Small Time Asymptotics And Expansions Of Option Prices Under Levy Based Models

Download Small Time Asymptotics And Expansions Of Option Prices Under Levy Based Models full books in PDF, epub, and Kindle. Read online Small Time Asymptotics And Expansions Of Option Prices Under Levy Based Models ebook anywhere anytime directly on your device. Fast Download speed and no annoying ads. We cannot guarantee that every ebooks is available!

Small-time Asymptotics and Expansions of Option Prices Under Levy-based Models

Author : Ruoting Gong
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (825 download)

DOWNLOAD NOW!

Book Synopsis Small-time Asymptotics and Expansions of Option Prices Under Levy-based Models by : Ruoting Gong

Download or read book Small-time Asymptotics and Expansions of Option Prices Under Levy-based Models written by Ruoting Gong and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log return processes of the underlying stock prices follow several Levy-based models. To be specific, we derive the time-to-maturity asymptotoc behavior for both at-the-money (ATM, out-of-the-money (OTM) and in-the-money (ITM) call option prices under several jump diffusion models and stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component. An accurate modeling of the option market and asset prices requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a fisk asset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic volatility model, we derive the small-time expansions, of arbitrary polynomial order, in time-t, for the tail distribution of the log-return process, and for the call-option price which is not at-the-money. Moreover, our approach allows for a unified treatment of more general payoff functions. As a consequence of our tail expansions, the polynomial expansion in t of the transition density is also obtained under mild conditions. The asymptotic behavior of the ATM call-option prices is more complicated to obtain, and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel second-order approximation for ATM option prices under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed.

Asymptotic Chaos Expansions in Finance

Author : David Nicolay
Publisher : Springer
ISBN 13 : 1447165063
Total Pages : 503 pages
Book Rating : 4.4/5 (471 download)

DOWNLOAD NOW!

Book Synopsis Asymptotic Chaos Expansions in Finance by : David Nicolay

Download or read book Asymptotic Chaos Expansions in Finance written by David Nicolay and published by Springer. This book was released on 2014-11-25 with total page 503 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic instantaneous volatility models such as Heston, SABR or SV-LMM have mostly been developed to control the shape and joint dynamics of the implied volatility surface. In principle, they are well suited for pricing and hedging vanilla and exotic options, for relative value strategies or for risk management. In practice however, most SV models lack a closed form valuation for European options. This book presents the recently developed Asymptotic Chaos Expansions methodology (ACE) which addresses that issue. Indeed its generic algorithm provides, for any regular SV model, the pure asymptotes at any order for both the static and dynamic maps of the implied volatility surface. Furthermore, ACE is programmable and can complement other approximation methods. Hence it allows a systematic approach to designing, parameterising, calibrating and exploiting SV models, typically for Vega hedging or American Monte-Carlo. Asymptotic Chaos Expansions in Finance illustrates the ACE approach for single underlyings (such as a stock price or FX rate), baskets (indexes, spreads) and term structure models (especially SV-HJM and SV-LMM). It also establishes fundamental links between the Wiener chaos of the instantaneous volatility and the small-time asymptotic structure of the stochastic implied volatility framework. It is addressed primarily to financial mathematics researchers and graduate students, interested in stochastic volatility, asymptotics or market models. Moreover, as it contains many self-contained approximation results, it will be useful to practitioners modelling the shape of the smile and its evolution.

Statistical Analysis of Short-time Option Prices Based on a Lévy Model

Author : Weiliang Wang (Mathematician)
Publisher :
ISBN 13 :
Total Pages : 39 pages
Book Rating : 4.:/5 (14 download)

DOWNLOAD NOW!

Book Synopsis Statistical Analysis of Short-time Option Prices Based on a Lévy Model by : Weiliang Wang (Mathematician)

Download or read book Statistical Analysis of Short-time Option Prices Based on a Lévy Model written by Weiliang Wang (Mathematician) and published by . This book was released on 2018 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Black-Scholes model has been widely used to find the prices of option, while several generalizations have been made due to its limitation. In this thesis, we consider one of the generalizations---the exponential Lévy model with a mixture of CGMY process and Brownian motion. We state the main results of the first-, second- and third-order expansions for close-to-the-money call option prices under this model. Using importance sampling based on Monte Carlo method, a dataset of call option prices can be simulated. Comparing the simulated true prices with the three different order approximations, we find that the higher-order approximation is more accurate than the lower-order in most cases, which can be used for calibrating the parameters in the model. In order to verify these results, we use call option prices obtained from the Standard & Poor's 500 index options. The third-order approximation of this real dataset is not as accurate as before.

