The aim of this diploma thesis is to analyze and implement selected option pricing models using statistical software. The first chapter introduces theoretical basics of options as financial instruments ideal for hedging and speculation. The second chapter constitutes the core part of this thesis since it unveils theoretical concepts of risk-neutral pricing and at the same time analyze some basic, as well as highly sophisticated option pricing models. In addition, each model is accompanied by a practical example of their effective implementation. The final chapter characterize the most widely used option trading strategies and defines the ideal expected market development linked to each strategy.

Theaim of this master-thesis is to study the impact of uncertainty in the local-and implied volatility surfaces when pricing certain structured products suchas capital protected notes and autocalls. Due to their long maturities, limitedavailability of data and liquidity issue, the uncertainty may have a crucialimpact on the choice of valuation model. The degree of sensitivity andreliability of two different valuation models are studied. The valuation models chosen for this thesis are the local volatility model of Dupire and the implied volatility model of Black&Scholes. The two models are stress tested with varying volatilities within an uncertainty interval chosen to be the volatilities obtained from Bid and Ask market prices. The volatility surface of the Mid market prices is set as the relative reference and then successively scaled up and down to measure the uncertainty.The results indicates that the uncertainty in the chosen interval for theDupire model is of higher order than in the Black&Scholes model, i.e. thelocal volatility model is more sensitive to volatility changes. Also, the pricederived in the Black&Scholes modelis closer to the market price of the issued CPN and the Dupire price is closer tothe issued Autocall. This might be an indication of uncertainty in thecalibration method, the size of the chosen uncertainty interval or the constantextrapolation assumption.A further notice is that the prices derived from the Black&Scholes model areoverall higher than the prices from the Dupire model. Another observation ofinterest is that the uncertainty between the models is significantly greaterthan within each model itself. / Syftet med dettaexamensarbete är att studera inverkan av osäkerhet, i prissättningen av struktureradeprodukter, som uppkommer på grund av förändringar i volatilitetsytan. I dennastudie värderas olika slags autocall- och kapitalskyddade struktureradeprodukter. Strukturerade produkter har typiskt långa löptider vilket medförosäkerhet i värderingen då mängden data är begränsad och man behöver ta tillextrapolations metoder för att komplettera. En annan faktor som avgörstorleksordningen på osäkerheten är illikviditeten, vilken mäts som spreadenmellan listade Bid och Ask priset. Dessa orsaker ligger bakom intresset attstudera osäkerheten för långa löptider över alla lösenpriser och dess inverkanpå två olika värderingsmodeller.Värderingsmodellerna som används i denna studie är Dupires lokala volatilitetsmodell samt Black&Scholes implicita volatilitets modell. Dessa ställs motvarandra i en jämförelse gällande stabilitet och förmåga att fånga uppvolatilitets ändringar. Man utgår från Mid volatilitetsytan som referens ochuppmäter prisändringar i intervallet från Bid upp till Ask volatilitetsytornagenom att skala Mid ytan. Resultaten indikerar på större prisskillnader inom Dupires modell i jämförelsemot Black&Scholes. Detta kan tolkas som att Dupires modell är mer känslig isammanhanget och har en starkare förmåga att fånga upp förändringar isvansarna. Vidare notering är att priserna beräknade i Dupire är relativtbilligare än motsvarande från Black&Scholes modellen. En ytterligareobservation är att osäkerheten mellan värderingsmodellerna är av högre ordningän inom var modell för sig. Ett annat resultat visar att CPN priset beräknat iBlack&Scholes modell ligger närmast marknadspriset medans marknadsprisetför Autocallen ligger närmare Dupires. Detta kan vara en indikation påosäkerheten i kalibreringsmetoden eventuellt det valda osäkerhetsintervalletoch konstanta extrapolations antagandet.

