Description:
by SER-HUANG POON and CLIVE W. J. GRANGER1 VOLATILITY FORECASTING IS AN important task in financial markets, and it has held the attention of academics and practitioners over the last two decades. At the time of writing, there are at least 93 published and working papers that study forecasting performance of various volatility models, and several times that number have been written on the subject of volatility modelling without the forecasting aspect. This extensive research reflects the importance of volatility in investment, security valuation, risk management, and monetary policy making. Discuss this paper
Description:
by Roberto Reno The aim of this Thesis is to study some selected topics on volatility estimation and modeling. Recently, these topics received great attention in the financial literature, since volatility modeling is crucial in practically all financial applications, including derivatives pricing, portfolio selection and risk management. Specifically, we focus on the concept of realized volatility, which became important in the last decade mainly thanks to the increased availability of high-frequency data on practically every financial asset traded in the main marketplaces. The concept of realized volatility traces back to an early idea of Merton (1980), and basically consists in the estimation of the daily variance via the ssum of squared intraday returns, see Andersen et al. (2003). The work presented here is linked to this strand of literature but an alternative estimator is adopted. This is based on Fourier analysis of the time series, hence the term Fourier estimator, which has been recently proposed by Malliavin and Mancino (2002). Moreover, we start from this result to introduce a nonparametric estimator of the diffusion coefficient. Discuss this paper
Description:
Xin Huang This dissertation consists of three related chapters that study financial market volatility, jumps and the economic factors behind them. Each of the chapters analyzes a different aspect of this problem. The first chapter examines tests for jumps based on recent asymptotic results. Monte Carlo evidence suggests that the daily ratio z-statistic has appropriate size, good power, and good jump detection capabilities revealed by the confusion matrix comprised of jump classification probabilities. Theoretical and Monte Carlo analysis indicate that microstructure noise biases the tests against detecting jumps, and that a simple lagging strategy corrects the bias. Empirical work documents evidence for jumps that account for seven percent of stock market price variance. Building on realized variance and bi-power variation measures constructed from high-frequency financial prices, the second chapter proposes a simple reduced form framework for modelling and forecasting daily return volatility. The chapter first decomposes the total daily return variance into three components, and proposes different models for the different variance components: an approximate long-memory HAR-GARCH model for the daytime continuous variance, an ACH model for the jump occurrence hazard rate, a log-linear structure for the conditional jump size, and an augmented GARCH model for the overnight variance. Then the chapter combines the different models to generate an overall forecasting framework, which improves the volatility forecasts for the daily, weekly and monthly horizons. The third chapter studies the economic factors that generate financial market volatility and jumps. It extends the recent literature by separating market responses into continuous variance and discontinuous jumps, and differentiating the market Discuss this paper
Description:
Thesis by By Lars Stentoft Summary of the Dissertation This dissertation is concerned with the area of Financial Econometrics dealing with the pricing of options. An option is a financial asset giving the owner the right, but not the obligation, to buy or sell another financial asset (usually stocks, indices, bonds or commodities) for a given price (the strike price) at some future point in time. Options have been traded on exchanges for decades, and the volume seems to grow almost exponentially over time. As a matter of fact, the number of trades and quotes for the IBM stock options recorded at the Chicago Board Options Exchange (CBOE) during 1995 was more than four times higher than the number of trades and quotes for the stock itself recorded at the New York Stock Exchange (NYSE) and NASDAQ together. It is thus understandable that ever since the seminal work of Black & Scholes (1973) and Merton (1973) the derivation of option pricing formulas has received a large amount of interest. Indeed, the Black-Scholes formula used to price European style options is by many practitioners as well as academics considered the most important breakthrough in modern finance, and the work has been recognized as having such importance that it earned Myron S. Scholes and Robert C. Merton the Nobel Prize in 1997. The formula is extensively used, although empirical analysis has pointed towards several systematic pricing errors. At the same time, the assumptions underlying the Black-Scholes-Merton framework have been widely criticized, and much work effort has been put into extending the valuation framework with the systematic pricing errors in mind. The first part of this