This thesis examines the empirical performance of four Affine Jump Diffusion models in pricing and hedging S&P 500 Index options: the Black Scholes (BS) model, Heston???s Stochastic Volatility (SV) model, a Stochastic Volatility Price Jump (SVJ) model and a Stochastic Volatility Price-Volatility Jump (SVJJ) model. The SVJJ model structure allows for simultaneous jumps in price and volatility processes, with correlated jump size distributions. To the best of our knowledge this is the first empirical study to test the hedging performance of the SVJJ model. As part of our research we derive the SVJJ model minimum variance hedge ratio. We find the SVJ model displays the best price prediction. The SV model lacks the structural complexity to eliminate Black Scholes pricing biases, whereas our results indicate the SVJJ model suffers from overfitting. Despite significant evidence from in and out-of-sample pricing that the SV and SVJ models were better specified than the BS model, this did not result in an improvement in dynamic hedging performance. Overall the BS delta hedge and SV minimum variance hedge produced the lowest errors, although their performance across moneyness-maturity categories differed greatly. The SVJ model???s results were surprisingly poor given its superior performance in out-of-sample pricing. We attribute the inadequate performance of the jump models to the lower hedging ratios these models provided, which may be a result of the negative expected jump sizes.

Diss. Stockholm : Handelshögskolan, 2003.

Dynamic covariance estimation for multivariate time series suffers from the curse of dimensionality. This renders parsimonious estimation methods essential for conducting reliable statistical inference. In this paper, the issue is addressed by modeling the underlying co-volatility dynamics of a time series vector through a lower dimensional collection of latent time-varying stochastic factors. Furthermore, we apply a Normal-Gamma prior to the elements of the factor loadings matrix. This hierarchical shrinkage prior effectively pulls the factor loadings of unimportant factors towards zero, thereby increasing parsimony even more. We apply the model to simulated data as well as daily log-returns of 300 S&P 500 stocks and demonstrate the effectiveness of the shrinkage prior to obtain sparse loadings matrices and more precise correlation estimates. Moreover, we investigate predictive performance and discuss different choices for the number of latent factors. Additionally to being a stand-alone tool, the algorithm is designed to act as a "plug and play" extension for other MCMC samplers; it is implemented in the R package factorstochvol. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

In this paper, we provide a parsimonious means of estimating panel VARs with stochastic volatility. We assume that coefficients associated with domestic lagged endogenous variables arise from a finite mixture of Gaussian distribution. Shrinkage on the cluster size is introduced through suitable priors on the component weights and cluster-relevant quantities are identified through novel normal-gamma shrinkage priors. To assess whether dynamic interdependencies between units are needed, we moreover impose shrinkage priors on the coefficients related to other countries' endogenous variables. Finally, our model controls for static interdependencies by assuming that the reduced form shocks of the model feature a factor stochastic volatility structure. We assess the merits of the proposed approach by using synthetic data as well as a real data application. In the empirical application, we forecast Eurozone unemployment rates and show that our proposed approach works well in terms of predictions. / Series: Department of Economics Working Paper Series

Due to recent research disproving old claims in financial mathematics such as constant volatility in option prices, new approaches have been incurred to analyze the implied volatility, namely stochastic volatility models. The use of stochastic volatility in option pricing is a relatively new and unexplored field of research with a lot of unknowns, where new answers are of great interest to anyone practicing valuation of derivative instruments such as options. With both single and two-factor stochastic volatility models containing various correlation structures with respect to the asset price and differing mean-reversions of variance the question arises as to how these values change their more observable counterpart: the implied volatility. Using the semi-analytical formula derived by Chiarella and Ziveyi, we compute European call option prices. Then, through the Black–Scholes formula, we solve for the implied volatility by applying the bisection method. The implied volatilities obtained are then approximated using various models of regression where the models’ coefficients are determined through the Moore–Penrose pseudo-inverse to produce implied volatility surfaces for each selected pair of correlations and mean-reversion rates. Through these methods we discover that for different mean-reversions and correlations the overall implied volatility varies significantly and the relationship between the strike price, time to maturity, implied volatility are transformed.

