Black-Scholes kısmi diferensiyel denklemin sonlu eleman ve sonlu fark yöntemleri ile çözüm analizi
Solution analysis of Black-Scholes partial differential equation by finite element and finite difference methods
Yazar: HAYATİ ÜNSAL ÖZER
Danışman: DOÇ. DR. AHMET DURAN
İstanbul Teknik Üniversitesi / Bilişim Enstitüsü / Hesaplamalı Bilimler ve Mühendislik Ana Bilim Dalı
Derece: Yüksek Lisans, 2016
Konu: Ekonomi = Economics ; Matematik = Mathematics

Özet:
Black-Scholes kısmi diferansiyel denklemi (Black & Scholes, 1973) matematiksel finans alanında ve finans endüstrisinde en çok kullanılan ünlü denklemlerden birisidir. Bu tez çalışmasında, Black-Scholes kısmi diferansiyel denkleminin sonlu eleman lineer yaklaşımı ve sonlu fark yöntemi kullanılarak sayısal çözüm analizi yapılmıştır. Literatürde bilinen uygun sayısal yöntemler kullanılmıştır. Opsiyon sözleşmelerinin fiyatlandırılması için, Black-Scholes kısmi diferansiyel denklemiyle elde edilen sayısal çözüm ile Black-Scholes formülü ile bulunan sonuç karşılaştırılmaktadır. Çeşitli finansal koşullarda analizler yapılmaktadır. Ayrıca, sonlu eleman ve sonlu fark sayısal yöntemlerinin Black-Scholes kısmi diferansiyel denklemine uygulanmasında ortaya çıkan sayısal hataların davranışı ve kaynağı araştırılmaktadır.
Çalışmanın ilk bölümünde, (Hull, 2008) ve (Wilmott, 2007) temel alınıp, Black-Scholes kısmi diferansiyel denkleminin elde ediliş yöntemlerinden biri gösterilerek denklem tanıtılmaktadır. Sonrasında sonlu fark yöntemleri ve sonlu eleman yöntemleri hakkında genel bilgiler verilmektedir. İkinci bölümde sonlu fark yöntemlerinden açık (explicit) yöntem, kapalı (implicit) yöntem ve Crank-Nicolson yöntemi kısaca anlatılmaktadır (Willmott, Howison and Dewynne, 1995). Uygulamada Crank-Nicolson yöntemi kullanıldı. Farklı koşullar altında alım (call) ve satım (put) opsiyonları sözleşmesi fiyatları Crank-Nicolson yöntemi ile elde edilerek Black-Scholes formülüne karşı gelen fiyatlar ile karşılaştırıldı. Elde edilen hataların çizelgelerine ve grafiklerine yer verilmiştir. Üçüncü bölümde Black-Scholes denkleminin çözümü için sonlu eleman yöntemi kullanıldı. Yöntemin kullanılabilmesi için denklemin zayıf formu elde edildi. Elde edilen bu zayıf form üzerinden konumsal ve zamansal yaklaşım uygulandı. Zamansal yaklaşım için de Crank-Nicolson yöntemi seçildi. Yaklaşımda elde edilecek cebirsel denklemler için lineer (doğrusal) eleman yaklaşım fonksiyonları kullanıldı. Uygulama yine farklı koşullardaki alım ve satım opsiyonları üzerinden yapıldı. Elde edilen sözleşme fiyatları Black-Scholes formülü ile karşılaştırılarak elde edilen hatalar çizelgeler ve grafikler üzerinden yorumlandı. Son olarak sonlu eleman yöntemi lineer eleman yaklaşımı için ağ iyileştirilmesi varlığı araştılıp bununla ilgili örnekler çizelge ve grafik üzerinden verildi. 
Sonuç olarak, başlangıç hisse fiyatı ile hisse işlem fiyatının yakın olduğu durumlarda hata miktarında salınımlar gözlemlendi. Sonlu eleman yöntemi lineer eleman yaklaşımı ile elde edilen hataların sonlu fark yöntemi Crank-Nicolson yaklaşımına göre daha büyük olduğu görüldü

Anahtar Sözcükler: Black-Scholes kısmi diferansiyel denklemi, sonlu eleman yöntemi, sonlu fark yöntemi, hata analizi, matematiksel finans, opsiyon sözleşmesi, Avrupa alım opsiyonu, Avrupa satım opsiyonu

