technique and provide the details for a Gamma change. The main feature of the model is the fact that—opposite to other, non-jointly Gaussian settings—its risk-neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit.     

Time consistency of Lévy models

Time consistency of the models used is an important ingredient to improve risk management. The empirical investigation in this article gives evidence for some models driven by Lévy processes to be highly consistent. This means that they provide a good statistical fit of empirical distributions of returns not only on the timescale used for calibration but on various other timescales as well. As a result these models produce more reliable risk numbers and derivative prices.     

STATISTICAL PROPERTIES OF DAILY RETURNS: EVIDENCE FROM EUROPEAN STOCK MARKETS

This paper attempts to model the distributional properties of daily stock returns on several European Stock Exchanges. The empirical findings reveal the presence of non-linear dependencies that cannot be captured by the random walk model. A model of return-generating process that fit the data empirically is the Generalized Autoregressive Conditional Heteroskedastic GARCH (1,1) process with a conditional student- t distribution.     

Non-linear dynamics in international stock market returns

Recent empirical finance research has reported non-linear dynamics within asset returns. However, much of this extant research has focussed upon asset markets within the US and UK. This paper examines whether such dynamics are also present in a series of six international equity index returns. Using empirical models which are consistent that the theoretical behavioural finance noise trader motivation of non-linearity, whereby market dynamics differ between small and large returns, our results suggest these models improve the in-sample fit and out-of-sample forecast over linear alternatives. Further, the point of regime transition differs between positive and negative returns indicating that noise traders are more likely to engage in trend-chasing behaviour in up markets and anchoring behaviour in down markets. Finally, the forecast gain in the Asia-Pacific markets is greater than in the European markets suggestive that limits to arbitrage are greater perhaps as fundamental traders knowledge of market dynamics and noise trader behaviour is still evolving.     

Measuring large comovements in financial markets

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearman's rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstrate that MCS has a number of advantages over conventional measures of tail dependence, both in theory and in practical applications. In the analysis of univariate financial series, data are filtered to remove temporal dependence as a matter of routine. We show that standard filtering procedures may strongly influence the conclusions drawn concerning tail dependence. We give empirical applications to two large data sets of high-frequency asset returns. Our results have immediate implications for portfolio risk management, derivative pricing and portfolio selection. In this context we address portfolio tail diversification and tail hedging. Amongst other aspects, it is shown that the proposed modeling framework improves the estimation of portfolio risk measures such as the value at risk.     

Extreme dependence in investor attention and stock returns – consequences for forecasting stock returns and measuring systemic risk

We characterize co-movements in investor attention by modeling multivariate internet search volume data. Using a variety of copula models that can capture both asymmetric and skewed dependence, we find empirical evidence of strong non-linear and asymmetric dependence in the attention investors give to companies. Modeling three years of daily stock returns and search volumes from Google Trends for 29 bank names, we find a striking similarity between the dependence structure inherent in stock returns and the dependence in the corresponding time series of search queries. We then document the existence of significant asymmetric and skewed tail dependence in the joint distribution of stock returns and investor attention. Finally, stock returns and internet search volumes appear to evolve concurrently in real time with neither one leading the other. Our findings have important implications, e.g. for the analysis of banks' interconnectedness based on equity data and the pricing of investor attention in the cross-section of stock returns.     

On estimating the expected return on the market: An exploratory investigation

The expected market return is a number frequently required for the solution of many investment and corporate finance problems, but by comparison with other financial variables, there has been little research on estimating this expected return. Current practice for estimating the expected market return adds the historical average realized excess market returns to the current observed interest rate. While this model explicitly reflects the dependence of the market return on the interest rate, it fails to account for the effect of changes in the level of market risk. Three models of equilibrium expected market returns which reflect this dependence are analyzed in this paper. Estimation procedures which incorporate the prior restriction that equilibrium expected excess returns on the market must be positive are derived and applied to return data for the period 1926–1978. The principal conclusions from this exploratory investigation are: (1) in estimating models of the expected market return, the non-negativity restriction of the expected excess return should be explicity included as part of the specification: (2) estimators which use realized returns should be adjusted for heteroscedasticity.     

Are international stock returns predictable?: An examination of linear and non-linear predictability using generalized spectral tests

This paper uses generalized spectral tests to examine whether international stock index returns are predictable using the history of the series. Unlike many other testing procedures, the generalized spectral tests used in this paper are robust to distributional assumptions, the presence of time-varying volatility, and allow for various forms of non-linear predictability. We find evidence of predictability in mean for over half of the international returns examined. In addition, we find most of the predictability to be non-linear in nature. The patterns of predictability are consistent with calendar effects and in some cases long-run dependence. Regardless of the implications of predictability of returns, this study is important because the generalized spectrum is defined for a range of different frequencies (corresponding to cycles of 2 days and greater), and we can therefore examine at what frequencies predictability occurs. This provides insight into whether there exists short-run, long-run, or both types of dependence.     

Quadratic Hawkes processes for financial prices

We introduce and establish the main properties of QHawkes (‘Quadratic’ Hawkes) models. QHawkes models generalize the Hawkes price models introduced in Bacry and Muzy [Quant. Finance, 2014, 14(7), 1147–1166], by allowing feedback effects in the jump intensity that are linear and quadratic in past returns. Our model exhibits two main properties that we believe are crucial in the modelling and the understanding of the volatility process: first, the model is time-reversal asymmetric, similar to financial markets whose time evolution has a preferred direction. Second, it generates a multiplicative, fat-tailed volatility process, that we characterize in detail in the case of exponentially decaying kernels, and which is linked to Pearson diffusions in the continuous limit. Several other interesting properties of QHawkes processes are discussed, in particular the fact that they can generate long memory without necessarily being at the critical point. A non-parametric fit of the QHawkes model on NYSE stock data shows that the off-diagonal component of the quadratic kernel indeed has a structure that standard Hawkes models fail to reproduce. We provide numerical simulations of our calibrated QHawkes model which is indeed seen to reproduce, with only a small amount of quadratic non-linearity, the correct magnitude of fat-tails and time reversal asymmetry seen in empirical time series.     

Modeling exchange rate dependence dynamics at different time horizons

Despite an extensive body of research, the best way to model the dependence of exchange rates remains an open question. In this paper we present a new approach which employs a flexible time-varying copula model. It allows the conditional correlation between exchange rates to be both time-varying and modeled independently from the marginal distributions. We introduce a dynamic specification for the correlation using the Fisher transformation. Applied to Euro/US dollar and Japanese Yen/US dollar, our results reveal a significantly time-varying correlation, dependent on the past return realizations. We find that a time-varying copula with the proposed correlation specification gives better results than alternative dynamic benchmark models. The dynamic copula model outperforms at six different time horizons, ranging from hourly to daily, confirming the model specification.