This paper introduces an analytical formula for the fractional-order conditional moments of nonlinear drift constant elasticity of variance (NLD-CEV) processes under regime switching, governed by continuous-time finite-state irreducible Markov chains. By employing a hybrid system approach, we derive exact closed-form expressions for these moments across arbitrary fractional orders and regime states, thereby enhancing the analytical tractability of NLD-CEV models under stochastic regimes. Our methodology hinges on formulating and solving a complex system of interconnected partial differential equations derived from the Feynman-Kac formula for switching diffusions. To illustrate the practical relevance of our approach, Monte Carlo simulations for process with Markovian switching are applied to validate the accuracy and computational efficiency of the analytical formulas. Furthermore, we apply our findings for the valuation of financial derivatives within a dynamic nonlinear mean-reverting regime-switching framework, which demonstrates significant improvements over traditional methods. This work offers substantial contributions to financial modeling and derivative pricing by providing a robust tool for practitioners and researchers who are dealing with complex stochastic environments.

An analytical approach to solve a time-fractional Cauchy problem of order 0<α≤1 based on the Ornstein–Uhlenbeck (OU), Cox–Ingersoll–Ross (CIR) and Jacobi processes with time-dependent parameters by transforming it into a system of linear fractional differential equations is established. We consider the process as an inhomogeneous Pearson diffusion and derive the analytical formulas for conditional expectations via the Volterra fractional integral equation. We also provide the β-conditional moments of the OU, CIR and Jacobi processes where β∈R. Finally, we illustrate with examples of the first and second moments of the extended OU and extended CIR processes by obtaining solutions with different α values and comparing to α=1.

An analytical derivation of the conditional moment-generating function (MGF) for a regime-switching nonlinear drift constant elasticity of variance process is established. The proposed model incorporates both regime-switching mechanisms and nonlinear drift components to better capture market phenomena such as volatility smiles and leverage effects. Regime-switching models can match the tendency of financial markets to often change their behavior abruptly and the phenomenon that the new behavior of financial variables often persists for several periods after such a change. Closed-form formulas for the MGF under various conditions, which are then applied for option pricing, are also derived. The efficacy and accuracy of the results are validated through a discrete Markov chain simulation. The results obtained from the proposed formulas completely match with those from MC simulations, while requiring significantly less computational time.

This paper studied a generalized case of the constant elasticity of variance diffusion (CEV) process whereas the drift term is substantially nonlinear in the short rate. Well-known instances deduced by this process are the extended Cox-Ingersoll-Ross (ECIR) process and the extended inverse Feller (EIF) process or 3/2-stochastic volatility model (SVM). We found particular sufficient conditions of existence and uniqueness of a positive pathwise strong solution for time-dependent parameter functions, and obtained closed-form formulas for conditional moments based on Feynman-Kac theorem. The accuracy and validity of the formulas were further investigated based on Monte Carlo simulations. (c) 2022 Elsevier Inc. All rights reserved.

Closed-form formulas representing conditional moments of two models of stochastic behavior of commodity prices are established. The obtained formulas are combinatorial in nature, involving the Faà Bruno's formula, and are derived through the use of an exponential-affine transform, which are solutions of partial differential equations arising from infinitesimal generators. Applications including pricing nonlinear payoff derivatives, such as the actual return-based realized and log-return realized variance swaps, are theoretically illustrated so that practitioners in commodity markets can adopt the formulas when they need to take into account the hedging of volatility risk. Compared with earlier approaches, this work provides more concise and simpler explicit forms of solutions, and avoids solving systems of recurrent ordinary differential equations. The validity of the proposed formulas are further investigated based on Monte Carlo simulations, and various mathematical properties of the formulas are also discussed.

Moment swaps are essentially forward contracts on realized higher moments of log-returns of a specified underlying asset, which play an important role in protection against different kinds of market shocks, and variance, skewness, and kurtosis swaps are examples of moment swaps currently traded in markets. To facilitate market practitioners, this work provides a simple and easy-to-use pricing formula of moment swaps on discrete sampling under the Black-Scholes model with time-dependent parameters. The formula is investigated for validity and compared with the fair delivery prices of moment swaps. Furthermore, a closed-form formula for hedging moment swaps on futures is deduced. Finally, Monte Carlo simulations are performed to support the accuracy of the pricing formula and numerical examples are provided to check the sensitivity of the parameters and relationships of calculated prices between moment swaps.

