Explicit Closed-form Solution Research Articles

Robust mean variance portfolio selection model in the jump-diffusion financial market with an intractable claim

In this paper, a continuous-time robust mean variance model in the jump-diffusion financial market with an intractable claim is considered, in which the price processes of the assets not only are driven by the Brownian motion, but also have the Poisson jumps. By combining the martingale representation theorem and the quantile formulation method, an explicit closed-form solution of the robust mean-variance portfolio selection model is given under some suitable assumptions.

Skew-elliptical distributions with applications in risk theory

In this paper we derive important properties of the well-known skew-elliptical (SE) distributions which was introduced in Azzalini and Capitanio (J R Stat Soc Ser B (Stat Methodol) 65:367–389, 2003), and includes the more familiar skew-normal, skew-Student-t and skew-logistic distributions. We then derive the tail value at risk (TVaR) for a portfolio of SE risks. We provide the portfolio risk decomposition with TVaR. Furthermore, we obtain the Esscher premium principle, the weighted-premium principle, and the entropic risk measure with the underlying SE distributions. We also provide an explicit closed-form solution to the optimal portfolio selection with the SE distributions, and provide a numerical simulation of the results.

Crack localization in beams by frequency shifts due to roving mass with rotary inertia

The problem of crack localization has been tackled in the past by detecting frequency changes due to the application of a mass appended to the structure. However the effect of the rotary inertia of the mass on the natural frequencies of a cracked beam has not been deeply investigated. In this paper a novel explicit closed form solution of the governing equation of a beam with a concentrated mass, with rotary inertia, in the presence of multiple cracks is proposed. Furthermore, an analytical proof to show that the natural frequencies of a cracked beam with a roving body with a rotary inertia will generally change abruptly as the body passes over a crack, provided that the crack permits differential flexural rotations, is presented. Numerical results in terms of natural frequencies are provided and the procedure to exploit the occurrence of frequency shifts to detect and locate each crack is described.

Wealth and price distribution by diffusive approximation in a repeated prediction market

The approximate agents’ wealth and price invariant densities of a repeated prediction market model is derived using the Fokker–Planck equation of the associated continuous-time jump process. We show that the approximation obtained from the evolution of log-wealth difference can be reliably exploited to compute all the quantities of interest in all the acceptable parameter space. When the risk aversion of the trader is high enough, we are able to derive an explicit closed-form solution for the price distribution which is asymptotically correct.

Closed-Form Solutions for Eigenbuckling of Rectangular Mindlin Plate

This paper studies the eigenbuckling of Mindlin plate with two adjacent edges clamped and the remaining edges simply supported or clamped by using the separation of variables method, and the concise and explicit closed-form solutions are obtained for the first time. The cases involving free edges can also be dealt with if there are two opposite edges simply supported. The closed-form solutions are in good agreement with the existing solutions, thus the validity of present method and accuracy of the obtained solutions are verified. This paper proves to be a major development of analytical method since it has long been acknowledged that the eigenbuckling of rectangular plates without two parallel edges simply supported are not amenable to analytical solutions.

Love's rectangular contact problem revisited: A complete solution

Love's rectangular contact solution was recognized as the key ingredient in developing fast Fourier Transform related algorithms for computational contact analyses. This paper proposes an effective notation to simplify the analytical derivations, which are only carried out on the primitive functions. The complete solution of the stresses and displacements, together with the surface deflection, produced by the both uniform normal and tangential loadings over a rectangular patch are solved in a more compact and consistent way, with explicit closed-form solutions optimized for computational efficiency and numerical stability. The correlation to the Green's functions due to Boussinesq and Cerruti is also noted. The present work complements the existing literature and provides a complete reference to the classical contact solution.

