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]

Esta dissertação revisa a literatura acadêmica existente sobre a teoria de opções utilizando os modelos de precificação com saltos. Os conceitos foram equalizados, a nomenclatura foi padronizada, sendo gerado um material de referência sobre o assunto. O pressuposto de lognormalidade com volatilidade constante não é aceito pelo mercado financeiro. É freqüente, no meio acadêmico, a busca de modelos que reproduzam os fenômenos observados de leptocurtose ou assimetria dos log-retornos financeiros e que possuam a mesma robustez e facilidade para manipulação analítica do consagrado modelo de Black-Scholes. Os modelos com saltos são uma alternativa para esse problema. Avaliou-se o modelo de Kou no mercado acionário brasileiro composto por um componente de difusão que segue um movimento browniano geométrico e um componente de saltos que segue um processo de Poisson com intensidade do salto descrito por uma distribuição duplamente exponencial. A simulação histórica do modelo aponta, em geral, uma superioridade preditiva do modelo, porém as dificuldades de calibração dos parâmetros e de hedge em mercados incompletos são as principais deficiências para o uso dos modelos com saltos. / This master dissertation reviews the academic literature about option pricing and hedging with jumps. The theory was equalized and the notation was standardized, becoming this document a reference document about this subject. The log-normality with constant volatility is not accepted by the market. Academics search consistent models with the same analytical capabilities like Black-Scholes? model which can support the observed leptokurtosis or asymmetry of the financial daily log-returns behavior. The jump models are an alternative to these issues. The Kou?s model was evaluated and this one consists of two parts: the first part being continuous and following a geometric Brownian motion and the second being a jump process with its jump intensity defined by a double exponential distribution. The model backtesting showed a better predictive performance of the Kou´s model against other models. However, there are some handicaps regarding to the parameters calibration and hedging.

<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>

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

In the literature, many methods are proposed to value American options. However, due to computational difficulty, there are only approximate solution or numerical method to evaluate American options. It is not easy for general investors either to understand nor to apply. In this thesis, we build up an option pricing and virtual asset model system, which provides a friendly environment for general public to calculate early exercise boundary of an American option. This system modularize the well-handled pricing models to provide the investors an easy way to value American options without learning difficult financial theories. The system consists two parts: the first one is an option pricing system, the other one is an asset model simulation system. The option pricing system provides various option pricing methods to the users; the virtual asset model system generates virtual asset prices for different underlying models.

In this thesis, we expand the option pricing and virtual asset model system by Cheng (2005) and include new simulations and maximum likelihood estimation of the parameter of the stochastic differential equations. For easy manipulation of general users, the interface of original option pricing system is modified. In addition, in order to let the system more completely, some stochastic models and methods of pricing and estimation are added. This system can be divided into three major parts. One is an option pricing system; The second is an asset model simulation system; The last is estimation system of the parameter of the model. Finally, the analysis for the data of network are carried out. The differences of the prices between estimator of this system and real market are compared.

Esta dissertação revisa a literatura acadêmica existente sobre a teoria de opções utilizando os modelos de precificação com saltos. Os conceitos foram equalizados, a nomenclatura foi padronizada, sendo gerado um material de referência sobre o assunto. O pressuposto de lognormalidade com volatilidade constante não é aceito pelo mercado financeiro. É freqüente, no meio acadêmico, a busca de modelos que reproduzam os fenômenos observados de leptocurtose ou assimetria dos log-retornos financeiros e que possuam a mesma robustez e facilidade para manipulação analítica do consagrado modelo de Black-Scholes. Os modelos com saltos são uma alternativa para esse problema. Avaliou-se o modelo de Kou no mercado acionário brasileiro composto por um componente de difusão que segue um movimento browniano geométrico e um componente de saltos que segue um processo de Poisson com intensidade do salto descrito por uma distribuição duplamente exponencial. A simulação histórica do modelo aponta, em geral, uma superioridade preditiva do modelo, porém as dificuldades de calibração dos parâmetros e de hedge em mercados incompletos são as principais deficiências para o uso dos modelos com saltos. / This master dissertation reviews the academic literature about option pricing and hedging with jumps. The theory was equalized and the notation was standardized, becoming this document a reference document about this subject. The log-normality with constant volatility is not accepted by the market. Academics search consistent models with the same analytical capabilities like Black-Scholes? model which can support the observed leptokurtosis or asymmetry of the financial daily log-returns behavior. The jump models are an alternative to these issues. The Kou?s model was evaluated and this one consists of two parts: the first part being continuous and following a geometric Brownian motion and the second being a jump process with its jump intensity defined by a double exponential distribution. The model backtesting showed a better predictive performance of the Kou´s model against other models. However, there are some handicaps regarding to the parameters calibration and hedging.