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 Title
 A Spectral Element Method to Price Single and MultiAsset European Options.
 Creator

Zhu, Wuming, Kopriva, David A., Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec N., Okten, Giray, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

We develop a spectral element method to price European options under the BlackScholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a GaussLobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the nonsmooth initial condition. For options with one asset under...
Show moreWe develop a spectral element method to price European options under the BlackScholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a GaussLobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the nonsmooth initial condition. For options with one asset under the jump diffusion model, the convolution integral is approximated by high order GaussLobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The use of the IMEX approximation in time means that only a block diagonal, rather than full, system of equations needs to be solved at each time step. For options with two variables, i.e., two assets under the BlackScholes model or one asset under the stochastic volatility model, the domain is subdivided into quadrilateral elements. Within each element, the expansion basis functions are chosen to be tensor products of the Legendre polynomials. Three iterative methods are investigated to solve the system of equations at each time step with the corresponding second order time integration schemes, i.e., IMEX and CrankNicholson. Also, the boundary conditions are carefully studied for the stochastic volatility model. The method is spectrally accurate (exponentially convergent) in space and second order accurate in time for European options under all the three models. Spectral accuracy is observed in not only the solution, but also in the Greeks.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0513
 Format
 Thesis
 Title
 Variance Gamma Pricing of American Futures Options.
 Creator

Yoo, Eunjoo, Nolder, Craig A., Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec N., Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

In financial markets under uncertainty, the classical BlackScholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and stochastic volatility models were introduced to financial modeling. Today crude oil futures markets are highly volatile. It is the purpose of this dissertation to develop a mathematical framework in which American options on crude oil futures contracts are priced...
Show moreIn financial markets under uncertainty, the classical BlackScholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and stochastic volatility models were introduced to financial modeling. Today crude oil futures markets are highly volatile. It is the purpose of this dissertation to develop a mathematical framework in which American options on crude oil futures contracts are priced more effectively than by current methods. In this work, we use the Variance Gamma process to model the futures price process. To generate the underlying process, we use a random tress method so that we evaluate the option prices at each tree node. Through fifty replications of a random tree, the averaged value is taken as a true option price. Pricing performance using this method is accessed using American options on crude oil commodity contracts from December 2003 to November 2004. In comparison with the Variance Gamma model, we price using the BlackScholes model as well. Over the entire sample period, a positive skewness and high kurtosis, especially in the shortterm options, are observed. In terms of pricing errors, the Variance Gamma process performs better than the BlackScholes model for the American options on crude oil commodities.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0691
 Format
 Thesis
 Title
 Numerical Methods for Portfolio Risk Estimation.
 Creator

Zhang, Jianke, Kercheval, Alec, Huﬀer, Fred, Gallivan, Kyle, Beaumont, Paul, Nichols, Warren, Department of Mathematics, Florida State University
 Abstract/Description

In portfolio risk management, a global covariance matrix forecast often needs to be adjusted by changing diagonal blocks corresponding to specific submarkets. Unless certain constraints are obeyed, this can result in the loss of positive definiteness of the global matrix. Imposing the proper constraints while minimizing the disturbance of offdiagonal blocks leads to a nonconvex optimization problem in numerical linear algebra called the Weighted Orthogonal Procrustes Problem. We analyze...
Show moreIn portfolio risk management, a global covariance matrix forecast often needs to be adjusted by changing diagonal blocks corresponding to specific submarkets. Unless certain constraints are obeyed, this can result in the loss of positive definiteness of the global matrix. Imposing the proper constraints while minimizing the disturbance of offdiagonal blocks leads to a nonconvex optimization problem in numerical linear algebra called the Weighted Orthogonal Procrustes Problem. We analyze and compare two local minimizing algorithms and offer an algorithm for global minimization. Our methods are faster and more effective than current numerical methods for covariance matrix revision.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd0542
 Format
 Thesis
 Title
 Calibration of Multivariate Generalized Hyperbolic Distributions Using the EM Algorithm, with Applications in Risk Management, Portfolio Optimization and Portfolio Credit Risk.
 Creator

Hu, Wenbo, Kercheval, Alec, Huﬀer, Fred, Case, Bettye, Nichols, Warren, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

The distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical...
Show moreThe distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical difficulty of calibrating the distributional parameters to the data. In this dissertation we describe a way to stably calibrate GH distributions for a wider range of parameters than has previously been reported. In particular, we develop a version of the EM algorithm for calibrating GH distributions. This is a modification of methods proposed in McNeil, Frey, and Embrechts (2005), and generalizes the algorithm of Protassov (2004). Our algorithm extends the stability of the calibration procedure to a wide range of parameters, now including parameter values that maximize loglikelihood for our real market data sets. This allows for the first time certain GH distributions to be used in modeling contexts when previously they have been numerically intractable. Our algorithm enables us to make new uses of GH distributions in three financial applications. First, we forecast univariate ValueatRisk (VaR) for stock index returns, and we show in outofsample backtesting that the GH distributions outperform the Gaussian distribution. Second, we calculate an efficient frontier for equity portfolio optimization under the skewedt distribution and using Expected Shortfall as the risk measure. Here, we show that the Gaussian efficient frontier is actually unreachable if returns are skewed t distributed. Third, we build an intensitybased model to price Basket Credit Default Swaps by calibrating the skewed t distribution directly, without the need to separately calibrate xi the skewed t copula. To our knowledge this is the first use of the skewed t distribution in portfolio optimization and in portfolio credit risk.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3694
 Format
 Thesis
 Title
 Option Pricing with Selfsimilar Additive Processes.
 Creator

Galloway, Mack L. (Mack Laws), Nolder, Craig, Huﬀer, Fred, Beaumont, Paul, Case, Bettye Anne, Quine, John R., Department of Mathematics, Florida State University
 Abstract/Description

The use of timeinhomogeneous additive models in option pricing has gained attention in recent years due to their potential to adequately price options across both strike and maturity with relatively few parameters. In this thesis two such classes of models based on the selfsimilar additive processes of Sato are developed. One class of models consists of the riskneutral exponentials of a selfsimilar additive process, while the other consists of the riskneutral exponentials of a Brownian...
Show moreThe use of timeinhomogeneous additive models in option pricing has gained attention in recent years due to their potential to adequately price options across both strike and maturity with relatively few parameters. In this thesis two such classes of models based on the selfsimilar additive processes of Sato are developed. One class of models consists of the riskneutral exponentials of a selfsimilar additive process, while the other consists of the riskneutral exponentials of a Brownian motion timechanged by an independent, increasing, selfsimilar additive process. Examples from each class are constructed in which the time one distributions are Variance Gamma or Normal Inverse Gaussian distributed. Pricing errors are assessed for the case of Standard and Poor's 500 index options from the year 2005. Both sets of timeinhomogeneous additive models show dramatic improvement in pricing error over their associated Lévy processes. Furthermore, with regard to the average of the pricing errors over the quote dates studied, the selfsimilar Normal Inverse Gaussian model yields a mean pricing error significantly less than that implied by the bidask spreads of the options, and also significantly less than that given by its associated, less parsimonious, Lévy stochastic volatility model.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd4372
 Format
 Thesis
 Title
 Partial Differential Equation Methods to Price Options in the Energy Market.
 Creator

Yan, Jinhua, Kopriva, David, Huﬀer, Fred, Case, Bettye Anne, Nolder, Craig, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

We develop partial differential equation methods with wellposed boundary conditions to price average strike options and swing options in the energy market. We use the energy method to develop boundary conditions that make a two space variable model of Asian options wellposed on a finite domain. To test the performance of wellposed boundary conditions, we price an average strike call. We also derive new boundary conditions for the average strike option from the putcall parity. Numerical...
Show moreWe develop partial differential equation methods with wellposed boundary conditions to price average strike options and swing options in the energy market. We use the energy method to develop boundary conditions that make a two space variable model of Asian options wellposed on a finite domain. To test the performance of wellposed boundary conditions, we price an average strike call. We also derive new boundary conditions for the average strike option from the putcall parity. Numerical results show that wellposed boundary conditions are working appropriately and solutions with new boundary conditions match the similarity solution significantly better than those provided in the existing literature. To price swing options, we develop a finite element penalty method on a one factor mean reverting diffusion model. We use the energy method to find wellposed boundary conditions on a finite domain, derive formulas to estimate the size of the numerical domain, develop a priori error estimates for both Dirichlet boundary conditions and Neumann boundary conditions. We verify the results through numerical experiments. Since the optimal exercise price is unknown in advance, which makes the swing option valuation challenging, we use a penalty method to resolve the difficulty caused by the early exercise feature. Numerical results show that the finite element penalty method is thousands times faster than the Binomial tree method at the same level of accuracy. Furthermore, we price a multiple right swing option with different strike prices. We find that a jump discontinuity can occur in the initial condition of a swing right since the exercise of another swing right may force its optimal exercise region to shrink. We develop an algorithm to identify the optimal exercise boundary at each time level, which allows us to record the optimal exercise time. Numerical results are accurate to one cent comparing with the benchmark solutions computed by a Binomial tree method. We extend applications to multiple right swing options with a waiting period restriction. A waiting period exists between two swing rights to be exercised successively, so we cannot exercise the latter right when we see an optimal exercise opportunity within the waiting period, but have to wait for the first optimal exercise opportunity after the waiting period. Therefore, we keep track of the optimal exercise time when pricing each swing right. We also verify an extreme case numerically. When the waiting time decreases, the value of M right swing option price increases to the value of M times an American option price as expected.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd7673
 Format
 Thesis
 Title
 Stochastic Volatility Extensions of the Swap Market Model.
 Creator

