Variance Gamma Model Research Articles

Closed‐form approximations for spread options in Lévy markets

AbstractWe provide new closed‐form approximations for the pricing of spread options in three specific instances of exponential Lévy markets, ie, when log‐returns are modeled as Brownian motions (Black‐Scholes model), variance gamma processes (VG model), or normal inverse Gaussian processes (NIG model). For the specific case of exchange options (spread options with zero strike), we generalize the well‐known Margrabe formula (1978) that is valid in a Black‐Scholes model to the VG model under a homogeneity assumption.

Bilateral Multiple Gamma Returns: Their Risks and Rewards

The bilateral gamma model for returns is naturally derived from the lognormal model. Maximizing entropy in a random time change delivers the symmetric variance gamma model. The asymmetric variance gamma follows on incorporating skewness. Differential speeds for the upward and downward motions lead to the bilateral gamma. A further generalizations results in the bilateral double gamma model when the speed parameter of the bilateral gamma model is itself taken to be gamma distributed on entropy maximization. A rich five to seven parameter specification of preferences renders possible the extraction of physical densities from option prices. The quality of such extraction is measured by examining the uniformity of the estimated distribution functions evaluated at realized forward returns.The economic value of risky returns is seen to embed three/five risk premia for the bilateral gamma/bilateral double gamma. For the bilateral gamma they are up and down side volatilities compensated in up and down side drifts, and the down side drift compensated in the up side drift. For the bilateral double gamma one adds in addition compensations for skewness. Results reveal a drop in the down side risk premium since the crisis with an increase in the recent period.

Efficient control variate methods with applications to exotic options pricing under subordinated Brownian motion models

In this paper we apply both biased and multivariate control variate methods to some exotic option pricing problems with exponential subordinated Brownian motion models for the underlying asset prices. For both arithmetic Asian and basket options, control variates conditional on geometric means of asset prices are constructed. To reduce biases, we derive expressions of expectations in terms of expectations of subordinators for relevant random variables. Numerical results show that the constructed control variates are more efficient than the classical control variates in reducing variances when pricing Asian and basket options under the normal inverse Gaussian and variance gamma models. We believe that the great variance reduction gains are due to the high correlations between the option playoffs and the constructed control variates. The efficiencies realized by the control variates are even much more significant when they are combined with quasi-Monte Carlo methods. Details are given in Section 6.

Closed-Form Approximations for Spread Options in LLvy Markets

We provide new closed-form approximations for the pricing of spread options in three specific instances of exponential Levy markets, i.e., when log-returns are modeled as Brownian motions (Black-Scholes model), Variance Gamma processes (VG model) or Normal Inverse Gaussian processes (NIG model). For the specific case of exchange options (spread options with zero strike), we generalize the well-known Margrabe formula (1978) that is valid in a Black-Scholes model to the VG model under a homogeneity assumption.

OPTION PRICING IN THE VARIANCE-GAMMA MODEL UNDER THE DRIFT JUMP

This paper continues elements of the research direction of the work of Madan et al. [(1998) The variance gamma process and option pricing, European Finance Review 2, 79–105] and gives analytical expressions for the prices of digital and European call options in the variance-gamma model under the assumption that the linear drift rate of stock log-returns can suddenly jump downwards. The time of the jump is taken to be exponentially distributed. The formulas obtained require the computation of some generalized hyperbolic functions.

Marginal consistent dependence modelling using weak subordination for Brownian motions

We present an approach for modelling dependencies in exponential Lévy market models with arbitrary margins originated from time changed Brownian motions. Using weak subordination of Buchmann et al. [Bernoulli, 2017], we face a new layer of dependencies, superior to traditional approaches based on pathwise subordination, since weakly subordinated processes are not required to have independent components considering multivariate stochastic time changes. We apply a subordinator being able to incorporate any joint or idiosyncratic information arrivals. We emphasize multivariate variance gamma and normal inverse Gaussian processes and state explicit formulae for the Lévy characteristics. Using maximum likelihood, we estimate multivariate variance gamma models on various market data and show that these models are highly preferable to traditional approaches. Consistent values of basket-options under given marginal pricing models are achieved using the Esscher transform, generating a non-flat implied correlation surface.

Marginal Consistent Dependence Modeling Using Weak Subordination for Brownian Motions

We present an approach for modeling dependencies in exponential Levy market models with arbitrary margins originated from time changed Brownian motions. Using weak subordination of Buchmann et al. (2016), we face a new layer of dependencies, superior to traditional approaches based on pathwise subordination, since weakly subordinated processes are not required to have independent components considering multivariate stochastic time changes. We apply a subordinator being able to incorporate any joint or idiosyncratic information arrivals. We emphasize multivariate variance gamma and normal inverse Gaussian processes and state explicit formulae for the Levy characteristics. Using maximum likelihood, we estimate a multivariate variance gamma model on various market data and show that the model is highly preferable to traditional approaches. Consistent values of basket-options under given marginal pricing models are achieved using Esscher transform, generating a non-flat implied correlation surface.

