Pricing derivatives with stochastic volatility

This Ph.D. thesis contains 4 essays in mathematical finance with a focus on pricing Asian option (Chapter 4), pricing futures and futures option (Chapter 5 and Chapter 6) and time dependent volatility in futures option (Chapter 7). In Chapter 4, the applicability of the Albrecher et al.(2005)'s comonotonicity approach was investigated in the context of various benchmark models for equities and com- modities. Instead of classical Levy models as in Albrecher et al.(2005), the focus is the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and the Schwartz (1997) two-factor model. It is shown that the method delivers rather tight upper bounds for the prices of Asian Options in these models and as a by-product delivers super-hedging strategies which can be easily implemented. In Chapter 5, two types of three-factor models were studied to give the value of com- modities futures contracts, which allow volatility to be stochastic. Both these two models have closed-form solutions for futures contracts price. However, it is shown that Model 2 is better than Model 1 theoretically and also performs very well empiri- cally. Moreover, Model 2 can easily be implemented in practice. In comparison to the Schwartz (1997) two-factor model, it is shown that Model 2 has its unique advantages; hence, it is also a good choice to price the value of commodity futures contracts. Fur- thermore, if these two models are used at the same time, a more accurate price for commodity futures contracts can be obtained in most situations. In Chapter 6, the applicability of the asymptotic approach developed in Fouque et al.(2000b) was investigated for pricing commodity futures options in a Schwartz (1997) multi-factor model, featuring both stochastic convenience yield and stochastic volatility. It is shown that the zero-order term in the expansion coincides with the Schwartz (1997) two-factor term, with averaged volatility, and an explicit expression for the first-order correction term is provided. With empirical data from the natural gas futures market, it is also demonstrated that a significantly better calibration can be achieved by using the correction term as compared to the standard Schwartz (1997) two-factor expression, at virtually no extra effort. In Chapter 7, a new pricing formula is derived for futures options in the Schwartz (1997) two-factor model with time dependent spot volatility. The pricing formula can also be used to find the result of the time dependent spot volatility with futures options prices in the market. Furthermore, the limitations of the method that is used to find the time dependent spot volatility will be explained, and it is also shown how to make sure of its accuracy.

Myocardial Extracellular Volume Expansion And The Risk Of Recurrent Atrial Fibrillation After Pulmonary Vein Isolation.

This study tested whether myocardial extracellular volume (ECV) is increased in patients with hypertension and atrial fibrillation (AF) undergoing pulmonary vein isolation and whether there is an association between ECV and post-procedural recurrence of AF. Hypertension is associated with myocardial fibrosis, an increase in ECV, and AF. Data linking these findings are limited. T1 measurements pre-contrast and post-contrast in a cardiac magnetic resonance (CMR) study provide a method for quantification of ECV. Consecutive patients with hypertension and recurrent AF referred for pulmonary vein isolation underwent a contrast CMR study with measurement of ECV and were followed up prospectively for a median of 18 months. The endpoint of interest was late recurrence of AF. Patients had elevated left ventricular (LV) volumes, LV mass, left atrial volumes, and increased ECV (patients with AF, 0.34 ± 0.03; healthy control patients, 0.29 ± 0.03; p < 0.001). There were positive associations between ECV and left atrial volume (r = 0.46, p < 0.01) and LV mass and a negative association between ECV and diastolic function (early mitral annular relaxation [E'], r = -0.55, p < 0.001). In the best overall multivariable model, ECV was the strongest predictor of the primary outcome of recurrent AF (hazard ratio: 1.29; 95% confidence interval: 1.15 to 1.44; p < 0.0001) and the secondary composite outcome of recurrent AF, heart failure admission, and death (hazard ratio: 1.35; 95% confidence interval: 1.21 to 1.51; p < 0.0001). Each 10% increase in ECV was associated with a 29% increased risk of recurrent AF. In patients with AF and hypertension, expansion of ECV is associated with diastolic function and left atrial remodeling and is a strong independent predictor of recurrent AF post-pulmonary vein isolation.

