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by Luc Bauwens, Fausto Galli and Pierre Giot Abstract We provide existence conditions and analytical expressions of the moments of logarithmic autoregressive conditional duration (Log-ACD) models. We focus on the dispersion index and the autocorrelation function and compare them with those of ACD (Engle and Russell 1998) and SCD models. Using duration data for several stocks traded on the New York Stock Exchange, we compare the models in terms of their ability at fitting some stylized facts. Keywords: Duration model, overdispersion, autocorrelation function, high frequency financial data. Discuss this paper
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TAQ dataset is one of the most important inputs to consider for high frequency analysis. This link shows example data to help get a feel of how real data looks like! Discuss this paper
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This is the online version of The C Book, second edition by Mike Banahan, Declan Brady and Mark Doran, originally published by Addison Wesley in 1991. This version is made freely available. Discuss this paper
Abstract: The author considers SABR model which is a two factor stochastic volatility model and gives an asymptotic expansion formula of implied volatilities for this model. His approach is based on infinite dimensional analysis on the Malliavin calculus and large deviation.
Furthermore, he applies the approach to a foreign exchange model where interest rates and the FX volatilities are stochastic and gives an asymptotic expansion formula of implied volatilities of foreign exchange options.
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Thesis by Besiana Rexhepi Abstract It is well-known that the fair value of options can be determined by using the Black-Scholes model. However, for liquidly traded options, i.e. instruments for which the market price is known, there is clear evidence that the Black-Scholes model is not correct. This is reflected in the existence of the volatility smile phenomenon which is one the most challenging problems in financial economics. Recently, a rigorous analysis of the time evolution of the empirically observed volatility smile, i.e. smile dynamics, has been reported by (CdF02) and (Fen05). However, the quantification of the volatility smile dynamics as implied by smile-consistent models has not been done rigorously, so far. People have addressed this by looking at the evolution of the smile, based on asymptotic analysis and qualitative investigations. In this work, we use similar statistical techniques as employed in the empirical studies, to quantify the smile dynamics that is implied by the following smile-consistent models: Displaced Diffusion, Constant Elasticity of Variance (CEV) and SABR Stochastic Volatility Models. We find that in markets where options exhibit extreme skew, e.g. equity options markets, the displaced diffusion and CEV models should be used with care, since these models have poor fitting capabilities to market prices and impose inaccurate smile dynamics. The SABR model on the other hand, was shown to be able to capture the smile dynamics very closely to the empirically observed dynamics.
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by A. G. (Tassos) Malliaris This chapter introduces the reader to the Black-Scholes -Merton model by identifying its assumptions and illustrating its mathematical derivation using intuitive financial reasoning. Numerical examples are also presented to help the reader understand practical aspects of this celebrated model. The analytical power of the Black-Scholes-Merton model comes from the brilliant assumption that the returns of the underlying asset follow an Ito process. This assumption allowed financial theorists to use financial reasoning with an extensive inventory of mathematical techniques to solve successfully for the pricing of contingent claims. Unlike many other scientific discoveries that are not often easily modified, the Black-Scholes-Merton model has been successfully extended and adapted to numerous underlying assets, thus offering pricing solutions as benchmark prices. This in turn has encouraged the development and implementation of numerous trading strategies that involved hedging, speculation and arbitrage. Discuss this paper
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by Jianqing Fan, Jiancheng Jiang, Chunming Zhang and Zhenwei Zhou Abstract: In an effort to capture the time variation on the instantaneous return and volatility functions, a family of time-dependent diffusion processes is introduced to model the term structure dynamics. This allows one to examine how the instantaneous return and price volatility change over time and price level. Nonparametric techniques, based on kernel regression, are used to estimate the time-varying coefficient functions in the drift and diffusion. The newly proposed semiparametric model includes most of the well-known short-term interest rate models, such as those proposed by Cox, Ingersoll and Ross (1985) and Chan, Karolyi, Longstaff and Sanders (1992). It can be used to test the goodness-of-fit of these famous time-homogeneous short rate models. The newly proposed method complements the time-homogeneous nonparametric estimation techniques of Stanton (1997) and Fan and Yao (1998), and is shown through simulations to truly capture the heteroscedasticity and time-inhomogeneous structure in volatility. A family of new statistics is introduced to test whether the time-homogeneous models adequately fit interest rates for certain periods of the economy. We illustrate the new methods by using weekly three-month treasury bill data. Discuss this paper
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These are the independent sites who have offered to mirror the books Thinking in C++, 2nd Edition, Thinking in Java, 1st and 2nd Editions, and Thinking in Patterns, all of which include source code. Make sure you check the contents of the sites against the Master Download Site (below); there is no guarantee that the mirror sites have been updated to include the most recent files.
