Η διατριβή χωρίζεται σε δύο μέρη. Στο Μέρος Α' χρησιμοποιούνται μαθηματικές μέθοδοι της Θεωρίας Παιγνίων και των Δυναμικών Συστημάτων για να μελετηθεί η κανονική και χαοτική δυναμική διαφόρων μοντέλων της Μικροοικονομίας. Βασικά αποτελέσματα είναι η μετάβαση σε συνθήκες πλήρους ανταγωνισμού και η διαφοροποίηση του παραγόμενου προιόντος σε ένα δυοπώλιο-τριοπώλιο. Στο Μέρος Β', κύριος στόχος της έρευνας ήταν να συνδεθούν ορισμένες από τις πλέον γνωστές μερικές διαφορικές εξισώσεις (ΜΔΕ) που χρησιμοποιούνται στα Οικονομικά Μαθηματικά και Χρηματοοικονομικά, με την εξίσωση της θερμότητας της Μαθηματικής Φυσικής, εφαρμόζοντας την κατά Lie συμμετρίες ανάλυση. Επίσης η ανάλυση αυτή αποδείχθηκε ιδιαίτερα ισχυρή για την εύρεση αλγεβρικών δομών εξισώσεων που περιγράφουν την τιμολόγηση αγαθών. Έτσι, οδηγούμαστε με συστηματικό τρόπο όχι μόνο στην εύρεση νέων λύσεων αλλά και στην ανακάλυψη κομψών γενικεύσεων των εξισώσεων αυτών. / The thesis is divided into two parts. In Part One we use the mathematical methods of Game Theory and Dynamical Systems to study the stable and chaotic dynamics of various models in Microeconomics. Some of our main results are the route to perfect competition and the differentiation of goods in a duopoly and in a triopoly. In Part Two, our main concern was to link some of the most well-known partial differential equations that are encountered in Economics and Financial Mathematics, with the heat equation of Mathematical Physics, using Lie symmetry analysis. More to that, this analysis proved extremely powerful to the finding of interesting algebraic properties for equations that describe the pricing of commodities. In such way, we succeed in presenting, in a systematic fashion, not only new solutions, but also elegant generalisations of the equations under investigation.

The implied volatility surface plays an important role for Front office and Risk Management functions at Nasdaq and other financial institutions which require mark-to-market of derivative books intraday in order to properly value their instruments and measure risk in trading activities. Based on the aforementioned business needs, being able to calibrate an end of day implied volatility surface based on new market information is a sought after trait. In this thesis a statistical learning approach is used to calibrate the implied volatility surface intraday. This is done by using OMXS30-2019 implied volatility surface data in combination with market information from close to at the money options and feeding it into 3 Machine Learning models. The models, including Feed Forward Neural Network, Recurrent Neural Network and Gaussian Process, were compared based on optimal input and data preprocessing steps. When comparing the best Machine Learning model to the benchmark the performance was similar, indicating that the calibration approach did not offer much improvement. However the calibrated models had a slightly lower spread and average error compared to the benchmark indicating that there is potential of using Machine Learning to calibrate the implied volatility surface. / Implicita volatilitetsytor är ett viktigt vektyg för front office- och riskhanteringsfunktioner hos Nasdaq och andra finansiella institut som behöver omvärdera deras portföljer bestående av derivat under dagen men också för att mäta risk i handeln. Baserat på ovannämnda affärsbehov är det eftertraktat att kunna kalibrera de implicita volatilitets ytorna som skapas i slutet av dagen nästkommande dag baserat på ny marknadsinformation. I denna uppsats används statistisk inlärning för att kalibrera dessa ytor. Detta görs genom att uttnytja historiska ytor från optioner i OMXS30 under 2019 i kombination med optioner nära at the money för att träna 3 Maskininlärnings modeller. Modellerna inkluderar Feed Forward Neural Network, Recurrent Neural Network och Gaussian Process som vidare jämfördes baserat på data som var bearbetat på olika sätt. Den bästa Maskinlärnings modellen jämfördes med ett basvärde som bestod av att använda föregående dags yta där resultatet inte innebar någon större förbättring. Samtidigt hade modellen en lägre spridning samt genomsnittligt fel i jämförelse med basvärdet som indikerar att det finns potential att använda Maskininlärning för att kalibrera dessa ytor.

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them. / Mathematical Sciences / Applied Mathematics / D. Phil. (Applied Mathematics)

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.

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them. / Mathematical Sciences / Applied Mathematics / D. Phil. (Applied Mathematics)

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.

The Black-Scholes model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used today, and regarded as one of the best ways of determining fair prices of options. In the classical Black-Scholes model for the market, it consists of an essentially riskless bond and a single risky asset. So far there is a number of straightforward extensions of the Black-Scholes analysis. Here we consider more complex products where each component in a portfolio entails several variables with constraints. This leads to elegant models based on multivariable stochastic integration, and describing several securities simultaneously. We derive a general asymptotic solution in a short time interval using the heat kernel expansion on a Riemannian metric. We then use our formula to predict the better price of options on multiple underlying assets. Especially, we apply our method to the case known as the one of two-color rainbow ptions, outperformance option, i.e., the special case of the model with two underlying assets. This asymptotic solution is important, as it explains hidden effects in a class of financial models.

<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>

<p>In this work we consider the nonlinear case of Black-Scholes equation and apply it to American options. Also, method of Liao and Khaliq of high order was applied to nonlinear Black-Scholes equation in case of American options. Here, we use this method oh fourth order in time and space to raise American option price accuracy.</p>

In this work we consider the nonlinear case of Black-Scholes equation and apply it to American options. Also, method of Liao and Khaliq of high order was applied to nonlinear Black-Scholes equation in case of American options. Here, we use this method oh fourth order in time and space to raise American option price accuracy.