dissertation examines a flexible valuation method based on simulation. The method is denoted the Least Squares Monte-Carlo (LSM) method by Longstaff & Schwartz (2001), and in Chapter 1 the LSM method is assessed through an extensive numerical analysis. Thus, the particular specification used in Longstaff & Schwartz (2001) for the case of a simple Black-Scholes model and the estimates obtained using different specifications of the cross-sectional regressions are compared. Based on the numerical stability a computationally simpler specification for the cross-sectional regression is suggested. The value of using this suggestion is further emphasized when the trade-off between computational time and precision is examined. Next, it is shown that the LSM method is easily extended, and the results from an application with multiple stochastic factors are compared to those from existing numerical methods. It is shown that as the number of stochastic factors increase the LSM method becomes superior to the Binomial Model in terms of the trade-off between computational time and precision In Chapter 2 the theoretical foundation for the use of the LSM method to American option pricing is provided. A central part of the LSM method is the approximation of a set of conditional expectation functions, and in this chapter it is shown that these approximations converge in a general multiperiod setting. Furthermore, actual rates of v convergence are derived in the two period case and the implications are discussed. For obvious reasons the chapter is quite technical. Hence a section is added to the chapter where the results are illustrated through a number of numerical results. With the mathematical foundation for the use of the LSM method in place, the second part of this dissertation, that is Chapters 3 and 4, provides applications of the method. In this section the aim is to extend the known valuation framework in search of explanations to the systematic pricing errors of the existing methods. In Chapter 3 the assumption of independently identically normally distributed returns underlying the standard option pricing models is relaxed. As an alternative to having constant volatility, the return process is modelled using the generalized autoregressive conditional heteroskedastic (GARCH) processes of Engle (1982) and Bollerslev (1986). Also the assumption of conditional normality is relaxed. Since no closed form solutions exist for the option prices when the volatility is time-varying an extensive Monte-Carlo study is performed. Using the option pricing model of Duan (1999) this study indicates that incorporating GARCH type features in the option pricing model can potentially help explain some empirically well documented systematic pricing errors even when the assumption of conditional normality is maintained. The chapter also introduces a GARCH model where the conditional distribution is of the Normal Inverse Gaussian type, and the chapter shows how this model can be implemented. It is illustrated that incorporating excess kurtosis and skewness in the option pricing model has important additional implications for the option prices. In Chapter 4 an extensive empirical analysis is performed, which confirms that the models with time-varying volatility can explain, at least part of, the systematic pricing errors for the CV model. Also, the models with time-varying volatility have smaller
Description:
CAROL ALEXANDER EMESE LAZAR GARCH processes constitute the major area of time series variance analysis, hence the limit of these processes is of considerable interest for continuous time volatility modelling. The continuous time limit of the GARCH(1,1) model is fundamental for limits of other GARCH processes, yet it has been the point of much debate between econometricians. The seminal work of Nelson (1990) derived the GARCH(1,1) limit as a stochastic volatility process, uncorrelated with the price process. But then a subsequent paper of Corradi (2000) that derives the limit as a deterministic volatility process and several other contradictory papers followed. We reconsider this continuous limit, arguing that because the strong GARCH model is not aggregating in time it is incorrect to consider its limit. Instead it is legitimate to use the weak definition of GARCH that is aggregating in time. This model differs from strong GARCH by defining the discrete time process on the best linear predictor of the squared errors, rather than the conditional variance itself. We prove that its continuous limit is a stochastic volatility model with correlated Brownian motions in which both the variance diffusion coefficient and the price-volatility correlation are related to the skewness and kurtosis of the physical returns density. Under certain assumptions our limit model reduces to Nelson Discuss this paper
Abstract This paper develops closed-form solutions for options on credit spreads with GARCH models. We extend the mean-reverting model proposed in Longstaff and Schwartz (1995) and we use the Heston and Nandi's (1999) GARCH specification rather than the traditional lognormal. Our model, being more flexible, captures better the empirical properties of observed credit spreads and contains Longstaff and Schwartz (1995) model as a special case. GARCH coefficients are estimated using spread levels for corporate bonds.
