PRICING CURRENCY FUTURES OPTIONS WITH LOGNORMALLY DISTRIBUTED JUMPS

We find that a mixed diffusion-jump process fits most daily currency futures price series better than a mixture of normal densities and, especially, an asymmetric stable Paretian model. We also find that Merton's (1976) mixed diffusion-jump option pricing model outperforms Black's (1 976) model for valuing currency futures options. Our results suggest that researchers should begin to consider the possibility of jump processes as time-independent models of other futures price series.     

Models of asset returns: changes of pattern from high to low event frequency

In this paper we have analysed asset returns of the New York Stock Exchange and the Helsinki Stock Exchange using various time-independent models such as normal, general stable, truncated Lévy, mixed diffusion-jump, compound normal, Student t distribution and power exponential distribution and the time-dependent GARCH(1, 1) model with Gaussian and Student t distributed innovations. In order to study changes of pattern at different event horizons, as well as changes of pattern over time for a given event horizon, we have analysed high-frequency or short-horizon intraday returns up from 20 s intervals to a full trading day, medium-frequency or medium-horizon daily returns and low-frequency or long-horizon returns with holding period up to 30 days. As for changes of pattern over time, we found that for medium-frequency returns there are relatively long periods of business-as-usual when the return-generating process is well-described by a normal distribution. We also found periods of ferment, when the volatility grows and complex time dependences tend to emerge, but the known time dependences cannot explain the variability of the distribution. Such changes of pattern are also observed for high-frequency or short-horizon returns, with the exception that the return-generating process never becomes normal. It also turned out that the time dependence of the distribution shape is far more prominent at high frequencies or short horizons than the time dependence of the variance. For long-horizon or low-frequency returns, the distribution is found to converge towards a normal distribution with the time dependences vanishing after a few days.     

NONLINEAR DYNAMICS AND THE DISTRIBUTION OF DAILY STOCK INDEX RETURNS

Three alternative models of daily stock index returns are considered: (1) a diffusion-jump process; (2) an extended generalized autoregressive conditional heteroskedasticity (GARCH) process; and (3) a combination of the GARCH and jump processes. Non-nested tests between the diffusion-jump process and a GARCH(1.1) process with t-distributed errors reject the diffusion-jump process, but do not always reject the GARCH process. Kolmogorov-Smirnov tests of fit, however, reject the GARCH(1,1)-t process for all cases. Nonlinear dependence is not removed for the value-weighted index and the S&P 500 stock index; therefore, deterministic chaos cannot be dismissed.     

Mixed tempered stable distribution

In this paper, we introduce a new parametric distribution, the mixed tempered stable. It has the same structure of the normal variance–mean mixtures but the normality assumption gives way to a semi-heavy tailed distribution. We show that, by choosing appropriately the parameters of the distribution and under the concrete specification of the mixing random variable, it is possible to obtain some well-known distributions as special cases. We employ the mixed tempered stable distribution which has many attractive features for modelling univariate returns. Our results suggest that it is flexible enough to accommodate different density shapes. Furthermore, the analysis applied to statistical time series shows that our approach provides a better fit than competing distributions that are common in the practice of finance.     

Alternatives to the normal model of stock returns: Gaussian mixture, generalised logF and generalised hyperbolic models

Simple parametric models of the marginal distribution of stock returns are an essential building block in many areas of applied finance. Even though it is well known that the normal distribution fails to represent most of the “stylised” facts characterising return distributions, it still dominates much of the applied work in finance. Using monthly S&P 500 stock index returns (1871–2005) as well as daily returns (2001–2005), we investigate the viability of three alternative parametric families to represent both the stylised and empirical facts: the generalised hyperbolic distribution, the generalised logF distribution, and finite mixtures of Gaussians. For monthly return data, all three alternatives give reasonable fits for all sub-periods. However, the generalised hyperbolic distribution fails to describe some features of the marginal distributions in some sub-periods. The daily return data are much more symmetric and expose another problem for all three distributions: the parameters describing the behaviour of the tails also influence the scale so that simpler alternatives or restricted parameterisations are called for.      

Multivariate asset return prediction with mixture models

The use of mixture distributions for modeling asset returns has a long history in finance. New methods of demonstrating support for the presence of mixtures in the multivariate case are provided. The use of a two-component multivariate normal mixture distribution, coupled with shrinkage via a quasi-Bayesian prior, is motivated, and shown to be numerically simple and reliable to estimate, unlike the majority of multivariate GARCH models in existence. Equally important, it provides a clear improvement over use of GARCH models feasible for use with a large number of assets, such as constant conditional correlation, dynamic conditional correlation, and their extensions, with respect to out-of-sample density forecasting. A generalization to a mixture of multivariate Laplace distributions is motivated via univariate and multivariate analysis of the data, and an expectation–maximization algorithm is developed for its estimation in conjunction with a quasi-Bayesian prior. It is shown to deliver significantly better forecasts than the mixed normal, with fast and numerically reliable estimation. Crucially, the distribution theory required for portfolio theory and risk assessment is developed.     