Maxillary protraction after surgically assisted maxillary expansion

This case report describes the orthodontic treatment of a 32-year-old woman with a Class III malocclusion, whose chief compliant was her dentofacial esthetics. The pretreatment lateral cephalometric tracings showed the presence of a Class III dentoskeletal malocclusion with components of maxillary deficiency. After discussion with the patient, the treatment option included surgically assisted rapid maxillary expansion (SARME) followed by orthopedic protraction (Sky Hook) and Class III elastics. Patient compliance was excellent and satisfactory dentofacial esthetics was achieved after treatment completion.

Skeletal alterations associated with the use of bonded rapid maxillary expansion appliance

Bonded maxillary expansion appliances have been suggested to control increases in the vertical dimension of the face after rapid maxillary expansion (RME). However, there is still no consensus in the literature about its real skeletal effects. The purpose of this prospective study was to evaluate, longitudinally, the vertical and sagittal cephalometric alterations after RME performed with bonded maxillary expansion appliance. The sample consisted of 26 children, with a mean age of 8.7 years (range: 6.9-10.9 years), with posterior skeletal crossbite and indication for RME. After maxillary expansion, the bonded appliance was used as a fixed retention for 3.4 months, being replaced by a removable retention subsequently. The cephalometric study was performed onto lateral radiographs, taken before treatment was started, and again 6.3 months after removing the bonded appliance. Intra-group comparison was made using paired t test. The results showed that there were no significant sagittal skeletal changes at the end of treatment. There was a small vertical skeletal increase in five of the eleven evaluated cephalometric measures. The maxilla displaced downward, but it did not modify the facial growth patterns or the direction of the mandible growth. Under the specific conditions of this research, it may be concluded that RME with acrylic bonded maxillary expansion appliance did promote signifciant vertical or sagittal cephalometric alterations. The vertical changes found with the use of the bonded appliance were small and probably transitory, similar to those occurred with the use of banded expansion appliances.

Correlation between metal-ceramic bond strength and coefficient of linear thermal expansion difference

The purpose of this study was to evaluate the metal-ceramic bond strength (MCBS) of 6 metal-ceramic pairs (2 Ni-Cr alloys and 1 Pd-Ag alloy with 2 dental ceramics) and correlate the MCBS values with the differences between the coefficients of linear thermal expansion (CTEs) of the metals and ceramics. Verabond (VB) Ni-Cr-Be alloy, Verabond II (VB2), Ni-Cr alloy, Pors-on 4 (P), Pd-Ag alloy, and IPS (I) and Duceram (D) ceramics were used for the MCBS test and dilatometric test. Forty-eight ceramic rings were built around metallic rods (3.0 mm in diameter and 70.0 mm in length) made from the evaluated alloys. The rods were subsequently embedded in gypsum cast in order to perform a tensile load test, which enabled calculating the CMBS. Five specimens (2.0 mm in diameter and 12.0 mm in length) of each material were made for the dilatometric test. The chromel-alumel thermocouple required for the test was welded into the metal test specimens and inserted into the ceramics. ANOVA and Tukey's test revealed significant differences (p=0.01) for the MCBS test results (MPa), with PI showing higher MCBS (67.72) than the other pairs, which did not present any significant differences. The CTE (10-6 oC-1) differences were: VBI (0.54), VBD (1.33), VB2I (-0.14), VB2D (0.63), PI (1.84) and PD (2.62). Pearson's correlation test (r=0.17) was performed to evaluate of correlation between MCBS and CTE differences. Within the limitations of this study and based on the obtained results, there was no correlation between MCBS and CTE differences for the evaluated metal-ceramic pairs.