Cette thèse explore théoriquement et empiriquement les implications de la dynamique jointe action/option sur divers problématiques liées au trading d’options. Dans un premier temps, nous commençons par l’étude de la dynamique jointe entre une option sur un stock et une option sur l’indice de marché. Le modèle CAPM fournit un cadre mathématique adéquat pour cette étude car il permet de modéliser la dynamique jointe d’un stock et son indice de marché. En passant aux prix d’options, nous montrons que le beta et la volatilité idiosyncratique, paramètres du modèle, permettent de caractériser la relation entre les surfaces de volatilité implicite du stock et de l’indice. Nous nous penchons alors sur l’estimation du paramètre beta sous la probabilité risque-neutre en utilisant les prix d’options. Cette mesure, appelée beta implicite, représente l’information contenue dans les prix d’options sur la réalisation du paramètre beta dans le futur.Pour cette raison, nous essayons de voir, si le beta implicite a un pouvoir prédictif du beta futur.En menant une étude empirique, nous concluons que le beta implicite n’améliore pas la capacité de prédiction en comparaison avec le beta historique qui est calculé à travers la régression linéaire des rendements du stock sur ceux de l’indice. Mieux encore, nous remarquons que l’oscillation du beta implicite autour du beta futur peut entraîner des opportunités d’arbitrage, et nous proposons une stratégie d’arbitrage qui permet de monétiser cet écart. D’un autre côté, nous montrons que l’estimateur du beta implicite pourrait être utilisé pour la couverture d’options sur le stock en utilisant des instruments sur l’indice, cette couverture concerne notamment le risque de volatilité et aussi le risque de delta. Dans la deuxième partie de notre travail, nous nous intéressons au problème de market making sur options. Dans cette étude, nous supposons que le modèle de dynamique du sous-jacent sous la probabilité risque-neutre pourrait être mal spécifié ce qui traduit un décalage entre la distribution implicite du sous-jacent et sa distribution historique.Dans un premier temps, nous considérons le cas d’un market maker risque neutre qui vise à maximiser l’espérance de sa richesse future. A travers l’utilisation d’une approche de contrôle optimal stochastique, nous déterminons les prix optimaux d’achat et de vente sur l’option et nous interprétons l’effet de présence d’inefficience de prix sur la stratégie optimale. Dans un deuxième temps, nous considérons que le market maker est averse au risque et essaie donc de réduire l’incertitude liée à son inventaire. En résolvant un problème d’optimisation basé sur un critère moyenne-variance, nous obtenons des approximations analytiques des prix optimaux d’achat et de vente. Nous montrons aussi les effets de l’inventaire et de l’inefficience du prix sur la stratégie optimale. Nous nous intéressons par la suite au market making d’options dans une dimension plus élevée. Ainsi, en suivant le même raisonnement, nous présentons un cadre pour le market making de deux options ayant des sous-jacents différents avec comme contrainte la réduction de variance liée au risque d’inventaire détenu par le market-maker. Nous déterminons dans ce cas la stratégie optimale et nous appuyons les résultats théoriques par des simulations numériques.Dans la dernière partie de notre travail, nous étudions la dynamique jointe entre la volatilité implicite à la monnaie et le sous jacent, et nous essayons d’établir le lien entre cette dynamique jointe et le skew implicite. Nous nous intéressons à un indicateur appelé "Skew Stickiness Ratio"qui a été introduit dans la littérature récente. Cet indicateur mesure la sensibilité de la volatilité implicite à la monnaie face aux mouvements du sous-jacent. Nous proposons une méthode qui permet d’estimer la valeur de cet indicateur sous la probabilité risque-neutre sans avoir besoin d’admettre des hypothèses sur la dynamique du sous-jacent. [...] / This thesis explores theoretically and empirically the implications of the stock/option joint dynamics on applications related to option trading. In the first part of the thesis, we look into the relations between stock options and index options under the risk-neutral measure. The Capital Asset Pricing Model offers an adequate mathematical framework for this study as it provides a modeling approach for the joint dynamics between the stock and the index. As we compute option prices according to this model, we find out that the beta and the idiosyncratic volatility of the stock, which are parameters of the model, characterize the relation between the implied volatility surface of the stock and the one of the index. For this reason, we focus on the estimation of the parameter beta under the risk-neutral measure through the use of option prices.This measure, that we call implied beta, is the information contained in option prices concerning the realization of the parameter beta in the future. Trying to use this additional information, we carry out an empirical study in order to investigate whether the implied beta has a predictive power of the forward realized beta. We conclude that the implied beta doesn’t perform better than the historical beta which is estimated using the linear regression of the stock’s returns onthe index returns. We conclude also that the oscillation of the implied beta around the forward realized beta can engender arbitrage opportunities, and we propose an arbitrage strategy which enables to monetize this difference. In addition, we show that the implied beta is useful to hedge stock options using instruments on the index. In the second part of our work, we consider the problem of option market making. We suppose that the model used to describe the dynamics of the underlying under the risk-neutral probability measure can be misspecified which means thatthe implied distribution of the underlying may be different from its historical one. We consider first the case of a risk neutral market maker who aims to maximize the expectation of her final wealth. Using a stochastic control approach, we determine the optimal bid and ask prices on the option and we interpret the effect of price inefficiency on the optimal strategy. Next to that, we suppose that the market maker is risk averse as she tries to minimize the variance of her finalwealth. We solve a mean-variance optimization problem and we provide analytic approximations for the optimal bid and ask prices. We show the effects of option inventory and price inefficiency on the optimal strategy. We try then to extrapolate the study to a higher dimension in order to see the effect of joint dynamics of the different underlyings on the optimal strategy. Thus, we study market making strategies on a pair of options having different underlyings with the aim to reduce the risk due to accumulated inventories in these two options. Through the resolution of the HJB equation associated to the new optimization problem, we determine the optimal strategy and we support our theoretical finding with numerical simulations. In the final part of the thesis, we study the joint dynamics of the at-the-money implied volatility and the spot process. We try to establish a relation between this joint dynamics and the implied skew through the use of a quantity called the Skew Stickiness Ratio which was introduced in the recent literature. The Skew Stickiness Ratio quantifies the effect of the log-return of the spot on the increment of theat-the-money volatility. We suggest a model-free approach for the estimation of the SSR (Skew Stickiness Ratio) under the risk-neutral measure, this approach doesn’t depend on hypothesis on the dynamics of the underlying. [...]

