Option market data are quoted in terms of option prices and are fragmented into over 100 individual contract files per day for each symbol. Traders and quantitative analysts compare values of options in terms of implied volatilities. The current project refactors fragmented option price data into implied volatility files organized by stock symbols and expiration dates. Each resulting file comprises the temporal evolution of daily volatility smile curves for every day prior to expiration. Possible analysis enabled by the refactored data is demonstrated.

The thesis describes and applies two parametric option pricing models which partially ease the well-known discrepancy between real world and Black-Scholes model. Stochastic volatility and jumps encompassed by Heston and SVJ models explain implied volatility smile and its heterogeneous term-structure. Both models are calibrated to market data observed for EURUSD currency options on January 23, 2015. While SVJ model provided a better fit for the market, especially for mid-term expiry smile curvature, its estimated risk-neutral parameters were unrealistic comparing with their counterparts under statistical measure. Estimations suggest zero long term price volatility and 2 jumps during the year with average magnitude of 6 \%. Both models failed to match curvature of short time to expiry smile and provided a good fit of term-structure and long-expiry smile. Analysing delta ratios adjusted for non-constant volatility as a possible alternatives the study considered minimum variance delta estimated with Heston model, delta ratio recommended by Nassim Taleb and two deltas adjusted for local volatility assuming sticky moneyness and sticky tree dynamics of implied volatility. On data set of EURUSD options from 1.1.2014 to 30.5.2015, our research did not find any alternative which would be more reliable than common Black-Scholes delta.

The Black-Scholes option pricing model has several well recognized deficiencies, one of which is its assumption of a constant and time-homogeneous stock return volatility term. The implied volatility smile has been studied by subsequent researchers and various models have been developed in an attempt to reproduce this phenomenon from within the models. However, few of these models yield closed-form pricing formulas that are easy to implement in practice. In this thesis, we study a Mixture Lognormal model (MLN) for European option pricing, which assumes that future stock prices are conditionally described by a mixture of lognormal distributions. The ability of mixture models in generating volatility smiles as well as delivering pricing improvement over the traditional Black-Scholes framework have been much researched under multi-component mixtures for many derivatives and high-volatility individual stock options. In this thesis, we investigate the performance of the model under the simplest two-component mixture in a market characterized by relative tranquillity and over a relatively stable period for broad-based index options. A careful interpretation is given to the model and the results obtained in the thesis. This di erentiates our study from many previous studies on this subject. Throughout the thesis, we establish the unique advantage of the MLN model, which is having closed-form option pricing formulas equal to the weighted mixture of Black-Scholes option prices. We also propose a robust calibration methodology to fit the model to market data. Extreme market states, in particular the so-called crash-o-phobia effect, are shown to be well captured by the calibrated model, albeit small pricing improvements are made over a relatively stable period of index option market. As a major contribution of this thesis, we extend the MLN model to price exotic options including binary, Asian, and barrier options. Closed-form formulas are derived for binary and continuously monitored barrier options and simulation-based pricing techniques are proposed for Asian and discretely monitored barrier options. Lastly, comparative results are analysed for various strike-maturity combinations, which provides insights into the formulation of hedging and risk management strategies.

Problem: The standard Black-Scholes framework cannot incorporate the volatility smiles usually observed in the markets. Instead, one must consider alternative stochastic volatility models such as the SABR. Little research about the suitability of the SABR model for Swedish market (swaption) data has been found. Purpose: The purpose of this paper is to account for and to calibrate the SABR model for swaptions trading on the Swedish market. We intend to alter the calibration techniques and parameter values to examine which method is the most consistent with the market. Method: In MATLAB, we investigate the model using two different minimization techniques to estimate the model’s parameters. For both techniques, we also implement refinements of the original SABR model. Results and Conclusion: The quality of the fit relies heavily on the underlying data. For the data used, we find superior fit for many different swaption smiles. In addition, little discrepancy in the quality of the fit between methods employed is found. We conclude that estimating the α parameter from at-the-money volatility produces slightly smaller errors than using minimization techniques to estimate all parameters. Using refinement techniques marginally increase the quality of the fit.

