Option Pricing For Models With Known Characteristic Function
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| Author | : Claudia Szytniewski |
| Publisher | : |
| Total Pages | : 116 |
| Release | : 2018-08-11 |
| Genre | : |
| ISBN | : 9781718121898 |
This thesis is a comprehensive review of option pricing when the characteristic function is known. It describes how characteristic functions are derived and it contains a full description of the Fourier Inversion technique that is used to retrieve option prices from characteristic functions. It also includes a comprehensive review of problems that have appeared throughout the last ten years when implementing those models.
| Author | : Andreas Kyprianou |
| Publisher | : John Wiley & Sons |
| Total Pages | : 344 |
| Release | : 2006-06-14 |
| Genre | : Business & Economics |
| ISBN | : 0470017201 |
Since around the turn of the millennium there has been a general acceptance that one of the more practical improvements one may make in the light of the shortfalls of the classical Black-Scholes model is to replace the underlying source of randomness, a Brownian motion, by a Lévy process. Working with Lévy processes allows one to capture desirable distributional characteristics in the stock returns. In addition, recent work on Lévy processes has led to the understanding of many probabilistic and analytical properties, which make the processes attractive as mathematical tools. At the same time, exotic derivatives are gaining increasing importance as financial instruments and are traded nowadays in large quantities in OTC markets. The current volume is a compendium of chapters, each of which consists of discursive review and recent research on the topic of exotic option pricing and advanced Lévy markets, written by leading scientists in this field. In recent years, Lévy processes have leapt to the fore as a tractable mechanism for modeling asset returns. Exotic option values are especially sensitive to an accurate portrayal of these dynamics. This comprehensive volume provides a valuable service for financial researchers everywhere by assembling key contributions from the world's leading researchers in the field. Peter Carr, Head of Quantitative Finance, Bloomberg LP. This book provides a front-row seat to the hottest new field in modern finance: options pricing in turbulent markets. The old models have failed, as many a professional investor can sadly attest. So many of the brightest minds in mathematical finance across the globe are now in search of new, more accurate models. Here, in one volume, is a comprehensive selection of this cutting-edge research. Richard L. Hudson, former Managing Editor of The Wall Street Journal Europe, and co-author with Benoit B. Mandelbrot of The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward
| Author | : Yannick Dillschneider |
| Publisher | : |
| Total Pages | : 12 |
| Release | : 2019 |
| Genre | : |
| ISBN | : |
In this paper, we derive explicit expressions for certain joint moments of stock prices and option prices within a generic affine stochastic volatility model. Evaluation of each moment requires weighted inverse Fourier transformation of a function that is determined by the risk-neutral and real-world characteristic functions of the state vector. Explicit availability of such moment expressions allows to devise a novel GMM approach to jointly estimate real-world and risk-neutral parameters of affine stochastic volatility models using observed individual option prices. Moreover, the moment expressions may be used to include option price information into other existing moment-based estimation approaches.
| Author | : Roger Lord |
| Publisher | : Rozenberg Publishers |
| Total Pages | : 211 |
| Release | : 2008 |
| Genre | : |
| ISBN | : 9051709099 |
| Author | : Stefano M. Iacus |
| Publisher | : John Wiley & Sons |
| Total Pages | : 402 |
| Release | : 2011-02-23 |
| Genre | : Business & Economics |
| ISBN | : 1119990203 |
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.
| Author | : Alexey Polishchuk |
| Publisher | : |
| Total Pages | : 16 |
| Release | : 2015 |
| Genre | : |
| ISBN | : |
We describe an optimal numerical procedure for computing Fourier inversion integrals necessary for pricing vanilla options in models where the characteristic function is known in closed form. Our approach is motivated by a need for a procedure that works across all model parameters and all maturities. The procedure consists of regularizing the pricing integrand to render it finite in the region where the characteristic function is known to be analytic and then choosing the integration contour which is optimal for inverting Fourier integrals. Regularization of the pricing integrand is achieved by subtracting the Black-Scholes model with a specifically chosen volatility parameter.
