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Pricing American Options Using a Spacetime Adaptive Finite Difference Method
"... American options are priced numerically using a space and timeadaptive finite difference method. The generalized BlackScholes operator is discretized on a Cartesian structured but nonequidistant grid in space. The space and timediscretizations are adjusted such that a predefined tolerance level ..."
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American options are priced numerically using a space and timeadaptive finite difference method. The generalized BlackScholes operator is discretized on a Cartesian structured but nonequidistant grid in space. The space and timediscretizations are adjusted such that a predefined tolerance level on the local discretization error is met. An operator splitting technique is used to separately handle the early exercise constraint and the solution of linear systems of equations from the finite difference discretization of the linear complementarity problem. In numerical experiments three variants of the adaptive timestepping algorithm with and without local timestepping are compared.
An iterative method for pricing American options under jumpdiffusion models,
 Appl. Numer. Math.
, 2011
"... Abstract We consider the numerical pricing of American options under the Bates model which adds lognormally distributed jumps for the asset value to the Heston stochastic volatility model. A linear complementarity problem (LCP) is formulated where partial derivatives are discretized using finite d ..."
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Abstract We consider the numerical pricing of American options under the Bates model which adds lognormally distributed jumps for the asset value to the Heston stochastic volatility model. A linear complementarity problem (LCP) is formulated where partial derivatives are discretized using finite differences and the integral resulting from the jumps is evaluated using simple quadrature. A rapidly converging fixed point iteration is described for the LCP, where each iterate requires the solution of an LCP. These are easily solved using a projected algebraic multigrid (PAMG) method. The numerical experiments demonstrate the efficiency of the proposed approach. Furthermore, they show that the PAMG method leads to better scalability than the projected SOR (PSOR) method when the discretization is refined.
HighOrder Splitting Methods for Forward PDEs and PIDEs.∗
"... This paper is dedicated to the construction of highorder (in both space and time) finitedifference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andrease ..."
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This paper is dedicated to the construction of highorder (in both space and time) finitedifference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreasen & Huge (2011) who reported a pair of consistent finitedifference schemes of firstorder approximation in time for an uncorrelated local stochastic volatility model. We extend their approach by constructing schemes that are secondorder in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue. 1
The Evaluation of American Compound Option Prices under Stochastic Volatility using the Sparse Grid Approach
, 2009
"... A compound option (the mother option) gives the holder the right, but not the obligation, to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we consider the problem of pricing Americantype compound options when the underlying dynamics follow Heston's sto ..."
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A compound option (the mother option) gives the holder the right, but not the obligation, to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we consider the problem of pricing Americantype compound options when the underlying dynamics follow Heston's stochastic volatility and with stochastic interest rate driven by CoxIngersollRoss processes. We use a partial differential equation (PDE) approach to obtain a numerical solution. The problem is formulated as the solution to a twopass freeboundary PDE problem, which is solved via a sparse grid approach and is found to be accurate and efficient compared with the results from a benchmark solution based on a leastsquares Monte Carlo simulation combined with the projected successive overrelaxation method.
Accurate Numerical Method for Pricing TwoAsset American Put Options
"... We develop an accurate finite difference scheme for pricing twoasset American put options. We use the central difference method for space derivatives and the implicit Euler method for the time derivative. Under certain mesh step size limitations, the matrix associated with the discrete operator is ..."
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We develop an accurate finite difference scheme for pricing twoasset American put options. We use the central difference method for space derivatives and the implicit Euler method for the time derivative. Under certain mesh step size limitations, the matrix associated with the discrete operator is an Mmatrix, which ensures that the solutions are oscillationfree. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. It is shown that the scheme is secondorder convergent with respect to the spatial variables. Numerical results support the theoretical results.
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, 2006
"... ���������������������� � ����������������������������������������������������� � Think different, think finite differences. From Sloganizer.netList of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I J. Persson and L. von Sydow (2003). ..."
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���������������������� � ����������������������������������������������������� � Think different, think finite differences. From Sloganizer.netList of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I J. Persson and L. von Sydow (2003). Pricing European Multiasset
American Options under Stochastic Volatility: Control Variates, Maturity Randomization & Multiscale Asymptotics
, 2014
"... American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. ..."
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American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate.
USLV: Unspanned Stochastic Local Volatility Model ∗
, 2013
"... We propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Our model extends the linearitygenerating unspanned volatility term structure model by Ca ..."
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We propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Our model extends the linearitygenerating unspanned volatility term structure model by Carr et al. (2011) by adding a local volatility layer to it. We outline efficient numerical schemes for pricing derivatives in this framework for a particular fourfactor specification (two “curve ” factors plus two “volatility ” factors). We show that the dynamics of such a system can be approximated by a Markov chain on a twodimensional space (Zt, Yt), where coordinates Zt and Yt are given by direct (Kroneker) products of values of pairs of curve and volatility factors, respectively. The resulting Markov chain dynamics on such partly “folded ” state space enables fast pricing by the standard backward induction. Using a nonparametric specification of the Markov chain generator, one can accurately match arbitrary sets of vanilla option quotes with different strikes and maturities. Furthermore, we consider an alternative formulation of the model in terms of an implied time change process. The latter is specified nonparametrically, again enabling accurate calibration to arbitrary sets of vanilla option quotes. ∗Opinions expressed in this paper are those of the authors, and do not necessarily reflect the view of JPMorgan Chase and Numerix. 1 ar