Modeling and Forecasting Implied Volatility: Implications for Trading, Pricing, and Risk Management

Modeling and forecasting of implied volatility (IV) is important to both practitioners and academics, especially in trading, pricing, hedging, and risk management activities, all of which require an accurate volatility. However, it has become challenging since the 1987 stock market crash, as implied volatilities (IVs) recovered from stock index options present two patterns: volatility smirk(skew) and volatility term-structure, if the two are examined at the same time, presents a rich implied volatility surface (IVS). This implies that the assumptions behind the Black-Scholes (1973) model do not hold empirically, as asset prices are mostly influenced by many underlying risk factors. This thesis, consists of four essays, is modeling and forecasting implied volatility in the presence of options markets’ empirical regularities. The first essay is modeling the dynamics IVS, it extends the Dumas, Fleming and Whaley (DFW) (1998) framework; for instance, using moneyness in the implied forward price and OTM put-call options on the FTSE100 index, a nonlinear optimization is used to estimate different models and thereby produce rich, smooth IVSs. Here, the constant-volatility model fails to explain the variations in the rich IVS. Next, it is found that three factors can explain about 69-88% of the variance in the IVS. Of this, on average, 56% is explained by the level factor, 15% by the term-structure factor, and the additional 7% by the jump-fear factor. The second essay proposes a quantile regression model for modeling contemporaneous asymmetric return-volatility relationship, which is the generalization of Hibbert et al. (2008) model. The results show strong negative asymmetric return-volatility relationship at various quantiles of IV distributions, it is monotonically increasing when moving from the median quantile to the uppermost quantile (i.e., 95%); therefore, OLS underestimates this relationship at upper quantiles. Additionally, the asymmetric relationship is more pronounced with the smirk (skew) adjusted volatility index measure in comparison to the old volatility index measure. Nonetheless, the volatility indices are ranked in terms of asymmetric volatility as follows: VIX, VSTOXX, VDAX, and VXN. The third essay examines the information content of the new-VDAX volatility index to forecast daily Value-at-Risk (VaR) estimates and compares its VaR forecasts with the forecasts of the Filtered Historical Simulation and RiskMetrics. All daily VaR models are then backtested from 1992-2009 using unconditional, independence, conditional coverage, and quadratic-score tests. It is found that the VDAX subsumes almost all information required for the volatility of daily VaR forecasts for a portfolio of the DAX30 index; implied-VaR models outperform all other VaR models. The fourth essay models the risk factors driving the swaption IVs. It is found that three factors can explain 94-97% of the variation in each of the EUR, USD, and GBP swaption IVs. There are significant linkages across factors, and bi-directional causality is at work between the factors implied by EUR and USD swaption IVs. Furthermore, the factors implied by EUR and USD IVs respond to each others’ shocks; however, surprisingly, GBP does not affect them. Second, the string market model calibration results show it can efficiently reproduce (or forecast) the volatility surface for each of the swaptions markets.

Essays on the Role of Time in Price Discovery (summary section only)

The purpose of this thesis is to examine the role of trade durations in price discovery. The motivation to use trade durations in the study of price discovery is that durations are robust to many microstructure effects that introduce a bias in the measurement of returns volatility. Another motivation to use trade durations in the study of price discovery is that it is difficult to think of economic variables, which really are useful in the determination of the source of volatility at arbitrarily high frequencies. The dissertation contains three essays. In the first essay, the role of trade durations in price discovery is examined with respect to the volatility pattern of stock returns. The theory on volatility is associated with the theory on the information content of trade, dear to the market microstructure theory. The first essay documents that the volatility per transaction is related to the intensity of trade, and a strong relationship between the stochastic process of trade durations and trading variables. In the second essay, the role of trade durations in price discovery is examined with respect to the quantification of risk due to a trading volume of a certain size. The theory on volume is intrinsically associated with the stock volatility pattern. The essay documents that volatility increases, in general, when traders choose to trade with large transactions. In the third essay, the role of trade durations in price discovery is examined with respect to the information content of a trade. The theory on the information content of a trade is associated with the theory on the rate of price revisions in the market. The essay documents that short durations are associated with information. Thus, traders are compensated for responding quickly to information

The expiration day effect of indexoptions and index futures on the underlying shares

This paper studies the effect of the expiration day of index options and futures on the trading volume, variance and price of the underlying shares. The data consists of all trades for the underlying shares in the FOX-index for expiration days during the period October 1995 to the mid of yer 1999. The main results seem to support the findings of Kan 2001, i.e. no manipulation on a larger scale. However, some indication of manipulation could be found if certain characteristics are favorable. These characteristics include: a) a large quantity of outstanding futures or at/in the money options contracts, b) there exists shares with high index weight but fairly low trading volume. Lastly, there is some indication that manipulation might be more popular towards the end of the examined time period.