In this thesis a general framework and accompanying guidelines for how to classify financial instruments within the fair value hierarchy (included within IFRS 13) is presented. IFRS 13 introduces a broad and loosely defined regulation of how to classify a financial instrument which leaves room for misinterpretation and uncertainties. In this thesis the pricing of financial instruments and behaviour of the market data used as inputs in the models has been investigated. This is to give better insight into what is classified as significant market data, how it is used and how it is approximated. Instruments that have been investigated are autocalls, swaps, European options and Asian options. The result is presented as general recommendations for how to classify the specified instruments with clearer boarders introduced between the levels in the hierarchy. Methods and deductions introduced in the thesis could also further be implemented in classification of closely related financial instruments but has been limited in this thesis due to time restrictions.     Nyckelord på svenska IFRS, Finansiella instrument, Klassificering, Fair value, Fair value hierarchy, Autocall, Swap, Europeisk option, Asiatisk option, Implicit volatilitet, Korrelation, Marknadsaktivitet, Räntesatser / I denna uppsats är ett generellt ramverk och medföljande riktlinjer för hur man klassificerar finansiella instrument inom fair value hierarkin (inkluderad i IFRS 13) presenterat. IFRS 13 introducerar en bred och löst definierad regulation om hur klassificering finansiella instrument ska gå till som lämnar rum för feltolkningar och oklarheter. I denna uppsats har prissättningen av finansiella instrument och beteende av marknadsdata som används i modellerna undersökts. Detta ger en bättre inblick i vad som klassificeras som signifikant marknadsdata, hur den används och hur den kan approximeras. Instrument som har undersökts är autocalls, swaps, europeiska optioner och asiatiska optioner. Resultatet presenteras som allmänna rekommendationer för hur man klassificerar de angivna instrumenten med tydligare gränser som införts mellan nivåerna i hierarkin. Metoder och slutsatser som är presenterade i uppsatsen kan även vidare användas vid klassifikation av liknande finansiella instrument men har i denna avhandling begränsats på grund av tidsskäl.

An vielen verschiedenen Stellen der angewandten Statistik sind die zu untersuchenden Objekte abhängig von stetigen Parametern. Typische Beispiele in Finanzmarktapplikationen sind implizierte Volatilitäten, risikoneutrale Dichten oder Zinskurven. Aufgrund der Marktkonventionen sowie weiteren technisch bedingten Gründen sind diese Objekte nur an diskreten Punkten, wie zum Beispiel an Ausübungspreise und Maturitäten, für die ein Geschäft in einem bestimmten Zeitraum abgeschlossen wurde, beobachtbar. Ein funktionaler Datensatz ist dann vorhanden, wenn diese Funktionen für verschiedene Zeitpunkte (z.B. Tage) oder verschiedene zugrundeliegende Aktiva gesammelt werden. Das erste Thema, das in dieser Dissertation betrachtet wird, behandelt die nichtparametrischen Methoden der Schätzung dieser Objekte (wie z.B. implizierte Volatilitäten) aus den beobachteten Daten. Neben den bekannten Glättungsmethoden wird eine Prozedur für die Glättung der implizierten Volatilitäten vorgeschlagen, die auf einer Kombination von nichtparametrischer Glättung und den Ergebnissen der arbitragefreien Theorie basiert. Der zweite Teil der Dissertation ist der funktionalen Datenanalyse (FDA), speziell im Zusammenhang mit den Problemen, der empirischen Finanzmarktanalyse gewidmet. Der theoretische Teil der Arbeit konzentriert sich auf die funktionale Hauptkomponentenanalyse -- das funktionale Ebenbild der bekannten Dimensionsreduktionstechnik. Ein umfangreicher überblick der existierenden Methoden wird gegeben, eine Schätzmethode, die von der Lösung des dualen Problems motiviert ist und die Zwei-Stichproben-Inferenz basierend auf der funktionalen Hauptkomponentenanalyse werden behandelt. Die FDA-Techniken sind auf die Analyse der implizierten Volatilitäten- und Zinskurvendynamik angewandt worden. Darüber hinaus, wird die Implementation der FDA-Techniken zusammen mit einer FDA-Bibliothek für die statistische Software Xplore behandelt. / In many different fields of applied statistics an object of interest is depending on some continuous parameter. Typical examples in finance are implied volatility functions, yield curves or risk-neutral densities. Due to the different market conventions and further technical reasons, these objects are observable only on a discrete grid, e.g. for a grid of strikes and maturities for which the trade has been settled at a given time-point. By collecting these functions for several time points (e.g. days) or for different underlyings, a bunch (sample) of functions is obtained - a functional data set. The first topic considered in this thesis concerns the strategies of recovering the functional objects (e.g. implied volatilities function) from the observed data based on the nonparametric smoothing methods. Besides the standard smoothing methods, a procedure based on a combination of nonparametric smoothing and the no-arbitrage-theory results is proposed for implied volatility smoothing. The second part of the thesis is devoted to the functional data analysis (FDA) and its connection to the problems present in the empirical analysis of the financial markets. The theoretical part of the thesis focuses on the functional principal components analysis -- functional counterpart of the well known multivariate dimension-reduction-technique. A comprehensive overview of the existing methods is given, an estimation method based on the dual problem as well as the two-sample inference based on the functional principal component analysis are discussed. The FDA techniques are applied to the analysis of the implied volatility and yield curve dynamics. In addition, the implementation of the FDA techniques together with a FDA library for the statistical environment XploRe are presented.