Convertible bond pricing with partial integro-differential equation model

In this paper, we introduce the concept of Exponential Variance Gamma (EVG) model to the valuation of convertible bond (CB). Rather than evaluating derivatives with standard Black–Scholes approach, we describe the dynamic underlying asset log price with VG process, which is one of classical Lévy processes with non-normal distribution but skewness and leptokurtosis. For numerical purpose, we develop a discrete scheme with stability and convergence, which combines so-called multi-stage compound-option model (MCO) and explicit–implicit difference method (EXIM) to discretize the partial integro-differential equation (PIDE). By comparing our results with Black–Scholes approach, we can show that because of the ability to capture skewness and leptokurtosis features, the new approach does provide a lower price for the valuation of CB.

Financial density forecasts: A comprehensive comparison of risk‐neutral and historical schemes

AbstractWe investigate the forecasting ability of the most commonly used benchmarks in financial economics. We approach the usual caveats of probabilistic forecasts studies—small samples, limited models, and nonholistic validations—by performing a comprehensive comparison of 15 predictive schemes during a time period of over 21 years. All densities are evaluated in terms of their statistical consistency, local accuracy and forecasting errors. Using a new composite indicator, the integrated forecast score, we show that risk‐neutral densities outperform historical‐based predictions in terms of information content. We find that the variance gamma model generates the highest out‐of‐sample likelihood of observed prices and the lowest predictive errors, whereas the GARCH‐based GJR‐FHS delivers the most consistent forecasts across the entire density range. In contrast, lognormal densities, the Heston model, or the nonparametric Breeden–Litzenberger formula yield biased predictions and are rejected in statistical tests.

Gas storage valuation under multifactor Lévy processes

A practical 1 The author's views are his own and do not necessarily reflect those of Gazprom Marketing & Trading Limited or any of its affiliates. problem for energy companies is instituting a consistent framework across its supply and trading activities to deliver on all-important P&L and at-Risk reporting requirements. With a focus on storage assets and wider natural gas market exposures, we present a gas storage valuation methodology, which uniquely uses a flexible multifactor Lévy process setting that allows for consistent valuation and risk management reporting across a general derivative book. Our approach is capable of replicating the complex covariance structure of the natural gas forward curve and capturing time spread volatility, a key driver of extrinsic storage value, while being simultaneously capable of accurately calibrating to market traded options. We begin by extending a single factor Mean Reverting Variance Gamma process to an arbitrary number of dimensions and, by way of specific examples, show how the traditional Principal Component Analysis based view of gas forward curve dynamics can be incorporated into a primarily market based valuation. We develop in the process an innovative implied moments based calibration technique, which allows for efficient calibration of general multifactor forward curve models to delivery period options common in energy and commodity markets. Furthermore, to accommodate the forward curve and traded options market consistency, we propose an appropriate joint market based calibration and historical estimation methodology. Through a formal model specification analysis, we provide evidence that the multifactor Lévy models we propose provide a better joint fit to NBP natural gas options-forward market data, relative to comparative benchmark models. Finally, we develop a novel multidimensional fast Fourier transform based storage valuation algorithm and provide empirical evidence that the multifactor Lévy model suite is better specified to more accurately capture extrinsic value.

Instantaneous portfolio theory

Instantaneous risk is described by the arrival rate of jumps in log price relatives. As a consequence there is then no concept of a mean return compensating risk exposures, as zero is the only instantaneous risk-free return. From this perspective, all portfolios are subject to risk and there are only bad and better ways of holding risk. For the purpose of analysing portfolios, the univariate variance gamma model is extended to higher dimensions with an arrival rate function with full high-dimensional support and independent levels of marginal skewness and excess kurtosis. Investment objectives are given by concave lower price functionals formulated as measure distorted variations. Specific measure distortions are calibrated to data on S&P 500 index options and the time series of the index. The time series estimation is conducted by digital moment matching applied to uncentred data and it is shown that data centring is a noisy activity to be generally avoided. The evaluation of the instantaneous investment objective requires the computation of measure distorted integrals. This is done using Monte Carlo applied to gamma distributed ellipitical radii with a low shape parameter. The resulting risk reward frontiers are between finite variation as the reward and measure distorted variations as risk. In the absence of an instantaneous risk-free return, portfolios on the efficient frontier are characterized by differences in asset variations being given by differences in asset covariations with the risk charge differential of the efficient portfolio. Portfolio variations seen as the equivalent of excess returns, may optimally be negative. Lower price maximizing portfolios are presented in two, six and twenty five dimensions.

Multinomial method for option pricing under Variance Gamma

ABSTRACTThis paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma (VG) process. The continuous time VG process is approximated by a continuous time process with the same first four cumulants and then discretized in time and space. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised before the expiration date. Numerical computations of European and American options are presented and compared with results obtained with finite differences method and with the Black–Scholes prices.

