共查询到20条相似文献,搜索用时 31 毫秒
1.
Marcin Magdziarz 《Journal of statistical physics》2009,136(3):553-564
In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider
a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive
geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find the corresponding fractional
Fokker-Planck equation governing the dynamics of the probability density function of the introduced process. We prove that
the considered model is arbitrage-free and incomplete. We find the corresponding subdiffusive Black-Scholes formula for the
fair prices of European options and show how these prices can be evaluated using Monte-Carlo methods. We compare the obtained
results with the classical ones. 相似文献
2.
A quantum model of option pricing: When Black-Scholes meets Schrödinger and its semi-classical limit
The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ2 with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing. 相似文献
3.
Analytical bounds for Asian options are almost exclusively available in the Black-Scholes framework. In this paper we derive bounds for the price of a discretely monitored arithmetic Asian option when the underlying asset follows an arbitrary Lévy process. Explicit formulas are given for Kou’s model, Merton’s model, the normal inverse Gaussian model, the CGMY model and the variance gamma model. The results are compared with the comonotonic upper bound, existing numerical results, Monte carlo simulations and in the case of the variance gamma model with an existing lower bound. The method outlined here provides lower and upper bounds that are quick to evaluate, and more accurate than existing bounds. 相似文献
4.
We consider pricing options in a jump-diffusion model which requires solving
a partial integro-differential equation. Discretizing the spatial direction
with a fourth order compact scheme leads to a linear system of ordinary
differential equations. For the temporal direction, we utilize the favorable
boundary value methods owing to their advantageous stability properties. In
addition, the resulting large sparse system can be solved rapidly by the
GMRES method with a circulant Strang-type preconditioner. Numerical results
demonstrate the high order accuracy of our scheme and the efficiency of the
preconditioned GMRES method. 相似文献
5.
随着金融市场的不断发展, 期权作为一种能够规避风险的金融衍生产品越来越引起投资者的青睐, 成交量呈逐年上升的趋势, 期权定价问题已经成为金融数学领域中一个重要的研究课题. 本文主要研究Black-Scholes模型下美式回望期权定价问题的数值解法. 美式回望期权定价问题是一个二维非线性抛物问题, 难以直接应用数值方法进行求解. 通过分析该问题的求解难点, 本文给出解决该困难的有效方法. 首先利用计价单位变换将定价问题转换为一维自由边值问题, 并采用Landau's变换将求解区域规范化; 而后针对问题的非线性特点,利用有限体积法和Newton法交替迭代求解期权价格和最佳实施边界, 并对数值解的非负性进行了分析. 最后, 通过与二叉树方法进行比较, 验证了本文方法的正确性和有效性, 为实际应用提供了理论基础. 相似文献
6.
提出了无界区域波动方程的区域分解算法,基于自然边界归化,分别研究了重叠型与非重叠型区域分解算法,首先将控制方程对时间进行离散化,得到关于时间步长离散化格式,对每一时间步长给出了Dirichlet-Neumann和Schwartz交替算法,对Schwartz交替算法,给出了算法的收敛性,对圆外区域研究了压缩因子,并给出了数值例子。 相似文献
7.
Mai Huong Nguyen & Matthias Ehrhardt 《advances in applied mathematics and mechanics.》2012,4(3):259-293
In this work we investigate the pricing of swing options in a model
where the underlying asset follows a jump diffusion process. We focus on the
derivation of the partial integro-differential equation (PIDE) which will be applied
to swing contracts and construct a novel pay-off function from a tree-based pay-off
matrix that can be used as initial condition in the PIDE formulation. For valuing
swing type derivatives we develop a theta implicit-explicit finite difference scheme
to discretize the PIDE using a Gaussian quadrature method for the integral part.
Based on known results for the classical theta-method the existence and uniqueness
of solution to the new implicit-explicit finite difference method is proven. Various
numerical examples illustrate the usability of the proposed method and allow us
to analyse the sensitivity of swing options with respect to model parameters. In
particular, the effects of number of exercise rights, jump intensities and dividend
yields will be investigated in depth. 相似文献
8.
YANG Xuan-Liu ZHANG Shun-Li QU Chang-Zheng 《理论物理通讯》2007,47(6):995-1000
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained. 相似文献
9.
L. Z.J. Liang D. Lemmens J. Tempere 《The European Physical Journal B - Condensed Matter and Complex Systems》2010,75(3):335-342
Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a
Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present
a method to adapt formulas for both the path-integral propagators and the option prices themselves, so that jump processes
are taken into account in conjunction with the usual drift and diffusion terms. In particular, we focus on stochastic volatility
models, such as the exponential Vasicek model, and extend the pricing formulas and propagator of this model to incorporate
jump diffusion with a given jump size distribution. This model is of importance to include non-Gaussian fluctuations beyond
the Black-Scholes model, and moreover yields a lognormal distribution of the volatilities, in agreement with results from
superstatistical analysis. The results obtained in the present formalism are checked with Monte Carlo simulations. 相似文献
10.
In this study, we prove that modified diffusion equations, including the generalized Burgers' equation with variable coefficients, can be derived from the Black-Scholes equation with a time-dependent parameter based on the propagator method known in quantum and statistical physics.The extension for the case of a local fractal derivative is also addressed and analyzed. 相似文献
11.
