The paper explores the properties of a class of multivariate Lévy processes, used for asset returns, with a focus on describing in an economic sensible and empirically appropriate way both linear and nonlinear dependence. The processes are subordinated Brownian motions. The subordinator has a common and an idiosyncratic component, to reflect the properties of trade, which it represents. A calibration to a portfolio of ten US stock indices returns over the period 2009-2013 shows that the hyperbolic specification fits very well marginal distributions, the overall correlation matrix and the return distribution of both long-only and long-short random portfolios, which incorporate also nonlinear dependence. Their tail behavior is well captured also by the variance gamma specification. The main message is not only the goodness of fit, but also the flexibility in capturing dependence and the easiness of calibration on large sets of returns.
We compute an analytical expression for the moment generating function of the joint random vector consisting of a spot price and its discretely monitored average for a large class of square-root price dynamics. This result, combined with the Fourier transform pricing method proposed by Carr and Madan [Carr, P., Madan D., 1999. Option valuation using the fast Fourier transform. Journal of Computational Finance 2(4), Summer, 61-73] allows us to derive a closed-form formula for the fair value of discretely monitored Asian-style options. Our analysis encompasses the case of commodity price dynamics displaying mean reversion and jointly fitting a quoted futures curve and the seasonal structure of spot price volatility. Four tests are conducted to assess the relative performance of the pricing procedure stemming from our formulae. Empirical results based on natural gas data from NYMEX and corn data from CBOT show a remarkable improvement over the main alternative techniques developed for pricing Asian-style options within the market standard framework of geometric Brownian motion.
We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using di¤erent parametric Lévy processes. We also discuss model-risk issues.JEL Classi…cation: G13, C63
The paper explores the fit properties of a class of multivariate Lévy processes, which are characterized as time-changed correlated Brownian motions. The time-change has a common and an idiosyncratic component, to reflect the properties of trade, which it represents. The resulting process may provide Variance-Gamma, Normal-Inverse-Gaussian or Generalized-Hyperbolic margins. A non-pairwise calibration to a portfolio of ten US daily stock returns over the period 2009-2013 shows that fit of the Hyperbolic specification is very good, in terms of marginal distributions and overall correlation matrix. It succeeds in explaining the return distribution of both long-only and longshort random portfolios better than competing models do. Their tail behavior is well captured also by the Variance-Gamma specification.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.