Photographic assessment of nasal morphology following rapid maxillary expansion in children

OBJECTIVE: The aim of the present study was to use facial analysis to determine the effects of rapid maxillary expansion (RME) on nasal morphology in children in the stages of primary and mixed dentition, with posterior cross-bite. MATERIAL AND METHODS: Facial photographs (front view and profile) of 60 patients in the pre-expansion period, immediate post-expansion period and one year following rapid maxillary expansion with a Haas appliance were evaluated on 2 occasions by 3 experienced orthodontists independently, with a 2-week interval between evaluations. The examiners were instructed to assess nasal morphology and had no knowledge regarding the content of the study. Intraexaminer and interexaminer agreement (assessed using the Kappa statistic) was acceptable. RESULTS: From the analysis of the mode of the examiners' findings, no alterations in nasal morphology occurred regarding the following aspects: dorsum of nose, alar base, nasal width of middle third and nasal base. Alterations were only detected in the nasolabial angle in 1.64% of the patients between the pre-expansion and immediate post-expansion photographs. In 4.92% of the patients between the immediate post-expansion period and 1 year following expansion; and in 6.56% of the patients between the pre-expansion period and one year following expansion. CONCLUSIONS: RME performed on children in stages of primary and mixed dentition did not have any impact on nasal morphology, as assessed using facial analysis.

Reaction-diffusion stochastic lattice model for a predator-prey system

We have the purpose of analyzing the effect of explicit diffusion processes in a predator-prey stochastic lattice model. More precisely we wish to investigate the possible effects due to diffusion upon the thresholds of coexistence of species, i. e., the possible changes in the transition between the active state and the absorbing state devoid of predators. To accomplish this task we have performed time dependent simulations and dynamic mean-field approximations. Our results indicate that the diffusive process can enhance the species coexistence.

Parametric model for the analysis of concrete expansion due to alkali-aggregate reaction

The alkali-aggregate reaction (AAR) is a chemical reaction that provokes a heterogeneous expansion of concrete and reduces important properties such as Young's modulus, leading to a reduction in the structure's useful life. In this study, a parametric model is employed to determine the spatial distribution of the concrete expansion, combining normalized factors that influence the reaction through an AAR expansion law. Optimization techniques were employed to adjust the numerical results and observations in a real structure. A three-dimensional version of the model has been implemented in a finite element commercial package (ANSYS(C)) and verified in the analysis of an accelerated mortar test. Comparisons were made between two AAR mathematical descriptions for the mechanical phenomenon, using the same methodology, and an expansion curve obtained from experiment. Some parametric studies are also presented. The numerical results compared very well with the experimental data validating the proposed method.

Crossover between the extended and localized regimes in stochastic partially self-avoiding walks in one-dimensional disordered systems

Consider N sites randomly and uniformly distributed in a d-dimensional hypercube. A walker explores this disordered medium going to the nearest site, which has not been visited in the last mu (memory) steps. The walker trajectory is composed of a transient part and a periodic part (cycle). For one-dimensional systems, travelers can or cannot explore all available space, giving rise to a crossover between localized and extended regimes at the critical memory mu(1) = log(2) N. The deterministic rule can be softened to consider more realistic situations with the inclusion of a stochastic parameter T (temperature). In this case, the walker movement is driven by a probability density function parameterized by T and a cost function. The cost function increases as the distance between two sites and favors hops to closer sites. As the temperature increases, the walker can escape from cycles that are reminiscent of the deterministic nature and extend the exploration. Here, we report an analytical model and numerical studies of the influence of the temperature and the critical memory in the exploration of one-dimensional disordered systems.

Role of noise in population dynamics cycles

Noise is an intrinsic feature of population dynamics and plays a crucial role in oscillations called phase-forgetting quasicycles by converting damped into sustained oscillations. This function of noise becomes evident when considering Langevin equations whose deterministic part yields only damped oscillations. We formulate here a consistent and systematic approach to population dynamics, leading to a Fokker-Planck equation and the associate Langevin equations in accordance with this conceptual framework, founded on stochastic lattice-gas models that describe spatially structured predator-prey systems. Langevin equations in the population densities and predator-prey pair density are derived in two stages. First, a birth-and-death stochastic process in the space of prey and predator numbers and predator-prey pair number is obtained by a contraction method that reduces the degrees of freedom. Second, a van Kampen expansion in the inverse of system size is then performed to get the Fokker-Planck equation. We also study the time correlation function, the asymptotic behavior of which is used to characterize the transition from the cyclic coexistence of species to the ordinary coexistence.

