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Author: Peter Bank
Publisher:
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Languages : en
Pages : 0
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In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.
Author: Peter Bank
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Languages : en
Pages : 0
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Book Description
In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.
Author: Antoine (Jack) Jacquier
Publisher:
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Languages : en
Pages : 21
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We introduce the notion of rough local stochastic volatility models, extending the classical concept to the case where volatility is driven by some Volterra process. In this setting, we show that the pricing function is the solution to a path-dependent PDE, for which we develop a numerical scheme based on Deep Learning techniques. Numerical simulations suggest that the latter is extremely efficient, and provides a good alternative to classical Monte Carlo simulations.
Author: Nikolaos Kolliopoulos
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Languages : en
Pages : 0
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Author: Ghislain Vong
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Category :
Languages : en
Pages : 12
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This article presents an alternative formulation of the standard Local Stochastic Volatility model (LSV) widely used for the pricing and risk-management of FX derivatives and to a lesser extent of equity derivatives. In the standard model, calibration is achieved by solving a non-linear two-factor Kolmogorov forward PDE, where a minimum number of vol points is required to achieve convergence of a finite difference implementation. In contrast, we propose to model the volatility process by a Markov chain defined over an arbitrary small number of states, so that calibration and pricing are achieved by solving a coupled system of one-factor PDEs. The practical benefits are twofolds: existing one-factor PDE solvers can be recycled to model stochastic volatility, while the reduction in number of discretisation points implies a speedup in execution time that enables real-time risk-management of large derivatives position.
Author: Cristian Homescu
Publisher:
ISBN:
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Languages : en
Pages : 57
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Book Description
We analyze in detail calibration and pricing performed within the framework of local stochastic volatility LSV models, which have become the industry market standard for FX and equity markets. We present the main arguments for the need of having such models, and address the question whether jumps have to be included. We include a comprehensive literature overview, and focus our exposition on important details related to calibration procedures and option pricing using PDEs or PIDEs derived from LSV models. We describe calibration procedures, with special attention given to usage and solution of corresponding forward Kolmogorov PDE/PIDE, and outline powerful algorithms for estimation of model parameters. Emphasis is placed on presenting practical details regarding the setup and the numerical solution of both forward and backward PDEs/PIDEs obtained from the LSV models. Consequently we discuss specifics (based on our experience and best practices from literature) regarding choice of boundary conditions, construction of nonuniform spatial grids and adaptive temporal grids, selection of efficient and appropriate finite difference schemes (with possible enhancements), etc. We also show how to practically integrate specific features of various types of financial instruments within calibration and pricing settings. We consider all questions and topics identified as most relevant during the selection, calibration and pricing procedures associated with local stochastic volatility models, providing answers (to the best of our knowledge), and present references for deeper understanding and for additional perspectives. In a nutshell, it is our intention to present here an effective roadmap for a successful LSV journey.
Author: Yu Tian
Publisher:
ISBN:
Category :
Languages : en
Pages : 146
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Book Description
This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.
Author: Peter K. Friz
Publisher: Springer Nature
ISBN: 3030415562
Category : Mathematics
Languages : en
Pages : 346
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Book Description
With many updates and additional exercises, the second edition of this book continues to provide readers with a gentle introduction to rough path analysis and regularity structures, theories that have yielded many new insights into the analysis of stochastic differential equations, and, most recently, stochastic partial differential equations. Rough path analysis provides the means for constructing a pathwise solution theory for stochastic differential equations which, in many respects, behaves like the theory of deterministic differential equations and permits a clean break between analytical and probabilistic arguments. Together with the theory of regularity structures, it forms a robust toolbox, allowing the recovery of many classical results without having to rely on specific probabilistic properties such as adaptedness or the martingale property. Essentially self-contained, this textbook puts the emphasis on ideas and short arguments, rather than aiming for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis and probability courses, with little more than Itô-integration against Brownian motion required for most of the text. From the reviews of the first edition: "Can easily be used as a support for a graduate course ... Presents in an accessible way the unique point of view of two experts who themselves have largely contributed to the theory" - Fabrice Baudouin in the Mathematical Reviews "It is easy to base a graduate course on rough paths on this ... A researcher who carefully works her way through all of the exercises will have a very good impression of the current state of the art" - Nicolas Perkowski in Zentralblatt MATH
Author: Christian Bayer
Publisher: SIAM
ISBN: 1611977789
Category : Mathematics
Languages : en
Pages : 292
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Book Description
Volatility underpins financial markets by encapsulating uncertainty about prices, individual behaviors, and decisions and has traditionally been modeled as a semimartingale, with consequent scaling properties. The mathematical description of the volatility process has been an active topic of research for decades; however, driven by empirical estimates of the scaling behavior of volatility, a new paradigm has emerged, whereby paths of volatility are rougher than those of semimartingales. According to this perspective, volatility behaves essentially as a fractional Brownian motion with a small Hurst parameter. The first book to offer a comprehensive exploration of the subject, Rough Volatility contributes to the understanding and application of rough volatility models by equipping readers with the tools and insights needed to delve into the topic, exploring the motivation for rough volatility modeling, providing a toolbox for computation and practical implementation, and organizing the material to reflect the subject’s development and progression. This book is designed for researchers and graduate students in quantitative finance as well as quantitative analysts and finance professionals.
Author: Wahid Khosrawi-Sardroudi
Publisher:
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Category :
Languages : en
Pages :
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Abstract: Financial markets have experienced a precipitous increase in complexity over the past decades, posing a significant challenge from a risk management point of view. This complexity motivates the application and development of sophisticated models based on the theory of stochastic processes and in particular stochastic calculus. In this regard, the contribution of this thesis is twofold, namely by extending the class if tractable stochastic processes in form of polynomial processes and polynomial semimartingales and by showing how efficient calibration of local stochastic volatility models is possible by applying machine learning techniques. In the first part - the main part - we extend the class of polynomial processes that has previously been established to include beyond stochastic discontinuity. This extension is motivated by the fact that certain events in financial markets take place at a deterministic time point but without foreseeable outcome. Such events consist e.g. of decisions regarding interest rates of central banks or political elections/votes. Since the outcome has a significant impact on markets, it is therefore desirable to consider stochastic processes, that can reproduce such jumps at previously specified time points. Such an extension has already been introduced in the affine framework. We will show that similar modifications hold true in the polynomial case. In particular, we will show how after this extension, computation of mixed moments in a multivariate setting reduces to solving a measure ordinary differential equation, posing a significant reduction in complexity to the measure partial differential case in the context of Kolmogorow equations. A central role in the theory of time-homogeneous polynomial processes is played by the theory of one parameter matrix semigroups. Hence, we will develop a two parameter version of the matrix semigroup theory under lower regularity then what exists in the literature. This accounts for time-inhomogeneity of the stochastic processes we consider. While in the one parameter case, full regularity follows already from very mild assumptions, we will see that this is not the case anymore in the two parameter case. In the second part of this thesis we investigate a more applied topic, namely the exact calibration of local stochastic volatility models to financial data. We show how this computationally challenging problem can be efficiently solved by applying machine learning te ...
Author: Andrea Pascucci
Publisher:
ISBN:
Category :
Languages : en
Pages : 27
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Book Description
We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.