Intraday and Weekend Volatility Patterns- Implications for Option Pricing

This study examines the intraday and weekend volatility on the German DAX. The intraday volatility is partitioned into smaller intervals and compared to a whole day’s volatility. The estimated intraday variance is U-shaped and the weekend variance is estimated to 19 % of a normal trading day. The patterns in the intraday and weekend volatility are used to develop an extension to the Black and Scholes formula to form a new time basis. Calendar or trading days are commonly used for measuring time in option pricing. The Continuous Time using Discrete Approximations model (CTDA) developed in this study uses a measure of time with smaller intervals, approaching continuous time. The model presented accounts for the lapse of time during trading only. Arbitrage pricing suggests that the option price equals the expected cost of hedging volatility during the option’s remaining life. In this model, time is allowed to lapse as volatility occurs on an intraday basis. The measure of time is modified in CTDA to correct for the non-constant volatility and to account for the patterns in volatility.

Pricing Index Options with Stochastic Volatility

The objective of this paper is to investigate the pricing accuracy under stochastic volatility where the volatility follows a square root process. The theoretical prices are compared with market price data (the German DAX index options market) by using two different techniques of parameter estimation, the method of moments and implicit estimation by inversion. Standard Black & Scholes pricing is used as a benchmark. The results indicate that the stochastic volatility model with parameters estimated by inversion using the available prices on the preceding day, is the most accurate pricing method of the three in this study and can be considered satisfactory. However, as the same model with parameters estimated using a rolling window (the method of moments) proved to be inferior to the benchmark, the importance of stable and correct estimation of the parameters is evident.

The Impact of Emissions Trading on the Price of Electricity in Nord Pool : Market Power and Price Determination in the Nordic Electricity Market

The objective of this thesis is to find out how dominant firms in a liberalised electricity market will react when they face an increase in the level of costs due to emissions trading, and how this will effect the price of electricity. The Nordic electricity market is chosen as the setting in which to examine the question, since recent studies on the subject suggest that interaction between electricity markets and emissions trading is very much dependent on conditions specific to each market area. There is reason to believe that imperfect competition prevails in the Nordic market, thus the issue is approached through the theory of oligopolistic competition. The generation capacity available at the market, marginal cost of electricity production and seasonal levels of demand form the data based on which the dominant firms are modelled using the Cournot model of competition. The calculations are made for two levels of demand, high and low, and with several values of demand elasticity. The producers are first modelled under no carbon costs and then by adding the cost of carbon dioxide at 20€/t to those technologies subject to carbon regulation. In all cases the situation under perfect competition is determined as a comparison point for the results of the Cournot game. The results imply that the potential for market power does exist on the Nordic market, but the possibility for exercising market power depends on the demand level. In season of high demand the dominant firms may raise the price significantly above competitive levels, and the situation is aggravated when the cost of carbon dioixide is accounted for. Under low demand leves there is no difference between perfect and imperfect competition. The results are highly dependent on the price elasticity of demand.

Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

Modeling Asymmetries in Financial Data with Multiplicative Error Models

This thesis addresses modeling of financial time series, especially stock market returns and daily price ranges. Modeling data of this kind can be approached with so-called multiplicative error models (MEM). These models nest several well known time series models such as GARCH, ACD and CARR models. They are able to capture many well established features of financial time series including volatility clustering and leptokurtosis. In contrast to these phenomena, different kinds of asymmetries have received relatively little attention in the existing literature. In this thesis asymmetries arise from various sources. They are observed in both conditional and unconditional distributions, for variables with non-negative values and for variables that have values on the real line. In the multivariate context asymmetries can be observed in the marginal distributions as well as in the relationships of the variables modeled. New methods for all these cases are proposed. Chapter 2 considers GARCH models and modeling of returns of two stock market indices. The chapter introduces the so-called generalized hyperbolic (GH) GARCH model to account for asymmetries in both conditional and unconditional distribution. In particular, two special cases of the GARCH-GH model which describe the data most accurately are proposed. They are found to improve the fit of the model when compared to symmetric GARCH models. The advantages of accounting for asymmetries are also observed through Value-at-Risk applications. Both theoretical and empirical contributions are provided in Chapter 3 of the thesis. In this chapter the so-called mixture conditional autoregressive range (MCARR) model is introduced, examined and applied to daily price ranges of the Hang Seng Index. The conditions for the strict and weak stationarity of the model as well as an expression for the autocorrelation function are obtained by writing the MCARR model as a first order autoregressive process with random coefficients. The chapter also introduces inverse gamma (IG) distribution to CARR models. The advantages of CARR-IG and MCARR-IG specifications over conventional CARR models are found in the empirical application both in- and out-of-sample. Chapter 4 discusses the simultaneous modeling of absolute returns and daily price ranges. In this part of the thesis a vector multiplicative error model (VMEM) with asymmetric Gumbel copula is found to provide substantial benefits over the existing VMEM models based on elliptical copulas. The proposed specification is able to capture the highly asymmetric dependence of the modeled variables thereby improving the performance of the model considerably. The economic significance of the results obtained is established when the information content of the volatility forecasts derived is examined.

Elements of Volatility at High Frequency

Volatility is central in options pricing and risk management. It reflects the uncertainty of investors and the inherent instability of the economy. Time series methods are among the most widely applied scientific methods to analyze and predict volatility. Very frequently sampled data contain much valuable information about the different elements of volatility and may ultimately reveal the reasons for time varying volatility. The use of such ultra-high-frequency data is common to all three essays of the dissertation. The dissertation belongs to the field of financial econometrics. The first essay uses wavelet methods to study the time-varying behavior of scaling laws and long-memory in the five-minute volatility series of Nokia on the Helsinki Stock Exchange around the burst of the IT-bubble. The essay is motivated by earlier findings which suggest that different scaling laws may apply to intraday time-scales and to larger time-scales, implying that the so-called annualized volatility depends on the data sampling frequency. The empirical results confirm the appearance of time varying long-memory and different scaling laws that, for a significant part, can be attributed to investor irrationality and to an intraday volatility periodicity called the New York effect. The findings have potentially important consequences for options pricing and risk management that commonly assume constant memory and scaling. The second essay investigates modelling the duration between trades in stock markets. Durations convoy information about investor intentions and provide an alternative view at volatility. Generalizations of standard autoregressive conditional duration (ACD) models are developed to meet needs observed in previous applications of the standard models. According to the empirical results based on data of actively traded stocks on the New York Stock Exchange and the Helsinki Stock Exchange the proposed generalization clearly outperforms the standard models and also performs well in comparison to another recently proposed alternative to the standard models. The distribution used to derive the generalization may also prove valuable in other areas of risk management. The third essay studies empirically the effect of decimalization on volatility and market microstructure noise. Decimalization refers to the change from fractional pricing to decimal pricing and it was carried out on the New York Stock Exchange in January, 2001. The methods used here are more accurate than in the earlier studies and put more weight on market microstructure. The main result is that decimalization decreased observed volatility by reducing noise variance especially for the highly active stocks. The results help risk management and market mechanism designing.

On measurements of formation, growth, hygroscopicity, and volatility of atmospheric ultrafine particles

Atmospheric aerosol particles affect the global climate as well as human health. In this thesis, formation of nanometer sized atmospheric aerosol particles and their subsequent growth was observed to occur all around the world. Typical formation rate of 3 nm particles at varied from 0.01 to 10 cm-3s-1. One order of magnitude higher formation rates were detected in urban environment. Highest formation rates up to 105 cm-3s-1 were detected in coastal areas and in industrial pollution plumes. Subsequent growth rates varied from 0.01 to 20 nm h-1. Smallest growth rates were observed in polar areas and the largest in the polluted urban environment. This was probably due to competition between growth by condensation and loss by coagulation. Observed growth rates were used in the calculation of a proxy condensable vapour concentration and its source rate in vastly different environments from pristine Antarctica to polluted India. Estimated concentrations varied only 2 orders of magnitude, but the source rates for the vapours varied up to 4 orders of magnitude. Highest source rates were in New Delhi and lowest were in the Antarctica. Indirect methods were applied to study the growth of freshly formed particles in the atmosphere. Also a newly developed Water Condensation Particle Counter, TSI 3785, was found to be a potential candidate to detect water solubility and thus indirectly composition of atmospheric ultra-fine particles. Based on indirect methods, the relative roles of sulphuric acid, non-volatile material and coagulation were investigated in rural Melpitz, Germany. Condensation of non-volatile material explained 20-40% and sulphuric acid the most of the remaining growth up to a point, when nucleation mode reached 10 to 20 nm in diameter. Coagulation contributed typically less than 5%. Furthermore, hygroscopicity measurements were applied to detect the contribution of water soluble and insoluble components in Athens. During more polluted days, the water soluble components contributed more to the growth. During less anthropogenic influence, non-soluble compounds explained a larger fraction of the growth. In addition, long range transport to a measurement station in Finland in a relatively polluted air mass was found to affect the hygroscopicity of the particles. This aging could have implications to cloud formation far away from the pollution sources.

