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1. The Fourier Transform and its Applications Stanford / Engineering (Electrical) Brad G. Osgood 2. Highlights of Calculus Course Course MIT / Mathematics Gilbert Strang 3. Computational Science and Engineering I Course Course MIT / Mathematics Gilbert Strang 5. Multivariable Calculus Course Course MIT / Mathematics Denis Auroux 6. Linear Algebra Course Course MIT / Mathematics Gilbert Strang 8. Differential Equations Course Course Differential Equations MIT / Mathematics Arthur Mattuck 9. 10. Introduction to Linear Dynamical Systems Course Course Introduction to Linear Dynamical Systems Stanford / Engineering (Except Electrical) Stephen Boyd 11. Convex Optimization II Course Course Convex Optimization II Stanford / Mathematics Stephen Boyd 12. 13. Math and Probability for Life Sciences Course Course Math and Probability for Life Sciences UCLA / Mathematics Herbert Enderton 14. Convex Optimization I Course Course Convex Optimization I Stanford / Engineering (Except Electrical) Stephen Boyd 15. 16. Mathematical Methods for Engineers II Course Course Mathematical Methods for Engineers II MIT / Mathematics Gilbert Strang 17. Multivariable Calculus Course Course Multivariable Calculus Berkeley / Mathematics Michael Hutch Discuss this paper
1. Introduction: Probability and Counting 2. Probability Functions 3. Permutations 4. Probability Functions (continued) 5. Conditional Probability 6. Conditional Probability (continued) 7. Independent Events 8. Random Variables 9. Expected Values 10. Binomial Distributions 11. Midterm Review 12. Multinomial Distributions 13. Geometric Distributions 14. Poisson Distributions 15. Poisson Distributions (continued) 16. Density Function 17. Exponential Distributions 18. Normal Distributions 19. Normal Distributions (continued) 20. Standard Normal Distributions 21. Central Limit Theorem 22. Hitstogram Correction 23. Midterm Review 2 24. Analyzing Data in Probability 25. Analyzing Data in Probability (continued) 26. Limit Theorems 27. Limit Theorems (continued) 28. Course Review Discuss this paper
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Overview Of Linear Dynamical Systems Linear Functions (Continued) Linearization (Continued) Nullspace Of A Matrix (Continued) Orthonormal Set Of Vectors Least-Squares Least-Squares Polynomial Fitting Multi-Objective Least-Squares Lecture Least-Norm Solution Examples Of Autonomous Linear Dynamical Systems Lecture Solution Via Laplace Transform And Matrix Exponential Lecture Time Transfer Property Lecture Markov Chain (Example) Lecture Jordan Canonical Form Lecture LU Factorization (Cont.) Lecture Continue On Unconstrained Minimization Lecture Newton's Method (Cont.) Lecture
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by Torben G. Andersena, Tim Bollerslevb, Peter F. Christoffersenc and Francis X. Dieboldd Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3, 4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.
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Authors: Galda, Galina Issue Date: 25-Jun-2008 Abstract: Asian options are an important family of derivative contracts with a wide variety of applications in commodity, currency, energy, interest rate, equity and insurance markets. In this master's thesis, we investigate methods for evaluating the price of the Asian call options with a fixed strike. One of them is the Monte Carlo method. The accurancy of this method can be observed through variance of the price. We will see that the variance with using Monte Carlo method has to be decreased. The Variance Reduction technique is useful for this aim. We will give evidence of the efficiency of one of the Variance Reduction thechniques - Control Variate method - in a mathematical context and a numerical comparison with the ordinary Monte Carlo method. Keywords: Asian options,Monte Carlo method, Variance Reduction techniques, Control Variate Discuss this paper
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Thesis by Simona Svoboda-Greenwood The LMM is an effective framework for the pricing of interest rate derivatives, not least because it models observable market quantities. In its lognormal form, calibration to market implied volatilities is intuitive and fast. The amendments required to incorporate a monotonically decreasing implied volatility skew are fairly straightforward and do not significantly reduce the ease and speed of calibration. However, the incorporation of a full implied volatility smile is significantly more challenging,from both a mathematical and computational perspective. There exist three main techniques for incorporating a volatility smile/skew in any modelling framework: allowing a local volatility function, stochastic volatility and jump dynamics. In this thesis various ways to incorporate smile/skew are studied, loosely based on the above three approaches. Both the constant-elasticity-of-variance and displaced-diffusion processes give rise to an implied volatility skew. In fact it has been experimentally shown that, for a certain parameterisation, the two processes produce closely matching prices for European call options over a variety of strikes and maturities. Here, this similarity in prices is analytically quantified, not only via an asymptotic expansion of the call prices, but also via expansion of the conditional probability density functions and a comparison of the raw and central moments of the two distributions.
