Pricing Average Options Under Time Changed Levy Processes
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Author | : Akira Yamazaki |
Publisher | : |
Total Pages | : 31 |
Release | : 2014 |
Genre | : |
ISBN | : |
This paper presents an approximate formula for pricing average options when the underlying asset price is driven by time-changed Levy processes. Time-changed Levy processes are attractive to use for a driving factor of underlying prices because the processes provide a flexible framework for generating jumps, capturing stochastic volatility as the random time change, and introducing the leverage effect. There have been very few studies dealing with pricing problems of exotic derivatives on time-changed Levy processes in contrast to standard European derivatives. Our pricing formula is based on the Gram-Charlier expansion and the key of the formula is to find analytic treatments for computing the moments of the normalized average asset price. In numerical examples, we demonstrate that our formula give accurate values of average call options when adopting Heston's stochastic volatility model, VG-CIR, and NIG-CIR models.
Author | : Peter Carr |
Publisher | : |
Total Pages | : 35 |
Release | : 2001 |
Genre | : |
ISBN | : |
We apply stochastic time change to Levy processes to generate a wide variety of tractable option pricing models. In particular, we prove a fundamental theorem that transforms the characteristic function of the time-changed Levy process into the Laplace transform of the stochastic time under appropriate measure change. We extend the traditional measure theory into the complex domain and define the measure change by a class of complex valued exponential martingales. We provide extensive examples to illustrate its applications and its link to existing models in the literature.
Author | : Jing-Zhi Huang |
Publisher | : |
Total Pages | : 48 |
Release | : 2008 |
Genre | : |
ISBN | : |
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.
Author | : Yoshio Miyahara |
Publisher | : World Scientific |
Total Pages | : 200 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 1848163479 |
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 Lvy 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 problems.
Author | : Philipp Beyer |
Publisher | : |
Total Pages | : 0 |
Release | : 2009 |
Genre | : |
ISBN | : |
Options depending on the forward skew are very popular. One such option is the forward starting call option - the basic building block of a cliquet option. Widely applied models to account for the forward skew dynamics to price such options include the Heston model, the Heston-Hull-White model and the Bates model. Within these models solutions for options including forward start features are available using (semi) analytical formulas. Today exponential (subordinated) Levy models being increasingly popular for modelling the asset dynamics. While the simple exponential Levy models imply the same forward volatily surface for all future times the subordinated models do not. Depending on the subordinator the dynamic of the forward volatility surface and therefore stochastic volatility can be modelled. Analytical pricing formulas based on the characteristic function and Fourier transform methods are available for the class of these models. We extend the applicability of analytical pricing to options including forward start features. To this end we derive the forward characteristic functions which can be used in Fourier transform based methods. As examples we consider the Variance Gamma model and the NIG model subordinated by a Gamma Ornstein Uhlenbeck process and respectively by an Cox-Ingersoll-Ross process. We check our analytical results by applying Monte Carlo methods. These results can for instance be applied to calibration of the forward volatility surface.
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 | : Peter Tankov |
Publisher | : CRC Press |
Total Pages | : 552 |
Release | : 2003-12-30 |
Genre | : Business & Economics |
ISBN | : 1135437947 |
WINNER of a Riskbook.com Best of 2004 Book Award! During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematic
Author | : Conall O'Sullivan |
Publisher | : |
Total Pages | : 24 |
Release | : 2005 |
Genre | : |
ISBN | : |
A model is developed that can price path dependent options when the underlying process is an exponential Levy process with closed form conditional characteristic function. The model is an extension of a recent quadrature option pricing model so that it can be applied with the use of Fourier and Fast Fourier transforms. Thus the model possesses nice features of both transform and quadrature option pricing techniques since it can be applied for a very general set of underlying Levy processes and can handle exotic path dependent features. The model is applied to European and Bermudan options for geometric Brownian motion, a jump-diffusion process, a variance gamma process and a normal inverse Gaussian process. However it must be noted that the model can also price other path dependent exotic options such as lookback and Asian options.
Author | : Jiayao Xie |
Publisher | : |
Total Pages | : |
Release | : 2012 |
Genre | : |
ISBN | : |
ABSTRACT. We develop a new method for pricing options on discretely sampled arithmetic average in exponential Levy models. The main idea is the reduction to a backward in- duction procedure for the difference Wn between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This al- lows for an efficient truncation of the state space. At each step of backward induction, Wn is calculated accurately and fast using a piece-wise interpolation or splines, fast convolu- tion and either flat iFT and (refined) iFFT or the parabolic iFT. Numerical results demonstrate the advantages of the method. Keywords: Option pricing, flat iFT method, parabolic iFT method, FFT, refined and enhanced FFT, Levy processes, KoBoL, CGMY, BM, Asian options.
Author | : Jasper Valstar |
Publisher | : |
Total Pages | : 56 |
Release | : 2008 |
Genre | : |
ISBN | : |