The effects of the 2004 reduction in the price of alcohol on alcohol-related harm in Finland : A natural experiment based on register data

Changes in alcohol pricing have been documented as inversely associated with changes in consumption and alcohol-related problems. Evidence of the association between price changes and health problems is nevertheless patchy and is based to a large extent on cross-sectional state-level data, or time series of such cross-sectional analyses. Natural experimental studies have been called for. There was a substantial reduction in the price of alcohol in Finland in 2004 due to a reduction in alcohol taxes of one third, on average, and the abolition of duty-free allowances for travellers from the EU. These changes in the Finnish alcohol policy could be considered a natural experiment, which offered a good opportunity to study what happens with regard to alcohol-related problems when prices go down. The present study investigated the effects of this reduction in alcohol prices on (1) alcohol-related and all-cause mortality, and mortality due to cardiovascular diseases, (2) alcohol-related morbidity in terms of hospitalisation, (3) socioeconomic differentials in alcohol-related mortality, and (4) small-area differences in interpersonal violence in the Helsinki Metropolitan area. Differential trends in alcohol-related mortality prior to the price reduction were also analysed. A variety of population-based register data was used in the study. Time-series intervention analysis modelling was applied to monthly aggregations of deaths and hospitalisation for the period 1996-2006. These and other mortality analyses were carried out for men and women aged 15 years and over. Socioeconomic differentials in alcohol-related mortality were assessed on a before/after basis, mortality being followed up in 2001-2003 (before the price reduction) and 2004-2005 (after). Alcohol-related mortality was defined in all the studies on mortality on the basis of information on both underlying and contributory causes of death. Hospitalisation related to alcohol meant that there was a reference to alcohol in the primary diagnosis. Data on interpersonal violence was gathered from 86 administrative small-areas in the Helsinki Metropolitan area and was also assessed on a before/after basis followed up in 2002-2003 and 2004-2005. The statistical methods employed to analyse these data sets included time-series analysis, and Poisson and linear regression. The results of the study indicate that alcohol-related deaths increased substantially among men aged 40-69 years and among women aged 50-69 after the price reduction when trends and seasonal variation were taken into account. The increase was mainly attributable to chronic causes, particularly liver diseases. Mortality due to cardiovascular diseases and all-cause mortality, on the other hand, decreased considerably among the-over-69-year-olds. The increase in alcohol-related mortality in absolute terms among the 30-59-year-olds was largest among the unemployed and early-age pensioners, and those with a low level of education, social class or income. The relative differences in change between the education and social class subgroups were small. The employed and those under the age of 35 did not suffer from increased alcohol-related mortality in the two years following the price reduction. The gap between the age and education groups, which was substantial in the 1980s, thus further broadened. With regard to alcohol-related hospitalisation, there was an increase in both chronic and acute causes among men under the age of 70, and among women in the 50-69-year age group when trends and seasonal variation were taken into account. Alcohol dependence and other alcohol-related mental and behavioural disorders were the largest category in both the total number of chronic hospitalisation and in the increase. There was no increase in the rate of interpersonal violence in the Helsinki Metropolitan area, and even a decrease in domestic violence. There was a significant relationship between the measures of social disadvantage on the area level and interpersonal violence, although the differences in the effects of the price reduction between the different areas were small. The findings of the present study suggest that that a reduction in alcohol prices may lead to a substantial increase in alcohol-related mortality and morbidity. However, large population group differences were observed regarding responsiveness to the price changes. In particular, the less privileged, such as the unemployed, were most sensitive. In contrast, at least in the Finnish context, the younger generations and the employed do not appear to be adversely affected, and those in the older age groups may even benefit from cheaper alcohol in terms of decreased rates of CVD mortality. The results also suggest that reductions in alcohol prices do not necessarily affect interpersonal violence. The population group differences in the effects of the price changes on alcohol-related harm should be acknowledged, and therefore the policy actions should focus on the population subgroups that are primarily responsive to the price reduction.