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Algorithms Behind Term Structure Models of Interest Rates II: The Hull-White Trinomial Tree of Interest Rates by Markus Leippold and Zvi Wiener In this article we implement the trinomial tree of the Hull-White model, which can be easily extended to allow different assumptions about the dynamics of the short rate process. We present the Mathematica algorithm for the extended Vasicek and the Black-Karasinski model. Whenever negative interest rates are generated with a positive probability, we make use of alternative branching processes, which guarantee the positivity of interest rates. Finally we show how to price simple options such as caplets, and compare the convergence of trinomial trees with different geometries.
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by Angelos Dassiosy and Jayalaxshmi Nagaradjasarma Although the square-root process has long been used as an alternative to the Black-Scholes geometric Brownian motion model for option valuation, the pricing of Asian options on this diffusion model has never been studied analytically. However, the additivity property of the square-root process makes it a very suitable model for the analysis of Asian options. In this paper, we develop explicit prices for digital and regular Asian options. We also obtain distributional results concerning the square-root process and its average over time, including analytic formulae for their joint density and moments. We also show that the distribution is actually determined by those moments.
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Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli Abstract
In [1] we study the problem of obtaining accurate estimates of the parameters, of the initial stochastic variance and of the risk premium parameter of the risk neutral measure of the Heston stochastic volatility model from the observation at discrete times of the stock log-returns and of the prices of a European call option on the stock. This problem is an inverse problem known in the literature as calibration problem. As a byproduct of the solution of the calibration problem we develop a tracking procedure that can be used to forecast the stochastic variance and the stock price. From a mathematical point of view the problem considered is formulated as a constrained optimization problem where the objective function is the logarithm of the likelihood associated to the parameter, the initial stochastic variance and the risk premium parameter values given the observed stock log-returns, option prices and observation times, this function is called (log-)likelihood function. The evaluation of the (log-)likelihood function associated to a given choice of the parameter, of the initial stochastic variance and of the risk premium parameter values requires the solution of a filtering problem for the Heston model. An accurate and computationally efficient solution of this filtering problem is necessary for a satisfactory solution of the calibration problem for the Heston model. A similar problem has been considered in [2] and [3]. The aim of this paper is to extend and improve the results obtained in [2] with respect to the formulation of the problem, the accuracy of the solution obtained and the computational efficiency of the solution method. In particular in comparison with [2] we reformulate the problem adding to the quantities that must be estimated the risk premium parameter and to the observations the option price at the initial time. The addition of the risk premium parameter to the quantities that must be estimated makes the problem considered more realistic and makes interesting the analysis of time series of real financial data using the method proposed. The addition of the option price at the initial time to the data used in the solution of the problem is natural and improves significantly the estimate of the initial stochastic variance. Moreover we simplify the expression of some formulae used in [2] in the computation of the (log-)likelihood function and we improve the optimization method employed to solve the maximum likelihood problem introducing some ad hoc preliminary optimization steps. Finally a new, easy to compute, formula that gives the ``Heston" option price is derived. The use of this new formula reduces substantially the computational cost of evaluating the (log-)likelihood function when compared to the cost of the (log-)likelihood function evaluation in [2]. Some numerical examples of the calibration problem using synthetic and real data are presented. As real data we use the 2005 historical data (precisely the data of the period Jan. 3, 2005, May 11, 2005) of the US S&P500 index and of the corresponding option prices. Very good results are obtained forecasting future values of the S&P500 index and of the corresponding option price using the solution of the calibration problem and the forecasting and tracking procedure mentioned above. In this website some auxiliary material useful in the understanding of [1] including some animations and some numerical experiments is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
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Kevin McNew and Paul L. Fackler Cointegration methods are increasingly used to test for market efficiency and integration. The economic rationale for these tests, however, is generally unclear. Using a simple spatial equilibrium model to simulate equilibrium price behavior, it is shown that prices in a well-integrated, efficient market need not be cointegrated. Furthermore, the number of cointegrating relationships among prices is not a good indicator of the degree to which a market is integrated. Key words: market integration, spatial markets, time-series analysis
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Manuel Ammann, Axel H. Kind, Christian Wilde We investigate the pricing performance of three convertible bond pricing models on the French convertible bond market using daily market prices. We examine a component model separating the convertible bond into a bond and option component, a method based on the Margrabe model for pricing exchange options, and a binomial-tree model with exogenous credit risk. All three models are found to deliver theoretical values for the analyzed convertible bonds that tend to be higher than the observed market prices. The prices obtained by the binomial-tree model are nearest to market prices and the mispricing is no longer statistically significant for the majority of bonds in our sample. For all models, the difference between market and model prices is greater for out-of-the money convertibles than for at- or in-the-money convertibles.
