Description:
by Tim Bollerslev Julia Litvinova George Tauchen Abstract We examine the relationship between volatility and past and future returns in high- frequency equity market data. Consistent with a prolonged "leverage" e®ect, we ¯nd the correlations between absolute high-frequency returns and current and past high-frequency returns to be signi¯cantly negative for several days, while the reverse cross-correlations between absolute returns and future returns are generally negligible. Based on a simple aggregation formula, we demonstrate how the high-frequency data may similarly be used in more e®ectively assessing volatility asymmetries over longer daily return horizons. Mo- tivated by the striking cross-correlation patterns uncovered in the high-frequency data, we investigate the ability of some popular continuous-time stochastic volatility models for ex- plaining the observed asymmetries. Our results clearly highlight the importance of allowing for multiple latent volatility factors at very ¯ne time scales in order to adequately describe and understand the patterns in the data. JEL Classi¯cation: G12, C51, C22, C13 Keywords: Volatility Asymmetry, Leverage E®ect, Volatility Feedback E®ect, Temporal Aggregation, High-Frequency Data, Stochastic Volatility Models, EMM Estimation
Description:
by Silvia Golia Summary: In the present paper, tick-by-tick, or ultra-high frequency data are analized. An Autoregressive Conditional Duration (ACD)-type process for the durations between consecutive events, the Fractional Integrated Autoregressive Conditional Duration process (Jasiak, 1999), is reviewed and discussed in order to admit the presence of long memory patterns. This process, as the ACD, is based on the assumption that the temporal dependence in the durations is captured by the mean function. The long term dependency is examined on the Italian stock Tiscali time series recorded from May 2000.
Abstract: These are my Lecture Notes for a course in Discrete Time Finance which I taught in the Winter term 2005 at the University of Leeds. I am aware that the notes are not yet free of error and the manuscrip needs further improvement. I am happy about any comment on the notes. Please send your comments via e-mail to ce16@st-andrews.ac.uk.
Keywords: Discrete Time Finance, Mathematical Finance Discuss this paper
Description:
I am not able to get the main page. There are around 20 lectures in this directory, as well as when you click on each of the links with [DIR] Icon. Discuss this paper
1. Binomial model. 2. Greeks. 3. Random number generators. 4. Correlated Brownian motions. 5. Monte Carlo simulation. 6. Crash course in stochastic analysis. 7. Numerical schemes for SDEs. 8. Black-Scholes PDE. 9. Numerical methods for PDEs. 10. Numerical methods for American options. 11. Implied volatility. The vanna-volga method and beyond.
Description:
Peter Jackel Riccardo Rebonato Abstract We present and approximation for the volatility of European swaptions in a forward rate based Brace-Gatarek-Musiela/Jamshidian framework [BGM97, Jam97] which enables us to calculate prices for swaptions without the need for Monte Carlo simulations. Also, we explain the mechanism behind the remarkable accuracy of these approximate prices. For cases where the yield curve varies noticeably as a function of maturity, a second, and even more accurate formula is derived.
Lectures Lecture 1: The Principles of Valuation Lecture 2: Dynamic Replication: Myths and Realities of Options Pricing Lecture 3: The Smile: Constraints and Problems Lecture 4: Arbitrage Bounds, Problems with Valuation, Models Lecture 5: Static Hedging and Implied Distributions Lecture 6: Extending Black Scholes & Local Volatility Models Lecture 7: More about Local Volatility Models Lecture 8: Implications of Local Volatility Models Lecture 9: Regimes of Volatility Lecture 10: Stochastic Volatility Models of the Smile Lecture 11: More on Stochastic Volatility Models of the Smile Lecture 12: Jump Diffusion Models of the Smile Lecture 13: Conclusions
Description:
by Patrick Hagan and Andrew Lesniewski We propose and study the SABR/LMM model. This is a term structure model of interest rates with stochastic volatility that is a natural extension of both the LIBOR market model and the SABR model. The key result of the paper is a closed form asymptotic formula for swaption volatility in the SABR/LMM model which allows for rapid and accurate valuation of European swaptions.
