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The 2008 financial crisis provides a valuable opportunity to study empirical data of market volatility during severe financial crisis. In this thesis, we study the implied volatility of VIX options during the crisis (2008) and a relatively calm period (2011). We present a method of calculating the implied volatility of VIX options and fit the implied volatilities using a 4th degree spline interpolation and propose method of extracting risk neutral density from fitted data. We analyze the slope and the level of the fitted implied volatility of VIX options during those periods. The results show that the level of the implied volatility of VIX options is higher and the slope is flatter during the distressed market compared to the relative calm periods.

Yes / This paper studies the predictive ability of corridor implied volatility (CIV) measure. It is motivated by the fact that CIV is measured with better precision and reliability than the model-free implied volatility due to the lack of liquid options in the tails of the risk-neutral distribution. By adding CIV measures to the modified GARCH specifications, the out-of-sample predictive ability of CIV is measured by the forecast accuracy of conditional volatility. It finds that the narrowest CIV measure, covering about 10% of the RND, dominate the 1-day ahead conditional volatility forecasts regardless of the choice of GARCH models in high volatile period; as market moves to non volatile periods, the optimal width broadens. For multi-day ahead forecasts narrow and mid-range CIV measures are favoured in the full sample and high volatile period for all forecast horizons, depending on which loss functions are used; whereas in non turbulent markets, certain mid-range CIV measures are favoured, for rare instances, wide CIV measures dominate the performance. Regarding the comparisons between best performed CIV measures and two benchmark measures (market volatility index and at-the-money Black–Scholes implied volatility), it shows that under the EGARCH framework, none of the benchmark measures are found to outperform best performed CIV measures, whereas under the GARCH and NAGARCH models, best performed CIV measures are outperformed by benchmark measures for certain instances.

This thesis investigates implied volatility in general classes of stock price models. To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage. We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality, assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call surface satisfies the no-arbitrage bounds (S-K)+≤ C(K, τ)≤ S. We used S to denote the current stock price, K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry. We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in a variety of models. We consider exponential L??vy models, obtaining new and somewhat surprising results. We then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case. Our results generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local volatility models using classical results of Varadhan. In obtaining the asymptotics for local volatility models, we use a representation of the European call as an integral over time to expiry. We devote an entire chapter to representations of the European call option; a key role is played by local time and the argument of Klebaner. A novel alternative that is especially useful in the local volatility case is also presented.

This thesis investigates implied volatility in general classes of stock price models. To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage. We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality, assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call surface satisfies the no-arbitrage bounds (S-K)+≤ C(K, τ)≤ S. We used S to denote the current stock price, K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry. We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in a variety of models. We consider exponential L??vy models, obtaining new and somewhat surprising results. We then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case. Our results generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local volatility models using classical results of Varadhan. In obtaining the asymptotics for local volatility models, we use a representation of the European call as an integral over time to expiry. We devote an entire chapter to representations of the European call option; a key role is played by local time and the argument of Klebaner. A novel alternative that is especially useful in the local volatility case is also presented.

This thesis tests the hypothesis that the qualities of different forecasts of exchange rate volatility depend on the underlying exchange rate regime. By examining the British pound during and after its withdrawal from the European Monetary System (EMS), this analysis compares "backward-looking" historical forecasts of future volatility with the "forward-looking" forecast of volatility reflected in current option prices. Because option implied volatility contains the market's most current expectations about future prices, theory and much previous evidence suggests this should be the superior predictor of future volatility. In contrast to previous research by findings, this study concludes that option implied volatility is not superior. During the time when the pound was in the EMS, implied volatility provided reasonably good forecasts of future volatility. However, after the pound withdrew from the EMS, various statistical measures of historical volatility are found to have greater informational content and predictive power about future actual volatility than implied volatility. In particular, a time series estimate, specifically a GARCH(1,1) model, had the most informational content and predictive power about realized pound volatility, especially in the period following sterling's withdrawal from the EMS. / Master of Arts

