由於忽略員工認股選擇權的稀釋性會造成偏誤的企業評價，本研究利用以認購權證為基礎的改良評價模型，並配合會計研究的剩餘淨利模型，欲探討Warrant-Pricing Model與 Black-Scholes-Model之差異。由於現行國際會計準則與美國會計準則都已明確規定員工認股選擇權需依公平價值認列為費用，我國會計公報未來必定朝此方向修改，為因應使用公平價值法對員工認股權評價，本文對於財報附註揭露之表達提出建議，以提供會計人員與審計人員進行財務報表編製與查核工作時為參考。 / Because employee stock option (ESO) has some special conditions which make them different from all the options transferring in markets, we can not use the general option pricing model, such as: Black-Scholes-Model, to price ESO. By using Warrant-Pricing Model and the residual income model, this research introduces us the differences between Warrant-Pricing Model and Black-Scholes-Model. Moreover this research leads to the conclusion that Warrant-Pricing Model can price ESO more properly, and it is helpful in evaluating company equity and pricing stock. This research also provide some advice to auditors and accountants on financial statement disclosures.

O presente trabalho pretende introduzir uma visão das Finanças sob o aspecto da Filosofia da Ciência. Para permitir um estudo mais detalhado, optou-se por utilizar um dos modelos mais utilizados em Finanças, o modelo Black-Scholes de apreçamento de opções, e situá-lo dentro do campo de aplicação da Filosofia da Ciência. Primeiramente buscou-se, antes de entrar numa análise do texto original que apresentou o modelo, contextualizá-lo no campo da Economia e das Finanças e reconstruir historicamente suas bases conceituais. Em seguida são apresentados alguns dos elementos principais que caracterizam os modelos filosóficos de mudança científica posteriores à posição definida pelo positivismo lógico. Especial atenção é dada às concepções Realista e Anti-Realista da Ciência. Ao final, é feita uma descrição de algumas peculiaridades empíricas do modelo Black-Scholes e é analisada a função do modelo dentro do campo da Economia e das Finanças. / This work is an introduction of a Philosophy of Science view of the Finance. We choose the Black-Scholes option valuation model, one of the most famous models of finance, and we submet it of an analysis in the Philosophy of Science point of view. At first, we present an historical reconstruction of Black-Scholes model conceptual basis, using the original text of 1973. After this, we show some aspects of philosophical models of scientific change after the position defined by Positivism. Special attention is given to Realism and Anti-Realismo conception of science. At the end, we describe some empirical aspects of Black-Scholes model and its correlation inside the Economy and Modern Theory of Finance.

In the present study I deal with a pricing of derivatives especially with the European option. In the first chapter there are described basic principles of pricing financial derivatives. I focus on the options strategies from the simplest to the more difficult one. The second chapter is dedicated to the Binomial pricing model. It is introduced its derivation, application, its pro and con. Next chapter contains a description of Black-Scholes model. Again it is explained derivation of this model and its properties. At the end of this chapter it is described relationship between Binomial and Black-Scholes models. The forth chapter is consisted of an analysis of real data of stocks company Philip Morris International, Lehman brothers Holding and American Insurance Group. I focus on the relationship between shares and options in time of the financial crisis. Last chapter is dedicated to the description of software concerning options which was created in Microsoft Excel and which is part of this study.

The purpose of this paper is solving the European option pricing problem under the Black–Scholes model. Our approach is to use the so-called stochastic Runge–Kutta (SRK) numericalscheme to find the corresponding expectation of the functional to the stochastic differentialequation under the Black–Scholes model. Several numerical solutions were made to study howquickly the result converges to the theoretical value. Then, we study the order of convergenceof the SRK method with the help of MATLAB.