Seit der Entdeckung der arbitragefreien Bewertung hat sich das Gebiet finance grundlegend geändert - sowohl in der Theorie als auch in der Anwendung. Märkte für Derivate haben sich entwickelt und Optionen dienen heutzutage als Basis- und als Absicherungsinstrumente. In dieser Dissertation betrachten wir einige Märkte für Aktienderivate. Wir beginnen mit statistischen Analysen des Marktes für europäische Optionen und des Marktes für Varianzswaps, weil diese Produkte die hauptsächlichen Absicherungsinstrumente für komplexe Optionen sind. Dann betrachten wir verschiedene Optionspreismodelle und ihre Kalibrierung an beobachtete Preisoberflächen. Schließlich untersuchen wir die Verbindung zwischen Optionspreisen und dem grundlegenden ökonomischen Konzept der Risikoaversion anhand des empirischen Preiskernes. / Since the ideas of arbitrage free pricing were born, finance has changed radically - both in theory and practice. Derivatives markets have evolved and options serve nowadays as underlyings and as hedging instruments. In this thesis, we consider some markets for equity derivatives. We start by statistical analysis of the markets for European options and variance swaps because these products are important for hedging more complex claims. Then we consider different option pricing models and their calibration to observed price surfaces. Finally, we investigate the connection between option prices and the fundamental economic concept of risk aversion by the empirical pricing kernel.

During the last few years, financial derivatives have been growing in trading volume. There seem to be a high demand and supply of derivatives on the market and one common derivative is the option contract. The option contract is frequently the subject of studies and many different pricing models have been created for options. One popular model is the Black-Scholes model, that prices European call options. This project will look closer at the Black-Scholes model and its assumption that volatility remains constant. The project will try to establish what predictive power the implied volatility has for the sign of the price changes in the option’s daily closing price. The implied volatility is defined as the value of volatility that can be used in an option pricing formula to yield the current market price and goes against the assumption of constant volatility. The model consists of a dependent variable that is the binary variable for the sign of the price changes, 1 if positive and 0 if negative. The independent variables are implied volatility, volume, price of the underlying, and VIX. The models used for testing are logistic regression, CART, random forest and XGBoost. The research question is: Can the sign of option price jumps be predicted with the implied volatility? The answer to the research question is that there are indications for the implied volatility having predictive power when predicting the sign of the price changes in the option, even though the results are not conclusive across all models. / Under de senaste åren har finansiella derivat ökat i handelsvolym. Det verkar finnas en hög efterfrågan och tillgång på derivat generellt på marknaden och ett vanligt sådant derivat är optionskontraktet. Optioner är ofta föremål för forskning och många olika prissättningsmodeller har skapats för optioner. En populär modell är Black-Scholes modellen som prissätter europeiska köpoptioner. Detta projekt kommer att titta närmare på Black-Scholes modellen och dess antagande om att volatiliteten förblir konstant. Projektet kommer att försöka fastställa vilken prediktiv kraft den implicita volatiliteten har för tecknet på prisförändringarna i optionens dagliga stängningskurs. Den implicita volatiliteten definieras som värdet av volatiliteten som kan användas i en optionsprissättningsformel för att ge det aktuella marknadspriset och går emot antagandet om konstant volatilitet. Modellen består av en beroende variabel som är en binär variabel för tecknet på prisförändringarna, 1 om positivt och 0 om negativt. De oberoende variablerna är implicit volatilitet, volym, pris på det underliggande instrumentet och VIX. Modellerna som används för att genomföra testen är logistisk regression, CART, random forest och XGBoost. Projektets frågeställning är: Kan tecknet på en options prisförändringar förutsägas med den implicita volatiliteten? Svaret som projektet kommit fram till är att det finns indikationer på att den implicita volatiliteten har prediktiv kraft när det gäller att förutsäga tecknet på prisförändringarna i optionen, även om resultaten inte är helt överensstämmande för alla modeller.

