The Generalized Local Volatility Modelling Framework

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The Generalized Local Volatility Modelling Framework

Author : Bernard Philippe Gourion
Publisher :
ISBN 13 :
Total Pages : 25 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis The Generalized Local Volatility Modelling Framework by : Bernard Philippe Gourion

Download or read book The Generalized Local Volatility Modelling Framework written by Bernard Philippe Gourion and published by . This book was released on 2018 with total page 25 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Local Volatility formula is a precious tool to identify the volatility process that will allow the perfect retrieval of all the market quotes of vanilla options on the market of equity and index derivatives. Since its first inception by Dupire, the local volatility model, applied initially in a modelling framework where only the underlying is stochastic, has been adapted to several more general cases to produce more generalized versions of the initial famous local volatility function. The most encountered one is the case where the interest rates are assumed to be stochastic. Another famous generalization is the case where the underlying has no longer continuous paths but also endures some jumps in its path over time: see for the case of an equity with a jump to ruin only, for a case where jumps are more general, and for the case where the dynamic of the underlying no longer embeds any continuity in its paths at all.Pursuing the idea of generalizing the Dupire's Local Volatility, the question of this short paper is simple: what is the local volatility formula when absolutely all goes stochastic: stochastic rate, stochastic repo, jumps due to proportionnal dividends, jump to ruin as well as more general jumps? The corresponding derivated formula becomes very difficult to handle in practical but it helps to identify what are the missing bricks of the markets and the datas to make this formula usable. In the end, a first application is developed for Jump To Ruin (JTR) models with stochastic rates and local intensity functions in the mood of and but in a more general case than these earlier works.

A General Valuation Framework for SABR and Stochastic Local Volatility Models

Author : Zhenyu Cui
Publisher :
ISBN 13 :
Total Pages : 44 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis A General Valuation Framework for SABR and Stochastic Local Volatility Models by : Zhenyu Cui

Download or read book A General Valuation Framework for SABR and Stochastic Local Volatility Models written by Zhenyu Cui and published by . This book was released on 2019 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Fitting Local Volatility: Analytic And Numerical Approaches In Black-scholes And Local Variance Gamma Models

Author : Andrey Itkin
Publisher : World Scientific
ISBN 13 : 9811212783
Total Pages : 205 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Fitting Local Volatility: Analytic And Numerical Approaches In Black-scholes And Local Variance Gamma Models by : Andrey Itkin

Download or read book Fitting Local Volatility: Analytic And Numerical Approaches In Black-scholes And Local Variance Gamma Models written by Andrey Itkin and published by World Scientific. This book was released on 2020-01-22 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: The concept of local volatility as well as the local volatility model are one of the classical topics of mathematical finance. Although the existing literature is wide, there still exist various problems that have not drawn sufficient attention so far, for example: a) construction of analytical solutions of the Dupire equation for an arbitrary shape of the local volatility function; b) construction of parametric or non-parametric regression of the local volatility surface suitable for fast calibration; c) no-arbitrage interpolation and extrapolation of the local and implied volatility surfaces; d) extension of the local volatility concept beyond the Black-Scholes model, etc. Also, recent progresses in deep learning and artificial neural networks as applied to financial engineering have made it reasonable to look again at various classical problems of mathematical finance including that of building a no-arbitrage local/implied volatility surface and calibrating it to the option market data.This book was written with the purpose of presenting new results previously developed in a series of papers and explaining them consistently, starting from the general concept of Dupire, Derman and Kani and then concentrating on various extensions proposed by the author and his co-authors. This volume collects all the results in one place, and provides some typical examples of the problems that can be efficiently solved using the proposed methods. This also results in a faster calibration of the local and implied volatility surfaces as compared to standard approaches.The methods and solutions presented in this volume are new and recently published, and are accompanied by various additional comments and considerations. Since from the mathematical point of view, the level of details is closer to the applied rather than to the abstract or pure theoretical mathematics, the book could also be recommended to graduate students with majors in computational or quantitative finance, financial engineering or even applied mathematics. In particular, the author used to teach some topics of this book as a part of his special course on computational finance at the Tandon School of Engineering, New York University.