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TIME SERIES ANALYSIS: FORECASTING PRODUCT DEMAND AND REVENUE
COPYRIGHT 1997 JAMES L. POWELL
OBJECTIVES OF THIS COURSE - Learn concepts (and technical jargon) for forecasting problems. - Get hands-on experience with leading time series methods, especially Box-Jenkins ARIMA methods and vector autoregressions (VARs). - See how alternative methods differ in their out-of-sample predictions. Discuss this paper
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There is a large and growing literature on how to model the dynamics of the default-free term structure to fit the observed historical data. Much less is known about how best to model the dynamics of defaultable yield curves. This paper develops a class of defaultable term structure models that is tractable enough to be empirically implemented and flexible enough to capture some important behaviors of the credit spreads in the data. We compare two non-nested models within this class using a Bayesian estimation technique, which helps to solve the problem of latent state variables. The Bayesian approach also enables us to test the two non-nested models on the basis of the Bayes factor. The results strongly suggest that models with constant transition probabilities will not be able to fit the observed dynamics of inter-rating spreads. Discuss this paper
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Anna Kalemanova Bernd Schmid Ralf Werner Abstract This paper presents an extension of the popular Large Homogeneous Portfolio (LHP) approach to the pricing of CDOs. LHP (which has already become a standard model in practice) assumes a flat default correlation structure over the reference credit portfolio and models default using a one factor Gaussian copula. However, this model fails to fit the prices of different CDO tranches simultaneously which leads to the well known implied correlation smile. Many researchers explain this phenomenon with the lack of tail dependence and propose to use a Student t copula. Incorporating the effect of tail dependence into the one factor portfolio credit model yields significant pricing improvement. However, the computation time increases dramatically as the Student t distribution is not stable under convolution. This makes it impossible to use the model for computationally intensive applications such as the determination of the optimal asset allocation in an investor Discuss this paper
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S. Hogana, R. Jarrowb, M. Teoc*, M. Warachkad Abstract This paper introduces the concept of statistical arbitrage, a long horizon trading opportunity that generates a riskless profit and is designed to exploit persistent anomalies. Statistical arbitrage circumvents the joint hypothesis dilemma of traditional market efficiency tests because its definition is independent of any equilibrium model and its existence is incompatible with market efficiency. We provide a methodology to test for statistical arbitrage and then empirically investigate whether momentum and value trading strategies constitute statistical arbitrage opportunities. Despite adjusting for transaction costs, the influence of small stocks, margin requirements, liquidity buffers for the marking-to-market of short-sales, and higher borrowing rates, we find evidence that these strategies generate statistical arbitrage. Furthermore, their profitability does not appear to decline over time.
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T.S. Ho Richard C. Stapleton Marti G. Subrahmanyam
Abstract In general, the risk of a financial instrument on a future valuation date depends on several stochastic variables. In the case of a currency swap, its value on a future date, can be modelled as a function of five stochastic variables. These represent the factors that determine the term structure of interest rates in the two currencies, and the foreign exchange rate between the currencies. The joint-probability distribution of the the relevant variables on the horizon date of is approximated by a multivariate-binomial distribution. The proposed methodology provides a fast and flexible alternative to Monte-Carlo simulation of the swap value. The distributions of value produced by the method can be employed to assist with both market and credit risk management.
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