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by Klebert Kentia Tonleu Volatility derivatives are products where the volatility is the main underlying notion. These products are particularly important for market investors as they use them to have insight into the level of volatility which empirical evidence show that it is stochastic. In this essay, we provide a short introduction to volatility derivatives. We start by motivating the change from constant volatility as assumed by the standard Black-Scholes model, to stochastic mean-reverting volatility Discuss this paper
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by DANIEL EGLOFF, MARKUS LEIPPOLD,AND LIUREN WU With increasing appreciation of the fact that stock return variance is stochastic and variance risk is heavily priced, the industry has created a series of variance derivative products to span variance risk. The variance swap contract is the most actively traded of these products. It pays at expiry the difference between the realized return variance and a fixed rate, called the variance swap rate, determined at the inception of the contract. We obtain a decade worth of variance swap rate quotes at five maturities. With the data, we first exploit the information in both the time series and the term structure of the variance swap rates to analyze the return variance rate dynamics and market pricing of variance risk. We then study both theoretically and empirically how investors can use variance swap contracts across different maturities to span the variance risk and to revise their dynamic asset allocation decisions. We find that with the swap contract to span the variance risk, an investor increases her investment in the underlying stock. In addition, the investor Discuss this paper
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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
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FRANCIS A. LONGSTAFF University of California, Los Angeles - Finance Area; National Bureau of Economic Research (NBER) EDUARDO S. SCHWARTZ University of California, Los Angeles - Finance Area; National Bureau of Economic Research (NBER)
This paper presents a simple yet powerful new approach for valuing American options by simulation. The key to this approach is to use least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference and binomial techniques cannot be used. We illustrate this technique with a series of realistic examples ranging from the valuation of an American put in a single-factor setting to the valuation of a deferred American swaption in a twenty-factor string model of the term structure.
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Thomas Lidebrandt In option price simulations, simulation-time is of great importance. Control variates is a variance reduction technique that can reduce simulation-time. Three approaches to the use of control variates in Monte Carlo option pricing are presented and evaluated. Employed methods include ordinary control variate implementation, a replicating delta hedge and re-simulation. Ordinary control variates uses a highly correlated random variable with known mean to reduce variance. The delta hedge tries to replicate the option and is constructed with an approximative delta formula, which is new to stock markets. The third method evaluated, called re-simulation, is a new method which use an earlier simulated option price as control variate. Applying an earlier option price as control variate results in a more generic method, since earlier simulated prices often exists. The three models are evaluated on Asian and Cliquet options, either in the standard Black and Scholes model or in Merton Discuss this paper
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Conditional variance swaps are claims on realised variance that is accumulated when the underlying asset price stays within a certain range. Being highly sensitive to movements in both asset price and its variance, they require a very reliable model for pricing and risk-managing. We apply the Heston stochastic volatility model to derive closed-form solutions for pricing and risk-managing of such swaps Discuss this paper
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Georgios Foufas and Mats G. Larson The main objective of this paper is to develop an adaptive nite element method for computation of the values and di erent sensitivity measures of ordinary European options, barrier options, and lookback options. The options are priced using the Black-Scholes PDE-model, and the resulting PDE:s are of parabolic type in one spatial dimension with different boundary conditions and jump conditions at monitoring dates. The adaptive nite element method is based on a posteriori estimates of the error in desired quantities, which we derive using duality techniques. The suggested adaptive nite element method is stable and gives fast and accurate results.
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Gunter Winkler Thomas Apel Uwe Wystup Quoted:
Introduction Due to the smile observed in options markets numerous authors have suggested different models such as generalized Levy processes, fractional Brownian motion, entropy based models [4], jump diffusions and stochastic volatility models. For vanilla options (put and call options) the dependence of the price on the volatility is monotone, whence using the Black-Scholes formula along with a volatility smile matrix is sufficient. Values of exotic options, however, do not always depend on the volatility in a monotone fashion, whence pricing consistently with the smile requires a more sophisticated model. Therefore, it is important to find efficient ways to calculate exotic option values in exotic models.
Abstract The stochastic Dirichlet problem computes values within a domain of certain functions with known values at the boundary of the domain. When applied to valuing barrier options, solutions are expressed as expected discounted payo Discuss this paper
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by John Hull of the University of Toronto, and Alan White of the University of Toronto
Abstract: In this paper we develop two fast procedures for valuing tranches of collateralized debt obligations and nth to default swaps. The procedures are based on a factor copula model of times to default and are alternatives to using fast Fourier transforms. One involves calculating the probability distribution of the number of defaults by a certain time using a recurrence relationship; the other involves using a Discuss this paper
Abstract The Variance-Gamma model has analytical formulae for the values of European calls and puts. These formulae have to be computed using numerical methods. In general, option valuation may require the use of numerical methods including PDE methods, lattice methods, and Monte Carlo methods. We investigate the use of Monte Carlo methods in the Variance-Gamma model. We demonstrate how a gamma bridge process can be constructed. Using the bridge together with stratified sampling we obtain considerablespeed improvements over a plain Monte Carlo method when pricing path-dependent options.The method is illustrated by pricing lookback, average rate and barrier options in the Variance-Gamma model. We find the method is up to around 400 times faster than plain Monte Carlo Discuss this paper
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John Hull and Alan White Abstract This paper provides a methodology for valuing credit default swaps when the payoff is contingent on default by a single reference entity and there is no counterparty default risk. The paper tests the sensitivity of credit default swap valuations to assumptions about the expected recovery rate. It also tests whether approximate no-arbitrage arguments give accurate valuations and provides an example of the application of the methodology to real data. In a companion paper entitled Valuing Credit Default Swaps II: Modeling Default Correlation, the analysis is extended to cover situations where the payo Discuss this paper
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John Hull and Alan White Abstract This paper extends the analysis in Valuing Credit Default Swaps I: No Counter-party Default Risk to provide a methodology for valuing credit default swaps that takes account of counterparty default risk and allows the payo Discuss this paper
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Fabio Mibielli Peixoto Abstract Monte Carlo simulation and a semi-analytical method are used to value a basket default swap and an homogeneous Collateralized Debt Obligation (CDO). The semianalytical technique is based on the one factor copula model proposed by J.P. Laurent and J. Gregory [1]. We study the properties of a CDO with Monte Carlo and compare the spread calculation with the one obtained by the factor model. Discuss this paper
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