Fitting Local Volatility

Author : Andrey Itkin
Publisher :
ISBN 13 : 9789811212772
Total Pages : 205 pages
Book Rating : 4.2/5 (127 download)

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Book Synopsis Fitting Local Volatility by : Andrey Itkin

Download or read book Fitting Local Volatility written by Andrey Itkin and published by . This book was released on 2020 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt:

From Moving Average Local and Stochastic Volatility Models to 2-Factor Stochastic Volatility Models

Author : Oleg Kovrizhkin
Publisher :
ISBN 13 :
Total Pages : 36 pages
Book Rating : 4.:/5 (129 download)

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Book Synopsis From Moving Average Local and Stochastic Volatility Models to 2-Factor Stochastic Volatility Models by : Oleg Kovrizhkin

Download or read book From Moving Average Local and Stochastic Volatility Models to 2-Factor Stochastic Volatility Models written by Oleg Kovrizhkin and published by . This book was released on 2008 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the following models:1. Generalization of a local volatility model rolled with a moving average of the spot: dS = mu Sdt + sigma(S/A)SdW$ where A(t) is a moving average of spot S.2. Generalization of Heston pure stochastic volatility model rolled with a moving average of the stochastic volatility: dS = mu Sdt + sigma SdW, dsigma^2 = k(theta - sigma^2)dt + gamma sigma dZ where theta(t) is a moving average of variance sigma^2.3. Generalization of a full stochastic volatility with the process for volatility depending on both sigma and S and rolled with a moving average of S: dS = mu Sdt + sigma SdW, dsigma = a(sigma, S/A)dt + b(sigma, S/A)dZ,corr(dW, dZ) = rho(sigma, S/A)$, where A(t) is a moving average of the spot S. We will generalize these and other ideas further and show that they lead to a 2-factor pure stochastic volatility model: dS = mu Sdt + sigma SdW$, sigma = sigma(v_1, v_2), dv_1 = a_1(v_1, v_2)dt + b_1(v_1, v_2)dZ_1,dv_2 = a_2(v_1, v_2)dt + b_2(v_1, v_2)dZ_2, corr(dW, dZ_1) = rho_1(v_1, v_2), corr(dW, dZ_2) = rho_2(v_1, v_2), corr(dZ_1, dZ_2) = rho_3(v_1, v_2) and give examples of analytically solvable models, applicable for multicurrency models consistent with cross currency pairs dynamics in FX. We also consider jumps and stochastic interest rates.

Local Volatility Model With Stochastic Interest Rate

Author : Bing Hu
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Local Volatility Model With Stochastic Interest Rate by : Bing Hu

Download or read book Local Volatility Model With Stochastic Interest Rate written by Bing Hu and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics

Author : Matthias R. Fengler
Publisher :
ISBN 13 :
Total Pages : 43 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics by : Matthias R. Fengler

Download or read book A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics written by Matthias R. Fengler and published by . This book was released on 2017 with total page 43 pages. Available in PDF, EPUB and Kindle. Book excerpt: A primary goal in modelling the implied volatility surface (IVS) for pricing and hedging aims at reducing complexity. For this purpose one fits the IVS each day and applies a principal component analysis using a functional norm. This approach, however, neglects the degenerated string structure of the implied volatility data and may result in a modelling bias. We propose a dynamic semiparametric factor model (DSFM), which approximates the IVS in a finite dimensional function space. The key feature is that we only fit in the local neighborhood of the design points. Our approach is a combination of methods from functional principal component analysis and backfitting techniques for additive models. The model is found to have an approximate 10% better performance than a sticky moneyness model. Finally, based on the DSFM, we devise a generalized vega-hedging strategy for exotic options that are priced in the local volatility framework. The generalized vega-hedging extends the usual approaches employed in the local volatility framework.

