[pt] Na tentativa de melhor capturar fatos estilizados do comportamento dos preços de opções financeiras, em especial para tratar a questão do sorriso da volatilidade, modelos de volatilidade estocástica têm sido objeto de estudo em diversos mercados. Neste contexto, o principal objetivo deste trabalho é avaliar os modelos de volatilidade estocástica de Heston (1993), Bates (1996) e Double Heston (2009) junto ao método de Lewis (2000) para precificar opções de ações no mercado brasileiro de derivativos, caracterizados por serem de curto prazo. Para isto foram precificadas opções de compra da Petrobrás e Vale. Os modelos foram comparados de acordo com a qualidade do ajuste aos dados in-sample e a capacidade preditiva com dados out-of-sample. Ademais, buscou-se verificar a volatilidade implícita gerada por cada um dos modelos. Ao fim, identificou-se que considerar a volatilidade como estocástica, mesmo quando é descrita por apenas um processo estocástico, é a decisão mais importante a ser tomada a fim de melhorar o apreçamento das opções. Além disso, adicionar saltos a um modelo de volatilidade estocástica parece ser mais relevante do que adicionar um segundo processo estocástico para modelar a volatilidade na precificação de opções de curto prazo. / [en] In an attempt to better capture stylized facts about financial option prices behavior, especially to address the issue of volatility smile, stochastic volatility models have been the object of study in several markets. In this context, the main purpose of this work is to assess the stochastic volatility models of Heston (1993), Bates (1996) and Double Heston (2009) along with the Lewis method (2000) for stock option pricing in Brazilian derivative market, featured by being short-term. Therefore, Petrobrás and Vale s call options were priced. The models were compared according to the in-sample fit skill and the out-of-sample forecasting power. Furthermore, it was verified the implied volatility begot by each model. In the end, it was figured out that consider the volatility as stochastic even when it is described by only one stochastic process is the preeminent matter to do in order to improve option pricing. Plus, adding jumps in a stochastic volatility model seems to be more important than adding a second stochastic process to model the volatility in short-term option pricing.

In dieser These untersuche ich die Wirkungsmechanismen stochastischer Volatilität in einem neoklassischem Wachstumsmodel mit Arbeitsmarktfriktionen, Anpassungskosten, variabler Kapitalintensität und kurzfristigen Einkommenseffekt. Nominale Rigiditäten werden in diesem Modell nicht betrachtet. Im gegebenen allgemeinen Gleichgewicht generiert stochastische Volatilität Konjunkturzyklen in den wesentlichen makroökonomischen Aggregaten. Dies ist das Resultat eines vorbeugenden Sparmotives der risiko-aversen Haushalte, dennoch sind die quantitativen Effekten auf die unbedingten Momente der makroökonomischen Aggregate vernachlässigbar. / In this thesis I examine the propagation mechanism of stochastic volatility in a neoclassical growth model that incorporates labor market search, adjustment cost to investment, variable capital utilization and a weak short-run wealth effect, but no nominal frictions such as sticky wage and price. In this general equilibrium environment, stochastic volatility generates business cycle fluctuations in major macroeconomic aggregates due to the precautionary motive of risk-averse agents, yet it has no significant effects on these major aggregates as suggested by the numerical analysis of the model.

<p>The present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility &ldquo / smile&rdquo / curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant.</p>

This thesis consists of four papers dealing with the futures price process. In the first paper, we propose a two-factor futures volatility model designed for the US natural gas market, but applicable to any futures market where volatility decreases with maturity and varies with the seasons. A closed form analytical expression for European call options is derived within the model and used to calibrate the model to implied market volatilities. The result is used to price swaptions and calendar spread options on the futures curve. In the second paper, a financial market is specified where the underlying asset is driven by a d-dimensional Wiener process and an M dimensional Markov process. On this market, we provide necessary and, in the time homogenous case, sufficient conditions for the futures price to possess a semi-affine term structure. Next, the case when the Markov process is unobservable is considered. We show that the pricing problem in this setting can be viewed as a filtering problem, and we present explicit solutions for futures. Finally, we present explicit solutions for options on futures both in the observable and unobservable case. The third paper is an empirical study of the SABR model, one of the latest contributions to the field of stochastic volatility models. By Monte Carlo simulation we test the accuracy of the approximation the model relies on, and we investigate the stability of the parameters involved. Further, the model is calibrated to market implied volatility, and its dynamic performance is tested. In the fourth paper, co-authored with Tomas Björk and Camilla Landén, we consider HJM type models for the term structure of futures prices, where the volatility is allowed to be an arbitrary smooth functional of the present futures price curve. Using a Lie algebraic approach we investigate when the infinite dimensional futures price process can be realized by a finite dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite dimensional realization. We study a number of concrete applications including the model developed in the first paper of this thesis. In particular, we provide necessary and sufficient conditions for when the induced spot price is a Markov process. We prove that the only HJM type futures price models with spot price dependent volatility structures, generically possessing a spot price realization, are the affine ones. These models are thus the only generic spot price models from a futures price term structure point of view. / Diss. Stockholm : Handelshögskolan, 2004