Tzigantcheva, Milena G. (Milena Gueorguieva), Nolder, Craig, Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec, Quine, Jack, Sumners, De Witt, Department of Mathematics, Florida...
Show moreTzigantcheva, Milena G. (Milena Gueorguieva), Nolder, Craig, Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec, Quine, Jack, Sumners, De Witt, Department of Mathematics, Florida State University
Show less  Abstract/Description

Two stochastic volatility extensions of the Swap Market Model, one with jumps and the other without, are derived. In both stochastic volatility extensions of the Swap Market Model the instantaneous volatility of the forward swap rates evolves according to a squareroot diffusion process. In the jumpdiffusion stochastic volatility extension of the Swap Market Model, the proportional lognormal jumps are applied to the swap rate dynamics. The speed, the flexibility and the accuracy of the fast...
Show moreTwo stochastic volatility extensions of the Swap Market Model, one with jumps and the other without, are derived. In both stochastic volatility extensions of the Swap Market Model the instantaneous volatility of the forward swap rates evolves according to a squareroot diffusion process. In the jumpdiffusion stochastic volatility extension of the Swap Market Model, the proportional lognormal jumps are applied to the swap rate dynamics. The speed, the flexibility and the accuracy of the fast fractional Fourier transform made possible a fast calibration to European swaption market prices. A specific functional form of the instantaneous swap rate volatility structure was used to meet the observed evidence that volatility of the instantaneous swap rate decreases with longer swaption maturity and with larger swaption tenors.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd1762
 Format
 Thesis
 Title
 Impulse Control Problems under NonConstant Volatility.
 Creator

Moreno, Juan F. (Juan Felipe), Kercheval, Alec, Huﬀer, Fred, Beaumont, Paul, Nichols, Warren, Nolder, Craig, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

The objective of this dissertation is to study impulse control problems in situations where the volatility of the underlying process is not constant. First, we explore the case where the dynamics of the underlying process are modified for a fixed (or random with known probability distribution) period of time after each intervention of the impulse control. We propose a modified intervention operator to be used in the QuasiVariational Inequalities approach for solving impulse control problems,...
Show moreThe objective of this dissertation is to study impulse control problems in situations where the volatility of the underlying process is not constant. First, we explore the case where the dynamics of the underlying process are modified for a fixed (or random with known probability distribution) period of time after each intervention of the impulse control. We propose a modified intervention operator to be used in the QuasiVariational Inequalities approach for solving impulse control problems, and we formulate and prove a verification theorem for finding the Value Function of the problem and the optimal control. Secondly, we use a perturbation approach to tackle impulse control problems when the volatility of the underlying process is stochastic but meanreverting. The perturbation method permits to approximate the Value Function and the parameters of the optimal control. Finally, we present a numerical scheme to obtain solutions to impulse control problems with constant and stochastic volatility. Throughout the thesis we find explicit solutions to practical applications in financial mathematics; specifically, in optimal central bank intervention of the exchange rate and in optimal policy dividend payments.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd2271
 Format
 Thesis
 Title
 Sensitivity Analysis of Options under Lévy Processes via Malliavin Calculus.
 Creator

Bayazit, Dervis, Nolder, Craig A., Huﬀer, Fred, Case, Bettye Anne, Kopriva, David, Okten, Giray, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

The sensitivity analysis of options is as important as pricing in option theory since it is used for hedging strategies, hence for risk management purposes. This dissertation presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite dimensional Malliavin calculus and the centered finite difference method via MonteCarlo...
Show moreThe sensitivity analysis of options is as important as pricing in option theory since it is used for hedging strategies, hence for risk management purposes. This dissertation presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite dimensional Malliavin calculus and the centered finite difference method via MonteCarlo simulations. We give explicit formulas that are used directly in MonteCarlo simulations. By using simulations, we show that a localized version of the Malliavin estimator outperforms others including the centered finite difference estimator for the call and digital options under Variance Gamma and Normal Inverse Gaussian processes driven option pricing models. In order to compare the performance of these methods we use an inverse Fourier transform method to calculate the exact values of the sensitivities of European call and digital options written on S&P 500 index. Our results show that a variation of localized Malliavin calculus approach gives a robust estimator while the convergence of centered finite difference method in MonteCarlo simulations varies with different Greeks and new sensitivities that we introduce. We also discuss an approximation method for the Variance Gamma process. We introduce new random number generators for the path wise simulations of the approximating process. We improve convergence results for a type of sensitivity by using a mixed Malliavin calculus on the increments of the approximating process.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1157
 Format
 Thesis
 Title
 Anova for Parameter Dependent Nonlinear PDEs and Numerical Methods for the Stochastic Stokes Equations.
 Creator