Calibrating the local volatility model

Author :
Publisher :
ISBN 13 :
Total Pages : 73 pages
Book Rating : 4.:/5 (113 download)

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Book Synopsis Calibrating the local volatility model by :

Download or read book Calibrating the local volatility model written by and published by . This book was released on 2012 with total page 73 pages. Available in PDF, EPUB and Kindle. Book excerpt:

A Hybrid Stochastic Volatility Model Incorporating Local Volatility

Author : Yu Tian
Publisher :
ISBN 13 :
Total Pages : 4 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis A Hybrid Stochastic Volatility Model Incorporating Local Volatility by : Yu Tian

Download or read book A Hybrid Stochastic Volatility Model Incorporating Local Volatility written by Yu Tian and published by . This book was released on 2014 with total page 4 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we present our study on a hybrid stochastic volatility model incorporating local volatility for pricing options in the foreign exchange (FX) market. The hybrid stochastic-local volatility model (SLV) could match the implied volatility surface well and meanwhile shows the flexibility for pricing exotic options. The difficulty in implementing the SLV model lies in the calibration of the leverage function, which can be roughly seen as a ratio between the local volatility and the conditional expectation of stochastic volatility. We will illustrate our implementation of the SLV model and show the pricing performance for exotic options.

Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models Based on Markov Chain Approximations

Author : Zhenyu Cui
Publisher :
ISBN 13 :
Total Pages : 49 pages
Book Rating : 4.:/5 (129 download)

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Book Synopsis Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models Based on Markov Chain Approximations by : Zhenyu Cui

Download or read book Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models Based on Markov Chain Approximations written by Zhenyu Cui and published by . This book was released on 2020 with total page 49 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Approximations for Asian Options in Local Volatility Models

Author : Paolo Foschi
Publisher :
ISBN 13 :
Total Pages : 30 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Approximations for Asian Options in Local Volatility Models by : Paolo Foschi

Download or read book Approximations for Asian Options in Local Volatility Models written by Paolo Foschi and published by . This book was released on 2016 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations. The Mathematica notebook with the implementation of the formulae is freely available in the authors' website.

A Threshold Model for Local Volatility: Evidence of Leverage and Mean Reversion Effects on Historical Data

Author : Antoine Lejay
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis A Threshold Model for Local Volatility: Evidence of Leverage and Mean Reversion Effects on Historical Data by : Antoine Lejay

Download or read book A Threshold Model for Local Volatility: Evidence of Leverage and Mean Reversion Effects on Historical Data written by Antoine Lejay and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In financial markets, low prices are generally associated with high volatilities and vice-versa, this well known stylized fact usually being referred to as leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts of leverage and mean-reversion effects in the dynamics of the prices. This model exhibits a regime switch in the dynamics accordingly to a certain threshold. It can be seen as a continuous time version of the Self-Exciting Threshold Autoregressive (SETAR) model. We propose an estimation procedure for the volatility and drift coefficients as well as for the threshold level. Tests are performed on the daily prices of 21 assets. They show empirical evidence for leverage and mean-reversion effects, consistent with the results in the literature.

Explicit Implied Volatilities for Multifactor Local-Stochastic Volatility Models

Author : Matthew Lorig
Publisher :
ISBN 13 :
Total Pages : 36 pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Explicit Implied Volatilities for Multifactor Local-Stochastic Volatility Models by : Matthew Lorig

Download or read book Explicit Implied Volatilities for Multifactor Local-Stochastic Volatility Models written by Matthew Lorig and published by . This book was released on 2014 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility, 3/2 stochastic volatility, and SABR local-stochastic volatility.