Local risk minimization for vulnerable European contingent claims on nontradable assets under regime switching models

ABSTRACTThis article focuses on an optimal hedging problem of the vulnerable European contingent claims. The underlying asset of the vulnerable European contingent claims is assumed to be nontradable. The interest rate, the appreciation rate and the volatility of risky assets are modulated by a finite-state continuous-time Markov chain. By using the local risk minimization method, we obtain an explicit closed-form solution for the optimal hedging strategies of the vulnerable European contingent claims. Further, we consider a problem of hedging for a vulnerable European call option. Optimal hedging strategies are obtained. Finally, a numerical example for the optimal hedging strategies of the vulnerable European call option in a two-regime case is provided to illustrate the sensitivities of the hedging strategies.

An explicit closed-form analytical solution for European options under the CGMY model

In this paper, we consider the analytical pricing of European path-independent options under the CGMY model, which is a particular type of pure jump Le´vy process, and agrees well with many observed properties of the real market data by allowing the diffusions and jumps to have both finite and infinite activity and variation. It is shown that, under this model, the option price is governed by a fractional partial differential equation (FPDE) with both the left-side and right-side spatial-fractional derivatives. In comparison to derivatives of integer order, fractional derivatives at a point not only involve properties of the function at that particular point, but also the information of the function in a certain subset of the entire domain of definition. This “globalness” of the fractional derivatives has added an additional degree of difficulty when either analytical methods or numerical solutions are attempted. Albeit difficult, we still have managed to derive an explicit closed-form analytical solution for European options under the CGMY model. Based on our solution, the asymptotic behaviors of the option price and the put-call parity under the CGMY model are further discussed. Practically, a reliable numerical evaluation technique for the current formula is proposed. With the numerical results, some analyses of impacts of four key parameters of the CGMY model on European option prices are also provided.

Unified formulation of the stress field of saint-Venant's flexure problem for symmetric cross-sections

The flexure problem of Saint-Venant's elastic beam under lateral traction is revisited. First an engineering stress field is proposed with the explicit closed form solution that results in shear stress distribution determined by mechanics of materials theory. Afterward a modified stress field is introduced for Saint-Venant's flexure problem with uniaxial symmetric cross-sections that recovers the solutions available from the theory of elasticity. The modified stress field which is a unified formulation of Saint-Venant's solution confirms the main features of shear stress distribution found in the earlier investigations and has an excellent agreement with the results of the theory of elasticity. Also the shear flexibility factor is comprehensively discussed and an explicit solution form is presented based on the modified stress field.

Best Bounds on Measures of Risk and Probability of Ruin for Alpha Unimodal Random Variables When There Is Limited Moment Information

This paper presents explicit formulae giving tight upper and lower bounds on the expectations of alpha-unimodal random variables having a known range and given set of moments. Such bounds can be useful in ordering of random variables in terms of risk and in PERT analysis where there is only incomplete stochastic information concerning the variables under investigation. Explicit closed form solutions are also given involving alpha-unimodal random variables having a known mean for two particularly important measures of risk—the squared distance or variance, and the absolute deviation. In addition, optimal tight bounds are given for the probability of ruin in the collective risk model when the severity distribution has an alpha-unimodal distribution with known moments.

A Robust Markowitz Mean-Variance Portfolio Selection Model with an Intractable Claim

This paper studies a robust Markowitz mean-variance model where an intractable claim is involved in the terminal wealth. The term “intractable claim” refers to claims (rewards or losses) that are completely irrelevant to the underlying market. The payoffs of such claims cannot be predicted or hedged based on the underlying financial market even if the information of the financial market is increasingly available to the investor over time. The target of the investor is to minimize the variance in the worst scenario over all the possible realizations of the underlying intractable claim. Because of the time-inconsistent nature of the problem, both the standard penalization approach and the duality method used to tackle robust stochastic control problems fail in solving our problem. Instead, the quantile formulation approach is adopted to tackle the problem and an explicit closed-form solution is obtained. The properties of the mean-variance frontier are also discussed.

