Feller processes of normal inverse Gaussian type

We consider the construction of normal inverse Gaussian (NIG) (and some related) Lévy processes from the probabilistic viewpoint and from that of the theory of pseudo-differential operators; we then introduce and analyse natural generalizations of these constructions. The resulting Feller processes are somewhat similar to the NIG Lévy process but may, for instance, possess mean-reverting features. Possible applications to financial mathematics are discussed, and approximations to solutions of corresponding generalizations of the Black-Scholes equation are derived.     

Predicting VaR for China's stock market: A score-driven model based on normal inverse Gaussian distribution

Under the framework of dynamic conditional score, we propose a parametric forecasting model for Value-at-Risk based on the normal inverse Gaussian distribution (Hereinafter NIG-DCS-VaR), which creatively incorporates intraday information into daily VaR forecast. NIG specifies an appropriate distribution to return and the semi-additivity of the NIG parameters makes it feasible to improve the estimation of daily return in light of intraday return, and thus the VaR can be explicitly obtained by calculating the quantile of the re-estimated distribution of daily return. We conducted an empirical analysis using two main indexes of the Chinese stock market, and a variety of backtesting approaches as well as the model confidence set approach prove that the VaR forecasts of NIG-DCS model generally gain an advantage over those of realized GARCH (RGARCH) models. Especially when the risk level is relatively high, NIG-DCS-VaR beats RGARCH-VaR in terms of coverage ability and independence.     

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.     

A fully parametric approach to return modelling and risk management of hedge funds

This paper examines the empirical properties of hedge fund returns and proposes a fully parametric model capable of adequately describing both univariate and multivariate return properties. The suggested model is based on the multivariate extension of the Normal Inverse Gaussian (NIG) distribution and will be shown to be capable of capturing the characteristic distributional features of hedge fund returns. Drawing on recent research in the area of Generalized Hyperbolic distributions and their calibration, we will elaborate on the application of the NIG-model for risk management purposes, and highlight the differences between the NIG and the Gaussian model.      

Risk premia: Exact solutions vs. log-linear approximations

We derive exact expressions for the risk premia for general distributions in a Lucas economy and show that the errors when using log-linear approximations can be economically significant when the shocks are nonnormal. Assuming growth rates are Normal Inverse Gaussian (NIG) and fitting the distribution to the data used in Mehra and Prescott (1985), the coefficient of relative risk aversion required to match the equity premium is more than halved compared to the finding in their article. We also consider a standard long-run risk model and, by comparing our exact solutions to the log-linear approximations, we show that the approximation errors are substantial, especially for high levels of risk aversion.     

Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform

The Lee-Carter mortality model provides a structure for stochastically modeling mortality rates incorporating both time (year) and age mortality dynamics. Their model is constructed by modeling the mortality rate as a function of both an age and a year effect. Recently the MBMM model (Mitchell et al. 2013) showed the Lee Carter model can be improved by fitting with the growth rates of mortality rates over time and age rather than the mortality rates themselves. The MBMM modification of the Lee-Carter model performs better than the original and many of the subsequent variants. In order to model the mortality rate under the martingale measure and to apply it for pricing the longevity derivatives, we adapt the MBMM structure and introduce a Lévy stochastic process with a normal inverse Gaussian (NIG) distribution in our model. The model has two advantages in addition to better fit: first, it can mimic the jumps in the mortality rates since the NIG distribution is fat-tailed with high kurtosis, and, second, this mortality model lends itself to pricing of longevity derivatives based on the assumed mortality model. Using the Esscher transformation we show how to find a related martingale measure, allowing martingale pricing for mortality/longevity risk–related derivatives. Finally, we apply our model to pricing a q-forward longevity derivative utilizing the structure proposed by Life and Longevity Markets Association.     

A transform approach to compute prices and Greeks of barrier options driven by a class of Lévy processes

In this paper we propose a transform method to compute the prices and Greeks of barrier options driven by a class of Lévy processes. We derive analytical expressions for the Laplace transforms in time of the prices and sensitivities of single barrier options in an exponential Lévy model with hyper-exponential jumps. Inversion of these single Laplace transforms yields rapid, accurate results. These results are employed to construct an approximation of the prices and sensitivities of barrier options in exponential generalized hyper-exponential Lévy models. The latter class includes many of the Lévy models employed in quantitative finance such as the variance gamma (VG), KoBoL, generalized hyperbolic, and the normal inverse Gaussian (NIG) models. Convergence of the approximating prices and sensitivities is proved. To provide a numerical illustration, this transform approach is compared with Monte Carlo simulation in cases where the driving process is a VG and a NIG Lévy process. Parameters are calibrated to Stoxx50E call options.     

