This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-based models. To be specific, we derive the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several jump-diffusion models and stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component. An accurate modeling of the option market and asset prices requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a risk asset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic volatility model, we derive the small-time expansions, of arbitrary polynomial order, in time-t, for the tail distribution of the log-return process, and for the call-option price which is not at-the-money. Moreover, our approach allows for a unified treatment of more general payoff functions. As a consequence of our tail expansions, the polynomial expansion in t of the transition density is also obtained under mild conditions. The asymptotic behavior of the ATM call-option prices is more complicated to obtain, and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel second-order approximation for ATM option prices under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed.

Η ακριβής πρόβλεψη της μελλοντικής μεταβλητότητας αποδεικνύεται ιδιαίτερα χρήσιμη για την τιμολόγηση παραγώγων προϊόντων και την αντιστάθμιση κινδύνων στη διαχείριση χαρτοφυλακίων. H τεκμαρτή μεταβλητότητα, όπως αυτή αντανακλάται στις τιμές των δικαιωμάτων προαίρεσης, αποτελεί την εκτίμηση της αγοράς για τη μελλοντική πραγματοποιηθείσα μεταβλητότητα και έχει αποδειχθεί ότι είναι πιο αποτελεσματική από την αντίστοιχη πρόβλεψη που προκύπτει από την ανάλυση ιστορικών χρονοσειρών. Η παρούσα διατριβή πραγματεύεται τη δημιουργία ενός δείκτη τεκμαρτής μεταβλητότητας για την Ελληνική χρηματιστηριακή αγορά, χρησιμοποιώντας έναν τρόπο υπολογισμού, ο οποίος είναι ανεξάρτητος από κάθε υπόδειγμα τιμολόγησης δικαιωμάτων προαίρεσης και βασίζεται σε ένα σταθμισμένο άθροισμα τιμών δικαιωμάτων. Η μεθοδολογία αυτή εφαρμόζεται για πρώτη φορά σε μια περιφερειακή, αναπτυσσόμενη αγορά, όπως το Χρηματιστήριο Αθηνών. Ο εν λόγω δείκτης τεκμαρτής μεταβλητότητας έχει τις προοπτικές να γίνει δείκτης αναφοράς των προσδοκιών για τη μελλοντική μεταβλητότητα στην Ελληνική μετοχική αγορά, καθώς αποδεικνύεται ότι υπερισχύει στατιστικά της ιστορικής μεταβλητότητας. Επίσης, οι επενδυτές του Χρηματιστηρίου Αθηνών μπορούν να χρησιμοποιούν το επίπεδό του και τις ημερήσιες μεταβολές του για να λάβουν επενδυτικές αποφάσεις, καθώς τα αποτελέσματα της οικονομετρικής ανάλυσης αποδεικνύουν ότι υπάρχει αρνητική και ασύμμετρη σχέση μεταξύ των μεταβολών του δείκτη τεκμαρτής μεταβλητότητας και των αποδόσεων του υποκείμενου μετοχικού δείκτη FTSE/Χ.Α.-20. Τέλος, η εμπειρική έρευνα καταγράφει την επιρροή της τεκμαρτής μεταβλητότητας των κυριοτέρων χρηματιστηρίων του εξωτερικού στην εγχώρια τεκμαρτή μεταβλητότητα, ενώ επιπλέον προσπαθεί να αναπτύξει ένα υπόδειγμα για την πρόβλεψη της τεκμαρτής μεταβλητότητας αυτής καθαυτής. / In this thesis a new measure of Greek stock market volatility based on the implied volatility of FTSE/ATHEX-20 index options is proposed. Greek Implied Volatility Index is calculated using the model-free methodology that involves option prices summations and is independent from the Black and Scholes pricing formula. The specific method is applied for the first time in a peripheral and illiquid market as the Athens Exchange. The empirical findings suggest that implied volatility includes information about future volatility beyond that contained in past realized volatility and in addition, prove that there is a statistically significant negative and asymmetric contemporaneous relationship between implied volatility changes and the underlying equity index returns. Finally, the volatility transmission effects on the Greek stock exchange from the major global exchanges are tested and documented. The basis of the international integration analysis, instead of the commonly used realized returns or variances, is the implied volatilities, as proxied by the corresponding implied volatility indices.

Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on " tube estimates " for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization. We provide and test the numerical application to the Heston model (without and with displacement), for which we generate semi-closed expressions of the smile

Derivatives have a large and significant role on the financial markets today and the popularity of options has increased. This has also increased the demand of finding a suitable option pricing model, since the ground-breaking model developed by Black & Scholes (1973) have poor pricing performance. Practitioners and academics have over the years developed different models with the assumption of non-constant volatility, without reaching any conclusions regarding which model is more suitable to use. This thesis examines four different models, the first model is the Practitioners Black & Scholes model proposed by Christoffersen & Jacobs (2004b). The second model is the Heston´s (1993) continuous time stochastic volatility model, a modification of the model is also included, which is called the Strike Vector Computation suggested by Kilin (2011). The last model is the Heston & Nandi (2000) Generalized Autoregressive Conditional Heteroscedasticity type discrete model. From a practical point of view the models are evaluated, with the goal of finding the model with the best pricing performance and the most practical usage. The model´s robustness is also tested to see how the models perform in out-of-sample during a high respectively low implied volatility market. All the models are effected in the robustness test, the out-sample ability is negatively affected by a high implied volatility market. The results show that both of the stochastic volatility models have superior performances in the in-sample and out-sample analysis. The Generalized Autoregressive Conditional Heteroscedasticity type discrete model shows surprisingly poor results both in the in-sample and out-sample analysis. The results indicate that option data should be used instead of historical return data to estimate the model’s parameters. This thesis also provides an insight on why overnight-index-swap (OIS) rates should be used instead of LIBOR rates as a proxy for the risk-free rate.

