Option Pricing under Randomised GBM Models

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

L 1 and L ∞ stability of transition densities of perturbed diffusions

In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{\infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.

Correspondence between isoscalar monopole strengths and α inelastic cross sections on 24Mg

The correspondence between the isoscalar monopole (IS0) transition strengths and α inelastic cross sections, the B(IS0)-(α,α′) correspondence, is investigated for 24Mg(α,α′) at 130 and 386 MeV. We adopt a microscopic coupled-channel reaction framework to link structural inputs, diagonal and transition densities, for 24Mg obtained with antisymmetrized molecular dynamics to the (α,α′) cross sections. We aim at clarifying how the B(IS0)-(α,α′) correspondence is affected by the nuclear distortion, the in-medium modification to the nucleon-nucleon effective interaction in the scattering process, and the coupled-channels effect. It is found that these effects are significant and the explanation of the B(IS0)-(α,α′) correspondence in the plane wave limit with the long-wavelength approximation, which is often used, makes no sense. Nevertheless, the B(IS0)-(α,α′) correspondence tends to remain because of a strong constraint on the transition densities between the ground state and the 0+ excited states. The correspondence is found to hold at 386 MeV with an error of about 20%–30%, while it is seriously stained at 130 MeV mainly by the strong nuclear distortion. It is also found that when a 0+ state that has a different structure from a simple α cluster state is considered, the B(IS0)-(α,α′) correspondence becomes less valid. For a quantitative discussion on the α clustering in 0+ excited states of nuclei, a microscopic description of both the structure and reaction parts will be necessary.

Explaining the density of post-communist interest group populations—resources, constituencies, and regime change

The article tests the energy–stability–area (ESA) model of interest group population density on a sample of different 2018 Czech, Hungarian, Polish and Slovenian energy, higher education and health care interest organisation populations. The unique context of recent simultaneous political, economic and in the cases of Czechia and Slovenia, national transitions present a hard test for population ecology theory. Besides the area (constituency size) and energy (resources, issue certainty) terms, the article brings the stability term back into the center of analysis. The stability term, that is, the effect of a profound change or shock to the polity is operationalised as Communist-era population densities. As all three policy domains are heavily state controlled and tightly regulated, the effect of neocorporatist interest intermediation is also tested. The article finds strong support for the energy and neocorporatism hypotheses and provides evidence for the effect of communist-era organisational population density on post-transition densities: The size of 2018 organisational populations is found to be dependent on pre-transition densities. The relationship is, however, not linear but curvilinear. Nevertheless, the analysis indicates that the effect of pre-transition population size is moderated by other environmental level factors.

Bayesian inference for diffusion processes: using higher-order approximations for transition densities

Modelling random dynamical systems in continuous time, diffusion processes are a powerful tool in many areas of science. Model parameters can be estimated from time-discretely observed processes using Markov chain Monte Carlo (MCMC) methods that introduce auxiliary data. These methods typically approximate the transition densities of the process numerically, both for calculating the posterior densities and proposing auxiliary data. Here, the Euler–Maruyama scheme is the standard approximation technique. However, the MCMC method is computationally expensive. Using higher-order approximations may accelerate it, but the specific implementation and benefit remain unclear. Hence, we investigate the utilization and usefulness of higher-order approximations in the example of the Milstein scheme. Our study demonstrates that the MCMC methods based on the Milstein approximation yield good estimation results. However, they are computationally more expensive and can be applied to multidimensional processes only with impractical restrictions. Moreover, the combination of the Milstein approximation and the well-known modified bridge proposal introduces additional numerical challenges.

ASYMPTOTIC EXPANSION FOR THE TRANSITION DENSITIES OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY THE GAMMA PROCESSES

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.