Evaluation of the MEMM, Parameter Estimation and Option Pricing for Geometric Lévy Processes

In this paper, we shall propose a useful approach to evaluate concretely the MEMM (minimal entropy martingale measure) for the typical geometric Lévy processes such as compound Poisson, stable, VG (Variance Gamma), CGMY (Carr-Geman-Madan-Yor), NIG (Normal Inverse Gaussian), etc. In addition, we shall estimate the parameters of geometric Lévy processes and value the European call option and Asian call option using the Nikkei financial data.     

Copula dynamics in CDOs

Values of tranche spreads of collateralized debt obligations (CDOs) are driven by the joint default performance of the assets in the collateral pool. The dependence between the entities in the portfolio mainly depends on current economic conditions. Therefore, a correlation implied from tranches can be seen as a measure of the general situation of the credit market. We analyse the European market of standardized CDOs using tranches of the iTraxx index in the periods before and during the global financial crisis. We investigate the evolution of the correlations using different copula models: the standard Gaussian, the NIG, the double-t, and the Gumbel copula model. After calibration of these models, one obtains a time varying vector of parameters. We analyse the dynamic pattern of these coefficients. That enables us to forecast future parameters and consequently calculate Value-at-Risk measures for iTraxx Europe tranches.     

A note on “Modelling exchange rate returns: which flexible distribution to use?”

Corlu and Corlu [Quant. Finance, 2014, doi: 10.1080/14697688.2014.942231] provided a novel modelling of exchange rate data for nine currencies using five flexible distributions. They stated that the generalized lambda, skew t and normal inverse Gaussian distributions ‘do a good job’. Here, we reanalyse the data and show that a distribution simpler than all of these fits at least as well as these distributions. We also find that the normal inverse Gaussian distribution provides good fits for only one of the data-sets.     

Firm ratings, momentum strategies, and crises: evidence from the US and Taiwanese stock markets

This paper investigates whether there is a link between momentum profitability and firm ratings. We follow traditional and practical (non-) investment-grade classifications to divide into three rating groups, high, median, and non investment-grade group (HIG, MIG, and NIG) since firm ratings express risk in relative rank order to contain valuable information. This study considers the US and Taiwanese stock markets. We find that firm ratings momentum strategies can even earn positive profits, larger than na?ve momentum, supporting that firm ratings can be used to strengthen naive momentum effects. By comparisons, the US firm ratings momentum with NIG produces larger profits than HIG but opposite in direction and V-shaped pattern in Taiwan. With an examination of crises on firm ratings momentum, we find that firm ratings momentum indeed helps increase the payoff during (non-)crises although firm ratings momentum profits should be strong following non-crises states and weak following crises states. However, firm ratings momentum profits partially result from the predictability of business cycle, calendar months, and information asymmetries. Our results highlight the critical importance of using firm ratings screens in empirical momentum studies.     

Using local Gaussian correlation in a nonlinear re-examination of financial contagion

This paper examines financial contagion, that is, whether the cross-market linkages in financial markets increase after a shock to a country. We use a new measure of local dependence (introduced by Tjøstheim and Hufthammer (2013)) to study the contagion effect. The central idea of the new approach is to approximate an arbitrary bivariate return distribution by a family of Gaussian bivariate distributions. At each point of the return distribution there is a Gaussian distribution that gives a good approximation at that point. The correlation of the approximating Gaussian distribution is taken as the local correlation in that neighbourhood. By examining the local Gaussian correlation before the shock (in a stable period) and after the shock (in the crisis period), we are able to test whether contagion has occurred by a bootstrap testing procedure. The use of local Gaussian correlation is compared to other methods of studying contagion. Further, the bootstrap test is examined in a Monte Carlo study, and shows good level and power properties. We illustrate our approach by re-examining the Mexican crisis of 1994, the Asian crisis of 1997–1998 and the financial crisis of 2007–2009. We find evidence of contagion based on our new procedure and are able to describe the nonlinear dependence structure of these crises.     