Asymptotic Methods for Option Pricing in Finance

Author : David Krief
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (114 download)

DOWNLOAD NOW!

Book Synopsis Asymptotic Methods for Option Pricing in Finance by : David Krief

Download or read book Asymptotic Methods for Option Pricing in Finance written by David Krief and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we study several mathematical finance problems, related to the pricing of derivatives. Using different asymptotic approaches, we develop methods to calculate accurate approximations of the prices of certain types of options in cases where no explicit formulas are available.In the first chapter, we are interested in the pricing of path-dependent options, with Monte-Carlo methods, when the underlying is modelled as an affine stochastic volatility model. We prove a long-time trajectorial large deviations principle. We then combine it with Varadhan's Lemma to calculate an asymptotically optimal measure change, that allows to reduce significantly the variance of the Monte-Carlo estimator of option prices.The second chapter considers the pricing with Monte-Carlo methods of options that depend on several underlying assets, such as basket options, in the Wishart stochastic volatility model, that generalizes the Heston model. Following the approach of the first chapter, we prove that the process verifies a long-time large deviations principle, that we use to reduce significantly the variance of the Monte-Carlo estimator of option prices, through an asymptotically optimal measure change. In parallel, we use the large deviations property to characterize the long-time behaviour of the Black-Scholes implied volatility of basket options.In the third chapter, we study the pricing of options on realized variance, when the spot volatility is modelled as a diffusion process with constant volatility. We use recent asymptotic results on densities of hypo-elliptic diffusions to calculate an expansion of the density of realized variance, that we integrate to obtain an expansion of option prices and their Black-Scholes implied volatility.The last chapter is dedicated to the pricing of interest rate derivatives in the Levy Libor market model, that generaliszes the classical (log-normal) Libor market model by introducing jumps. Writing the first model as a perturbation of the second and using the Feynman-Kac representation, we calculate explicit expansions of the prices of interest rate derivatives and, in particular, caplets and swaptions.

Option Pricing in Incomplete Markets

Author : Yoshio Miyahara
Publisher : World Scientific
ISBN 13 : 1848163487
Total Pages : 200 pages
Book Rating : 4.8/5 (481 download)

DOWNLOAD NOW!

Book Synopsis Option Pricing in Incomplete Markets by : Yoshio Miyahara

Download or read book Option Pricing in Incomplete Markets written by Yoshio Miyahara and published by World Scientific. This book was released on 2012 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume offers the reader practical methods to compute the option prices in the incomplete asset markets. The [GLP & MEMM] pricing models are clearly introduced, and the properties of these models are discussed in great detail. It is shown that the geometric L(r)vy process (GLP) is a typical example of the incomplete market, and that the MEMM (minimal entropy martingale measure) is an extremely powerful pricing measure. This volume also presents the calibration procedure of the [GLP \& MEMM] model that has been widely used in the application of practical problem

Asymptotic Distribution Expansions in Option Pricing

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

DOWNLOAD NOW!

Book Synopsis Asymptotic Distribution Expansions in Option Pricing by : Daniel Giamouridis

Download or read book Asymptotic Distribution Expansions in Option Pricing written by Daniel Giamouridis and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This article extends an option pricing model and studies the properties of implied probability distribution functions (PDFs) recovered from American interest rate futures options. The model relaxes the restricting assumption of lognormally distributed returns accommodating a wide variety of implied PDFs shapes. Non-normal skewness and kurtosis are found to contribute significantly to estimating more precise implied densities. The model achieves good in-sample accuracy, similar to that achieved by alternative approaches. The recovered implied PDFs are, finally, found to be closer to a median PDF estimated using a number of alternative techniques.