Abstract:
Black-Scholes partial differential equation (Black & Scholes, 1973) is one of the most famous equations in mathematical finance and financial industry. In this thesis, numerical solution analysis is done for Black-Scholes partial differential equation using finite element method with linear approach and finite difference  methods. The methods which are used, are known in literature. The numerical solutions are compared with Black-Scholes formula for option pricing. The numerical errors are determined for the finite element and finite difference applications to Black-Scholes partial differential equation. The aim of this thesis is to examine the behavior and source of the corresponding errors under various market situations.
Black-Scholes partial differential equation is introduced in the first chapter of this thesis. One of the derivations of the Black-Scholes PDE is showed based on (Hull, 2008) and (Wilmott, 2007). The derivation starts with establishing a model to stock price. Then, it continues with developping a portfolio which consist of one long option and one short stock. Also, Itô's lemma is needed to reach the equation. In the final part of the first chapter, general informations are given about finite difference and finite element methods. The basic informations about the finite difference method are given in the light of (Willmott, Howison and Dewynne, 1995). 
In chapter two, the approximated solution of Black-Scholes PDE is calculated by the finite difference method. The Crank-Nicolson method is used because it is more efficient than the forward difference and the bacward difference methods. We used MATLAB software to compute the approximated solution for the call and put options. The approximated solution is compared with exact solution using Black-Scholes formula. We examine the behavior of the corresponding errors for various market situations such as stable market and high volatility conditions. In the graphs, the oscillations are observed when the initial stock price is close to the exercise price. 
In chapter three, the basic informations about the finite element method are given in the light of (Reddy, 1993). The finite element method divides a given problem into a set of simple subproblems, then finds the approximated solution assembling solutions produced for each subproblem. 
The finite element method has three basic features. First, the set of geometrically simple subdomains of the problem gives a geometrically complex domain. Second, any continuous function which can be showed by a linear combination of algebraic polynomials derives the approximation function over each finite element. Third, satisfying the governing equations gives algebraic relations among the undetermined coefficients. 
Given a problem, the approximated solution with finite element method consists of these step: 
- Discretization of the given domain into a set of finite elements 
- Differential equation is expressed in a weak form
- Derivation of element equation for all elements. Construct finite element interpolation functions and compute element matrices
- Assembly of element equation to obtain general equation system
- Impositon of the boundary conditions
- Solution of the assembled equations
The solution of time-dependent diferential equations involves two steps: Spatial approximation and temporal approximation.  The spatial finite element model results in a set of time-dependent ordinary differential equation. In temporal approximation step, finite difference method is used to obtain algebraic equations from time-dependent ordinary differential equation.
In chapter three, the approximated solution of Black-Scholes PDE is calculated by the finite element method. First, weak form of differential equation is derived. After that, spatial approximation and temporal approximation are implemented to obtain algebraic equations. Linear-element approximation functions are used to construct the algebraic equations. The approximated solution is computed for the call and put options using  MATLAB software. The approximated solution is compared with the exact solution coming from Black-Scholes formula. Like the behavior of errors coming from the finite difference method, the oscillations are observed when the initial stock price is close to the exercise price. In addition, mesh refinement is examined for finite element method with linear-element approximation. Domain is divided in different spatial step intervals and error values are computed corresponding to each different spatial step interval then computed error values are compared in the tables and the graphs.
In finite difference method, the oscillations are observed when the initial stock price is close to exercise price under low volatility conditions. This result is compatible with (Hackmann, 2009). We observe that the absence of limit on normal distribution function N(d_1) as exercise time approaches causes the oscillations when the initial stock price is close to exercise price.
Generally, Black-Scholes model is more useful when financial market is stable. We observe that the number of oscillations increases under high volatility conditions. Moreover, the cumulative rounding error rose when the number of steps and the remaining time for exercise increased during the application of finite difference method to Black-Scholes partial differention equation. These observations are compatible with literature.
Likewise, the oscillations in finite element method are similar to finite difference method as the initial stock price is close to exercise price.  The errors in finite element method are greater than those of finite difference method. The reason for this is thought as related to the chosen linear element approximation functions in the finite element method. Also, better results may be obtained for selected different element approximation functions (See: Topper, 2005). 
We tested the effectiveness of the mesh refinement for the finite element method with linear-element approximation. We divided the domain set for different spatial step intervals (ds) for various initial stock prices, given a fixed exercise price. As a result of the mesh refinement, it is seen that there are different cases for the finite elemet method with linear approximation and more efficient results are obtained in the case of low exercise price

Key words: Black-Scholes partial differential equation, finite element method, finite difference method, error analysis, mathematical finance, option contract, European call option, European put option