CALL OPTION VALUATION FOR DISCRETE NORMAL MIXTURES

In this study a mixture call option pricing model is derived to examine the impact of non-normal underlying returns densities. Observed fat-tailed and skewed distributions are assumed to be the result of independent Gaussian processes with nonstationary parameters, modeled by discrete k-component independent normal mixtures. The mixture model provides an exact solution with intuitive appeal using weighted sums of Black-Scholes (B-S) solutions. Simulating returns densities representative of equity securities, significant mispricing by B-S is found in low-priced at- and out-of-the-money near-term options. The lower the variance and the higher the leptokurtosis and positive skewness of the underlying returns, the more pronounced is this mispricing. Values of in-the-money options and options with several weeks or more to expiration are closely approximated by B-S.     

On the distribution and conditional heteroscedasticity in Taiwan stock prices

This paper studies the distribution and conditional heteroscedasticity in stock returns on the Taiwan stock market. Apart from the normal distribution, in order to explain the leptokurtosis and skewness observed in the stock return distribution, we also examine the Student-t, the Poisson–normal, and the mixed-normal distributions, which are essentially a mixture of normal distributions, as conditional distributions in the stock return process. We also use the ARMA (1,1) model to adjust the serial correlation, and adopt the GJR–generalized autoregressive conditional heteroscedasticity (GARCH (1,1)) model to account for the conditional heterscedasticity in the return process. The empirical results show that the mixed–normal–GARCH model is the most probable specification for Taiwan stock returns. The results also show that skewness seems to be diversifiable through portfolio. Thus the normal–GARCH or the Student-t–GARCH model which involves symmetric conditional distribution may be a reasonable model to describe the stock portfolio return process¹.     

Normality of stock returns with event time clocks

We introduce a different way to measure time using event clocks, with which we can observe a normal distribution of intraday stock returns. Most finance studies employ a ‘default’ time measurement that uses a calendar clock. Cumulative evidence from prior literature shows that returns with a calendar clock follow a distribution with an excess kurtosis and a heavier tail, relative to a normal distribution. We examine the distribution of intraday stock returns using different clocks. We find that returns do not follow a normal distribution with a traditional calendar clock, but do follow a normal distribution when event clocks are applied.     

Forecasting VaR using analytic higher moments for GARCH processes

It is widely accepted that some of the most accurate Value-at-Risk (VaR) estimates are based on an appropriately specified GARCH process. But when the forecast horizon is greater than the frequency of the GARCH model, such predictions have typically required time-consuming simulations of the aggregated returns distributions. This paper shows that fast, quasi-analytic GARCH VaR calculations can be based on new formulae for the first four moments of aggregated GARCH returns. Our extensive empirical study compares the Cornish–Fisher expansion with the Johnson SU distribution for fitting distributions to analytic moments of normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets, for the purpose of deriving accurate GARCH VaR forecasts over multiple horizons and significance levels.     

The Distributional Characteristics of a Selection of Contracts Traded on the London International Financial Futures Exchange

This study examines the distributional properties of futures prices for contracts traded on LIFFE. A filtering process is employed to remove day of the week and holiday effects, a maturity effect, moving average effects and the influence of an asset's conditional variance from the raw returns series. Alternative distributional models from the stable paretian and ARCH families are examined for their applicability to futures data using a stability under additions. The results conclusively reject the hypothesis that futures returns are normally distributed with findings in favour of two related hypotheses – the mixtures of stable distribution and the ordinary stable distribution.     

The Distributional Characteristics of a Selection of Contracts Traded on the London International Financial Futures Exchange

This study examines the distributional properties of futures prices for contracts traded on LIFFE. A filtering process is employed to remove day of the week and holiday effects, a maturity effect, moving average effects and the influence of an asset's conditional variance from the raw returns series. Alternative distributional models from the stable paretian and ARCH families are examined for their applicability to futures data using a stability under additions. The results conclusively reject the hypothesis that futures returns are normally distributed with findings in favour of two related hypotheses – the mixtures of stable distribution and the ordinary stable distribution.     

Dynamic principal component CAW models for high-dimensional realized covariance matrices

We propose a new dynamic principal component CAW model (DPC-CAW) for time-series of high-dimensional realized covariance matrices of asset returns (up to 100 assets). The model performs a spectral decomposition of the scale matrix of a central Wishart distribution and assumes independent dynamics for the principal components' variances and the eigenvector processes. A three-step estimation procedure makes the model applicable to high-dimensional covariance matrices. We analyze the finite sample properties of the estimation approach and provide an empirical application to realized covariance matrices for 100 assets. The DPC-CAW model has particularly good forecasting properties and outperforms its competitors for realized covariance matrices.     