Closed-Form Approximation of Timer Option Prices Under General Stochastic Volatility Models

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

DOWNLOAD NOW!

Book Synopsis Closed-Form Approximation of Timer Option Prices Under General Stochastic Volatility Models by : Minqiang Li

Download or read book Closed-Form Approximation of Timer Option Prices Under General Stochastic Volatility Models written by Minqiang Li and published by . This book was released on 2013 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop an asymptotic expansion technique for pricing timer options under general stochastic volatility models around small volatility of variance. Closed-form approximation formulas have been obtained for the Heston model and the 3/2-model. The approximation has an easy-to-understand Black-Scholes-like form and many other attractive properties. Numerical analysis shows that the approximation formulas are very fast and accurate.

Essays on American Options Pricing Under Levy Models with Stochastic Volatility and Jumps

Author : Ye Chen
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (114 download)

DOWNLOAD NOW!

Book Synopsis Essays on American Options Pricing Under Levy Models with Stochastic Volatility and Jumps by : Ye Chen

Download or read book Essays on American Options Pricing Under Levy Models with Stochastic Volatility and Jumps written by Ye Chen and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In ``A Multi-demensional Transform for Pricing American Options Under Stochastic Volatility Models", we present a new transform-based approach for pricing American options under low-dimensional stochastic volatility models which can be used to construct multi-dimensional path-independent lattices for all low-dimensional stochastic volatility models given in the literature, including SV, SV2, SVJ, SV2J, and SVJ2 models. We demonstrate that the prices of European options obtained using the path-independent lattices converge rapidly to their true prices obtained using quasi-analytical solutions. Our transform-based approach is computationally more efficient than all other methods given in the literature for a large class of low-dimensional stochastic volatility models. In ``A Multi-demensional Transform for Pricing American Options Under Levy Models", We extend the multi-dimensional transform to Levy models with stochastic volatility and jumps in the underlying stock price process. Efficient path-independent tree can be constructed for both European and American options. Our path-independent lattice method can be applied to almost all Levy models in the literature, such as Merton (1976), Bates (1996, 2000, 2006), Pan (2002), the NIG model, the VG model and the CGMY model. The numerical results show that our method is extemly accurate and fast. In ``Empirical performance of Levy models for American Options", we investigate in-sample fitting and out-of-sample pricing performance on American call options under Levy models. The drawback of the BS model has been well documented in the literatures, such as negative skewness with excess kurtosis, fat tail, and non-normality. Therefore, many models have been proposed to resolve known issues associated the BS model. For example, to resolve volatility smile, local volatility, stochastic volatility, and diffusion with jumps have been considered in the literatures; to resolve non-normality, non-Markov processes have been considered, e.g., Poisson process, variance gamma process, and other type of Levy processes. One would ask: what is the gain from each of the generalized models? Or, which model is the best for option pricing? We address these problems by examining which model results in the lowest pricing error for American style contracts. For in-sample analysis, the rank (from best to worst) is Pan, CGMYsv, VGsv, Heston, CGMY, VG and BS. And for out-of-sample pricing performance, the rank (from best to worst) is CGMYsv, VGsv, Pan, Heston, BS, VG, and CGMY. Adding stochastic volatility and jump into a model improves American options pricing performance, but pure jump models are worse than the BS model in American options pricing. Our empirical results show that pure jump model are over-fitting, but not improve American options pricing when they are applied to out-of-sample data.

Option Pricing in the Moderate Deviations Regime

Author : Peter Friz
Publisher :
ISBN 13 :
Total Pages : 20 pages
Book Rating : 4.:/5 (13 download)

DOWNLOAD NOW!

Book Synopsis Option Pricing in the Moderate Deviations Regime by : Peter Friz

Download or read book Option Pricing in the Moderate Deviations Regime written by Peter Friz and published by . This book was released on 2016 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider call option prices in diffusion models close to expiry, in an asymptotic regime ("moderately out of the money") that interpolates between the well-studied cases of at-the-money options and out-of-the-money fixed-strike options. First and higher order small-time moderate deviation estimates of call prices and implied volatility are obtained. The expansions involve only simple expressions of the model parameters, and we show in detail how to calculate them for generic local and stochastic volatility models. Some numerical examples for the Heston model illustrate the accuracy of our results.

