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Numerical Valuation of European and American Options under Kou’s JumpDiffusion Model
"... Numerical methods are developed for pricing European and American options under Kou’s jumpdiffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is logdoubleexponentially distributed. The price of a European o ..."
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Numerical methods are developed for pricing European and American options under Kou’s jumpdiffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is logdoubleexponentially distributed. The price of a European option is given by a partial integrodifferential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are described: an operator slitting method and a penalty method. Numerical experiments confirm that the developed methods are very efficient as fairly accurate option prices can be computed in a few milliseconds on a PC.
The discontinuous Galerkin method for fractal conservation laws
, 2009
"... We propose, analyze, and demonstrate a discontinuous Galerkin method for fractional conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and when piecewise constant elements are utilized, ..."
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We propose, analyze, and demonstrate a discontinuous Galerkin method for fractional conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and when piecewise constant elements are utilized, we prove a rate of convergence toward the unique entropy solution. We present numerical results for different types of solutions of linear and nonlinear fractional conservation laws.
Pricing methods and hedging strategies for volatility derivatives. Working Paper
, 2003
"... In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the effects of variations in the contract design, such as the differences between specifying log returns or actual returns, taking into consideration the impact of possible j ..."
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Cited by 10 (0 self)
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In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the effects of variations in the contract design, such as the differences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the difficulties associated with hedging these products. Naive deltahedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and outofthemoney strangles at each volatility observation. 1
The Effect of Modelling Parameters on the Value of GMWB Guarantees
, 2007
"... In this article, an extensive study of the noarbitrage fee for Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuity riders is carried out. Taking into account such contractual features as the separation of mutual fund fees and the fees earmarked for hedging the guarantee, as well as the po ..."
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Cited by 8 (2 self)
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In this article, an extensive study of the noarbitrage fee for Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuity riders is carried out. Taking into account such contractual features as the separation of mutual fund fees and the fees earmarked for hedging the guarantee, as well as the possibility of jumps in the value of the underlying asset, increases the value of the GMWB guarantee considerably. We also explore the effects of various modelling assumptions on the optimal withdrawal strategy of the contract holder, as well as the impact on the guarantee value of suboptimal withdrawal behaviour. Our general conclusions are that only if several unrealistic modelling assumptions are made is it possible to obtain GMWB fees in the same range as is normally charged. In all other cases, it would appear that typical fees are not enough to cover the cost of hedging these guarantees.
Vetzal: Dynamic Hedging under Jump Diffusion with Transaction Costs
 Journal of Finance
, 1998
"... If the price of an asset follows a jump diffusion process, the market is in general incomplete. In this case, hedging a contingent claim written on the asset is not a trivial matter, and other instruments besides the underlying must be used to hedge in order to provide adequate protection against ju ..."
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Cited by 7 (2 self)
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If the price of an asset follows a jump diffusion process, the market is in general incomplete. In this case, hedging a contingent claim written on the asset is not a trivial matter, and other instruments besides the underlying must be used to hedge in order to provide adequate protection against jump risk. We devise a dynamic hedging strategy that uses a hedge portfolio consisting of the underlying asset and liquidly traded options, where transaction costs are assumed present due to a relative bidask spread. At each rebalance time, the hedge weights are chosen to simultaneously (i) eliminate the instantaneous diffusion risk by imposing delta neutrality; and (ii) minimize an objective that is a linear combination of a jump risk and transaction cost penalty function. Since reducing the jump risk is a competing goal visàvis controlling for transaction cost, the respective components in the objective must be appropriately weighted. Hedging simulations of this procedure are carried out, and our results indicate that the proposed dynamic hedging strategy provides sufficient protection against the diffusion and jump risk while not incurring large transaction costs.
Differencequadrature schemes for nonlinear degenerate parabolic integroPDE
 SIAM J. Numer. Anal
, 2010
"... Abstract. We derive and analyze monotone differencequadrature schemes for Bellman equations of controlled Lévy (jumpdiffusion) processes. These equations are fully nonlinear, degenerate parabolic integroPDEs interpreted in the sense of viscosity solutions. We propose new “direct ” discretization ..."
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Abstract. We derive and analyze monotone differencequadrature schemes for Bellman equations of controlled Lévy (jumpdiffusion) processes. These equations are fully nonlinear, degenerate parabolic integroPDEs interpreted in the sense of viscosity solutions. We propose new “direct ” discretizations of the nonlocal part of the equation that give rise to monotone schemes capable of handling singular Lévy measures. Furthermore, we develop a new general theory for deriving error estimates for approximate solutions of integroPDEs, which thereafter is applied to the proposed differencequadrature schemes. Contents
Numerical Methods for Nonlinear Equations in Option Pricing
, 2003
"... This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illust ..."
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This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options.
Calibration and hedging under jump diffusion
 Review of Derivatives Research
, 2006
"... A jump diffusion model coupled with a local volatility function has been suggested by Andersen and Andreasen (2000). This model is attractive in that it shows promise in terms of being able to capture observed market crosssectional implied volatilities, without being unduly complex. By generating a ..."
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A jump diffusion model coupled with a local volatility function has been suggested by Andersen and Andreasen (2000). This model is attractive in that it shows promise in terms of being able to capture observed market crosssectional implied volatilities, without being unduly complex. By generating a discrete set of American option prices assuming a jump diffusion with known parameters (i.e. in a synthetic market), we investigate two crucial challenges intrinsic to this type of model: calibration of parameters and hedging of jump risk. Our investigation suggests that it can be difficult to estimate the model parameters that govern the jump size distribution. However, the local volatility function is easier to estimate when an appropriate regularization (e.g. splines) is used to avoid overfitting. In general, even though the estimation problem is illposed, it appears that combining jump diffusion with a local volatility function produces a model which can be calibrated with sufficient accuracy to prices of liquid vanilla options. With regard to hedging jump risk, two different hedging strategies are explored: a semistatic approach which uses a portfolio of the underlying and traded short maturity options to hedge a long maturity option, and a dynamic technique which involves frequent trading of options and the underlying. Simulation experiments in the synthetic market suggest that both of these methods can be used to sharply reduce the standard deviation of the hedging portfolio relative profit and loss distribution.
METHODS FOR PRICING AMERICAN OPTIONS UNDER REGIME SWITCHING ∗
"... Abstract. We analyze a number of techniques for pricing American options under a regime switching stochastic process. The techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable. In the case of implicit methods we also c ..."
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Abstract. We analyze a number of techniques for pricing American options under a regime switching stochastic process. The techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable. In the case of implicit methods we also compare a number of iterative procedures for solving the associated nonlinear algebraic equations. Numerical tests indicate that a fixed point policy iteration, coupled with a direct control formulation, is a reliable general purpose method. Finally, we remark that we formulate the American problem as an abstract optimal control problem; hence our results are applicable to more general problems as well.
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.