Minimization of a Function of a Quadratic Functional with Application to Optimal Portfolio Selection

We present an explicit closed-form solution to the problem of minimizing the combination of linear functional and a function of quadratic functional, subject to a system of affine constraints. This...

Effect of wedge disclination dipole on dislocation emission from a surface crack tip in nanocrystalline materials

The interaction between a wedge disclination dipole and a linear crack of the surface in nanocrystalline materials is investigated. Utilizing the conformal mapping technique and the Muskhelishvili complex variable method, the explicit closed-form solutions are derived for complex potentials and stress fields. The critical stress intensity factors (SIFs) for the first dislocation emission from the crick tip are also calculated. The effects of the relative parameters on the normalized critical SIFs, such as the disclination strength, the edge dislocation emission angle, the dipole size, the distance between the crack tip and the center of the disclination dipole, the direction angle of the disclination dipole and the crack length, are discussed in detail as well. The results reveal that the shielding effect produced by disclination dipole increases with the increasing disclination strength, which means it can prevent the dislocation emission from the crack tip to decrease material toughness.

ANALYSIS ON GENERALIZED IMPACT ANGLE CONTROL GUIDANCE LAW

In this paper, a generalized guidance law with an arbitrary pair of guidance coefficients for impact angle control is proposed. Under the assumptions of a stationary target and a lag-free missile with constant speed, necessary conditions for the guidance coefficients to satisfy the required terminal constraints are obtained by deriving an explicit closed-form solution. Moreover, optimality of the generalized impact-angle control guidance law is discussed. By solving an inverse optimal control problem for the guidance law, it is found that the generalized guidance law can minimize a certain quadratic performance index. Finally, analytic solutions of the generalized guidance law for a first-order lag system are investigated. By solving a third-order linear time-varying ordinary differential equation, the blowing-up phenomenon of the guidance loop as the missile approaches the target is mathematically proved. Moreover, it is found that terminal misses due to the system lag are expressed in terms of the guidance coefficients, homing geometry, and the ratio of time-to-go to system time constant.

Lie group analysis of non-linear dynamic of micro structures under electrostatic field

This paper presents an analytical solution of nonlinear differential equation of micro-structures subjected to electrostatic fields. The constitutive equation of such a model is a second order differential equation (ODE). The problem is solved when the assumption of linear deflection is considered. However, deflection of micro cantilevers in practical applications is non-linear. Moreover, the constitutive ODE is stiff and various numerical algorithms used to solve it yield non-consistent numerical solutions. A deduction order method - Lie group symmetry is employed to reduce the order of the ODE. Although the resulting first order ODE has no symmetry that would guarantee an explicit close form solution, it enables an analytical formulation for the no-damping assumption only. The restoring force term in the first order ODE reveals the pull-in voltage as expressed in classical MEMS textbooks. It is shown that the numerical solution for the second order ODE and the reduced first order ODE are same. Finding any symmetry other than translation, scaling or rotation will enable the reduction of the first order ODE and thus, the formulation of an analytical solution to this highly non-linear problem.

Analytical solutions for elastic fields caused by eigenstrains in two joined and perfectly bonded half-spaces and related problems

This paper reports the derivation of the explicit integral kernels for the elastic fields due to eigenstrains in two joined and perfectly bonded half-space solids or bimaterials. The domain integrations of these kernels result in the analytical solutions to displacements and stresses. When the eigenstrains are all in solid I, the kernel for the elastic fields has four groups in this solid and two groups in the other joined solid. Explicit closed-form solutions for a cuboidal inclusion with uniform eigenstrains are derived as the basic solutions, i.e., the influence coefficients. When the computational domain is meshed into cuboidal elements of the same sizes, the total elastic fields can be quickly obtained by implementing three-dimensional fast Fourier transform-based algorithms. The results for the fields due to a cuboidal, a spherical, and a cylindrical inclusion, as well as multiple cuboidal inclusions are presented and discussed. Residual stresses in an elasto-plastic contact involved a coated substrate is further analyzed with the new solution approach.