In this study, we show that realized bipower variation which is an extension of realized power variation is an alternative method that estimates integrated variance like realized variance. It is seen that realized bipower variation is robust to rare jumps. Robustness means that if we add rare jumps to a stochastic volatility process, realized bipower variation process continues to estimate integrated variance although realized variance estimates integrated variance plus the quadratic variation of the jump component. This robustness is crucial since it separates the discontinuous component of quadratic variation which comes from the jump part of the logarithmic price process. Thus, we demonstrate that if the logarithmic price process is in the class of stochastic volatility plus rare jumps processes then the difference between realized variance and realized bipower variation process estimates the discontinuous component of the quadratic variation. So, quadratic variation of the jump component can be estimated and jump detection can be achieved.

In this thesis, we present two different approaches for the stochastic volatility and stochastic interest rate model with jump and analyze the performance of four alternative models. In the first approach, suggested by Scott, the closed form solution for prices on European call stock options are developed by deriving characteristic functions with the help of martingale methods. Here, we study the asset price process and give in detail the derivation of the European call option price process. The second approach, suggested by Bashki-Cao-Chen, describes the closed form solution of European call option by deriving the partial integro-differential equation. In this one we g ive the derivations of both asset price dynamics and the European call option price process. Finally, in the application part of the thesis, we examine the performance of four alternative models using General Electric Stock Option Data. These models are constructed by using the theoretical results of the second approach.

此研究主要研究壽險業的系統性風險與違約風險之評價，基於投資組合的波動度去建立隨機過程模型。特別是那些隱含無法被多角化的財務風險、系統性風險，透過研究，使用Heston(1993)模型去描述標的資產的隨機波動程度比以往使用Black-Scholes(1973)模型描述股價的波動變化更能反映實際的風險狀況，並透過CIR過程來表示瞬間的波動程度。在這個模型之中，把過去以平賭測度決定違約選擇權的方法延伸。此外透過探討違約價值之敏感度，根據不同的情境測試對於壽險公司負債的影響。最後透過數值的結果與敏感度分析隨機波動模型與確定性的模型之差異。 當資本準備增加時，資產與負債比提高，因負債仍固定承諾予保戶之利率增長，而資產因應系統性風險的發生而減損仍能支付負債，致使違約風險降低，進而使得評價時點的違約金額降低。當系統風險發生時，風險值上升，違約價值為右偏分布，代表在極端條件下有可能有極大的損失；反之，當整個金融體系經濟情勢良好，公司擁有足夠的經濟資本時，風險值下降，滿足VaR75與CTE65的法規限制，此時公司的清償能力足以反映系統性風險。 / This paper considers the problem of valuating the default option of the life insurers that are subject to systematic financial risk in the sense that the volatility of the investment portfolio is modeled through stochastic processes. In particular, this implies that the financial risk cannot be eliminated through diversifying the asset portfolio. In our work, Heston (1993) model is employed in describing the evolution of the volatility of an underlying asset, while the instantaneous variance is a CIR process. Within this model, we study a general set of equivalent martingale measures, and determine the default option by applying these measures. In addition, we investigate the sensitivity of the default values given regulatory forbearance for the life insurance liabilities considered. Numerical examples are included, and the use of the stochastic volatility model is compared with deterministic models. As reserve of capital is increasing, asset-liability ratio is also increasing. The liability grew up with promised interest rate, and it could be covered by the asset when the systematic risk events happened. Therefore, the default risk was decreasing, that caused the default value decreasing. When the systematic risk events happened, the value of risk was increasing, and the default value was positive skew distribution. That means the maximum loss will be coming in the extreme case. On the other hand, when prosperity economy occurred, the value of risk was decreasing, which in compliance with the law of VaR75&CTE65 rules, and the insurance company had enough capital to face the systematic risk events.

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<p>The present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility &ldquo / smile&rdquo / curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant.</p>