Asset pricing models are well established and have been used extensively by practitioners both for pricing options as well as for hedging them. Though Black-Scholes is the original and most commonly communicated asset pricing model, alternative asset pricing models which incorporate additional features have since been developed. We present three asset pricing models here - the Black-Scholes model, the Heston model and the Merton (1976) model. For each asset pricing model we test the hedge effectiveness of delta hedging, minimum variance hedging and static hedging, where appropriate. The options hedged under the aforementioned techniques and asset pricing models are down-and-out call options, lookback options and cliquet options. The hedges are performed over three strikes, which represent At-the-money, Out-the-money and In-the-money options. Stock prices are simulated under the stochastic-volatility double jump diffusion (SVJJ) model, which incorporates stochastic volatility as well as jumps in the stock and volatility process. Simulation is performed under two ’Worlds’. World 1 is set under normal market conditions, whereas World 2 represents stressed market conditions. Calibrating each asset pricing model to observed option prices is performed via the use of a least squares optimisation routine. We find that there is not an asset pricing model which consistently provides a better hedge in World 1. In World 2, however, the Heston model marginally outperforms the Black-Scholes model overall. This can be explained through the higher volatility under World 2, which the Heston model can more accurately describe given the stochastic volatility component. Calibration difficulties are experienced with the Merton model. These difficulties lead to larger errors when minimum variance hedging and alternative calibration techniques should be considered for future users of the optimiser.

There exist many numerical methods for numerical solutions of the systems of stochastic differential equations. We choose the method of deterministic quadrature formulae proposed by Müller–Gronbach, and Yaroslavtseva in 2016. The idea is to apply a simplified version of the cubature in Wiener space. We explain the method and check how good it works in the simplest case of the classical Black–Scholes model.

In this thesis the Black Scholes and the Heston stock prices are investigated and the models are compared. The Black Scholes model assumes that the volatility is constant, while the Heston model allows stochastic volatility which is more flexible and can perform better with empirical data. Both models are analysed and simulated, and the parameters are estimated based on empirical data of S&P 500. Results are based on simulations and characteristic functions which are presented with figures of probability density functions.

Thesis (MSc)--Stellenbosch University, 2015 / AFRIKAANSE OPSOMMING : Die groeiende belangstelling in kalibrering en modelrisiko is ’n redelik resente ontwikkeling in finansiële wiskunde. Hierdie proefskrif fokusseer op hierdie sake, veral in verband met die prysbepaling van vanielje-en eksotiese opsies, en vergelyk die prestasie van verskeie Lévy modelle. ’n Nuwe metode om modelrisiko te meet word ook voorgestel (hoofstuk 6). Ons kalibreer eers verskeie Lévy modelle aan die log-opbrengs van die S&P500 indeks. Statistiese toetse en grafieke voorstellings toon albei aan dat suiwer sprongmodelle (VG, NIG en CGMY) die verdeling van die opbrengs beter beskryf as die Black-Scholes model. Daarna kalibreer ons hierdie vier modelle aan S&P500 indeks opsie data en ook aan "CGMY-wˆ ereld" data (’n gesimuleerde wÃłreld wat beskryf word deur die CGMY-model) met behulp van die wortel van gemiddelde kwadraat fout. Die CGMY model vaar beter as die VG, NIG en Black-Scholes modelle. Ons waarneem ook ’n effense verskil tussen die nuwe parameters van CGMY model en sy wisselende parameters, ten spyte van die feit dat CGMY model gekalibreer is aan die "CGMYwêreld" data. Versperrings-en terugblik opsies word daarna geprys, deur gebruik te maak van die gekalibreerde parameters vir ons modelle. Hierdie pryse word dan vergelyk met die "ware" pryse (bereken met die ware parameters van die "CGMY-wêreld), en ’n beduidende verskil tussen die modelpryse en die "ware" pryse word waargeneem. Ons eindig met ’n poging om hierdie modelrisiko te kwantiseer / ENGLISH ABSTRACT : The growing interest in calibration and model risk is a fairly recent development in financial mathematics. This thesis focussing on these issues, particularly in relation to the pricing of vanilla and exotic options, and compare the performance of various Lévy models. A new method to measure model risk is also proposed (Chapter 6). We calibrate only several Lévy models to the log-return of S&P500 index data. Statistical tests and graphs representations both show that pure jump models (VG, NIG and CGMY) the distribution of the proceeds better described as the Black-Scholes model. Then we calibrate these four models to the S&P500 index option data and also to "CGMY-world" data (a simulated world described by the CGMY model) using the root mean square error. Which CGMY model outperform VG, NIG and Black-Scholes models. We observe also a slight difference between the new parameters of CGMY model and its varying parameters, despite the fact that CGMY model is calibrated to the "CGMY-world" data. Barriers and lookback options are then priced, making use of the calibrated parameters for our models. These prices are then compared with the "real" prices (calculated with the true parameters of the "CGMY world), and a significant difference between the model prices and the "real" rates are observed. We end with an attempt to quantization this model risk.

