Mathematical Finance

Year 2002, Volume: 1 Issue: 1, 53 - 66, 15.04.2002

Ömer Önalan

Abstract

The introduction of financial derivatives such as futures and options on underlying as stock, currencies, interest rate etc. makes possible to decreasing of financial risks. The basic idea of a financial derivative production: to find participants in the market who are willing to share the risks and profits of future developments in the market which are subject to uncertainty. The pricing of these financial enstruments is based on an advenced mathematical theory, called Ito stochastics calculus. The basic model for a random price processes is described by Brownian motion and releted differential equations. An analitical formula for pricing of European call option was given by Black - Scholes (1973) and Merton (1973). Their approach has become the firm basis for modern financial mathematics which uses advenced tools such as martingale theory, Ito calculus and stochastics control, to find adequate solutions to the pricing of derivative securities.

Keywords

Financial Derivatives, Brownian Motion, Black-Scholes Formula, Option, Ito Stochastic Analysis, Change of Measure, Girsanov Transformation, Martingale, Equvalent Martingale Measure, Arbitrage, Interest Rate, Semimartingale Models

Matematiksel Finans

Abstract

Hisse senedi, döviz, faiz oranı, vb. gibi mallar üzerine futures ve options gibi finansal türevlerin oluşturulması, finansal risklerin düşürülmesine imkan vermektedir. Bir finansal türev oluşturmanın altında yatan temel fikir; belirsizliğin hakim olduğu piyasada gelecekteki gelişmelere bağlı olarak ortaya çıkacak, risk ve kazancı, paylaşmaya istekli piyasa katılımcılarını bulmaktır. Bu finansal enstrümanların fiyatlandırılması, ito stokastik analizi olarak adlandırılan bir teoriye dayanmaktadır. Rassal fiyat süreci için temel model Brownian hareket ve onunla ilişkili diferansiyel denklemlerle tanımlanmaktadır. Avrupa alım opsiyonunun fiyatlandırılması için Black-Scholes (1973) ve Merton (11973) tarafından bir analitik formül verilmiştir. Onların yaklaşımı, türev menkul kıymetlerin fiyatlanmasına uygun çözümler bulmak için martingale teori, ito analizi ve stokastik kontrol gibi ileri düzeydeki araçları kullanan, modern finansal matematik için sağlam bir zemin oluşturulmuştur.

Keywords

Finansal Türev, Brownian Hareket, Black-Scholes Formülü, Opsiyon, Ito Stokastik Analizi, Ölçümün Değişimi, Girsinov Dönüşümü, Martingale, Denk Martingale Ölçümü, Arbitraj, Faiz Oranı, Semimartingale Modelleri

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