Chen, Zheng, Gunzburger, Max, Huﬀer, Fred, Peterson, Janet, Wang, Xiaoqiang, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation includes the application of analysisofvariance (ANOVA) expansions to analyze solutions of parameter dependent partial differential equations and the analysis and finite element approximations of the Stokes equations with stochastic forcing terms. In the first part of the dissertation, the impact of parameter dependent boundary conditions on the solutions of a class of nonlinear PDEs is considered. Based on the ANOVA expansions of functionals of the solutions, the effects...
Show moreThis dissertation includes the application of analysisofvariance (ANOVA) expansions to analyze solutions of parameter dependent partial differential equations and the analysis and finite element approximations of the Stokes equations with stochastic forcing terms. In the first part of the dissertation, the impact of parameter dependent boundary conditions on the solutions of a class of nonlinear PDEs is considered. Based on the ANOVA expansions of functionals of the solutions, the effects of different parameter sampling methods on the accuracy of surrogate optimization approaches to PDE constrained optimization is considered. The effects of the smoothness of the functional and the nonlinearity in the PDE on the decay of the higherorder ANOVA terms are studied. The concept of effective dimensions is used to determine the accuracy of the ANOVA expansions. Demonstrations are given to show that whenever truncated ANOVA expansions of functionals provide accurate approximations, optimizers found through a simple surrogate optimization strategy are also relatively accurate. The effects of several parameter sampling strategies on the accuracy of the surrogate optimization method are also considered; it is found that for this sparse sampling application, the Latin hypercube sampling method has advantages over other wellknown sampling methods. Although most of the results are presented and discussed in the context of surrogate optimization problems, they also apply to other settings such as stochastic ensemble methods and reducedorder modeling for nonlinear PDEs. In the second part of the dissertation, we study the numerical analysis of the Stokes equations driven by a stochastic process. The random processes we use are white noise, colored noise and the homogeneous Gaussian process. When the process is white noise, we deal with the singularity of matrix Green's functions in the form of mild solutions with the aid of the theory of distributions. We develop finite element methods to solve the stochastic Stokes equations. In the 2D and 3D cases, we derive error estimates for the approximate solutions. The results of numerical experiments are provided in the 2D case that demonstrate the algorithm and convergence rates. On the other hand, the singularity of the matrix Green's functions necessitates the use of the homogeneous Gaussian process. In the framework of theory of abstract Wiener spaces, the stochastic integrals with respect to the homogeneous Gaussian process can be defined on a larger space than L2 . With some conditions on the density function in the definition of the homogeneous Gaussian process, the matrix Green's functions have well defined integrals. We have studied the probability properties of this kind of integral and simulated discretized colored noise.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd3851
 Format
 Thesis
 Title
 A Computational Study of Ion Conductance in the KcsA K⁺ Channel Using a NernstPlanck Model with Explicit Resident Ions.
 Creator

Jung, Yong Woon, Mascagni, Michael A., Huﬀer, Fred, Bowers, Philip, Klassen, Eric, Cogan, Nick, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation, we describe the biophysical mechanisms underlying the relationship between the structure and function of the KcsA K+ channel. Because of the conciseness of electrodiffusion theory and the computational advantages of a continuum approach, NernstPlanck (NP) type models such as the GoldmanHodgkinKatz (GHK) and PoissonNernstPlanck (PNP) models have been used to describe currents in ion channels. However, the standard PNP (SPNP) model is known to be inapplicable to...
Show moreIn this dissertation, we describe the biophysical mechanisms underlying the relationship between the structure and function of the KcsA K+ channel. Because of the conciseness of electrodiffusion theory and the computational advantages of a continuum approach, NernstPlanck (NP) type models such as the GoldmanHodgkinKatz (GHK) and PoissonNernstPlanck (PNP) models have been used to describe currents in ion channels. However, the standard PNP (SPNP) model is known to be inapplicable to narrow ion channels because it cannot handle discrete ion properties. To overcome this weakness, we formulated the explicit resident ions NernstPlanck (ERINP) model, which applies a local explicit model where the continuum model fails. Then we tested the effects of the ERI Coulomb potential, the ERI induced potential, and the ERI dielectric constant for ion conductance were tested in the ERINP model. Using the currentvoltage (IV ) and currentconcentration (IC) relationships determined from the ERINP model, we discovered biologically significant information that is unobtainable from the traditional continuum model. The mathematical analysis of the K+ ion dynamics revealed a tight structurefunction system with a shallow well, a deep well, and two K+ ions resident in the selectivity filter. We also demonstrated that the ERINP model not only reproduced the experimental results with a realistic set of parameters, it also reduced CPU costs.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd3741
 Format
 Thesis