A Closed-Form Pricing Formula for Log-Return Variance Swaps under Stochastic Volatility and Stochastic Interest Rate

At present, the study concerning pricing variance swaps under CIR the (Cox–Ingersoll–Ross)–Heston hybrid model has achieved many results ; however, due to the instantaneous interest rate and instantaneous volatility in the model following the Feller square root process, only a semi-closed solution can be obtained by solving PDEs. This paper presents a simplified approach to price log-return variance swaps under the CIR–Heston hybrid model. Compared with Cao’s work, an important feature of our approach is that there is no need to solve complex PDEs; a closed-form solution is obtained by applying the martingale theory and Ito^’s lemma. The closed-form solution is significant because it can achieve accurate pricing and no longer takes time to adjust parameters by numerical method. Another significant feature of this paper is that the impact of sampling frequency on pricing formula is analyzed; then the closed-form solution can be extended to an approximate formula. The price curves of the closed-form solution and the approximate solution are presented by numerical simulation. When the sampling frequency is large enough, the two curves almost coincide, which means that our approximate formula is simple and reliable.

Nonrelativistic Electron–Ion Bremsstrahlung: An Approximate Formula for All Parameters

The evaluation of the electron–ion bremsstrahlung cross section—exact to all orders in the Coulomb potential—is computationally expensive due to the appearance of hypergeometric functions. Therefore, tabulations are widely used. Here, we provide an approximate formula for the nonrelativistic dipole process valid for all applicable relative velocities and photon energies. Its validity spans from the Born to the classical regime and from soft-photon emission to the kinematic endpoint. The error remains below 3% (and widely below 1%) except at an isolated region of hard-photon emission at the quantum-to-classical crossover. We use the formula to obtain the thermally averaged emission spectrum and cooling function in a Maxwellian plasma and demonstrate that they are accurate to better than 2%.

Sequential Probabilistic Ratio Test for the Scale Parameter of the P -Norm Distribution

We consider a series of independent observations from a P -norm distribution with the position parameter μ and the scale parameter σ . We test the simple hypothesis H 0 : σ = σ 1 versus H 1 : σ = σ 2 . Firstly, we give the stop rule and decision rule of sequential probabilistic ratio test (SPRT). Secondly, we prove the existence of h σ which needs to satisfy the specific situation in SPRT method, and the approximate formula of the mean sample function is derived. Finally, a simulation example is given. The simulation shows that the ratio of sample size required by SPRT and the classic Neyman–Pearson N − P test is about 50.92 % at most, 38.30 % at least.

Exact and approximate similarities of non-necessarily rational planar, parametrized curves, using centers of gravity and inertia tensors

We provide an algorithm to detect whether two bounded, planar parametrized curves are similar, i.e. whether there exists a similarity transforming one of the curves onto the other. The algorithm is valid for completely general parametrizations, and can be adapted to the case when the input is given with finite precision, using the notion of approximate [Formula: see text]. The algorithm is based on the computation of centers of gravity and inertia tensors of the considered curves or of the planar regions enclosed by the curves, which have nice properties when a similarity transformation is applied. In more detail, the centers of gravity are mapped onto each other, and the matrices representing the inertia tensors satisfy a simple relationship: when the similarity is a congruence (i.e. distances are preserved) the matrices are congruent, and in the more general case the relationship is analogous, but involves the square of the scaling constant. Using both properties, and except for certain pathological cases, the similarities can be found. Additional ideas are presented for the case of closed, i.e. compact, curves.

Приближенное представление дилогарифмами решения одной вариационной краевой задачи для круга при граничном условии Неймана

It is known that boundary value problems for the Laplace and Poisson equations are equivalent to the problem of the calculus of variations – the minimum of an integral for which the given partial differential equation is the Euler – Lagrange equation. For example, the problem of the minimum of the Dirichlet integral in the unit disc centered at the origin on some admissible set of functions for given values of the normal derivative on the circle is equivalent to the Neimann boundary value problem for the Laplace equation in this domain. An effective approximate dilogarithm representation of the solution of the above equivalent variational boundary value problem is constructed on the basis of the known exact solution of the Neumann Boundary value problem for a circle using a special approximate formula for the Dini integral. The approximate formula is effective in the sense that it is quite simple in numerical implementation, stable, and the error estimation, which is uniform over a circle, allows calculations with the given accuracy. A special quadrature formula for the Dini integral has a remarkable property – its coefficients are non-negative. Quadrature formulas with non-negative coefficients occupy a special place in the theory of approximate calculations of definite integrals and its applications. Naturally, this property becomes even more significant when the coefficient are not number, but some functions. The performed numerical analysis of the approximate solution confirms its effectiveness.

An approximate formula for calculating the expectations of functionals from random processes based on using the Wiener chaos expansion

In this work, we propose a new method for calculating the mathematical expectation of nonlinear functionals from random processes. The method is based on using Wiener chaos expansion and approximate formulas, exact for functional polynomials of given degree. Examples illustrating approximation accuracy are considered.