Download or read book An Equity Interest Rate Hybrid Model with Stochastic Volatility and the Interest Rate Smile written by Lech A. Grzelak and published by . This book was released on 2014 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt: We define an equity-interest rate hybrid model in which the equity part is driven by the Heston stochastic volatility [Hes93], and the interest rate (IR) is generated by the displaced-diffusion stochastic volatility Libor Market Model [AA02]. We assume a non-zero correlation between the main processes. By an appropriate change of measure the dimension of the corresponding pricing PDE can be greatly reduced. We place by a number of approximations the model in the class of affine processes [DPS00], for which we then provide the corresponding forward characteristic function. We discuss in detail the accuracy of the approximations and the efficient calibration. Finally, by experiments, we show the effect of the correlations and interest rate smile/skew on typical equity-interest rate hybrid product prices. For a whole strip of strikes this approximate hybrid model can be evaluated for equity plain vanilla options in just milliseconds.
Download or read book An Equity interest Rate Hybrid Model with Stochastic Volatility and the Interest Rate Smile written by Lech Aleksander Grzelak and published by . This book was released on 2010 with total page 25 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Incorporating an Interest Rate Smile in an Equity Local Volatility Model written by Lech A. Grzelak and published by . This book was released on 2010 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this paper is on finding a connection between the interest rate and equity asset classes. We propose an equity interest rate hybrid model which preserves market observable smiles: the equity from plain vanilla products via a local volatility framework and the interest rate from caps and swaptions via the Stochastic Volatility Libor Market Model. We define a multi-factor short-rate process implied from the Libor Market Model via an arbitrage-free interpolation and combine it with the local volatility equity model for stochastic interest rates. We show that the interest rate smile has a significant impact on the equity local volatility. The model developed is intuitive and straightforward, enabling consistent pricing of related hybrid products. Moreover, it preserves the non-arbitrage Heath, Jarrow, Morton conditions.
Download or read book Equity Derivatives and Hybrids written by Oliver Brockhaus and published by Springer. This book was released on 2016-04-29 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the development of the Black-Scholes model, research on equity derivatives has evolved rapidly to the point where it is now difficult to cut through the myriad of literature to find relevant material. Written by a quant with many years of experience in the field this book provides an up-to-date account of equity and equity-hybrid (equity-rates, equity-credit, equity-foreign exchange) derivatives modeling from a practitioner's perspective. The content reflects the requirements of practitioners in financial institutions: Quants will find a survey of state-of-the-art models and guidance on how to efficiently implement them with regards to market data representation, calibration, and sensitivity computation. Traders and structurers will learn about structured products, selection of the most appropriate models, as well as efficient hedging methods while risk managers will better understand market, credit, and model risk and find valuable information on advanced correlation concepts. Equity Derivatives and Hybrids provides exhaustive coverage of both market standard and new approaches, including: -Empirical properties of stock returns including autocorrelation and jumps -Dividend discount models -Non-Markovian and discrete-time volatility processes -Correlation skew modeling via copula as well as local and stochastic correlation factors -Hybrid modeling covering local and stochastic processes for interest rate, hazard rate, and volatility as well as closed form solutions -Credit, debt, and funding valuation adjustment (CVA, DVA, FVA) -Monte Carlo techniques for sensitivities including algorithmic differentiation, path recycling, as well as multilevel. Written in a highly accessible manner with examples, applications, research, and ideas throughout, this book provides a valuable resource for quantitative-minded practitioners and researchers.
Download or read book On Cross Currency Models with Stochastic Volatility and Correlated Interest Rates written by Lech A. Grzelak and published by . This book was released on 2014 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semi-closed form approximations which lead to efficient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity-FX-interest rate hybrid payoffs.
Download or read book Hybrid Equity Interest Rate Model with Stochastic Interest Rate and Volatility written by Moez Mrad and published by . This book was released on 2014 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper presents an Equity-IR hybrid model that fits in the class of affine diffusion processes. In the absence of cash dividend payment, the moment generating function can be easily and rapidly computed. This allows for an efficient calibration of the model based on Vanilla European Options prices. In practice, cash dividend payments are common and are modeled via the piece-wise and escrowed approaches. We show how to compute Vanilla European Options prices efficiently in this case. The semi-analytical approximation pricing formulas obtained in the piece-wise approach are one of our primary results.In order to assess the semi-analytical pricing method, we compare the computed prices to those obtained via Monte Carlo simulation. Three discretization schemes are used: the two first ones, which have been introduced by Andersen, are the Quadratic Exponential (QE) and its extension with martingale correction QE-M. The last scheme is a new extension of the QE scheme denoted QE-I, that we introduced in order to allow the use of larger simulation time steps by comparison to QE and QE-M.
Download or read book Volatility Perturbations in Financial Markets written by and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book An Analytical Approximation for European Option Prices Under Stochastic Interest Rate Economy written by Hideharu Funahashi and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper extends the Wiener-Ito chaos expansion approach proposed by Funahashi and Kijima (2013) to an equity-interest-rate hybrid model for the pricing of European contingent claims with special emphasis on calibration to the option markets. Our model can capture the volatility skew and smile of option markets, as well as the stochastic nature of interest rates. Further, the proposed method is applicable to widely used option pricing models such as local volatility models, stochastic volatility models, and their combinations with the stochastic nature of interest rates; hence, it is suitable for practical purposes. Through numerical examples, we show that our approximation is quite accurate even for long-maturity and/or high-volatility cases.
