Math Finance Publications

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A. Itkin

Pricing options with VG model using FFT

We discuss various analytic and numerical methods that have been used to get option prices within a framework of the VG model. We show that some popular methods, for instance, Carr-Madan's FFT method could blow up for certain values of the model parameters even for an European vanilla option. Alternative methods - one originally proposed by Lewis, and Black-Scholes-wise method are considered that seem to work fine for any value of the VG parameters. Test examples are given to demonstrate efficiency of these methods. Convergency of all methods is also discussed

arXiv.org > physics > physics/0503137

Extended version prepared for the VG Conference (see Presentations)

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A. Itkin, P.Carr

Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models (draft version)

ArXiv.Org #1002.1995v1

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Articles in journals

Authors	Title	Abstract	Published	Link	Text
A. Itkin, P. Carr	Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case	We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As such a process we consider a class of L\'evy models with stochastic time change. Our analysis reveals a natural small parameter of the problem which allows a general asymptotic method to be developed in order to obtain a closed-form expression for the fair price of the above products. As examples, we consider the CIR clock change, general affine models of activity rates and the 3/2 power clock change, and give an analytical expression of the swap price. Comparison of the results obtained with a familiar log-contract approach is provided.	Review of Derivatives Research, Vol.13, N.2, 2010, p.141-176.	$Description: C:\AndreyItkin\My Documents\WebPage\rdr.jpg$	$Description: C:\AndreyItkin\My Documents\WebPage\readme.gif$
A. Itkin, P. Carr	Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models	In mathematical finance a popular approach for pricing options under some L\'evy model would be to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE) that usually does not allow an analytical solution, while a numerical solution also faces some problems. In this paper we develop a new approach on how to transform the PIDE into a class of so-called pseudo-parabolic equations which are well known in mathematical physics but are relatively new for mathematical finance. As an example we will discuss several jump-diffusion models which L\'evy measure allows such a transformation.	Computational Economics (forthcoming)		$Description: C:\AndreyItkin\My Documents\WebPage\readme.gif$
A. Itkin, P. Carr	Jumps without Tears: A New Splitting Technology for Barrier Options	The market pricing of OTC FX options displays both stochastic volatility and stochastic skewness in the risk-neutral distribution governing currency returns. To capture this unique phenomenon Carr and Wu developed a model (SSM) with three dynamical state variables. They then used Fourier methods to value simple European-style options. However pricing exotic options requires numerical solution of 3D unsteady PIDE with mixed derivatives which is expensive. In this paper to achieve this goal we propose a new splitting technique. Being combined with another method of the authors, which uses pseudo-parabolic PDE instead of PIDE, this reduces the original 3D unsteady problem to a set of 1D unsteady PDEs, thus allowing a significant computational speedup. We demonstrate this technique for single and double barrier options priced using the SSM.	International Journal of Numerical Analysis and Modeling, v.8, N.4, 2011, pp 667–704.	$Description: C:\AndreyItkin\My Documents\WebPage\ijnam.bmp$	$Description: C:\AndreyItkin\My Documents\WebPage\readme.gif$
A. Itkin	New solvable stochastic volatility models for pricing volatility	Classical solvable stochastic volatility models (SVM) use a CEV process for instantaneous variance where the CEV parameter gamma takes just few values: 0 - the Ornstein-Uhlenbeck process, 1/2 - the Heston (or square root) process, 1- GARCH, and 3/2 - the 3/2 model. Some other models were discovered in Labordere, 2009 by making connection between stochastic volatility and solvable diffusion processes in quantum mechanics. In particular, he used to build a bridge between solvable (super)potentials (the Natanzon (super)potentials, which allow reduction of a Schrodinger equation to a Gauss confluent hypergeometric equation) and existing SVM. In this paper we discuss another approach to extend the class of solvable SVM in terms of hypergeometric functions. Thus obtained new models could be useful for pricing volatility derivatives (variance and volatility swaps, moment swaps	Submitted to Review of Derivatives Research,	$Description: C:\AndreyItkin\My Documents\WebPage\rdr.jpg$	$Description: C:\AndreyItkin\My Documents\WebPage\readme.gif$

Math Finance Publications

Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models (draft version)

Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case

Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models

Jumps without Tears: A New Splitting Technology for Barrier Options