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

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.

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.

Master of Science / Department of Agricultural Economics / Ted C. Schroeder / In recent years, the agricultural markets have been subject to increased prices and unusual levels of elevated volatility. One likely driver of this is the mandated ethanol expansion in the Energy Policy Act of 2005. Previous research has identified relationships in market prices and variability between the energy and grain markets, but little has been done to evaluate volatility spillover across a broader spectrum of agricultural commodities. Additionally, few studies have assessed causal linkages across market implied volatilities. This research examines implied volatility spillover in futures markets across major agricultural commodities and energies. The analysis also determines the time path and magnitude of volatility translation across the markets and compares the causal relationships between pre-ethanol boom and post-ethanol boom time periods. Granger causality tests are conducted using multivariate and bivariate vector autoregressive modeling techniques, and impulse response functions are employed to obtain time paths of the reactions. Overall, results indicate that strong implied volatility spillover relationships exist between the grain markets and between the live cattle and feeder cattle markets. The analysis also finds that the agricultural markets have evolved from lean hogs being the primary volatility leader in the pre-ethanol boom era to corn being the primary volatility leader in the post-ethanol boom era. Despite a high correlation between crude oil and corn volatilities in the post-ethanol boom time period, the causal linkage between the two commodities’ volatilities may not be as definite as other literature suggests.

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.

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<p>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.</p><p>Purpose: The purpose of this thesis is to analyze differences and similarities between the issuers’ changes of their covered warrants implied volatility.</p><p>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.</p><p>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.</p>