L.G Suttorp 《Annals of Physics》1979,122(2):397-435
In thispaper three types of relativistic bound-state equations for a fermion pair with instantaneous interaction are studied, viz., the instantaneous Bethe-Salpeter equation, the quasi-potential equation, and the two-particle Dirac equation. General forms for the equations describing bound states with arbitrary spin, parity, and charge parity are derived. For the special case of spinless states bound by interactions with a Coulomb-type potential the properties of the ground-state solutions of the three equations are investigated both analytically and numerically. The coupling-constant spectrum turns out to depend strongly on the spinor structure of the fermion interaction. If the latter is chosen such that the nonrelativistic limits of the equations coincide, an analogous spectrum is found for the instantaneous Bethe-Salpeter and the quasi-potential equations, whereas the two-particle Dirac equation yields qualitatively different results. 相似文献
12.
We show that fermionic dark matter (DM) which communicates with the Standard Model (SM) via the Higgs portal is a viable scenario, even if a SM-like Higgs is found at around 125 GeV. Using effective field theory we show that for DM with a mass in the range from about 60 GeV to 2 TeV the Higgs portal needs to be parity violating in order to be in agreement with direct detection searches. For parity conserving interactions we identify two distinct options that remain viable: a resonant Higgs portal, and an indirect Higgs portal. We illustrate both possibilities using a simple renormalizable toy model. 相似文献
13.
A class of finite-difference interface schemes suitable for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution is presented. Grids of this type are generated by adaptive mesh refinement methods according to resolution needs dictated by the physics of the problem being modeled. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Three distinct geometries are identified at the interfaces of a domain with interior patch-refinement: edges, concave corners and convex corners. Asymptotic stability in time of the numerical scheme is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface stencils corresponding to an explicit fourth-order finite-difference scheme are presented for each geometry. To preserve stability, a reduction in local accuracy is required at the corner geometries. It is also found that no second-order accurate solution exists that satisfies the summation-by-parts condition. Tests using the 2-D scalar advection equation and an inviscid compressible vortex support the stability and accuracy of these stencils for both linear and nonlinear problems. 相似文献
14.
Quantum interferometric strategy with input two-mode squeezed vacuum[Phys.Rev.Lett.104 103602]is reexamined for both parity and S_z~2 measurements.Unlike the previous scheme,we find that phase sensitivity obtained with the S_z~2 measurement is minimized at phase origin,which may be useful to estimate a small phase shift at high precision.For the phase deviated from zero,the sensitivity increases more slowly than that of the parity detection. 相似文献
15.
16.
《Waves in Random and Complex Media》2013,23(1):147-168
Abstract In this paper, we conduct numerical experiments with matrix Riccati equations (MREs) which describe the reflection (R) and transmission (T) matrices of the specific intensities in a layer containing randomly distributed scattering particles. The theoretical formulation of MREs is discussed in our previous paper where we show that R and T for a thick layer can be efficiently computed by successively doubling R and T matrices for a thin layer (with small optical thickness τΔ). We can compute R(τΔ) and T(τΔ) very accurately using either a fourth-order Runge–Kutta scheme or the fourth-order iterative solution. The differences between these results and those computed by the eigenmode expansion technique (EMET) are very small (<0.1%). Although the MRE formulation cannot be extended to handle the inhomogeneous term (source term) in the differential equation, we show that the force term can be reformulated as an equivalent boundary condition which is consistent with MRE methods. MRE methods offer an alternative way of solving plane-parallel radiative transport problems. For large problems that do not fit into computer memory, the MRE method provides a significant reduction in computer memory and computational time. 相似文献
17.
We present a fully second order implicit/explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems. The idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations. In this scope, the hydrodynamics equations are discretized explicitly making use of the capability of well-understood explicit schemes. On the other hand, the nonlinear heat conduction is solved implicitly. Such methods are often referred to as IMEX methods [2], [1], [3]. The Jacobian-Free Newton Krylov (JFNK) method (e.g. [10], [9]) is applied to the problem in such a way as to render a nonlinearly iterated IMEX method. We solve three test problems in order to validate the numerical order of the scheme. For each test, we established second order time convergence. We support these numerical results with a modified equation analysis (MEA) [21], [20]. The set of equations studied here constitute a base model for radiation hydrodynamics. 相似文献
18.
We present an efficient numerical framework for analyzing spinodal decomposition described by the Cahn–Hilliard equation. We focus on the analysis of various implicit time schemes for two and three dimensional problems. We demonstrate that significant computational gains can be obtained by applying embedded, higher order Runge–Kutta methods in a time adaptive setting. This allows accessing time-scales that vary by five orders of magnitude. In addition, we also formulate a set of test problems that isolate each of the sub-processes involved in spinodal decomposition: interface creation and bulky phase coarsening. We analyze the error fluctuations using these test problems on the split form of the Cahn–Hilliard equation solved using the finite element method with basis functions of different orders. Any scheme that ensures at least four elements per interface satisfactorily captures both sub-processes. Our findings show that linear basis functions have superior error-to-cost properties. 相似文献
19.
We present a very simple scheme for generating four-qubit cluster
states with one step using parity measurement based on
dipole-induced transparency in a cavity-waveguide system. The scheme
only uses the photon detectors to check the parity of the spatially
separated dipole, which are the same (even parity) or different (odd
parity) through measuring the light fields in the waveguide. The
initial entangled states remain after nondetective identification
and they can be used for successive tasks. 相似文献