Chapter 1: Introduction to Finance Chapter 2: Present Value Introduction to Part B: Valuation Chapter 3: Fixed-Income Securities Chapter 4: Common Stocks Chapter 5: Forwards and Futures Chapter 6: Options Introduction to Part C: Time Value and Price of Risk Chapter 7: Historical Asset Returns Chapter 8: Time Value of Money Chapter 9: Risk Chapter 10: Portfolio Theory Chapter 11: Capital Asset Pricing Model (CAPM) Chapter 12: Arbitrage Pricing Theory (APT) Introduction to Part D: Coporate Finance Chapter 13: Efficient Market Hypothesis Chapter 14: Capital Budgeting Chapter 15: Real Options Chapter 16: Financing Decisions
Description:
by Jesper Lund Slides from the lectures February 3, Introduction and Bond Market Basics February 10, Estimation of the Term Structure, Part I February 17, Estimation of the Term Structure, Part II February 24, Introduction to Risk-Neutral Pricing and Binomial Models March 3, Risk-Neutral Pricing and Binomial Models (part II) March 24, Term-Structure Models in Continuous Time (one-factor models) March 31, Term-Structure Models in Continuous Time, Part II April 14, Term-Structure Models in Continuous Time, Part III (multi-factor models) April 21, Term-Structure Models in Continuous Time, Part IV (calibration, HJM) April 23, Calibration in Binomial Models April 28, Calibration in lattice models -- part II May 5, Pricing Term-Structure Derivatives May 12, Mortgage-Backed Securities May 19, MBS (part II) and Risk-Management Issues
Description:
by Nicolas Privault Stochastic interest rate models MA6627 - Semester A, 2007/2008. 1. A review of stochastic calculus. 2. A review of Black-Scholes pricing. 3. Short term interest rate modelling. 4. Pricing of zero coupon bonds. 5. Forward rate modelling. 6. The Heath-Jarrow-Morton (HJM) model. 7. The forward measure and derivative pricing. 8. Curve fitting and a two factor model. 9. Pricing of caps and swaptions on the LIBOR. 10. The Brace-Gatarek-Musiela (BGM) model. 11. Appendix: Mathematical tools.
Description:
by Jim Gatheral, Merrill Lynch In the course of the following lectures, we will study why equity options are priced as they are. In so doing, we will apply many of the techniques students will have learned in previous semesters and develop some intuition for the pricing of both vanilla and exotic equity options. By considering specific examples, we will see that in pricing options, it is often as important to take into account the dynamics of underlying variables as it is to match known market prices of other claims. My hope is that these lectures will prove particularly useful to those who end up specializing in the structuring, pricing, trading and risk management of equity derivatives
Description:
presentaiotn by Peter Jackel An approximation for the volatility of European swaptions is presented that makes it possible to calculate prices of swaptions without the need for numerical computations such as Monte Carlo simulations or lattice-based integration methods. The approximation can be used whenever the covariance matrix of an initial set of fixed income observables is known and thus applies to most interest rate models such as the extended Vasicek (also known as multi-factor Hull-White) model or the Brace- Gatarek-Musiela/Jamshidian framework. Also, the mechanism behind the remarkable accuracy of the approximation is explained.
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
The Derman and Kani Implied Binomial Tree Implied Trinomial Tree of Derman, Kani and Chriss Hull-White Model The Heston Model SABR Model Discuss this paper
Description:
Jesper Lund Part II: GMM for diffusion processes Nonparametric estimation for diffusions Reviews of GMM and kernel estimators Discuss this paper
Description:
Abstract: In this paper we investigate one factor models that extend the classical Gaussian copula model for pricing CDOs. The proposed models are very tractable and perform significantly better than the classical Gaussian copula model. Moreover, we introduce the concept of Levy base correlation. The obtained Levy base correlation curve is much flatter than the corresponding Gaussian one. This indicates that the models do fit the observed data much better. Additionally, flat base correlation curves are also much more reliable for pricing of bespoke tranches. Discuss this paper
Description:
Kazufumi Itoy Jari Toivanenz The deterministic numerical valuation of American options under Heston's stochastic volatility model is considered. The prices are given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. A new truncation of the domain is described for small asset values while for large asset values and variance a standard truncation is used. The nite difference discretization is constructed by numerically solving quadratic optimization problem aiming to minimize the truncation error at each grid point. A Lagrange approach is used to treat the linear complementarity problems. Numerical examples demonstrate the accuracy and effectiveness of the proposed approach. Keywords: American option pricing, stochastic volatility model, linear complementarity problem, nite difference method, quadratic programming, multigrid method, Lagrange method, penalty method
Description:
by Dr. Thomas Papenbrock Lecture 1 Introduction: Motivation and course overview 2 Computer use: Security, compiling, linking, graphics Fortran 90: data types 3 Fortran 90 cont Discuss this paper
Description:
Charlotte Christiansen The Aarhus School of Business, Denmark Abstract: The paper aims to improve the knowledge of the empirical properties of the long maturity region of the forward rate curve. Firstly, the theoretical negative correlation between the slope at the long end of the forward rate curve and the term structure variance is recovered empirically and found to be statistically significant. Secondly, the expectations hypothesis is analyzed for the long maturity region of the forward rate curve using Discuss this paper
Visit Link 
(2 Votes) 
(0 Votes) 
(1 Vote) 
(1 Vote) 
(1 Vote) 