1/N expansion in noncommutative quantum mechanics

We study the 1/N expansion in noncommutative quantum mechanics for the anharmonic and Coulombian potentials. The expansion for the anharmonic oscillator presented good convergence properties, but for the Coulombian potential, we found a divergent large N expansion when using the usual noncommutative generalization of the potential. We proposed a modified version of the noncommutative Coulombian potential which provides a well-behaved 1/N expansion.

CMB in a box: Causal structure and the Fourier-Bessel expansion

This paper makes two points. First, we show that the line-of-sight solution to cosmic microwave anisotropies in Fourier space, even though formally defined for arbitrarily large wavelengths, leads to position-space solutions which only depend on the sources of anisotropies inside the past light cone of the observer. This foretold manifestation of causality in position (real) space happens order by order in a series expansion in powers of the visibility gamma = e(-mu), where mu is the optical depth to Thomson scattering. We show that the contributions of order gamma(N) to the cosmic microwave background (CMB) anisotropies are regulated by spacetime window functions which have support only inside the past light cone of the point of observation. Second, we show that the Fourier-Bessel expansion of the physical fields (including the temperature and polarization momenta) is an alternative to the usual Fourier basis as a framework to compute the anisotropies. The viability of the Fourier-Bessel series for treating the CMB is a consequence of the fact that the visibility function becomes exponentially small at redshifts z >> 10(3), effectively cutting off the past light cone and introducing a finite radius inside which initial conditions can affect physical observables measured at our position (x) over right arrow = 0 and time t(0). Hence, for each multipole l there is a discrete tower of momenta k(il) (not a continuum) which can affect physical observables, with the smallest momenta being k(1l) similar to l. The Fourier-Bessel modes take into account precisely the information from the sources of anisotropies that propagates from the initial value surface to the point of observation-no more, no less. We also show that the physical observables (the temperature and polarization maps), and hence the angular power spectra, are unaffected by that choice of basis. This implies that the Fourier-Bessel expansion is the optimal scheme with which one can compute CMB anisotropies.

Shared information in stationary states of stochastic processes

We present four estimators of the shared information (or interdepency) in ground states given that the coefficients appearing in the wave function are all real non-negative numbers and therefore can be interpreted as probabilities of configurations. Such ground states of Hermitian and non-Hermitian Hamiltonians can be given, for example, by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models. We consider in detail the last case, the system being a classical not a quantum one. Using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive, conformal invariant and nonconformal invariant massless phases. We show that like in the case of the quantum problem, the estimators verify the area law with logarithmic corrections when phase transitions take place.

Directed Abelian algebras and their application to stochastic models

With each directed acyclic graph (this includes some D-dimensional lattices) one can associate some Abelian algebras that we call directed Abelian algebras (DAAs). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA, one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground-state wave functions (stationary state probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and chooses Hamiltonians linear in the generators, in finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent z=D. One possible application of the DAA is to sandpile models. In the paper we present this application, considering one- and two-dimensional lattices. In the one-dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent sigma(tau)=3/2). We study the local density of particles inside large avalanches, showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found sigma(tau)=1.780 +/- 0.005.

Random perturbations of stochastic processes with unbounded variable length memory

We consider binary infinite order stochastic chains perturbed by a random noise. This means that at each time step, the value assumed by the chain can be randomly and independently flipped with a small fixed probability. We show that the transition probabilities of the perturbed chain are uniformly close to the corresponding transition probabilities of the original chain. As a consequence, in the case of stochastic chains with unbounded but otherwise finite variable length memory, we show that it is possible to recover the context tree of the original chain, using a suitable version of the algorithm Context, provided that the noise is small enough.