Description:
Jin-Chuan Duan Abstract The main objective of this paper is to propose an alternative valuation framework for pricing foreign currency and cross-currency options, which is capable of accommodating existing empirical regularities. The paper generalizes the GARCH option pricing methodology of Duan (1995) to a twocountry setting. Specifically, we assume a bivariate nonlinear GARCH system for the exchange rate and the foreign asset price, and generalize the local risk-neutral valuation principle for pricing derivatives. We define an equilibrium price measure in the two-country economy and derive the locally risk-neutralized GARCH processes for the exchange rate and the foreign asset price. Foreign currency options and cross-currency options are then valued using Monte Carlo simulations. Our setup accommodates rich empirical regularities such as stochastic volatility, fat tailed distributions and leverage effect extensively documented for financial data series. Numerical results show that our proposed model exhibits properties that are consistent with the documented empirical regularities for foreign currency options and quanto options.
Description:
Tim Bollerslev Northwestern Unirwsity, Er,anston, IL 60208, USA Ray Y. Chou Georgia Institute of Technoloa, Atlanta, GA 30332, USA Kenneth F. Kroner University of Arizona, Tucson, AZ 85721, USA
Although volatility clustering has a long history as a salient empirical regularity characterizing high-frequency speculative prices, it was not until recently that applied researchers in finance have recognized the importance of explicitly modeling time-varying second-order moments. Instrumental in most of these empirical studies has been the Autoregressive Conditional Heteroskedasticity (ARCH) model introduced by Engle (1982). This paper contains an overview of some of the developments in the formulation of ARCH models and a survey of the numerous empirical applications using financial data. Several suggestions for future research, including the implementation and tests of competing asset pricing theories, market microstructure models, information transmission mechanisms, dynamic hedging strategies, and the pricing of derivative assets, are also discussed. Discuss this paper
This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single-factor (one-lag) version of this model contains Heston's (1993) stochastic volatility model as a diffusion limit and therefore unifies the discrete-time GARCH and continuous-time stochastic volatility literature of option pricing. The new model provides the first readily computed option formula for a random volatility model in which current volatility is easily estimated from historical asset prices observed at discrete intervals. Empirical analysis on S&P 500 index options shows the single-factor version of the GARCH model to be a substantial improvement over the Black-Scholes (1973) model. The GARCH model continues to substantially outperform the Black-Scholes model even when the Black-Scholes model is updated every period and uses implied volatilities from option prices, while the parameters of the GARCH model are held constant and volatility is filtered from the history of asset prices. The improvement is due largely to the ability of the GARCH model to describe the correlation of volatility with spot returns. This allows the GARCH model to capture strike-price biases in the Black-Scholes model that give rise to the skew in implied volatilities in the index options market. Discuss this paper
Description:
Andreas Heigl Abstract This master thesis describes how to price options by means of Genetic Programming. The underlying model is the Generalized Autoregressive Conditional Heteroskedastic (GARCH) asset return process. The goal of this master thesis is to nd a closed-form solution for the price of European call options where the underlying securities follow a GARCH process. The data are simulated over a wide range to cover a lot of existing options in one single equation. Genetic Programming is used to generate the pricing function from the data. Genetic Programming is a method of producing programs just by de ning a problemdependent tness function. The resulting equation is found via a heuristic algorithm inspired by natural evolution. Three di erent methods of bloat control are used. Additionally Automatic De ned Functions (ADFs) and a hybrid approach are tested, too. To ensure that a good con guration setting is used, preliminary testing of many di erent settings has been done, suggesting that simpler con gurations are more successful in this environment. The resulting equation can be used to calculate the price of an option in the given range with minimal errors. This equation is well behaved and can be used in standard spread sheet programs. It o ers a wider range of utilization or a higher accuracy, respectively than other existing approaches.