Download or read book Multiscale Stochastic Volatility for Equity Interest Rate and Credit Derivatives written by Jean-Pierre Fouque and published by . This book was released on 2014-05-14 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consolidate and extend ideas from their previous book. Ideal for practitioners and as a graduate-level textbook.
Download or read book Extension of Stochastic Volatility Equity Models with Hull White Interest Rate Process written by Lech A. Grzelak and published by . This book was released on 2014 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present an extension of stochastic volatility equity models by a stochastic Hull-White interest rate component while assuming non-zero correlations between the underlying processes. We place these systems of stochastic differential equations in the class of affine jump diffusion - linear quadratic jump-diffusion processes (Duffie, Pan and Singleton, Cheng and Scaillet) so that the pricing of European products can be efficiently done within the Fourier cosine expansion pricing framework. We compare the new stochastic volatility Schobel-Zhu-Hull-White hybrid model with a Heston-Hull-White model, and also apply the models to price some hybrid structured derivatives that combine the equity and interest rate asset classes.
Download or read book Get an Implied Correlation to Price Equity Interest Rates Hybrids written by Mathieu Hamel and published by . This book was released on 2008 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt: To price vanilla options, the market does not rely on empirical parameters. Why should it be the case when pricing hybrids? When pricing long dated equity or indexes linked derivatives, one cannot assume the interest rates to be constant. As a consequence the market, consciously or not, does not only make bets on the volatility but also on the potential drift of the underlying. Therefore long dated vanilla options incorporate information on the dynamic of the interest rates. Using market prices from variance swaps, caps/floors and long dated vanilla options, one proposes a simple way to extract implied correlation between indexes and interest rates through a stochastic interest rates-stochastic volatility model.
Download or read book On the Construction of Analytic Solutions for a Diffusion reaction Equation with a Discontinuous Switch Mechanism written by F.J. Vermolen and published by . This book was released on 2008 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Interest Rate Models Theory and Practice written by Damiano Brigo and published by Springer Science & Business Media. This book was released on 2007-09-26 with total page 1016 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced. The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. The fast-growing interest for hybrid products has led to a new chapter. A special focus here is devoted to the pricing of inflation-linked derivatives. The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.
Download or read book On Calibration and Simulation of Local Volatility Model with Stochastic Interest Rate written by Mingyang Xu and published by . This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Local volatility model is a relatively simple way to capture volatility skew/smile. In spite of its drawbacks, it remains popular among practitioners for derivative pricing and hedging. For long-dated options or interest rate/equity hybrid products, in order to take into account the effect of stochastic interest rate on equity price volatility stochastic interest rate is often modelled together with stochastic equity price. Similar to local volatility model with deterministic interest rate, a forward Dupire PDE can be derived using Arrow-Debreu price method, which can then be shown to be equivalent to adding an additional correction term on top of Dupire forward PDE with deterministic interest rate. Calibrating a local volatility model by the forward Dupire PDE approach with adaptively mixed grids ensures both calibration accuracy and efficiency. Based on Malliavin calculus an accurate analytic approximation is also derived for the correction term incorporating impacts from both interest rate volatility and correlation, which integrates along a more likely straight line path for better accuracy. Eventually, the hybrid local volatility model can be calibrated in a two-step process, namely, calibrate local volatility model with deterministic interest rate and add adjustment for stochastic interest rate. Due to the lack of analytic solution and path-dependency nature of some products, Monte Carlo is a simple but flexible pricing method. In order to improve its convergence, we develop a scheme to combine merits of different simulation schemes and show its effectiveness.
Download or read book Gigi Model written by Roberto Baviera and published by . This book was released on 2011 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt: A parsimonious model for interest rates' term structure is deviced to take into account both volatility smile and multifactor dynamics. We propose a stochastic volatility generalization of the Bond Market Model that allows to price caps and floors with one-dimensional easy-to-handle closed formulas. For each caplet/floorlet the model generalizes the classical Black formula with 1 free parameter (the implied volatility) to a scheme with 3 parameters, each one responsible for one characteristic of the implied curve (average volatility, vol-of-vol, skew). On a given set of reset dates, Monte Carlo simulations are straightforward even in the spot measure, due to the simplicity of the dynamics modeled as a Markov chain.A comparison with an implied volatility approach is discussed. Calibration issues are described in detail and a good agreement to the EURO cap/floor market is found.
Download or read book Alternative Models for Stochastic Volatility Corrections for Equity and Interest Rate Derivatives written by Tianyu Liang and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: ABSTRACT: A lot of attention has been paid to the stochastic volatility model where the volatility is randomly fluctuating driven by an additional Brownian motion. In our work, we change the mean level in the mean-reverting process from a constant to a function of the underlying process. We apply our models to the pricing of both equity and interest rate derivatives. Throughout the thesis, a singular perturbation method is employed to derive closed-form formulas up to first order asymptotic solutions. We also implement multiplicative noise to arithmetic Ornstein-Uhlenbeck process to produce a wider variety of effects. Calibration and Monte Carlo simulation results show that the proposed model outperform Fouque's original stochastic volatility model during some particular window in history. A more efficient numerical scheme, the heterogeneous multi-scale method (HMM), is introduced to simulate the multi-scale differential equations discussed over the chapters.
Download or read book Multiscale Stochastic Volatility for Equity Interest Rate and Credit Derivatives written by Jean-Pierre Fouque and published by Cambridge University Press. This book was released on 2011-09-29 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.