本文提出一個動態隱含波動度函數模型，以改善一般隱含波動度函數難以隨時間的經過而調整波動度曲線且無法描述資料的時間序列特性等缺點。本文模型為兩階段隱含波動度函數模型，分別配適隱含波動度函數的時間穩定(time-invariant)部分與時間不穩定(time-variant)部分。 本文模型在波動度的時間不穩定部分配適非對稱GARCH(1,1)過程，以描述隱含波動度的時間序列特性。本文使用的非對稱GARCH(1,1)過程將標的資產的正報酬與負報酬對價平隱含波動度的影響分別估計，並將蘊含於歷史價平隱含波動度中的訊息及標的資產報酬率與波動度之間的關連性藉由價平隱含波動度過程納入隱含波動度函數中，使隱含波動度函數能納入波動度的時間序列特性及資產報酬與波動度的相關性，藉此納入最近期的市場資訊，以增加隱含波動度模型的解釋及預測能力。時間穩定部分則根據Pena et al.(1999)的研究結果，取不對稱二次函數形式以配適實證上發現的笑狀波幅現象。時間穩定部分並導入相對價內外程度做為變數，以之描述價內外程度、距到期時間、及價平隱含波動度三者的交互關係；並以相對隱含波動度作為被解釋變數，使隱含波動度函數模型除理論上包含了比先前文獻提出的模型更多的訊息及彈性外，還能描繪「隱含波動度函數隨波動度的高低水準而變動」、「越接近到期日，隱含波動度對價內外程度的曲線越彎曲」、「隱含波動度函數為非對稱的曲線」、「波動度和資產價格有很高的相關性」等實證上常發現的現象。 本文以統計測度及交易策略之獲利能力檢定模型的解釋能力及預測能力是否具有統計與經濟上的顯著性。本文歸納之前文獻提出的不同隱含波動度函數模型，並以之與本文提出的模型做比較。本文以台指選擇權五分鐘交易頻率的成交價作為實證標的，以2003年1月1日~2006年12月31日作為樣本期間，並將模型解釋力及AIC作為模型樣本內配適能力之比較標準，我們發現本文提出的模型具有最佳的資料解釋能力。本文以2006年7月1日~2006年12月31日作為隱含波動度模型預測期間，以統計誤差及delta投資策略檢定模型的預測能力是否具有統計及經濟上的顯著性。實證結果指出，本文提出的模型對於預測下一期的隱含波動度及下一期的選擇權價格，皆有相當良好的表現。關於統計顯著性方面，我們發現本文提出的動態隱含波動度函數模型對於未來的隱含波動度及選擇權價格的預測偏誤約為其他隱含波動度函數模型的五分之一，而預測方向正確頻率亦高於預測錯誤的頻率且超過50%。關於經濟顯著性方面，本文使用delta投資組合進行經濟顯著性檢定，結果發現在不考慮交易成本下，本文提出的模型具有顯著的獲利能力。顯示去除標的資產價格變動對選擇權造成的影響後，選擇權波動度的預測準確性確實能經由delta投資組合捕捉；在考慮交易成本後，各模型皆無法獲得超額報酬。最後，本文提出的動態隱含波動度函數模型在考量非同步交易問題、30分鐘及60分鐘等不同的資料頻率、不同的投資組合交易策略後，整體的結論依然不變。 / This paper proposes a new implied volatility function to facilitate implied volatility forecasting and option pricing. This function specifically takes the time variation in the option implied volatility into account. Our model considers the time-variant part and fits it with an asymmetric GARCH(1,1) model, so that our model contains the information in the returns of spot asset and contains the relationship of the returns and the volatility of spot asset. This function also takes the time invariant in the option implied volatility into account. Our model fits the time invariant part with an asymmetric quadratic functional form to model the smile on the volatility. Our model describes the phenomena often found in the literature, such as the implied volatility level increases as time to maturity decreases, the curvature of the dependence of implied volatility on moneyness increases as options near maturity, the implied volatility curve changes as the volatility level changes, and the implied volatility function is an asymmetric curve. For the empirical results, we used a sample of 5 minutes transaction prices for Taiwan stock index options. For the in-sample period January 1, 2003–June 30, 2006, our model has the highest adjusted- and lowest AIC. For the out-of-sample period July 1, 2006–December 31, 2006, the statistical significance shows that our model substantially improves the forecasting ability and reduces the out-of-sample valuation errors in comparison with previous implied volatility functions. We conjecture that such good performance may be due to the ability of the GARCH model to simultaneously capture the correlation of volatility with spot returns and the path dependence in volatility. To test the economic significance of our model, we examine the profitability of the delta-hedged trading strategy based on various volatility models. We find that although these strategies are able to generate profits without transaction costs, their profits disappear quickly when the transaction costs are taken into consideration. Our conclusions were unchanged when we considered the non-synchronization problem or when we test various data frequency and different strategies.