Since the introduction of derivatives into the modern financial market, volatility based tradingstrategies have emerged as important tools for asset managers. Since the financial crisis apopular trading strategy has been dispersion trading, however few published studies ofdispersion trading exist. This thesis aim to perform a study of how dispersion strategies performand their characteristics. This is achieved by finding basic common dispersion trading strategies,isolate and evaluate their attributes to then draw conclusions in general about dispersion trading.Three basic dispersion strategies are found based on vanilla option spreads and their performanceis back-tested. It was found that the strategies delivered positive return with low marketcorrelation and acceptable risk. It is also found that transaction costs is a key-factors tosuccessfully use dispersion trading. Thus it is a vital factor to consider when creating adispersion based trading strategy. An interesting topic for further research is how trading signalssuch as the implied correlation and the implied volatility spread can be used to increaseprofitability. As well to model market impact from dispersion trading.

由於一般投資人在買進或賣出選擇權時，並不會同時買進多個履約價的選擇權，故會造成選擇權隱含波動度的微笑曲線出現有不連續的現象。本文嘗試運用台指選擇權建構一個日內的隱含波動度微笑曲線交易策略，利用曲線配適的方法來捕捉瞬間時點下隱含波動度曲線發生不連續的現象，雖然最後出來的損益並不如預期但還是驗證了台指選擇權市場有多次這種不連續的機會且價格失衡的狀態會回歸正常。 / Option’s implied volatility smile curves discontinuous phenomenon exists when general investors buy or sell options, they won’t buy in every strike’s options. This paper attempts to use Taiwan Index Options (trading code: TXO) to construct a trading strategy based on the implied volatility. We use curve fitting method to capture volatility smile curve’s instant discontinuous. Although we find out that the strategy won’t make a profit, there were several times when TXO market’s implied volatility smile curves were discontinuous, and the market option price will eventually go back to the theoretical price.

Des travaux récents dans le domaine des mathématiques financières ont fait émerger l'importance de l'étude de la régularité et du comportement fin des queues de distribution pour certaines classes de diffusions à coefficients non globalement réguliers. Dans cette thèse, nous traitons des problèmes issus de ce contexte. Nous étudions d'abord l'existence, la régularité et l'asymptotique en espace de densités pour les solutions d'équations différentielles stochastiques en n'imposant que des conditions locales sur les coefficients de l'équation. Notre analyse se base sur les outils du calcul de Malliavin et sur des estimations pour les processus d'Ito confinés dans un tube autour d'une courbe déterministe. Nous obtenons des estimations significatives de la fonction de répartition et de la densité dans des classes de modèles comprenant des généralisations du CIR et du CEV et des modèles à volatilité locale-stochastique : dans ce deuxième cas, les estimations entraînent l'explosion des moments du sous-jacent et ont ainsi un impact sur le comportement asymptotique en strike de la volatilité implicite. La modélisation paramétrique de la surface de volatilité, à son tour, fait l'objet de la deuxième partie. Nous considérons le modèle SVI de J. Gatheral, en proposant une nouvelle stratégie de calibration quasi-explicite, dont nous illustrons les performances sur des données de marché. Ensuite, nous analysons la capacité du SVI à générer des approximations pour les smiles symétriques, en le généralisant à un modèle dépendant du temps. Nous en testons l'application à un modèle de Heston (sans et avec déplacement), en générant des approximations semi-fermées pour le smile de volatilité / Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on « tube estimates » for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization. We provide and test the numerical application to the Heston model (without and with displacement), for which we generate semi-closed expressions of the smile

This thesis performs portfolio optimization using three allocation methods, Certainty Equivalence Tangency (CET), Global Minimum Variance (GMV) and Minimum Conditional Value-at-Risk (MinCVaR). We estimate expected returns and covariance matrices based on 7 stock market indices with a DCC-GARCH model including an ARMA (1.1) process and an external regressor of an implied volatility index (VIX). We then simulate returns using a rolling window of 500 daily observations and construct portfolios based on the allocation methods. The results suggest that the model can sufficiently estimate expected returns and covariance matrices and we can outperform benchmarks in form of equally weighted and historical portfolios in terms of higher returns and lower risk. Over the whole out-of-sample period the CET portfolio yields the highest mean returns and GMV and MinCVaR can significantly lower the variance. The inclusion of VIX has marginal effects on the forecasting accuracy and it seems to impair the estimation of risk.

No description available.

This diploma thesis describes problem of change-points in volatility of the time-series and their impact on price of nancial assets. Those change-points are estimated by using statistical methods and tests. Change-point estimation was tested on simulated datas and real world driven datas. Simulation helped to discover signi cant characteristics of change-point test, those data were simulated with using stochastic calculus. Google share prices and prices of call options were chosen to analyse impact of volatility change on those prices. Also implied volatility and its impact to call option price was analysed.