Author :
Publisher :
Page : 27 pages
File Size : 24,63 MB
Release : 1996
Category :
ISBN :
It is explored how incomplete markets can be studied with the help of asymptotics. A compound Poisson model for the stock price is assumed and an expansion for the price of a European option is obtained as the stock price process converges to a geometric Brownian motion. This formulation also permits one to confront statistical uncertainty in the volatility of the stock price, and we show how this uncertainty impacts on the value of the option.
Author : Yoshio Miyahara
Publisher : World Scientific
Page : 200 pages
File Size : 44,15 MB
Release : 2012
Category : Electronic books
ISBN : 1848163487
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
Author : David Krief
Publisher :
Page : 0 pages
File Size : 44,47 MB
Release : 2018
Category :
ISBN :
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.
Author : Jennifer (Bo) Liang
Publisher :
Page : 148 pages
File Size : 37,24 MB
Release : 1997
Category : Asymptotic expansions
ISBN :
Author :
Publisher :
Page : pages
File Size : 41,42 MB
Release : 2007
Category :
ISBN :
The thesis is divided into two parts: The first part deals with modelling long-range dependence in asset returns. Certain long-range dependence models, which have been suggested for financial modelling, fall outside the semimartingale set-up. We suggest Poisson shot noise processes as a skeleton of a long-range dependence model which provides an economic reasoning for long memory. We study weak convergence to a fractional Brownian motion. Whereas fractional Brownian motion allows for arbitrage, the shot noise processes themselves can be chosen arbitrage-free. As complement we also investigate shot noise processes which consist of shots with finite limits. They converge to a Brownian motion, i.e. they have the same asymptotic behaviour as compound Poisson processes. In the second part of the thesis we analyze American options and so-called "game options" in a general semimartingale setting. Game options naturally generalize American options by giving both counterparites the right to cancel the contract prematurely. Whereas in recent years various suggestions have been made how to price European-type contingent claims in incomplete markets, up to now there is only little corresponding literature dealing with American options. Pricing the latter is conceptually more involved: in addition to the uncertainty caused by the underlyings, one has to take the seller's ignorance of the buyer's exercise strategy into account. We generalize the "neutral derivative pricing" approach to American and game options which leads to unique "neutral" derivative price processes in incomplete markets. An alternative approach is "utility-based indifference pricing" which was firstly suggested by Hodges and Neuberger (1989) and which ist by now a standard concept to valuate European style derivatives in incomplete markets. We generalize this concept to American options and so-called "chooser options". It leads to a quite surprising result con.
Author : Julien Guyon
Publisher : CRC Press
Page : 486 pages
File Size : 22,83 MB
Release : 2013-12-19
Category : Business & Economics
ISBN : 1466570334
New Tools to Solve Your Option Pricing Problems For nonlinear PDEs encountered in quantitative finance, advanced probabilistic methods are needed to address dimensionality issues. Written by two leaders in quantitative research—including Risk magazine’s 2013 Quant of the Year—Nonlinear Option Pricing compares various numerical methods for solving high-dimensional nonlinear problems arising in option pricing. Designed for practitioners, it is the first authored book to discuss nonlinear Black-Scholes PDEs and compare the efficiency of many different methods. Real-World Solutions for Quantitative Analysts The book helps quants develop both their analytical and numerical expertise. It focuses on general mathematical tools rather than specific financial questions so that readers can easily use the tools to solve their own nonlinear problems. The authors build intuition through numerous real-world examples of numerical implementation. Although the focus is on ideas and numerical examples, the authors introduce relevant mathematical notions and important results and proofs. The book also covers several original approaches, including regression methods and dual methods for pricing chooser options, Monte Carlo approaches for pricing in the uncertain volatility model and the uncertain lapse and mortality model, the Markovian projection method and the particle method for calibrating local stochastic volatility models to market prices of vanilla options with/without stochastic interest rates, the a + bλ technique for building local correlation models that calibrate to market prices of vanilla options on a basket, and a new stochastic representation of nonlinear PDE solutions based on marked branching diffusions.
Author : Lishang Jiang
Publisher : World Scientific Publishing Company
Page : 343 pages
File Size : 42,78 MB
Release : 2005-07-18
Category : Business & Economics
ISBN : 9813106557
From the unique perspective of partial differential equations (PDE), this self-contained book presents a systematic, advanced introduction to the Black-Scholes-Merton's option pricing theory.A unified approach is used to model various types of option pricing as PDE problems, to derive pricing formulas as their solutions, and to design efficient algorithms from the numerical calculation of PDEs. In particular, the qualitative and quantitative analysis of American option pricing is treated based on free boundary problems, and the implied volatility as an inverse problem is solved in the optimal control framework of parabolic equations.
Author : Stefan Rostek
Publisher : Springer
Page : 137 pages
File Size : 22,43 MB
Release : 2009-08-29
Category : Business & Economics
ISBN : 9783642003707
Mandelbrot and van Ness (1968) suggested fractional Brownian motion as a parsimonious model for the dynamics of ?nancial price data, which allows for dependence between returns over time. Starting with Rogers(1997) there is an ongoing dispute on the proper usage of fractional Brownian motion in option pricing theory. Problems arise because fractional Brownian motion is not a semimartingale and therefore “no arbitrage pricing” cannot be applied. While this is consensus, the consequences are not as clear. The orthodox interpretation is simply that fractional Brownian motion is an inadequate candidate for a price process. However, as shown by Cheridito (2003) any theoretical arbitrage opportunities disappear by assuming that market p- ticipants cannot react instantaneously. This is the point of departure of Rostek’s dissertation. He contributes to this research in several respects: (i) He delivers a thorough introduction to fr- tional integration calculus and uses the binomial approximation of fractional Brownianmotion to give the reader a ?rst idea of this special market setting.
Author : Christophe Chorro
Publisher : Springer
Page : 202 pages
File Size : 30,89 MB
Release : 2014-12-04
Category : Business & Economics
ISBN : 3662450372
The current world financial scene indicates at an intertwined and interdependent relationship between financial market activity and economic health. This book explains how the economic messages delivered by the dynamic evolution of financial asset returns are strongly related to option prices. The Black Scholes framework is introduced and by underlining its shortcomings, an alternative approach is presented that has emerged over the past ten years of academic research, an approach that is much more grounded on a realistic statistical analysis of data rather than on ad hoc tractable continuous time option pricing models. The reader then learns what it takes to understand and implement these option pricing models based on time series analysis in a self-contained way. The discussion covers modeling choices available to the quantitative analyst, as well as the tools to decide upon a particular model based on the historical datasets of financial returns. The reader is then guided into numerical deduction of option prices from these models and illustrations with real examples are used to reflect the accuracy of the approach using datasets of options on equity indices.
Author :
Publisher :
Page : 382 pages
File Size : 41,50 MB
Release : 2007
Category :
ISBN :