We analyze the relation between volatility spillovers and jumps in financial markets. For this, we compared the volatility spillover index proposed by Diebold and Yilmaz (2009) with a global volatility component, estimated through a multivariate stochastic volatility model with jumps in the mean and in the conditional volatility. This model allows a direct dating of events that alter the global volatility structure, based on a permanent/transitory decomposition in the structure of returns and volatilities, and also the estimation of market risk measures. We conclude that the multivariate stochastic volatility model solves some limitations in the spillover index and can be a useful tool in measuring and managing risk in global financial markets. / Analisamos a relação existente entre spillovers e saltos na volatilidade nos mercados financeiros. Para isso, comparamos o índice de spillover de volatilidade proposto por Diebold and Yilmaz (2009), com um componente de volatilidade global, estimado através de um modelo multivariado de volatilidade estocástica com saltos na média e na volatilidade condicional. Este modelo permite uma datação direta dos eventos que alteram a estrutura de volatilidade global, baseando-se na decomposição das estruturas de retorno e volatilidade entre efeitos permanentes/transitórios, como também a estimação de medidas de risco de mercado. Concluímos que este modelo resolve algumas das limitações do índice de spillover além de fornecer um método prático para mensurar e administrar o risco nos mercados financeiros globais.

Submitted by Júlia Fonseca (julia.f.fonseca@gmail.com) on 2013-06-24T17:08:45Z No. of bitstreams: 1 Julia Fonseca.pdf: 901363 bytes, checksum: 4df97de4c5783eb1e3f3acea7c9e8e25 (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2013-06-27T12:57:48Z (GMT) No. of bitstreams: 1 Julia Fonseca.pdf: 901363 bytes, checksum: 4df97de4c5783eb1e3f3acea7c9e8e25 (MD5) / Made available in DSpace on 2013-06-27T12:58:06Z (GMT). No. of bitstreams: 1 Julia Fonseca.pdf: 901363 bytes, checksum: 4df97de4c5783eb1e3f3acea7c9e8e25 (MD5) Previous issue date: 2013-06-17 / We investigate the eff ect of aggregate uncertainty shocks on real variables. More speci fically, we introduce a shock in the volatility of productivity in an RBC model with long-run volatility risk and preferences that exhibit generalised disappointment aversion. We find that, when combined with a negative productivity shock, a volatility shock leads to further decline in real variables, such as output, consumption, hours worked and investment. For instance, out of the 2% decrease in output as a result of both shocks, we attribute 0.25% to the e ffect of an increase in volatility. We also fi nd that this e ffect is the same as the one obtained in a model with Epstein-Zin- Weil preferences, but higher than that of a model with expected utility. Moreover, GDA preferences yield superior asset pricing results, when compared to both Epstein-Zin-Weil preferences and expected utility.

Can we capture the explosive nature of volatility skew observed in the market, without resorting to non-Markovian models? We show that, in terms of skew, the Heston model cannot match the market at both long and short maturities simultaneously. We introduce Abi Jaber (2019)'s Lifted Heston model and explain how to price options with it using both the cosine method and standard Monte-Carlo techniques. This allows us to back out implied volatilities and compute skew for both models, confirming that the Lifted Heston nests the standard Heston model. We then produce and analyze the skew for Lifted Heston models with a varying number N of mean reverting terms, and give an empirical study into the time complexity of increasing N. We observe a weak increase in convergence speed in the cosine method for increased N, and comment on the number of factors to implement for practical use.

In this thesis, we propose a new and simple approach of extending the single-factor Heston stochastic volatility model to a more flexible one in solving option pricing problems.  In this approach, the volatility process for the underlying asset dynamics depends on the time to maturity of the option. As this idea is inspired by the Heath-Jarrow-Morton framework which models the evolution of the full dynamics of forward rate curves for various maturities, we name this approach as the HJM-type stochastic volatility (HJM-SV)  model. We conduct an empirical analysis by calibrating this model to real-market option data for underlying assets including an equity  (ABB stock) and a market index (EURO STOXX 50), for two separated time spans from Jan 2017 to Dec 2017 (before the COVID-19 pandemic) and from Nov 2019 to Nov 2020 (after the start of COVID-19 pandemic). We investigate the optimal way of dividing the set of option maturities into three classes, namely, the short-maturity, middle-maturity, and long-maturity classes. We calibrate our HJM-SV model to the data in the following way, for each class a single-factor Heston stochastic volatility model is calibrated to the corresponding market data. We address the question that how well the new HJM-SV model captures the feature of implied volatility surface given by the market data.