Description:
Christian Schmitt Abstract Various empirid studies have shown that the time-varying volatility of asset returns can be described by GARCH (generalized autoregressive conditional heteroskedasticity) models. The corresponding GARCH option pricing model of Duan (1995) is capable of depicting the "smile-effect" which often can be found in option prices. In some derivative markets, however, the slope of the smile is not symmetrical. In this paper an option pricing model in the context of the EGARCH (Exponential GARCH) process will be developed. Extensive numerical analyses suggest that the EGARCH option pricing model is able to explain the different slopes of the smile curve.
Description:
Abstract There is extensive empirical evidence that index option prices systematically differ from Black- Scholes prices. Out-of-the-money put prices (and in-the-money call prices) are relatively high compared to the Black-Scholes price. Motivated by these empirical facts, we develop a new discretetime dynamic model of stock returns with Inverse Gaussian innovations. The model allows for conditional skewness as well as conditional heteroskedasticity and a leverage effect. We present an analytic option pricing formula consistent with this stock return dynamic. An extensive empirical test of the model using S&P500 index options shows that the new Inverse Gaussian GARCH model Discuss this paper
Description:
Lars Stentoft Abstract In the option pricing literature it is well documented that the constant volatility model of Black & Scholes (1973) suffers from a number of biases. In particular, numerous studies have documented smiles in the implied volatility as a function of moneyness as well as a tendency for the constant volatility model to underprice in particular short term out of the money options. This has been linked to the assumption of identically normally distributed returns which empirically is found to be violated. Instead, evidence generally favors time-varying volatility of the returns and conditional distributions which are leptokurtic and potentially skewed. In this paper, we model this using generalized autoregressive conditional heteroskedastic (GARCH) processes and their diffusion limits together with the GARCH option pricing model of Duan (1995). When the volatility is allowed to vary through time it is generally not possible to derive an analytical formula for the option price. Simulation techniques are obvious alternative candidates, but application of the GARCH option pricing model has been hampered by the lack of simulation techniques able to incorporate early exercise features. In the present paper, we show how the Least Squares Monte Carlo technique of Longstaff & Schwartz (2001) can be used to price options which have the possibility of early exercise in a GARCH framework. We report the results from an extensive Monte Carlo study indicating that incorporating GARCH features in the option pricing model can help explain some empirically well documented systematic pricing errors even when the assumption of conditional normality is maintained. Empirically, we use the algorithm to price American style options on a number of individual stocks as well as a major US Index. The study indicates that allowing volatility to vary through time is very important when valuing individual options.
Description:
Giovanni Barone-Adesia, Henrik Rasmussenb and Claudia Ravanelli Abstract We derive analytically the first four conditional moments of the integrated variance implied by the GARCH diffusion process. From these moments we obtain an analytical closed-form approximation formula to price European options under the GARCH diffusion model. Using Monte Carlo simulations, we show that this approximation formula is accurate for a large set of reasonable parameters. Finally, we use the closed-form option pricing solution to shed light on the qualitative properties of implied volatility surfaces induced by GARCH diffusion models.
Description:
and Michel Lubrano (GREQAM-CNRS) July 1997 Last revision October 2001 Abstract This paper shows how one can compute option prices from a Bayesian inference view- point, using a GARCH model for the dynamics of the the volatility of the underlying asset. The proposed evaluation of an option is the predictive expectation of its payo function. The predictive distribution of this function provides a natural metric, provided it is neutralised with respect to risk, for gauging the predictive option price or other option evaluations. The proposed method is compared to the Black and Scholes evaluation, in which a marginal mean volatility is plugged, but which does not provide a natural metric. The methods are illustrated using symmetric, asymmetric and smooth transition GARCH models with data on a stock index in Brussels. JEL Classi cation: C11, C15, C22, G13 Keywords: Bayesian inference, GARCH, option pricing, simulation Discuss this paper
Description:
Chun-Yang Liu Abstract The GARCH model has been successful in describing the volatility dynamics of asset return series. However, tree-based GARCH option pricing algorithms su Discuss this paper
)
Date (
Visit Link 
(0 Votes) 
(1 Vote) 
(7 Votes) 
(3 Votes) 