Risk Adjustments of Option Prices Under Time-changed Dynamics

Author : Elisa Nicolato
Publisher :
ISBN 13 :
Total Pages : 31 pages
Book Rating : 4.:/5 (13 download)

DOWNLOAD NOW!

Book Synopsis Risk Adjustments of Option Prices Under Time-changed Dynamics by : Elisa Nicolato

Download or read book Risk Adjustments of Option Prices Under Time-changed Dynamics written by Elisa Nicolato and published by . This book was released on 2014 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt: We derive a closed-form expansion of option prices in terms of Black-Scholes prices and higher-order Greeks. We show how the true price of an option less its Black-Scholes price is given by a series of premiums on higher-order risks that are not priced under the Black-Scholes model assumptions. The expansion can be used for a broad class of option pricing models with dynamics governed by time-changed Brownian motions. Specifically, we study expansions for exponential Lévy models such as the Variance Gamma and the Normal Inverse Gaussian models as well as their stochastic volatility counterparts, e.g., the VGSV and NIGSV models. Moreover, we consider extensions of the expansion to a more general subclass of affine jump-diffusion models for which the pricing transform may not be known in closed form.

Specification Analysis of Option Pricing Models Based on Time-Changed Levy Processes

Author : Jing-Zhi Huang
Publisher :
ISBN 13 :
Total Pages : 48 pages
Book Rating : 4.:/5 (129 download)

DOWNLOAD NOW!

Book Synopsis Specification Analysis of Option Pricing Models Based on Time-Changed Levy Processes by : Jing-Zhi Huang

Download or read book Specification Analysis of Option Pricing Models Based on Time-Changed Levy Processes written by Jing-Zhi Huang and published by . This book was released on 2008 with total page 48 pages. Available in PDF, EPUB and Kindle. Book excerpt: We analyze the specifications of option pricing models based on time-changed Levy processes. We classify option pricing models based on the sucture of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the Samp;P 500 index options, we must incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component.

Topics in Numerical Methods for Finance

Author : Mark Cummins
Publisher : Springer Science & Business Media
ISBN 13 : 1461434335
Total Pages : 213 pages
Book Rating : 4.4/5 (614 download)

DOWNLOAD NOW!

Book Synopsis Topics in Numerical Methods for Finance by : Mark Cummins

Download or read book Topics in Numerical Methods for Finance written by Mark Cummins and published by Springer Science & Business Media. This book was released on 2012-07-15 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting state-of-the-art methods in the area, the book begins with a presentation of weak discrete time approximations of jump-diffusion stochastic differential equations for derivatives pricing and risk measurement. Using a moving least squares reconstruction, a numerical approach is then developed that allows for the construction of arbitrage-free surfaces. Free boundary problems are considered next, with particular focus on stochastic impulse control problems that arise when the cost of control includes a fixed cost, common in financial applications. The text proceeds with the development of a fear index based on equity option surfaces, allowing for the measurement of overall fear levels in the market. The problem of American option pricing is considered next, applying simulation methods combined with regression techniques and discussing convergence properties. Changing focus to integral transform methods, a variety of option pricing problems are considered. The COS method is practically applied for the pricing of options under uncertain volatility, a method developed by the authors that relies on the dynamic programming principle and Fourier cosine series expansions. Efficient approximation methods are next developed for the application of the fast Fourier transform for option pricing under multifactor affine models with stochastic volatility and jumps. Following this, fast and accurate pricing techniques are showcased for the pricing of credit derivative contracts with discrete monitoring based on the Wiener-Hopf factorisation. With an energy theme, a recombining pentanomial lattice is developed for the pricing of gas swing contracts under regime switching dynamics. The book concludes with a linear and nonlinear review of the arbitrage-free parity theory for the CDS and bond markets.

Switching Levy Models in Continuous Time

Author : Kyriakos Chourdakis
Publisher :
ISBN 13 :
Total Pages : 39 pages
Book Rating : 4.:/5 (129 download)

DOWNLOAD NOW!