Singular integrals in boundary elements for coupled stretching–bending analysis of unsymmetric laminates

When an unsymmetric laminated plate is considered, the coupling of in-plane and plate bending problems is unavoidable. The boundary element for the coupled stretching–bending analysis was developed previously with suitable complex form fundamental solution. Usually, the singular integrals involved in the boundary element are treated by the conventional methods such as Gaussian quadrature rule for regular functions, logarithmic Gaussian quadrature formulas for the function with logarithmic terms, and the use of finite part integrals for the evaluation in sense of Cauchy principal value, or calculated indirectly through the employment of rigid body movement. To avoid the complexity of the numerical integration with complex form fundamental solution, in this paper we provide the explicit closed-form solutions for the singular integrals, which simplify the computer programming and expedite the numerical computation. And hence, the accuracy and efficiency of the associated boundary elements are improved through the newly derived analytical solutions.

Multiperiod conditional valuation of barrier options with incomplete information

In this paper, we provide analytical valuation results for barrier options with three different types of information: continuous observations of the underlying asset value, delayed continuous observations and multiple discrete observations. Market, counterparty credit risk management and economic risk capital implications of the valuations with incomplete information are also discussed. We obtain precise analytical solutions using all the information generated by the previous discrete observations of the underlying asset, generated in a consistent fashion by path-dependent simulation. Our results have far-reaching implications for economic capital charges on path-dependent derivatives. We demonstrate that using non-conditional valuation mis-estimates the credit charges on the barrier options with respect to the precise credit charges computed with valuations conditional on the actual available information. The rest of this paper is organized as follows. Section 2 describes the classical case with full, continuous information about the underlying asset. Section 3 deals with the case of delayed continuous information. Section 4 investigates the case of discrete observations at multiple discrete times. Section 5 offers explicit closed-form solutions for the main types of single-barrier options with discrete information and parity results. Section 6 provides valuation results for double-barrier options. Section 7 presents risk management applications for foreign exchange (FX) barrier options, while section 8 concludes our paper.

Analytically pricing double barrier options based on a time-fractional Black–Scholes equation

This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. In this scenario, the option price is governed by a modified Black–Scholes equation with a time-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their “globalness”, i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. The existence of the time-fractional derivative, in conjunction with the presence of two barriers of the double-barrier options, has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Albeit difficult, we have managed to find an explicit closed-form analytical solution for double-barrier options, which has been taken to price the single barrier options and European path-independent options under the same framework as a special case of the current solution. In addition, not only have we provided a theoretical proof for the convergence of the newly-found analytic solution in series form, which is a vital step to show the closeness of our solution, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice.

FROM BOUNDARY CROSSING OF NON-RANDOM FUNCTIONS TO BOUNDARY CROSSING OF STOCHASTIC PROCESSES

One problem of wide interest involves estimating expected crossing-times. Several tools have been developed to solve this problem beginning with the works of Wald and the theory of sequential analysis. Deriving the explicit close form solution for the expected crossing times may be difficult. In this paper, we provide a framework that can be used to estimate expected crossing times of arbitrary stochastic processes. Our key assumption is the knowledge of the average behavior of the supremum of the process. Our results include a universal sharp lower bound on the expected crossing times. Furthermore, for a wide class of time-homogeneous, Markov processes, including Bessel processes, we are able to derive an upper bound E[a(Tr)]≤2r, which implies that sup r>0|((E[a(Tr)]−r)/r)|≤1, where a(t)=E[sup tXt] with {Xt}t≥0 be a non-negative, measurable process. This inequality motivates our claim that a(t) can be viewed as a natural clock for all such processes. The cases of multidimensional processes, non-symmetric and random boundaries are handled as well. We also present applications of these bounds on renewal processes in Example 10 and other stochastic processes.