Option pricing under non-normality: a comparative analysis

This paper carries out a comparative analysis of the calibration and performance of a variety of options pricing models. These include Black and Scholes (J Polit Econ 81:637–659, 1973), the Gram–Charlier (GC) approach of Backus et al. (1997), the stochastic volatility (HS) model of Heston (Rev Financ Stud 6:327–343, 1993), the closed-form GARCH process of Heston and Nandi (Rev Financ Stud 13:585–625, 2000) and a variety of Lévy processes including the Variance Gamma (VG), Normal Inverse Gaussian (NIG), and, CGMY and Kou (Manag Sci 48:1086–1101, 2002) jump-diffusion models. Unlike most studies of option pricing, we compare these models using a common point-in-time data which reflects the perspective of a new investor who wishes to choose between models using only the most minimal recent data set. For each of these models, we also examine the accuracy of delta and delta-gamma approximations to the valuation of both individual options and an illustrative option portfolio.     

A general class of distortion operators for pricing contingent claims with applications to CAT bonds

ABSTRACT

The current paper provides a general approach to construct distortion operators that can price financial and insurance risks. Our approach generalizes the (Wang 2000) transform and recovers multiple distortions proposed in the literature as particular cases. This approach enables designing distortions that are consistent with various pricing principles used in finance and insurance such as no-arbitrage models, equilibrium models and actuarial premium calculation principles. Such distortions allow for the incorporation of risk-aversion, distribution features (e.g. skewness and kurtosis) and other considerations that are relevant to price contingent claims. The pricing performance of multiple distortions obtained through our approach is assessed on CAT bonds data. The current paper is the first to provide evidence that jump-diffusion models are appropriate for CAT bonds pricing, and that natural disaster aversion impacts empirical prices. A simpler distortion based on a distribution mixture is finally proposed for CAT bonds pricing to facilitate the implementation.     

Multilevel Monte Carlo for exponential Lévy models

We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and \(\alpha\)-stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments.     

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.     

Market symmetry in time-changed Brownian models

In this paper we examine which Brownian subordination with drift exhibits the symmetry property introduced by Fajardo and Mordecki [2006. Quantitative Finance 6, 219–227]. We find that when the subordination results in a Lévy process, a necessary and sufficient condition for the symmetry to hold is that the drift must be equal to-1/2. Also, we derive explicit conditions to test whether the NIG, CGMY and Meixner processes are symmetric or not. Finally, we perform some tests with real financial data.     

Litteraturanmeldelse

This article examines the notion of distortion of copulas, a natural extension of distortion within the univariate framework. We study three approaches to this extension: (1) distortion of the margins alone while keeping the original copula structure; (2) distortion of the margins while simultaneously altering the copula structure; and (3) synchronized distortion of the copula and its margins. When applying distortion within the multivariate framework, it is important to preserve the properties of a copula function. For the first two approaches, this is a rather straightforward result; however, for the third approach, the proof has been exquisitely constructed in Morillas (2005). These three approaches unify the different types of multivariate distortion that have scarcely scattered in the literature. Our contribution in this paper is to further consider this unifying framework: we give numerous examples to illustrate and we examine their properties particularly with some aspects of ordering multivariate risks. The extension of multivariate distortion can be practically implemented in risk management where there is a need to perform aggregation and attribution of portfolios of correlated risks. Furthermore, ancillary to the results discussed in this article, we are able to generalize the formula developed by Genest &; Rivest (2001) for computing the distribution of the probability integral transformation of a random vector and extend it to the case within the distortion framework. For purposes of illustration, we applied the distortion concept to value excess of loss reinsurance for an insurance policy where the loss amount could vary by type of loss.     

Comparing alternative Lévy base correlation models for pricing and hedging CDO tranches

In this paper we investigate alternative Lévy base correlation models that arise from the Gamma, Inverse Gaussian and CMY distribution classes. We compare these models with the basic (exponential) Lévy base correlation model and the classical Gaussian base correlation model. For all investigated models, the Lévy base correlation curve is significantly flatter than the corresponding Gaussian curve, which indicates better correspondence of the Lévy models with reality. Furthermore, we present the results of pricing bespoke tranchlets and comparing deltas of both standard and custom-made tranches under all the considered models. We focus on deltas with respect to the CDS index and individual CDSs, and the hedge ratio for hedging the equity tranche with the junior mezzanine.     

On an inverse Gaussian process

Abstract

Tweedie [11] investigated properties of the Inverse Gaussian distribution. We define in this paper the Inverse Gaussian process. For the discrete case we find the density function of the functions of Inverse Gaussian variates. We look for covariance function and stochastic integral as well as conditional density functions of an Inverse Gaussian process.