Portfolio Selection and Optimization with Higher Moments: Evidence from the Indian Stock Market

The Markowitz portfolio optimization model, popularly known as the Mean-Variance model, assumes that stockreturns follow normal distribution. But when stock returns do not follow normal distribution, this model wouldbe inadequate as it would prescribe sub-optimal portfolios. Stock market literature often deliberates that stock returns are non-normal. In such context the Markowitz model would not be sufficient to estimate the portfolio risks. The purpose of this paper is to expand the original Markowitz portfolio theory (mean-variance) via adding the higher order moments like skewness (third moment about the mean) and kurtosis (fourth moment about the mean) in the return characteristics. The research paper investigates the impact of including higher moments using multi-objective programming model for portfolio stock selection and optimization. The empirical results indicate that the inclusion of higher moments had a considerable impact in estimating the returns behavior of portfolios. The portfolios optimized using all the four moments, generated higher returns for the given level of risk in comparison to the returns of the Markowitz model during the study period 2000–2011. The results of this study would be immensely useful to fund managers, portfolio managers and investors as it would help them in understanding the Indian stock market behavior better and also in selecting alternative portfolio selection models.     

Does the choice of performance measure influence the evaluation of hedge funds?

The Sharpe ratio is adequate for evaluating investment funds when the returns of those funds are normally distributed and the investor intends to place all his risky assets into just one investment fund. Hedge fund returns differ significantly from a normal distribution. For this reason, other performance measures for hedge fund returns have been proposed in both the academic and practice-oriented literature. In conducting an empirical study based on return data of 2763 hedge funds, we compare the Sharpe ratio with 12 other performance measures. Despite significant deviations of hedge fund returns from a normal distribution, our comparison of the Sharpe ratio to the other performance measures results in virtually identical rank ordering across hedge funds.     

An Analysis of the Distribution of Extreme Share Returns in the UK from 1975 to 2000

This paper seeks to characterise the distribution of extreme returns for a UK share index over the years 1975 to 2000. In particular, the suitability of the following distributions is investigated: Gumbel, Frechet, Weibull, Generalised Extreme Value, Generalised Pareto, Log‐Normal and Generalised Logistic. Daily returns for the FT All Share index were obtained from Datastream, and the maxima and minima of these daily returns over a variety of selection intervals were calculated. Plots of summary statistics for the weekly maxima and minima on statistical distribution maps suggested that the best fitting distribution would be either the Generalised Extreme Value or the Generalised Logistic. The results from fitting each of these two distributions to extremes of a series of UK share returns support the conclusion that the Generalised Logistic distribution best fits the UK data for extremes over the period of the study. The Generalised Logistic distribution has fatter tails than either the log‐normal or the Generalised Extreme Value distribution, hence this finding is of importance to investors who are concerned with assessing the risk of a portfolio.     

Fat tails in private equity fund returns: The smooth double Pareto distribution

Does the distribution of private equity returns have fat tails? A new smooth double Pareto distribution can explain the stationary distribution of private equity funds' valuation multiples. This distribution emerges from a random growth model with lognormally distributed initial fund valuations. This model endogenously generates power-law tails in the stationary cross-section. The new distribution fits the data better than competing distributions. Fat tails are particularly pronounced in venture capital funds and suggest returns with infinite variance over the lifetime of the fund. The smooth double Pareto distribution has wide applicability to growth processes with a random initial value.     

Implicit expectiles and measures of implied volatility

It is well known that the normal distribution is inadequate in capturing the skewed and heavy-tailed behaviour of exchange rate returns. To this end, various flexible distributions that are capable of modelling the asymmetric and tailed behaviour of returns have been proposed. In this paper, we investigate the performance of the generalized lambda distribution (GLD) to capture the skewed and leptokurtic behaviour of exchange rate returns. We do this by conducting a comprehensive numerical study to compare the performance of the GLD against the performances of the skewed t distribution, the unbounded Johnson family of distributions and the normal inverse Gaussian (NIG) distribution. Our results suggest that in terms of the value-at-risk and expected shortfall, the GLD shows at least similar performance to the skewed t distribution and the NIG distribution. Considering the ease in GLD’s use for random variate generation in Monte Carlo simulations, we conclude that the GLD can be a good alternative in various financial applications where modelling of the heavy tail behaviour is critical.     

Can Markov switching models predict excess foreign exchange returns?

This paper merges the literature on technical trading rules with the literature on Markov switching to develop economically useful trading rules. The Markov models’ out-of-sample, excess returns modestly exceed those of standard technical rules and are profitable over the most recent subsample. A portfolio of Markov and standard technical rules outperforms either set individually, on a risk-adjusted basis. The Markov rules’ high excess returns contrast with mixed performance on statistical tests of forecast accuracy. There is no clear source for the trends, but permitting the mean to depend on higher moments of the exchange rate distribution modestly increases returns.