On Numerical Methods for Spread Options

Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial variables. In most cases, analytically closed form solutions for pricing such payoffs are not available, and the application of numerical pricing methods turns out to be non-trivial. We consider several such non-trivial cases and explore the performance of the highly efficient numerical technique of Hurd and Zhou (2010), comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana and Fusai (2013). We show that the former is in essence an application of the two–dimensional Parseval Identity. As application examples, we price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process, in a threefactor stochastic volatility model, as well as in examples of models driven by other popular multivariate Levy processes such as the variance Gamma process, and discuss the price sensitivity with respect to volatility. We also consider examples in the fixed–income market, specifically, on cross–currency interest rate spreads and on LIBOR/OIS spreads. In terms of FFT computation, we have used the FFTW library (see Frigo and Johnson (2010)) and we document appropriate usage of this library to reconcile it with the MATLAB ifft2 counterpart.

Profit Testing of Profit Sharing Life Insurance Policies in the Presence of Extreme Risks

This paper examines the profit testing of life insurance companies that issue participating policies, type B and type A universal life policies, and variable annuities with guaranteed minimum maturity and death benefits, when investment returns are stochastic and modeled by normal or variance gamma distributions. We rely on the stochastic profit testing techniques introduced in Dickson, Hardy, and Waters (2013) to examine the influence of the models' parameters and of the models themselves on the profit testing indicators. We show that the variance gamma model results in more conservative predictions - in a smaller expected net present value and in larger risk indicators for the net present value - than the normal model for participating policies and universal life policies. The paper also shows that variable annuities with guaranteed minimum maturity benefits are little sensitive to the choice of model. For variable annuities with guaranteed minimum death benefits, we show that the expected net present value is higher but that the risk indicators built on the net present value remain higher in the variance gamma model.

A Flexible Generalized Hyperbolic Option Pricing Model and Its Special Cases*

We study a generalized hyperbolic (GH) time-changed Levy process for option pricing and examine six three-parameter special cases: the variance gamma (VG) model of Madan, Carr, and Chang (1998), t, hyperbolic (H), normal inverse Gaussian (NIG), reciprocal hyperbolic (RH), and normal reciprocal inverse Gaussian (NRIG) option pricing models. We study the GH model’s moment properties of the associated risk-neutral distribution of logarithmic spot returns, and obtain an explicit pricing formula for European options facilitated by the time-change Levy process construction. Using S&P 500 Index European options during low and high volatility sample periods, we compare the GH model empirically with existing benchmark models such as the finite-moment log-stable model and the Black–Scholes model. The GH model offers the best in- and out-of-sample performance overall, and a proposed t model special case generally outperforms the existing VG special case. We also present a stochastic volatility extension of the GH model.

On risk measuring in the variance-gamma model

Abstract In this paper, we discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.

On the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models

We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Levy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical examples for two typical exponential Levy models: Merton models and variance gamma models.

Random spectral measure for non Gaussian moving averages

ABSTRACTWe study the distribution of phases and amplitudes for the spectral representation of weighted moving averages of a general noise measure. The simple independent structure, known for the Gaussian case, and involving Rayleigh amplitude and uniform phase distributions, is lost for the non Gaussian noise case. We show that the amplitude/phase distributions exhibit a rich and more complex structure depending not just on the covariance of the process but specifically on the form of the kernel and the noise distribution. We present a theoretical tool for studying these distributions that follows from a proof of the spectral theorem that yields an explicit expression for the spectral measure. The main interest is in noise measures based on second-order Lévy motions since such measures are easily available through independent sampling. We approximate the spectral stochastic measure by independent noise increments which allows us to obtain amplitude/phase distributions that is of fundamental interest for analyzing processes in the frequency domain. For the purpose of approximating the moving average process through sums of trigonometric functions, we assess the mean square error of discretization of the spectral representation. For a specified accuracy, the approximation is explicitly given. We illustrate the method for the moving averages driven by the Laplace motion.

Financial Density Forecasts: A Comprehensive Comparison of Risk-Neutral and Historical Schemes

We investigate the forecasting ability of the most commonly used benchmarks in financial economics. We approach the main methodological caveats of probabilistic forecasts studies – small samples, limited models and non-holistic validations – by performing a comprehensive comparison of 15 predictive schemes during a time period of over 21 years. All densities are evaluated in terms of their statistical consistency, local accuracy and forecasting errors. Through the development of a new indicator, the Integrated Forecast Score (IFS), we show that risk-neutral densities outperform historical-based predictions in terms of information content. We find that the Variance Gamma model generates the highest out-of-sample likelihood of observed prices and the lowest predictive errors, whereas the ARCH-based GJR-FHS delivers the most consistent forecasts across the entire density range. In contrast, log-normal densities, the Heston model or the non-parametric Breeden-Litzenberger formula yield biased predictions and are rejected in statistical tests.

Multilevel Monte Carlo for exponential L\xe9vy 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.