Book Synopsis Switching Levy Models in Continuous Time by : Kyriakos Chourdakis

Download or read book Switching Levy Models in Continuous Time written by Kyriakos Chourdakis and published by . This book was released on 2008 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper introduces a general regime switching Levy process, and constructs the characteristic function in closed form. Correlations between the underlying Markov chain and the asset returns are also allowed, by imposing asset price jumps whenever a regime change takes place. Based on the characteristic function the conditional densities and vanilla option prices can be rapidly computed using FFT. It is shown that the regime switching model has the potential to capture a wide variety of implied volatility skews. The paper also discusses the pricing of exotic contracts, like barrier, Bermudan and American options, by implementation of a quadrature method. A detailed numerical experiment illustrates the application of the regime switching framework.

Stochastic Volatility Models: Option Price Approximation, Asymptotics and Maximum Likelihood Estimation

Author : Jian Yang
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (931 download)

DOWNLOAD NOW!

Book Synopsis Stochastic Volatility Models: Option Price Approximation, Asymptotics and Maximum Likelihood Estimation by : Jian Yang

Download or read book Stochastic Volatility Models: Option Price Approximation, Asymptotics and Maximum Likelihood Estimation written by Jian Yang and published by . This book was released on 2006 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

CEV Asymptotics of American Options

Author : Chi Seng Pun
Publisher :
ISBN 13 :
Total Pages : 31 pages
Book Rating : 4.:/5 (129 download)

DOWNLOAD NOW!

Book Synopsis CEV Asymptotics of American Options by : Chi Seng Pun

Download or read book CEV Asymptotics of American Options written by Chi Seng Pun and published by . This book was released on 2020 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt: The constant elasticity of variance (CEV) model is a practical approach to option pricing by fitting to the implied volatility smile. Its application to American-style derivatives, however, poses analytical and numerical challenges. By taking the Laplace-Carson transform (LCT) to the free-boundary value problem characterizing the option value function and the early exercise boundary, the analytical result involves confluent hyper-geometric functions. Thus, the numerical computation could be unstable and inefficient for certain set of parameter values. We solve this problem by an asymptotic approach to the American option pricing problem under the CEV model. We demonstrate the use of the proposed approach using perpetual and finite-time American puts.

Applied Partial Differential Equations

Author : J. R. Ockendon
Publisher :
ISBN 13 : 9780198527718
Total Pages : 466 pages
Book Rating : 4.5/5 (277 download)

DOWNLOAD NOW!

Book Synopsis Applied Partial Differential Equations by : J. R. Ockendon

Download or read book Applied Partial Differential Equations written by J. R. Ockendon and published by . This book was released on 2003 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: Partial differential equations are used in mathematical models of a huge range of real-world phenomena, from electromagnetism to financial markets. This new edition of Applied PDEs contains many new sections and exercises Including, American options, transform methods, free surface flows, linear elasticity and complex characteristics.

Option Pricing and Estimation of Financial Models with R

Author : Stefano M. Iacus
Publisher : John Wiley & Sons
ISBN 13 : 1119990203
Total Pages : 402 pages
Book Rating : 4.1/5 (199 download)

DOWNLOAD NOW!

Book Synopsis Option Pricing and Estimation of Financial Models with R by : Stefano M. Iacus

Download or read book Option Pricing and Estimation of Financial Models with R written by Stefano M. Iacus and published by John Wiley & Sons. This book was released on 2011-02-23 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents inference and simulation of stochastic process in the field of model calibration for financial times series modelled by continuous time processes and numerical option pricing. Introduces the bases of probability theory and goes on to explain how to model financial times series with continuous models, how to calibrate them from discrete data and further covers option pricing with one or more underlying assets based on these models. Analysis and implementation of models goes beyond the standard Black and Scholes framework and includes Markov switching models, Lévy models and other models with jumps (e.g. the telegraph process); Topics other than option pricing include: volatility and covariation estimation, change point analysis, asymptotic expansion and classification of financial time series from a statistical viewpoint. The book features problems with solutions and examples. All the examples and R code are available as an additional R package, therefore all the examples can be reproduced.