A boundary point lemma for Black-Scholes type operators

Group Classification of a Generalized Black--Scholes--Merton Equation

The complete group classification of a generalization of the Black-Scholes-Merton model is carried out by making use of the underlying equivalence and additional equivalence transformations. For each non linear case obtained through this classification, invariant solutions are given. To that end, two boundary conditions of financial interest are considered, the terminal and the barrier option conditions.

Application of Neural Network Machine Learning to Solution of Black-Scholes Equation

This paper presents a novel way to predict options price for one day in advance, utilizing the method of Quasi-Reversibility for solving the Black-Scholes equation. The Black-Scholes equation solved forwards in time with Tikhonov regularization as an ill-posed problem allows for extrapolation of option prices. This provides high-accuracy results, which can be further improved by applying Neural Network Machine Learning to the solution of the Black-Scholes equation as well as initial and boundary conditions and implied volatility. Using historical option and stock price data the results obtaine…

The Use of Numeraires in Multi-dimensional Black-Scholes Partial Differential Equations

The change of numeraire gives very important computational simplification in option pricing. This technique reduces the number of sources of risks that need to be accounted for and so it is useful in pricing complicated derivatives that have several sources of risks. In this article, we considered the underlying mathematical theory of numeraire technique in the viewpoint of PED theory and illustrated it with five concrete pricing problems. In the viewpoint of PED theory, the numeraire technique is a method of reducing the dimension of status spaces where PDE is defined.

Introducing and solving generalized Black-Scholes PDEs through the use of functional calculus

We introduce some families of generalized Black--Scholes equations which involve the Riemann-Liouville and Weyl space-fractional derivatives. We prove that these generalized Black--Scholes equations are well-posed in $(L^1-L^\infty)$-interpolation spaces. More precisely, we show that the elliptic type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial operators and sectorial operators in an abstract setting. Such a conn…

Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility, $σ$, in which the last is defined by a stochastic differential equation with an Orstein--Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, $S$, and a new variable, $y$. We find that for arbitrary functional form of the volatility, $σ(y)$, the (1 + 2) evolution equation always…

Removing non-smoothness in solving Black-Scholes equation using a perturbation method

Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, including homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by {first} performing variable transformations that push the point to infinity. As a…

Monotone methods in counterparty risk models with non-linear Black-Scholes-type equations

A non-linear Black-Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the non-linear reaction function allows for an equivalent transformation of the semi-linear Black-Scholes equation into a standard parabolic problem with a monotone non-linear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most co…

Control of the Black-Scholes equation

There are hundreds of papers dealing with the Black-Scholes equation. Yet, no one seems to have ever used the scaling invariance coming from the heat equation. And there appears to be very few studies on the "control aspect" of the equation. In previous works, J.Y. Chemin and Cl. David obtained new results for the mass critical nonlinear Schrödinger equation, with a simple principle: one uses the fact that for this nonlinear equation, solutions of scales which are different enough almost do not interact. The nonlinearity of the original equation just required to determine a condition about the…

Asian Option Pricing under Uncertain Volatility Model

In this paper, we study the asymptotic behavior of Asian option prices in the worst case scenario under an uncertain volatility model. We give a procedure to approximate the Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and cutting the obtained Black-Scholes-Barenblatt equation into two Black-Scholes-like equations, we obtain an approximation method to solve the fully nonlinear PDE.

Black-Scholes 方程的拉普拉斯变换方法

Laplace transformation method for the Black-Scholes equation

本文应用 Sheen、Sloan 和 Thomée 提出的创新拉普拉斯变换方法求解 Black-Scholes 方程，该方法具有任意高阶收敛性且天然可并行，并通过多种期权数值实验验证了其高效性。

Reducing time-dependent multifactor Black and Scholes equation with knock-out features to equivalent time-constant coefficient equation, and applications

We consider the multifactor Black and Scholes equation with time-dependent coefficients, and a knock-out feature contingent on the underlying asset values reaching a limit (reflected by a Dirichlet condition on the boundary). We prove that this equation, which has important applications in finance and insurance, can be reduced to an equivalent time-constant coefficient equation, with coefficients defined as averages of the original ones. Equivalent results are also valid for general second order parabolic equations, with applications in other fields in the natural sciences. The result establis…

Viscosity solution of a Delta Greek nonlinear Black-Scholes equation

In this paper, a class of nonlinear option pricing models involving transaction costs is considered. The diffusion coefficient of the nonlinear parabolic equation for the price $V$ is assumed to be a linear function of the option's underlying asset price and the Gamma Greek $V_{xx}$. The main aim of this work is to study the governing PDE of the Delta Greek. The existence of viscosity solutions is proved using the vanishing viscosity method. Regularizing the equation by adding a small perturbation to the initial problem, a sequence of approximate solutions $u^{\varepsilon}$ is constructed and …

使用径向基函数数值求解多维型 Black-Scholes 分数阶偏微分方程

Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black-Scholes

本文提出使用基于径向基函数的无网格方法数值求解多维型 Black-Scholes 时间-空间分数阶偏微分方程，展示了该方法处理多维问题的灵活性及降低矩阵条件数的有效途径。

评估线性和非线性Black-Scholes方程最优执行边界近似方法的变换技术

Transformation methods for evaluating approximations to the optimal exercise boundary for linear and nonlinear Black-Scholes equations

本文提出一种变换方法，将美式期权提前执行边界的自由边界问题转化为固定域上的非线性非局部抛物型方程，并设计迭代数值方案以近似求解边界，同时分析了线性和非线性Black-Scholes方程中边界对模型参数的依赖。

广义Black-Scholes方程的混沌性

Chaos for generalized Black-Scholes equations

本文通过广义算子 $A=x^a\partial_x$（$0<a<1$）统一研究Black-Scholes方程与热方程，证明相应退化抛物方程在加权连续函数Banach空间上具有混沌算子半群，从而推广并简化了先前结果。

从Black-Scholes视角看选举前联盟协议

Pre-electoral coalition agreement from the Black-Scholes point of view

本文类比期权定价，将政党支持率视为随机过程，研究主要政党与少数党联盟应达到的支持水平，以使少数党多获代表，并以黑山2023年选举为例说明。