Essays on the Modeling and Prediction of Volatility and Higher Moments of Stock Returns (summary section only)

One of the most fundamental and widely accepted ideas in finance is that investors are compensated through higher returns for taking on non-diversifiable risk. Hence the quantification, modeling and prediction of risk have been, and still are one of the most prolific research areas in financial economics. It was recognized early on that there are predictable patterns in the variance of speculative prices. Later research has shown that there may also be systematic variation in the skewness and kurtosis of financial returns. Lacking in the literature so far, is an out-of-sample forecast evaluation of the potential benefits of these new more complicated models with time-varying higher moments. Such an evaluation is the topic of this dissertation. Essay 1 investigates the forecast performance of the GARCH (1,1) model when estimated with 9 different error distributions on Standard and Poor’s 500 Index Future returns. By utilizing the theory of realized variance to construct an appropriate ex post measure of variance from intra-day data it is shown that allowing for a leptokurtic error distribution leads to significant improvements in variance forecasts compared to using the normal distribution. This result holds for daily, weekly as well as monthly forecast horizons. It is also found that allowing for skewness and time variation in the higher moments of the distribution does not further improve forecasts. In Essay 2, by using 20 years of daily Standard and Poor 500 index returns, it is found that density forecasts are much improved by allowing for constant excess kurtosis but not improved by allowing for skewness. By allowing the kurtosis and skewness to be time varying the density forecasts are not further improved but on the contrary made slightly worse. In Essay 3 a new model incorporating conditional variance, skewness and kurtosis based on the Normal Inverse Gaussian (NIG) distribution is proposed. The new model and two previously used NIG models are evaluated by their Value at Risk (VaR) forecasts on a long series of daily Standard and Poor’s 500 returns. The results show that only the new model produces satisfactory VaR forecasts for both 1% and 5% VaR Taken together the results of the thesis show that kurtosis appears not to exhibit predictable time variation, whereas there is found some predictability in the skewness. However, the dynamic properties of the skewness are not completely captured by any of the models.

Intraday Dynamics of International Equity Markets

The increased availability of high frequency data sets have led to important new insights in understanding of financial markets. The use of high frequency data is interesting and persuasive, since it can reveal new information that cannot be seen in lower data aggregation. This dissertation explores some of the many important issues connected with the use, analysis and application of high frequency data. These include the effects of intraday seasonal, the behaviour of time varying volatility, the information content of various market data, and the issue of inter market linkages utilizing high frequency 5 minute observations from major European and the U.S stock indices, namely DAX30 of Germany, CAC40 of France, SMI of Switzerland, FTSE100 of the UK and SP500 of the U.S. The first essay in the dissertation shows that there are remarkable similarities in the intraday behaviour of conditional volatility across European equity markets. Moreover, the U.S macroeconomic news announcements have significant cross border effect on both, European equity returns and volatilities. The second essay reports substantial intraday return and volatility linkages across European stock indices of the UK and Germany. This relationship appears virtually unchanged by the presence or absence of the U.S stock market. However, the return correlation among the U.K and German markets rises significantly following the U.S stock market opening, which could largely be described as a contemporaneous effect. The third essay sheds light on market microstructure issues in which traders and market makers learn from watching market data, and it is this learning process that leads to price adjustments. This study concludes that trading volume plays an important role in explaining international return and volatility transmissions. The examination concerning asymmetry reveals that the impact of the positive volume changes is larger on foreign stock market volatility than the negative changes. The fourth and the final essay documents number of regularities in the pattern of intraday return volatility, trading volume and bid-ask spreads. This study also reports a contemporaneous and positive relationship between the intraday return volatility, bid ask spread and unexpected trading volume. These results verify the role of trading volume and bid ask quotes as proxies for information arrival in producing contemporaneous and subsequent intraday return volatility. Moreover, asymmetric effect of trading volume on conditional volatility is also confirmed. Overall, this dissertation explores the role of information in explaining the intraday return and volatility dynamics in international stock markets. The process through which the information is incorporated in stock prices is central to all information-based models. The intraday data facilitates the investigation that how information gets incorporated into security prices as a result of the trading behavior of informed and uninformed traders. Thus high frequency data appears critical in enhancing our understanding of intraday behavior of various stock markets’ variables as it has important implications for market participants, regulators and academic researchers.