Doctor of Philosophy / Department of Agricultural Economics / Ted C. Schroeder / Risk is an inevitable part of agricultural production and all producers face various forms of risk. Output price has been shown to be the major contributor to the risk in cattle feeding, yet few choose to manage this risk. This study used subjective price expectations and price distributions of survey participants to determine how producer's expectations compare with that of the market. In addition, demographic information gathered from survey participants allowed for further examination as to how these factors effect price outlook and variability. Data used for this study were gathered through survey responses from Kansas State University Extension meeting and workshop participants and other meetings targeted to livestock producers. First, data were aggregated and analyzed at a group level. Only two of the twelve price forecast were significantly lower than the futures settlement price. On the other hand, all but one of the aggregated group volatility expectations was different. Typically nearby contract price risk expectation was underestimated and distant contract price risk expectation was overestimated. Individual respondent's discreet stated price and price distribution information was fitted to a continuous distribution and an implied mean and standard deviation were determined. These were compared to market price and price risk data. Respondent's expectation of price was significantly lower than the market for distant months for five of the six groups. Individual volatilities resulting from each fitted distribution were significantly lower from the volatility measure resulting from Black's model. Demographic data were estimated to show the impact of this information on overall error of price forecast and price risk expectations. Those living outside the Northeast and Northern Plains tended to have larger error in their expectation of price volatility. Larger backgrounding operations reported lower price variance error and selling more fed cattle each year increased price risk expectation error. Lastly, prior use of risk management tools for the most part did not have an impact on error in either price expectation or price volatility expectation.

The double-mean-reverting model by Gatheral is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci, we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

This thesis aims to back-test the ability of implied volatility carry trade strategies to outperform the carry trade strategies in the FX markets. Recent research has shown that the profitability of the strategies is partly attributable to the market mispricings of the forward volatility agreements and a tendency of the forward implied volatility to overestimate the future spot implied volatility. This thesis uses a similar approach to construct portfolios containing 10 developed as well as 9 emerging market currencies. Our approach is based on the assumption that Uncovered Interest rate Parity (UIP), Forward Unbiasedness Hypothesis (FUH) and Forward Volatility Unbiasedness Hypothesis (FVUH) do not hold and therefore providing investors with several opportunities to construct trading strategies taking advantage of these market mispricings. In this thesis, we show that the foreign exchange carry trade strategy composed of the specific developed and emerging country's currencies can be outperformed by portfolio consisting of the implied volatility carry trade strategies in the FX market over the analysed period. The portfolios are adjusted to the riskiness which is accounted for by the VIX and VXY-G7 index for developed and VIX and VXY-EM index for emerging economies. The strong performance of the strategies outlined in this thesis can be of significant value to FX traders and portfolio managers.

Problem: Investors are dependent on the issuers’ valuation of covered warrants because the issuers also act as market makers. Hence it is crucial that the issuers value each of the five variables used in the Black & Scholes pricing formula in the same way at both the buying and selling occasion. For a covered warrant investor the most important is-sue is the volatility and how it changes over time. This thesis will therefore search for differences in changes of implied volatility between the different issuers. Purpose: The purpose of this thesis is to analyze differences and similarities between the issuers’ changes of their covered warrants implied volatility. Method: The authors have calculated the implied volatility for a sample of warrants with H&M and Ericsson as underlying assets. Black & Scholes formula has been used and this part of the thesis is made with a quantitative approach. After the implied volatility had been calculated correlation tests to the mean as well as to the stock were made. When analyzing the results the authors, in addition to the calculation, used a qualitative method by interviewing market makers. This was made in order to find better explanations to the results. Conclusions: The differences in changes of implied volatility found between different warrants were small. In general, one warrant changed in the same way as the other ones from one day to another. These results reject the rumors that single issuers adjust their implied volatility in order to make more money. When single events in form of reports were analyzed, the authors found that the issuers changed their volatility in the same way to adjust for the changed uncertainty about the stocks future price. Further, these events clarifies that the basic dynamics of implied volatility is followed by the market. The analysis of how the implied volatility changes with respect to the stock price movements indicates a negative correlation. This implies that an increase in the stock price will lower the implied volatility and vice verse.