Avrupa Tipi Satış Opsiyonu Modeli için Nümerik bir Değerlendirme

The Black-Scholes equations have been increasingly popular over the last three decades since they provide more practical information for optional behaviours. Therefore, effective methods have been needed to analyse these models. This study will mainly focus on investigating the behaviour of the Black-Scholes equation for the European put option pricing model. To achieve this, numerical solutions of the Black-Scholes European option pricing model are produced by three combined methods. Spatial discretization of the Black-Scholes model is performed using a fourth-order finite difference (FD4) scheme that allows a highly accurate approximation of the solutions. For the time discretization, three numerical techniques are proposed: a strong-stability preserving Runge Kutta (SSPRK3), a fourth-order Runge Kutta (RK4) and a one-step method. The results produced by the combined methods have been compared with available literature and the exact solution. It has seen that the results with minimal computational effort are sufficiently accurate.Black-Scholes denklemleri opsiyon davranışlarında pratik bilgiler sağladığından son otuz yılda daha popüler hale gelmiştir. Bu nedenle, bu modelleri analiz etmek için etkili yöntemlere ihtiyaç duyulmaktadır. Bu çalışma temel olarak Avrupa tipi satış opsiyonu fiyatlama modeli için Black-Scholes denkleminin davranışını araştırmaya odaklanmıştır. Bunun için, Black-Scholes Avrupa tipi opsiyon fiyatlama modelinin sayısal çözümleri üç birleştirilmiş yöntem ile üretilmiştir. Black-Scholes modelinin uzaysal ayrıklaştırması, çözümlerin yüksek hassasiyetli yaklaşımlarına izin veren dördüncü mertebeden bir sonlu fark (FD4) şeması kullanılarak yapılmıştır. Zaman ayrıklaştırması için üç sayısal teknik kullanılmıştır: Kuvvetli kararlılık koruyan RungeKutta (SSPRK3), dördüncü mertebe Runge Kutta (RK4) ve tek adımlı bir yöntem. Birleştirilmiş yöntemlerle üretilen sonuçlar literatürde mevcut olan çözüm ve tam çözüm ile karşılaştırılmıştır. Sonuçların minimum hesaplama çabasıyla yeterince hassas olduğu görülmüştür

Método dos elementos finitos baseado em polinómios de Hermite cúbicos, para resolução da equação de Black-Scholes não linear com opções europeias

Foi desenvolvido um algoritmo numérico para resolver uma equação diferencial parcial generalizada de Black-Scholes, que surge na precificação de opções europeias, considerando os custos de transação. O método Crank-Nicolson é usado para discretizar no tempo e o método de interpolação cúbica de Hermite para discretizar no espaço. A eficiência e precisão do método proposto são testadas numericamente e, os resultados confirmam o comportamento teórico das soluções, que também se encontra em boa concordância com a solução exata.A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution

Mathematical Models and Numerical Methods for Pricing Options on Investment Projects under Uncertainties

In this work, we focus on establishing partial differential equation (PDE) models for pricing flexibility options on investment projects under uncertainties and numerical methods for solving these models. we develop a finite difference method and an advanced fitted finite volume scheme and combine with an interior penalty method, as well as their convergence analyses, to solve the PDE and LCP models developed. The MATLAB program is for implementing testing the models of numerical algorithms developed

An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering

In this work we study an interior penalty method for a finite-dimensional large-scale linear complementarity problem (LCP) arising often from the discretization of stochastic optimal problems in financial engineering. In this approach, we approximate the LCP by a nonlinear algebraic equation containing a penalty term linked to the logarithmic barrier function for constrained optimization problems. We show that the penalty equation has a solution and establish a convergence theory for the approximate solutions. A smooth Newton method is proposed for solving the penalty equation and properties of the Jacobian matrix in the Newton method have been investigated. Numerical experimental results using three non-trivial test examples are presented to demonstrate the rates of convergence, efficiency and usefulness of the method for solving practical problems

A finite difference method for pricing European and American options under a geometric Lévy process

In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then propose a finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process

IMPLEMENTATION OF MONTE CARLO MOMENT MATCHING METHOD FOR PRICING LOOKBACK FLOATING STRIKE OPTION

Monte Carlo method was a numerical method that was popular in finance. This method had disadvantages at convergences, so the moment matching was used to improve the efficiency from Monte Carlo method. The research has discussed about pricing of the lookback floating strike option using the Monte Carlo moment matching method. The monthly stock price of PT TELKOM from 2004 to 2021 that used in this research.  The results obtained by adding variance reduction moment matching in Monte Carlo method, which produces a relatively had smaller error when compared to the relative error of the standard Monte Carlo method. The orders of convergence from Monte Carlo method with variance reduction moment matching for call and put option are about 1.1 and 1.4. The conclusion that addition of the moment matching can increase the efficiency of the Monte Carlo method in determining the price of the lookback floating strike option

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.South Afric