This paper proposes an analytical formula for the conditional expectations of path-dependent product of polynomial and exponential function in the form of $$\begin{aligned} \left( \sum _{j=0}^{n}\lambda _j^{(l)}r_{t_l}^j\right) e\,^{\sum \limits _{k=1}^m\alpha _k^{(l)}r_{t_k}} \end{aligned}$$ for $$n,m \in \mathbb {N}$$ , $$l=1,2,...,m$$ , $$0 \le t_{1}<t_{2}< \cdots<t_{m} = T <\infty $$ and $$\lambda _{j}^{(l)}, \alpha _{k}^{(l)}\in \mathbb {R}$$ , where $$\{r_{t}\}_{t\in [0,T]}$$ corresponds to the extended Cox–Ingersoll–Ross (ECIR) process. The validation of the analytical formula is illustrated for several examples by comparing the results from the formula with those from Monte Carlo (MC) simulations. The efficiency of the formula is also presented via the computational run-times as compared with MC simulation. Moreover, the application of the analytical formula of this work is demonstrated for pricing arrears interest rate swaps under the ECIR process.

Diffusion models have been thoroughly studied for their use in seeking stochastic differential equation (SDE) solutions and investigating their properties, such as moments and conditional moments, which play significant roles in many real-world applications and are especially beneficial for estimating parameters. In fact, these moments can be directly calculated by applying the transition probability density function (PDF), which is often unknown or unavailable in closed form; the formulas for the conditional moments of the SDE may be unavailable in closed form, as well. In this work, we studied an extended case of Pearson diffusion processes, which are a class of diffusions that have squared diffusion coefficients with time-dependent parameter functions. A complete investigation was carried out for both light- and heavy-tailed Pearson diffusion processes, including Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, Fisher–Snedecor, reciprocal gamma, and Student. We introduce a simple but novel approach to closed-form formulas for conditional moments of inhomogeneous Pearson diffusion processes. The approach does not require any knowledge of eigenfunctions or the transition PDF. In each class of stationary distributions reduced from Pearson diffusions, the formula is explored and presented in a concise form. The closed-form formulas obtained are also numerically validated by MC simulations.

Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs depend on uncertain future real values of underlying assets which emphasize various real-world applications. In general, valuations for contingent claims can be derived from the conditional expectations of underlying assets. For a simple process, the moments are usually directly obtained from its transition probability density function (PDF). However, if the transition PDF is unavailable in simple form, the derivations of the moments and the contingent claim prices will not be accessible in closed forms. This paper provides a closed-form formula for pricing contingent claims with nonlinear payoff under a no-arbitrage principle when underlying assets follow the extended Cox–Ingersoll–Ross (ECIR) process with the symmetry properties of the Brownian motion. The formula proposed here is a consequence of successfully solving an explicit solution for a system of recurrence partial differential equations in which its solution subtly depends on the conditional moments. For the particular CIR process, we obtain simple closed-form formulas by solving the Riccati differential equation. Furthermore, we carry out a complete investigation of the convergent case for those formulas. In case such as that of the unsolvable Riccati differential equation, ECIR case, a numerical method for numerically evaluating the mentioned analytical formulas and numerical validations for the formulas are examined. The validity and efficiency of the formulas are numerically demonstrated by comparison with results from Monte Carlo simulations for various modeling parameters. Finally, the proposed formula is applied to the value contingent claims such as coupon bonds, interest rate swaps, and arrears swaps.

In this paper, analytical formulas for pricing discretely-sampled skewness and kurtosis swaps based on the Schwartz’s one-factor model is derived by applying the results of the conditional moments proposed by Chumpong, Mekchay, and Rujivan (2019). The results would be beneficial for market practitioners to describe commodity prices. The analytical pricing formulas for the skewness and kurtosis swaps of commodity will be useful for hedging against price volatility risks in commodity markets.

Abstract We present an analytical option pricing formula for the European options, in which the price dynamics of a risky asset follows a mean-reverting process with a time-dependent parameter. The process can be adapted to describe a seasonal variation in price such as in agricultural commodity markets. An analytical solution is derived based on the solution of a partial differential equation, which shows that a European option price can be decomposed into two terms: the payoff of the option at the initial time and the time-integral over the lifetime of the option driven by a time-dependent parameter. Finally, results obtained from the formula have been compared with Monte Carlo simulations and a Black–Scholes-type formula under various kinds of long-run mean functions, and some examples of option price behaviours have been provided.