The Black-Scholes option pricing model has several well recognized deficiencies, one of which is its assumption of a constant and time-homogeneous stock return volatility term. The implied volatility smile has been studied by subsequent researchers and various models have been developed in an attempt to reproduce this phenomenon from within the models. However, few of these models yield closed-form pricing formulas that are easy to implement in practice. In this thesis, we study a Mixture Lognormal model (MLN) for European option pricing, which assumes that future stock prices are conditionally described by a mixture of lognormal distributions. The ability of mixture models in generating volatility smiles as well as delivering pricing improvement over the traditional Black-Scholes framework have been much researched under multi-component mixtures for many derivatives and high-volatility individual stock options. In this thesis, we investigate the performance of the model under the simplest two-component mixture in a market characterized by relative tranquillity and over a relatively stable period for broad-based index options. A careful interpretation is given to the model and the results obtained in the thesis. This di erentiates our study from many previous studies on this subject. Throughout the thesis, we establish the unique advantage of the MLN model, which is having closed-form option pricing formulas equal to the weighted mixture of Black-Scholes option prices. We also propose a robust calibration methodology to fit the model to market data. Extreme market states, in particular the so-called crash-o-phobia effect, are shown to be well captured by the calibrated model, albeit small pricing improvements are made over a relatively stable period of index option market. As a major contribution of this thesis, we extend the MLN model to price exotic options including binary, Asian, and barrier options. Closed-form formulas are derived for binary and continuously monitored barrier options and simulation-based pricing techniques are proposed for Asian and discretely monitored barrier options. Lastly, comparative results are analysed for various strike-maturity combinations, which provides insights into the formulation of hedging and risk management strategies.

För att prissätta optioner är val av modell en viktig fråga. I denna kandidatuppsats beskrivs både Black & Scholes modell och SABR-modellen. Förstnämnda modell är enklare än SABR-modellen men bygger på antaganden som inte stämmer överens med verkligheten. Den ger heller inte någon explicit formel för den implicita volatiliteten och predikterar inte heller på ett korrekt sätt fenomenet volatility smile vilket observeras på marknaden. Syftet med uppsatsen är att utvärdera prestandan hos SABR-modellen och användarvänligheten, samt att undersöka lite av teorin bakom modellen och vissa av dess egenskaper. Till grund för beräkningarna ligger datamaterial hämtat från Nasdaq OMX Nordic. Enligt mina beräkningar är resultatet att SABR-modellen endast presterar marginellt bättre än Black & Scholes-modellen. Dock kan även små förbättringar spela stor roll i dessa sammanhang.

<p>För att prissätta optioner är val av modell en viktig fråga. I denna kandidatuppsats</p><p>beskrivs både Black & Scholes modell och SABR-modellen. Förstnämnda modell är</p><p>enklare än SABR-modellen men bygger på antaganden som inte stämmer överens med</p><p>verkligheten. Den ger heller inte någon explicit formel för den implicita volatiliteten</p><p>och predikterar inte heller på ett korrekt sätt fenomenet volatility smile vilket</p><p>observeras på marknaden.</p><p>Syftet med uppsatsen är att utvärdera prestandan hos SABR-modellen och</p><p>användarvänligheten, samt att undersöka lite av teorin bakom modellen och vissa av</p><p>dess egenskaper. Till grund för beräkningarna ligger datamaterial hämtat från Nasdaq</p><p>OMX Nordic.</p><p>Enligt mina beräkningar är resultatet att SABR-modellen endast presterar marginellt</p><p>bättre än Black & Scholes-modellen. Dock kan även små förbättringar spela stor roll i</p><p>dessa sammanhang.</p>