| Author | : |
| Publisher | : |
| Total Pages | : 111 |
| Release | : 2008 |
| Genre | : Characteristic functions |
| ISBN | : |
Pricing problems of financial derivatives are among the most important ones in Quantitative Finance. Since 1973 when a Nobel prize winning model was introduced by Black, Merton and Scholes the Brownian Motion (BM) process gained huge attention of professionals professionals. It is now known, however, that stock market log-returns do not follow the very popular BM process. Derivative pricing models which are based on more general Lévy processes tend to perform better. --Carr & Madan (1999) and Lewis (2001) (CML) developed a method for vanilla options valuation based on a characteristic function of asset log-returns assuming that they follow a Lévy process. Assuming that at least part of the problem is in adequate modeling of the distribution of log-returns of the underlying price process, we use instead a nonparametric approach in the CML formula and replaced the unknown characteristic function with its empirical version, the Empirical Characteristic Functions (ECF). We consider four modifications of this model based on the ECF. The first modification requires only historical log-returns of the underlying price process. The other three modifications of the model need, in addition, a calibration based on historical option prices. We compare their performance based on the historical data of the DAX index and on ODAX options written on the index between the 1st of June 2006 and the 17th of May 2007. The resulting pricing errors show that one of our models performs, at least in the cases considered in the project, better than the Carr & Madan (1999) model based on calibration of a parametric Lévy model, called a VG model. --Our study seems to confirm a necessity of using implied parameters, apart from an adequate modeling of the probability distribution of the asset log-returns. It indicates that to precisely reproduce behaviour of the real option prices yet other factors like stochastic volatility need to be included in the option pricing model. Fortunately the discrepancies between our model and real option prices are reduced by introducing the implied parameters which seem to be easily modeled and forecasted using a mixture of regression and time series models. Such approach is computationaly less expensive than the explicit modeling of the stochastic volatility like in the Heston (1993) model and its modifications.
| Author | : Jianwei Zhu |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 181 |
| Release | : 2013-04-17 |
| Genre | : Business & Economics |
| ISBN | : 3662043092 |
From a technical point of view, the celebrated Black and Scholes option pricing formula was originally developed using a separation of variables technique. However, already Merton mentioned in his seminal 1973 pa per, that it could have been developed by using Fourier transforms as well. Indeed, as is well known nowadays, Fourier transforms are a rather convenient solution technique for many models involving the fundamental partial differential equation of financial economics. It took the community nearly another twenty years to recognize that Fourier transform is even more useful, if one applies it to problems in financial economics without seeking an explicit analytical inverse trans form. Heston (1993) probably was the first to demonstrate how to solve a stochastic volatility option pricing model quasi analytically using the characteristic function of the problem, which is nothing else than the Fourier transform of the underlying Arrow /Debreu-prices, and doing the inverse transformation numerically. This opened the door for a whole bunch of new closed form solutions in the transformed Fourier space and still is one of the most active research areas in financial economics.
| Author | : Ali Hirsa |
| Publisher | : Academic Press |
| Total Pages | : 456 |
| Release | : 2013-12-18 |
| Genre | : Business & Economics |
| ISBN | : 0123846838 |
An Introduction to the Mathematics of Financial Derivatives is a popular, intuitive text that eases the transition between basic summaries of financial engineering to more advanced treatments using stochastic calculus. Requiring only a basic knowledge of calculus and probability, it takes readers on a tour of advanced financial engineering. This classic title has been revised by Ali Hirsa, who accentuates its well-known strengths while introducing new subjects, updating others, and bringing new continuity to the whole. Popular with readers because it emphasizes intuition and common sense, An Introduction to the Mathematics of Financial Derivatives remains the only "introductory" text that can appeal to people outside the mathematics and physics communities as it explains the hows and whys of practical finance problems. - Facilitates readers' understanding of underlying mathematical and theoretical models by presenting a mixture of theory and applications with hands-on learning - Presented intuitively, breaking up complex mathematics concepts into easily understood notions - Encourages use of discrete chapters as complementary readings on different topics, offering flexibility in learning and teaching
| Author | : Eric Jondeau |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 541 |
| Release | : 2007-04-05 |
| Genre | : Mathematics |
| ISBN | : 1846286964 |
This book examines non-Gaussian distributions. It addresses the causes and consequences of non-normality and time dependency in both asset returns and option prices. The book is written for non-mathematicians who want to model financial market prices so the emphasis throughout is on practice. There are abundant empirical illustrations of the models and techniques described, many of which could be equally applied to other financial time series.