The insight in structures of the blood vessels is a basis for study of blood flows to help understanding the abnormalities of blood vessels that can cause vascular diseases. Basic concept used for constructing structures of blood vessels in organs is arterial branching, which is usually characterized by fractal similarity in the bifurcation pattern. In this work, the concept of Lindenmayer system (L-system) is modified for three-dimensional (3D) tree-like structures to model structures of blood vessels in organs, and then, applied to construct and visualize structural blood vessels via our software created based on openGL and Lazarus program. The structure of blood vessels is constructed based on the physiological law of arterial branching proposed Murray (Murray’s law) under additional assumptions and constraints such as the spreading of blood vessels to cover all directions, the angle condition and the non-overlapping vessels condition. The concept is applied to simulate structures of blood vessels in 3 study cases, including symmetric arterial branching, non-symmetric arterial branching and structure of blood vessel on different domains. The results of structures of blood vessels generated from all cases are measured based on the number of segments, the total blood volume and the fractal dimension. The results of modeling and simulation in this work are illustrated by comparing with other results appeared literature. Moreover, the constructed structures of the blood vessels based on this 3D L-system could be useful for future research such as blood flow, pressure and other properties involving in structures of blood vessels in different organs of human and animals. HIGHLIGHTS A new 3D L-system is developed based on directional vectors for construction of 3D tree-like structures such as structures of blood vessels The model of structures of blood vessels is constructed based on the physiological laws of arterial branching (Murray’s law) with additional assumptions on the spreading of blood vessels, the angle condition, and the non-overlapping of blood vessels Algorithm and software are developed based on L-system to simulate and visualize 3D structures of blood vessels GRAPHICAL ABSTRACT

This paper derives a simple closed-form formula for the n th conditional moment of the Ornstein-Uhlenbeck (O-U) process, for any positive integer n. The system of recursive ordinary differential equations (ODEs) associated with the n th conditional moment of the O-U process is solved analytically. We also provide practitioners a pseudocode for an algorithm to compute the conditional moments and discuss the efficiency of our formula compared to solving the system of recursive ODEs using the direct method.

We describe and validate the simulation of a dam-break flow on natural topography by solving shallow water equations with a well-balanced, positivity-preserving first order finite volume scheme and a dynamically adaptive general rectangular tree grid method. The results were validated with close agreement with the experimental data. Moreover, the results for the simulation of dam-break floods in rugged terrain compared with other numerical results, computed by adaptive and non-adaptive grid second order schemes, show that the model is very efficient by reducing the number of computational cells and the computational time without too much loss of accuracy. The validation and comparisons indicate that the model has potential for practical usage.

Finite volume method with reconstruction and bottom modification techniques for simulating open channel flows in arbitrary cross-sectional areas is presented. These techniques are introduced to handle the difficulty in approximating water depth at wet/dry areas. Various numerical experiments with source terms are demonstrated to confirm the accuracy of numerical scheme. Further, we have applied the present method to simulate open channel flow in the Yom River, Phrae Province, Thailand. The simulation results are compared with measured data.

In this study, an explicit formula for conditional expectations of the product of polynomial and exponential function of an affine transform is derived under the extended Cox-Ingersoll-Ross (ECIR) process. Moreover, we simplify the result to derive an explicit formula for the CIR process.

The purpose of this paper is a computational algorithm for simulation and visualization of flood inundation on natural topography. The algorithm is constructed based on the shallow water equations, which are solved numerically using an adaptive tree grid finite volume method that is also equipped with the dynamic domain defining technique. The algorithm is tested to simulate the flood inundation in Thailand. The results are compared with the non-adaptive finest grid simulation. The comparison shows that the algorithm can reduce the number of grid cells and the computational times, without much loss of accuracy in the results.

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally $W^1_\infty$ and piecewise in a suitable Besov class embedded in $C^{1,\alpha}$ with $\alpha \in (0,1]$. The idea is to have the surface sufficiently well resolved in $W^1_\infty$ relative to the current resolution of the PDE in $H^1$. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in $W^1_\infty$ and PDE error in $H^1$.

A well-balanced finite volume method for solving two-dimensional shallow water equations with weighted average flux (WAF) is developed in this work to simulate flooding. Friction source terms are estimated with a semi-implicit scheme resulting in an efficient numerical method for simulating shallow water flows over irregular domains, for both wet and dry beds. A wet/dry cell tracking technique is also presented for reducing computational time. The accuracy of these methods are investigated by application to well-studied cases. For practical purposes, the developed scheme is applied to simulate the flooding of the Chao Phraya river from Chai Nat to Sing Buri provinces in Thailand during October 13–17, 2011. The numerical simulations yield results that agree with the existing data obtained from the satellite images.

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.