A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion

In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotically exact if an increasing number of terms is included, and provides remarkably accurate results even when truncated to the first few (say 3) terms. The coefficients of such expansion can be determined straightforwardly through a recursion, and involve simple one-dimensional integrals. We present several examples of financial interest, and we compare our results with the state-of-the-art approximation of discretely sampled diffusions [A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that the exponent expansion provides a similar accuracy in most of the cases, but a better behavior in the low-volatility regime. Furthermore the implementation of the present approach turns out to be simpler. Within the functional integration framework the exponent expansion allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. This is illustrated with the application to simple path-dependent interest rate derivatives. Finally we discuss how these results can also be used to increase the efficiency of numerical (both deterministic and stochastic) approaches to derivative pricing.Comment: 28 pages, 7 figure

Finite-size spin-wave theory of a collinear antiferromagnet

The ground-state and low-energy properties of the two-dimensional $J_1{-}J_2$ Heisenberg model in the collinear phase are investigated using finite-size spin-wave theory [Q. F. Zhong and S. Sorella, {\em Europhys. Lett.} {\bf 21}, 629 (1993)], and Lanczos exact diagonalizations. For spin one-half -- where the effects of quantization are the strongest -- the spin-wave expansion turns out to be quantitatively accurate for $J_2/J_1\gtrsim 0.8$. In this regime, both the magnetic structure factor and the spin susceptibility are very close to the spin-wave predictions. The spin-wave estimate of the order parameter in the collinear phase, $m^\dagger\simeq 0.3$, is in remarkable agreement with recent neutron scattering measurements on ${\rm Li_2VOSiO_4}$.Comment: 10 pages, 3 figure

Algorithmic differentiation and the calculation of forces by quantum Monte Carlo

We describe an efficient algorithm to compute forces in quantum Monte Carlo using adjoint algorithmic differentiation. This allows us to apply the space warp coordinate transformation in differential form, and compute all the 3M force components of a system with M atoms with a computational effort comparable with the one to obtain the total energy. Few examples illustrating the method for an electronic system containing several water molecules are presented. With the present technique, the calculation of finite-temperature thermodynamic properties of materials with quantum Monte Carlo will be feasible in the near future.Comment: 32 pages, 4 figure, to appear in The Journal of Chemical Physic

Quantum Phase Transition in Coupled Spin Ladders

The ground state of an array of coupled, spin-half, antiferromagnetic ladders is studied using spin-wave theory, exact diagonalization (up to 36 sites) and quantum Monte Carlo techniques (up to 256 sites). Our results clearly indicate the occurrence of a zero-temperature phase transition between a N\'eel ordered and a non-magnetic phase at a finite value of the inter-ladder coupling ($\alpha_c\simeq0.3$). This transition is marked by remarkable changes in the structure of the excitation spectrum.Comment: 4 pages, 6 postscript figures, to appear in Physical Review

Ising transition in the two-dimensional quantum J1−J2J_1-J_2 Heisenberg model

We study the thermodynamics of the spin-$S$ two-dimensional quantum Heisenberg antiferromagnet on the square lattice with nearest ($J_1$) and next-nearest ($J_2$) neighbor couplings in its collinear phase ($J_2/J_1>0.5$), using the pure-quantum self-consistent harmonic approximation. Our results show the persistence of a finite-temperature Ising phase transition for every value of the spin, provided that the ratio $J_2/J_1$ is greater than a critical value corresponding to the onset of collinear long-range order at zero temperature. We also calculate the spin- and temperature-dependence of the collinear susceptibility and correlation length, and we discuss our results in light of the experiments on Li$_2$VOSiO$_4$ and related compounds.Comment: 4 page, 4 figure

Finite size Spin Wave theory of the triangular Heisenberg model

We present a finite size spin wave calculation on the Heisenberg antiferromagnet on the triangular lattice focusing in particular on the low-energy part of the excitation spectrum. For s=1/2 the good agreement with the exact diagonalization and quantum Monte Carlo results supports the reliability of the spin wave expansion to describe the low-energy spin excitations of the Heisenberg model even in presence of frustration. This indicates that the spin susceptibility of the triangular antiferromagnet is very close to the linear spin wave result.Comment: 6 pages (LateX), 2 ps-figure

The Mott Metal-Insulator transition in the half-filled Hubbard model on the Triangular Lattice

We investigate the metal-insulator transition in the half-filled Hubbard model on a two-dimensional triangular lattice using both the Kotliar-Ruckenstein slave-boson technique, and exact numerical diagonalization of finite clusters. Contrary to the case of the square lattice, where the perfect nesting of the Fermi surface leads to a metal-insulator transition at arbitrarily small values of U, always accompanied by antiferromagnetic ordering, on the triangular lattice, due to the lack of perfect nesting, the transition takes place at a finite value of U, and frustration induces a non-trivial competition among different magnetic phases. Indeed, within the mean-field approximation in the slave-boson approach, as the interaction grows the paramagnetic metal turns into a metallic phase with incommensurate spiral ordering. Increasing further the interaction, a linear spin-density-wave is stabilized, and finally for strong coupling the latter phase undergoes a first-order transition towards an antiferromagnetic insulator. No trace of the intermediate phases is instead seen in the exact diagonalization results, indicating a transition between a paramagnetic metal and an antiferromagnetic insulator.Comment: 5 pages, 4 figure