Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model

We consider the stochastic volatility model d S _t = σ _t S _t d W _t ,d σ _t = ω σ _t d Z _t , with (W _t ,Z _t ) uncorrelated standard Brownian motions. This is a special case of the Hull-White and the β=1 (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the n→∞ limit of a very large number of time steps of size τ, at fixed \(\beta =\frac 12\omega ^{2}\tau n^{2}\) and \(\rho ={\sigma _{0}^{2}}\tau \), and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of S _t . Under the Euler-Maruyama discretization for (S _t ,logσ _t ), the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.     

Emergence of Heavy-Tailed Distributions in a Random Multiplicative Model Driven by a Gaussian Stochastic Process

We consider a random multiplicative stochastic process with multipliers given by the exponential of a Brownian motion. The positive integer moments of the distribution function can be computed exactly, and can be represented as the grand partition function of an equivalent lattice gas with attractive 2-body interactions. The numerical results for the positive integer moments display a sharp transition at a critical value of the model parameters, which corresponds to a phase transition in the equivalent lattice gas model. The shape of the terminal distribution changes suddenly at the critical point to a heavy-tailed distribution. The transition can be related to the position of the complex zeros of the grand partition function of the lattice gas, in analogy with the Lee, Yang picture of phase transitions in statistical mechanics. We study the properties of the equivalent lattice gas in the thermodynamical limit, which corresponds to the continuous time limit of the random multiplicative model, and derive the asymptotics of the approach to the continuous time limit. The results can be generalized to a wider class of random multiplicative processes, driven by the exponential of a Gaussian stochastic process.     

Excited baryons and heavy pentaquarks in large- N<Subscript>c</Subscript> QCD

We briefly discuss the large-N_c picture for excited baryons, present a new method for the calculation of matrix elements and illustrate it by computing the strong decays of heavy exotic states.     

Small-noise limit of the quasi-Gaussian log-normal HJM model

Quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation, which greatly simplifies their numerical implementation. We present a qualitative study of the solutions of the quasi-Gaussian log-normal HJM model. Using a small-noise deterministic limit we show that the short rate may explode to infinity in finite time. This implies the explosion of the Eurodollar futures prices in this model. We derive explicit explosion criteria under mild assumptions on the shape of the yield curve.     

The logistic-normal integral and its generalizations

We consider the solutions of the one-dimensional heat equation in an unbounded domain with initial conditions of the form f(x)/(1+exp(σx))$f\left(x\right)/(1+exp\left(\sigma x\right))$. This includes as a particular case the logistic-normal integral, which corresponds to f(x)=1$f\left(x\right)=1$. Such initial conditions appear in stochastic calculus problems, and the numerical simulation of short-rate interest rate models and credit models with log-normally distributed short rates and hazard rates respectively. We show that the solutions at time t$t$ can be computed exactly on a grid of equidistant points of width σt$\sigma t$ in terms of the solutions of the heat equation with initial condition f(x)$f\left(x\right)$. The exact results on the grid can be used as nodes for a precise interpolation. Series representation of the solutions can be obtained by an application of the Poisson summation formula.     

Parity doubling and SU(2)LxSU(2)R restoration in the hadron spectrum

We construct the most general nonlinear representation of chiral SU(2)LxSU(2)R broken down spontaneously to the isospin SU(2), on a pair of hadrons of same spin and isospin and opposite parity. We show that any such representation is equivalent, through a hadron field transformation, to two irreducible representations on two hadrons of opposite parity with different masses and axial-vector couplings. This implies that chiral symmetry realized in the Nambu-Goldstone mode does not predict the existence of degenerate multiplets of hadrons of opposite parity nor any relations between their couplings or masses.     

Eurodollar futures pricing in log-normal interest rate models in discrete time

We demonstrate the appearance of explosions in three quantities in interest rate models with log-normally distributed rates in discrete time. (1) The expectation of the money market account in the Black, Derman, Toy model, (2) the prices of Eurodollar futures contracts in a model with log-normally distributed rates in the terminal measure and (3) the prices of Eurodollar futures contracts in the one-factor log-normal Libor market model (LMM). We derive exact upper and lower bounds on the prices and on the standard deviation of the Monte Carlo pricing of Eurodollar futures in the one factor log-normal Libor market model. These bounds explode at a non-zero value of volatility, and thus imply a limitation on the applicability of the LMM and on its Monte Carlo simulation to sufficiently low volatilities.