Apresentamos duas abordagens para obter uma densidade de probabilidades para o preço futuro de um ativo: uma densidade preditiva, baseada em um modelo Bayesiano para série de tempo e uma densidade implícita, baseada na fórmula de precificação de opções de Black & Scholes. Considerando o modelo de Black & Scholes, derivamos as condições necessárias para obter a densidade implícita do preço do ativo na data de vencimento. Baseando-­se nas densidades de previsão, comparamos o modelo implícito com a abordagem histórica do modelo Bayesiano. A partir destas densidades, calculamos probabilidades de ordem e tomamos decisões de vender/comprar um ativo. Como exemplo, apresentamos como utilizar estas distribuições para construir uma fórmula de precificação. / We present two different approaches to obtain a probability density function for the stocks future price: a predictive distribution, based on a Bayesian time series model, and the implied distribution, based on Black & Scholes option pricing formula. Considering the Black & Scholes model, we derive the necessary conditions to obtain the implied distribution of the stock price on the exercise date. Based on predictive densities, we compare the market implied model (Black & Scholes) with a historical based approach (Bayesian time series model). After obtaining the density functions, it is simple to evaluate probabilities of one being bigger than the other and to make a decision of selling/buying a stock. Also, as an example, we present how to use these distributions to build an option pricing formula.

This report focus upon implied volatility for American styled Asian options, and a least squares approximation method as a way of estimating its magnitude. Asian option prices are calculated/approximated based on Quasi-Monte Carlo simulations and least squares regression, where a known volatility is being used as input. A regression tree then empirically builds a database of regression vectors for the implied volatility based on the simulated output of option prices. The mean squared errors between imputed and estimated volatilities are then compared using a five-folded cross-validation test as well as the non-parametric Kruskal-Wallis hypothesis test of equal distributions. The study results in a proposed semi-parametric model for estimating implied volatilities from options. The user must however be aware of that this model may suffer from bias in estimation, and should thereby be used with caution.

This thesis consists of a summary and five papers, dealing with financial applications of optimal stopping, optimal control and volatility. In Paper I, we present a method to recover a time-independent piecewise constant volatility from a finite set of perpetual American put option prices. In Paper II, we study the optimal liquidation problem under the assumption that the asset price follows a geometric Brownian motion with unknown drift, which takes one of two given values. The optimal strategy is to liquidate the first time the asset price falls below a monotonically increasing, continuous time-dependent boundary. In Paper III, we investigate the optimal liquidation problem under the assumption that the asset price follows a jump-diffusion with unknown intensity, which takes one of two given values. The best liquidation strategy is to sell the asset the first time the jump process falls below or goes above a monotone time-dependent boundary. Paper IV treats the optimal dividend problem in a model allowing for positive jumps of the underlying firm value. The optimal dividend strategy is of barrier type, i.e. to pay out all surplus above a certain level as dividends, and then pay nothing as long as the firm value is below this level. Finally, in Paper V it is shown that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Lévy models is the existence of jumps towards the strike price in the underlying process.