The Risk Measurement under the Variance-Gamma Process with Drift Switching

Abstract

The paper discusses an extension of the variance-gamma process with stochastic linear drift coefficient. It is assumed that the linear drift coefficient may switch to a different value at the exponentially distributed time. The size of the drift jump is supposed to have a multinomial distribution. We have obtained the distribution function, the probability density function and the lower partial expectation for the considered process in closed forms. The results are applied to the calculation of the value at risk and the expected shortfall of the investment portfolio in the related multivariate stochastic model.

The Semi-Hyperbolic Distribution and Its Applications

“…The derived formulas use the modified Bessel function of the second kind, the Appell function, and can be computed under one second with the modern software. As in the VG model (Madan et al [30], Ivanov [63]), the explicit solutions are obtained for the cumulative distribution function, the first and second lower partial moments of the SH distribution, and the received formulas are applied to the problem of computation of the VAR, the ES, and the semivariance in the related investment portfolio model with dependent asset returns.…”

“…The computations of the measures in semi-analytical forms by the Fourier transform technique are given in Armenti et al [60] and Drapeu et al [61]. Analytical formulas for some specific models are provided by Ivanov [62,63] and Rockafellar and Uryasev [64]. Monte Carlo simulations of the VAR and ES were made by Chun et al [65] and Mafusalov and Uryasev [66].…”

On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model

“…Finlay and Seneta (2008) develop different techniques for parameter estimation in the skew Student's t-model and discuss the modeling of the S&P500 index and the oil prices. The variance-gamma process (the variance-gamma distribution is the normal-inverse mixture with the gamma mixing density, and hence the variance-gamma process has the stochastic drift modeled by the gamma process) is considered in Madan et al (1998), Seneta (2004), Daal and Madan (2005), Ivanov (2018), Ivanov (2022), Linders andStassen (2016), andMozumder et al (2015), among others. Madan et al (1998) summarize the basic properties of the variance-gamma distribution and suggest the method of receiving analytical results in the variance-gamma model.…”

“…Mozumder et al (2015) study the S&P500 index options in the variance-gamma model. Ivanov (2022) proceeds from the ideas of Madan et al (1998) and obtains closed-form results for an extension of the variance-gamma model.…”