Mathematical Models of Financial Derivatives

Author : Yue-Kuen Kwok
Publisher : Springer Science & Business Media
ISBN 13 : 3540686886
Total Pages : 541 pages
Book Rating : 4.5/5 (46 download)

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Book Synopsis Mathematical Models of Financial Derivatives by : Yue-Kuen Kwok

Download or read book Mathematical Models of Financial Derivatives written by Yue-Kuen Kwok and published by Springer Science & Business Media. This book was released on 2008-07-10 with total page 541 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second edition, now featuring new material, focuses on the valuation principles that are common to most derivative securities. A wide range of financial derivatives commonly traded in the equity and fixed income markets are analysed, emphasising aspects of pricing, hedging and practical usage. This second edition features additional emphasis on the discussion of Ito calculus and Girsanovs Theorem, and the risk-neutral measure and equivalent martingale pricing approach. A new chapter on credit risk models and pricing of credit derivatives has been added. Up-to-date research results are provided by many useful exercises.

Estimation of Volatilities Under a Merton's Jump-diffusion Model and an Uncertain Volatility Model

Author : Changhong He
Publisher :
ISBN 13 :
Total Pages : 312 pages
Book Rating : 4.E/5 ( download)

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Book Synopsis Estimation of Volatilities Under a Merton's Jump-diffusion Model and an Uncertain Volatility Model by : Changhong He

Download or read book Estimation of Volatilities Under a Merton's Jump-diffusion Model and an Uncertain Volatility Model written by Changhong He and published by . This book was released on 2005 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Multi-Currency Local Volatility Model

Author : Daniel Alexandre Bloch
Publisher :
ISBN 13 :
Total Pages : 37 pages
Book Rating : 4.:/5 (129 download)

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Book Synopsis Multi-Currency Local Volatility Model by : Daniel Alexandre Bloch

Download or read book Multi-Currency Local Volatility Model written by Daniel Alexandre Bloch and published by . This book was released on 2008 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: We establish the need for local volatility coupled with domestic and foreign stochastic interest rates to properly manage some exotic hybrid options. We then compute such a local volatility and identify a bias with respect to the local volatility with deterministic rates. Performing variance-covariance analysis on the logarithm of the underlying price together with the domestic and foreign spot rates we estimate that bias by calculating the variances of the logarithm of the underlying price with and without stochastic rates at fixed points in time and in space. Equating the resulting variances we express the local volatility with stochastic rates in terms of the one with deterministic rates plus a bias obtaining an exact, fast and robust way of calibrating any local volatility with stochastic rates to market prices. We calculate it by using a bootstrapping method requiring solving a quadratic equation at each maturity and strike and present results on the Japanese market.

Credit Risk Frontiers

Author : Tomasz Bielecki
Publisher : John Wiley & Sons
ISBN 13 : 157660358X
Total Pages : 770 pages
Book Rating : 4.5/5 (766 download)

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Book Synopsis Credit Risk Frontiers by : Tomasz Bielecki

Download or read book Credit Risk Frontiers written by Tomasz Bielecki and published by John Wiley & Sons. This book was released on 2011-02-08 with total page 770 pages. Available in PDF, EPUB and Kindle. Book excerpt: A timely guide to understanding and implementing credit derivatives Credit derivatives are here to stay and will continue to play a role in finance in the future. But what will that role be? What issues and challenges should be addressed? And what lessons can be learned from the credit mess? Credit Risk Frontiers offers answers to these and other questions by presenting the latest research in this field and addressing important issues exposed by the financial crisis. It covers this subject from a real world perspective, tackling issues such as liquidity, poor data, and credit spreads, as well as the latest innovations in portfolio products and hedging and risk management techniques. Provides a coherent presentation of recent advances in the theory and practice of credit derivatives Takes into account the new products and risk requirements of a post financial crisis world Contains information regarding various aspects of the credit derivative market as well as cutting edge research regarding those aspects If you want to gain a better understanding of how credit derivatives can help your trading or investing endeavors, then Credit Risk Frontiers is a book you need to read.