[pt] A expansão do mercado de derivativos no mundo e principalmente no Brasil tem impulsionado seus usuários a aprimorar e desenvolver ferramentas de apreçamento mais eficientes. Com esse intuito, o presente trabalho tem por objetivo evidenciar qual janela de observações gera a curtose e a assimetria que mais suavize o sorriso da volatilidade utilizando-se do modelo Corrado-Su. Para tanto, as empresas escolhidas foram a Petrobrás PN e a Vale PNA, devido a suas ações e opções de compra serem as mais líquidas no mercado brasileiro. A análise dos dados apontou para uma maior suavização do sorriso da volatilidade por parte das janelas de dados de curto prazo sobre as longo prazo, e uma equivalência de desempenho das primeiras ao do modelo Black-Scholes. / [en] The expansion of the derivatives market in the world and especially in Brazil has driven its users to enhance and develop tools for more efficient pricing. With this purpose, this paper aims to point which window of observations generates the kurtosis and skewness that more soften the volatility smile using the Corrado-Su model. Therefore, the firms that were chosen were Petrobras PN and Vale PNA, because their stocks and options are the most liquid in Brazilian market. The data analysis indicated a greater smoothing volatility smile using the windows of observations of the short term instead of the long term, and a equivalent performance of the first ones to that of the Black-Scholes model.

This article presents a new approach for calculating sensitivities of European swaptions. The sensitivities are found by applying an adjoint method to a stochastic volatility model, namely the SABR/LIBOR market model. This market model predicts the volatility smile and follows the market fluctuations more accurately than earlier used deterministic volatility market models for complex derivatives. The new adjoint method involves not only sensitivity calculations, it also presents a way of estimating the time discretization error using an a posteriori approach. The error calculation is described in this document but not investigated further. The first step in order to calculate the sensitivities is to calibrate the SABR/LIBOR market model to some market data. In our calculations we used data from June 15 2011 with 6 month intervals between the maturity times. When this calibration is complete all of the parameters in the SABR/LIBOR market model are specified and we can continue with the sensitivity calculations using the new adjoint method. The results from these calculations show that the method is a good choice for estimating sensitivities if we consider a complex financial derivative like the European swaption. The method is quite computational so we recommend that it is only used on a small number of securities with respect to a large number of parameters. The method provides more market-driven price and sensitivity estimations than earlier used methods and can benefit hedging of portfolios.

被廣泛應用在選擇權定價的Black-Scholes 模型[3] 時常在深價內與深價外 的選擇權價格有錯價的現象，也就是理論價格估計實際市場價格的偏差。藉由 Black-Scholes 評價公式所反推出的隱含波動度往往不像我們所期待的在不同履約價格具有一致性，這種現象被稱為波動度的微笑曲線。在這份論文裡，我們參考Jarrow and Rudd [13] 提出的方法，將Edgeworth展開式套用在Black-Scholes模型作延伸應用，進而推導出偏態峰態修正後的的評價公式，再利用台指選擇權的市場資料作實證分析並與Filho and Rosenfeld [1] 的研究作比較。我們發現從台指選擇權的實證結果得到非常態分配的隱含偏態和隱含峰態。此外，理論價格的估計偏誤比例顯著的被新的模型改善且隱含波動度的微笑曲線也變的較為平坦，這個方法提供我們一個有效的方法，利用標的資產的偏態峰態得到該資產的近似分配。 / The Black-Scholes [3] option pricing model widely applied in option contracts frequently misprices deep-in-the-money and deep-out-of-the-money options. The implied volatilities computed by the Black-Scholes formula are not identical on each strike price as we expect. This phenomenon is called the volatility smile or skew. In this thesis, we derived a skewness- and kurtosis-adjusted option pricing model using an Edgeworth expansion constructed by Jarrow and Rudd [13] to an investigation of TAIEX option prices and compare the results with those in Filho and Rosenfeld [1]. We found that non-normal skewness and kurtosis are implied by TAIEX option returns. Moreover, the magnitude of price deviations were signicantly corrected and the volatility skew is attened. This approach provides an useful way to derive an approximate distribution of a underlying security with its skewness and kurtosis.