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This link shows real time order book position for any symbol, and is helpful in giving an idea of how orders are matched. It is updated every second. If you are not familiar with order matching process, looking at the book viewer window for like half an hour will give you an idea of how prices are determined
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by Francesco Audrino, Dominik Colangelo Abstract: This paper introduces a new semi-parametric methodology for the implied volatility surface, which incorporates machine learning algorithms. Given a starting model, a tree boosting algorithm sequentially minimizes the residuals of observed and estimated implied volatility. To overcome the poor predicting power of existing models, we include a grid in the region of interest, and implement a cross-validation strategy to find an optimal stopping value for the tree boosting. Back testing the out-of-sample performance on a large data set of implied volatilities on S&P 500 options, we provide empirical evidence of the strong predictive potential of our methodology.
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thesis by Sensen Lin Stochastic volatility is an interesting area in financial mathematics. Parabolic partial differential equations with mixed differentiation terms are the focus of numerical solution of Heston model. This document covers the numerical methods to Heston model. Chapter 1 is an introduction to the problem and my main interest. Chapter 2 is an overview of Heston model and its closed form solution. The closed form solution is a benchmark to test the numerical methods Chapter 3 talks about the explicit scheme which is a straightforward method in solving Heston model. The result and restriction of this model are illustrated. Chapter 4 discusses the ADI method dealing with special equations like Heston PDE. The details of this method are covered and comparison between schemes is given. Discuss this paper
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by Valdo Durrleman Given the quote price of a call or put option, the Black-Scholes implied volatility is the unique volatility parameter to be put into Black-Scholes formula to give the same price as the option quote price. This dissertation is concerned with the link between the implied volatility and the actual volatility of the underlying stock. Such a link is of particular practical interest since it relates the fundamental quantity for pricing financial derivatives (the actual volatility of the underlying stock), which is not observable, to directly observable quantities such as implied volatilities. The link that we derive in chapter 2 is a link between the dynamics of the two quantities. So far these quantities were mostly studied at a given time whereas we work at the level of processes. This is the main result of the dissertation. In chapter 1, we shall first review current practical problems in option pricing. Our aim there is twofold. First, we want to show that from a practical point of view, studying dynamics is very natural. Second, we shall identify two practical issues to which we shall propose answers in chapter 3. Although the main motivation of this dissertation comes from contemporary issues in the study of financial markets, chapter 2 also gives a solution to an inverse problem in the mathematical sense. One wishes to recover the structure of a stochastic process from a family of conditional expectations over its distribution. Besides the main result, this dissertation makes the following contributions. It brings new insights about implied volatility dynamics. In particular, it was observed that its motion was extremely Discuss this paper
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Giovanni Montana, Kostas Triantafyllopoulos, Theodoros Tsagaris A number of recent emerging applications call for studying data streams, potentially infinite flows of information updated in real-time. When multiple co-evolving data streams are observed, an important task is to determine how these streams depend on each other, accounting for dynamic dependence patterns without imposing any restrictive probabilistic law governing this dependence. In this paper we argue that flexible least squares (FLS), a penalized version of ordinary least squares that accommodates for time-varying regression coefficients, can be deployed successfully in this context. Our motivating application is statistical arbitrage, an investment strategy that exploits patterns detected in financial data streams. We demonstrate that FLS is algebraically equivalent to the well-known Kalman filter equations, and take advantage of this equivalence to gain a better understanding of FLS and suggest a more efficient algorithm. Promising experimental results obtained from a FLS-based algorithmic trading system for the S&P 500 Futures Index are reported
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by ilya, gikhman In this article we discuss the fundamentals of pricing of the popular financial instruments. The basic point of our approach is to extend the present value benchmark concept. The present value valuation approach plays the similar role as The Newton Laws in the Classic Mechanics. Thus our primary goal is to present a new outlook on valuation of the debt securities and its derivatives. We also, demonstrate why the present value is not a complete method of pricing either securities or derivatives. Then, as illustration we present a valuation of the floating rate, callable and convertible bonds. Next we discuss major drawbacks of the risk neutral interpretation of the derivatives pricing. At the end of the article we discuss interest rate swap and derivative valuation of some classes of the fixed income securities.
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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
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by Alireza Javaheri and Alain Galli, Abstract In this article we present an introduction to various Filtering algorithms and some of their applications to the world of Quantitative Finance. We shall first mention the fundamental case of Gaussian noises where we obtain the well-known Kalman Filter. Because of common nonlinearities, we will be discussing the Extended Kalman Filter Discuss this paper
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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
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John Hull and Alan White This paper is concerned with the implementation of the LIBOR market model and its extensions. It develops and tests a simple analytic approximation for calculating the volatilities used by the market to price European swap options from the volatilities used by the market to price interest rate caps. The approximation is found to be very accurate for the range of market parameters normally encountered. It enables swap option volatility skews to be implied from cap volatility skews. It also allows the LIBOR market model to be easily calibrated to broker quotes on caps and European swap options so that a wide range of non-standard interest rate derivatives can be valued.
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Abstract: I demonstrate four little-known properties of the Black-Scholes option pricing formula: (1) An easy way to find delta. (2) A quaint relation between call- and put-prices. (3) Why vega-hedging though non-sensical will help. (4) What happens if you take vega-hedging too far. Discuss this paper
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Sajib Barua The Fast Fourier Transform (FFT) has been used in many scientific and engineering applications. The use of FFT for financial derivatives has been gaining momentum in the recent past. In this thesis, i) we have improved a recently proposed model of FFT for pricing financial derivatives to help design an efficient parallel algorithm. The improved mathematical model put forth in our research bridges a gap between quantitative approaches for the option pricing problem and practical implementation of such approaches on modern computer architectures. The thesis goes further by proving that the improved model of fast Fourier transform for option pricing produces accurate option values. ii) We have developed a parallel algorithm for the FFT using the classical Cooley-Tukey algorithm and improved this algorithm by introducing a data swapping technique that brings data closer to the respective processors and hence reduces the communication overhead to a large extent leading to better performance of the parallel algorithm. We have tested the new algorithm on a node SunFire high performance computing system and compared the new algorithm with the traditional Cooley-Tukey algorithm. Option values are calculated for various strike prices with a proper selection of strike-price spacing to ensure fine-grid integration for FFT computation as well as to maximize the number of strikes lying in the desired region of the stock price. Compared to the traditional Cooley-Tukey algorithm, the current algorithm with data swapping performs better by more than for large data sizes. In the rapidly changing market place, these improvements could mean a lot for an investor or financial institution because obtaining faster results offers a competitive advantages.
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LIXIN WU DEPARTMENT OF MATHEMATICS Abstract. We claim to have developed the optimal methodology for non-parametric calibration of market model to the prices of at-the-money (ATM) caps/floors and swaptions, and to the historic correlations of the LIBOR rates. We take the approach of divide-andconquer: first fit the model to historic correlations, then to the implied Black volatilities of the input options. Regularization is adopted and the calibration is cast into minimizationmaximization problems by the method of Lagrange multiplier. By utilizing the quadratic functional form of both objective function and constraints, we solve the inner maximization problems with a single matrix eigenvalue decomposition, which renders the efficiency of our method. The outer minimization problems, meanwhile, are nicely subdued by gradient-based descending methods due to the convexity of the objective functions. The well-posedness of the Lagrange multiplier problems and the convergence of the descending methods are rigorously justified. Numerical results show that we have achieved very quality calibration. We have also developed a technique to calculate the hedging ratios of a derivative security with respect to the benchmark derivative instruments, using the auxiliary results of the calibration.
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