Finansų inžinerijos dėka buvo sukurti egzotiniai opcionai, kurie patrauklūs investuotojams dėl didesnio nei standartiniai opcionai pelningumo ir nestandartizacijos. Pastaraisiais metais padidėjo užbiržinėje rinkoje prekiaujamų egzotinių opcionų likvidumas, dėl ko investuotojams jie tapo dar patrauklesni. Finansų institucijos, norėdamos pasiūlyti investuotojams geriausiai jų lūkesčius atitinkančius finansinius instrumentus, konkuruoja tarpusavyje dėl naujų egzotinių opcionų kūrimo. Egzotiniai opcionai gali būti kuriami ne tik akcijų, indeksų, palūkanų normų ar valiutų pagrindu, bet netgi realiai neegzistuojančio turto pagrindu. Dėl tokios egzotinių opcionų įvairovės kyla egzotinių opcionų vertinimo problema. Darbo objektas – egzotiniai opcionai kaip kintamos vertės išvestinės finansinės priemonės. Darbo tikslas – išnagrinėjus egzotinių opcionų savybes ir įkainojimo metodus, suformuoti modelį egzotinių opcionų vertinimui ir atlikti modelio parametrų jautrumo analizę. Mokslinės finansų literatūros analizė parodė, kad opcionai gali būti naudojami apsidraudimo nuo rizikos arba spekuliaciniais tikslais. Išnagrinėjusi opcionų savybes ir egzotinių opcionų klasifikacijas, autorė pasiūlė savo sukurtą egzotinių opcionų klasifikaciją, kuri priklauso nuo opciono charakteristikų. Išnagrinėjus mokslinę literatūrą nustatyta, kad vertinant opcionus svarbiausia atsižvelgti į opcionų vertę sudarančius parametrus: bazinio turto rinkos kainą bei jos kintamumą, vykdymo kainą, nerizikingą palūkanų... [toliau žr. visą tekstą] / Financial engineering have created exotic options that are more attractive to investors for more profitability than plain-vanilla options and non-standartization. Recently years have grown liquidity on OTC tradable options, and they became even more attractive for investors. Financial institutions compete for new exotic option creation, because they want to offer investors the best financial instruments for their expectations. Exotic options could be created not only on stocks, index, interest rates or currency bases, but even on not real-existed asset. There exists a problem of exotic options valuation, because there are a big variety of exotic options. The object of the study – exotic options as variable value derivatives. The purpose of the study – after analyse of characteristics and pricing methods of options, create a model for exotic options evaluation and make model parameters sensitivity analysis. The findings of the scholar finance literature pointed, that options could be used for hedging from risks or speculation. After analysis of options characteristics and exotic options classifications, authoress offer new exotic options classification, which depends on option characteristics. To summarize of scolar literature pointed, that the most important for valuing options is their parameters: strike price, underlying spot price and volatility, risk free rate, maturity and, if it is, dividens. After comparable analysis it emerged, that exotic options greeks functions... [to full text]

<p>In the Black-Scholes model, the volatility considered being deterministic and it causes some</p><p>inefficiencies and trends in pricing options. It has been proposed by many authors that the</p><p>volatility should be modelled by a stochastic process. Heston Model is one solution to this</p><p>problem. To simulate the Heston Model we should be able to overcome the correlation</p><p>between asset price and the stochastic volatility. This paper considers a solution to this issue.</p><p>A review of the Heston Model presented in this paper and after modelling some investigations</p><p>are done on the applet.</p><p>Also the application of this model on some type of options has programmed by MATLAB</p><p>Graphical User Interface (GUI).</p>