A Numerical Discussion for the European Put Option Model

Year 2021, Volume: 14 Issue: 1, 132 - 140, 31.03.2021

Seda Gülen

https://doi.org/10.18185/erzifbed.758426

Abstract

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 analyze these models. This study will focus mainly on investigating the behavior 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.

Keywords

Black-Scholes equation, option pricing modeling, high-order finite difference, time discretization methods

Thanks

I would like to thank Prof. Dr. Murat Sari for his valuable suggestions to improve this article.

Avrupa Tipi Satış Opsiyonu Modeli İçin Nümerik Bir Tartışma

Abstract

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 hassasiyeli yaklaşımlaşımlarına izin veren dördüncü mertebeden bir sonlu fark (FD4) kullanılarak yapılmıştır. Zaman ayrıklaştırması için üç sayısal teknik kullanılmıştır: Güçlü kararlılık koruyan Runge Kutta (SSPRK3), dördüncü mertebe Runge Kutta (RK4) ve tek adımlı yöntem. Birleştirilmiş yöntemlerle üretilen sonuçlar mevcut literatür çözümü ve tam çözüm ile karşılaştırılmıştır.

Keywords

Black-Scholes denklemi, opsiyon fiyatlama modellemesi, yüksek mertebe sonlu fark, zaman ayrıklaştırma yöntemleri

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