Results 1  10
of
52
Penalty Methods For American Options With Stochastic Volatility
, 1998
"... The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. ..."
Abstract

Cited by 100 (20 self)
 Add to MetaCart
The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned PCG method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.
Option pricing under a double exponential jump diffusion model
 Management Science
, 2004
"... Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with ..."
Abstract

Cited by 95 (4 self)
 Add to MetaCart
Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with jumps. We demonstrate a double exponential jump diffusion model can lead to an analytic approximation for Þnite horizon American options (by extending the BaroneAdesi and Whaley method) and analytical solutions for popular pathdependent options (such as lookback, barrier, and perpetual American options). Numerical examples indicate that the formulae are easy to be implemented and accurate.
Numerical methods for controlled HamiltonJacobiBellman PDEs in finance
 Journal of Computational Finance
"... Many nonlinear option pricing problems can be formulated as optimal control problems, leading to HamiltonJacobiBellman (HJB) or HamiltonJacobiBellmanIsaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergenc ..."
Abstract

Cited by 31 (13 self)
 Add to MetaCart
(Show Context)
Many nonlinear option pricing problems can be formulated as optimal control problems, leading to HamiltonJacobiBellman (HJB) or HamiltonJacobiBellmanIsaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB type equations, we can guarantee convergence of a Newtontype (Policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newtontype iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees.
A semiLagrangian approach for natural gas storage valuation and optimal operation
, 2006
"... The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a HamiltonJacobiBellman (HJB) equation. In this paper, we present a semiLagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a ..."
Abstract

Cited by 30 (5 self)
 Add to MetaCart
The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a HamiltonJacobiBellman (HJB) equation. In this paper, we present a semiLagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a wide class of spot price models that exhibit meanreverting, seasonality dynamics and price jumps. We develop fully implicit and CrankNicolson timestepping schemes based on a semiLagrangian approach and prove the convergence of fully implicit timestepping to the viscosity solution of the HJB equation. We show that fully implicit timestepping is equivalent to a discrete control strategy, which allows for a convenient interpretation of the optimal controls. The semiLagrangian approach avoids the nonlinear iterations required by an implicit finite difference method without requiring additional cost. Numerical experiments are presented for several variants of the basic scheme.
A SemiLagrangian approach for American Asian options under jump diffusion
 SIAM Journal on Scientific Computing
, 2003
"... version 1.7 A semiLagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semiLagrangian timestepping. CrankNicolson ..."
Abstract

Cited by 22 (7 self)
 Add to MetaCart
(Show Context)
version 1.7 A semiLagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semiLagrangian timestepping. CrankNicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. With low volatility values, it is observed that the nonsmoothness at the strike in the payoff affects the convergence rate; subquadratic convergence rate is observed.
Pricing Options in JumpDiffusion Models: An Extrapolation Approach
, 2008
"... We propose a new computational method for the valuation of options in jumpdiffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicitexplicit (IMEX) time discretization sc ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
We propose a new computational method for the valuation of options in jumpdiffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicitexplicit (IMEX) time discretization schemes, where the differential (diffusion) term is treated implicitly, while the integral (jump) term is treated explicitly. In particular, the popular IMEX Euler scheme is firstorder accurate in time. Secondorder accuracy in time can be achieved by using the IMEX midpoint scheme. In contrast to the above approaches, we propose a new highorder time discretization scheme for the PIDE based on the extrapolation approach to the solution of ODEs that also treats the diffusion term implicitly and the jump term explicitly. The scheme is simple to implement, can be added to any PIDE solver based on the IMEX Euler scheme, and is remarkably fast and accurate. We demonstrate our approach on the examples of Merton’s and Kou’s jumpdiffusion models, the diffusionextended variance gamma model, as well as the twodimensional DuffiePanSingleton model with correlated and contemporaneous jumps in the stock price and its volatility. By way of example, pricing a oneyear doublebarrier option in Kou’s jumpdiffusion model, our scheme attains accuracy of 10−5 in 72 time steps (in 0.05 seconds). In contrast, it takes the firstorder IMEX Euler scheme more than 1.3 million time steps (in 873 seconds) and the secondorder IMEX midpoint scheme 768 time steps (in 0.49 seconds) to attain the same accuracy. Our scheme is also well suited for Bermudan options. Combining simplicity of implementation and remarkable gains in computational efficiency, we expect this method to be very attractive
Robust Numerical Valuation of European and American Options under the CGMY Process
, 2007
"... We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processe ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
(Show Context)
We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semiLagrangian scheme. The resulting Partial IntegroDifferential Equation (PIDE) is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.
Componentwise splitting methods for pricing American options under stochastic volatility
 Int. J. Theor. Appl. Finance
, 2007
"... Abstract A linear complementarity problem (LCP) is formulated for the price of American options under the Bates model which combines the Heston stochastic volatility model and the Merton jumpdiffusion model. A finite difference discretization is described for the partial derivatives and a simple qu ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
(Show Context)
Abstract A linear complementarity problem (LCP) is formulated for the price of American options under the Bates model which combines the Heston stochastic volatility model and the Merton jumpdiffusion model. A finite difference discretization is described for the partial derivatives and a simple quadrature is used for the integral term due to jumps. A componentwise splitting method is generalized for the Bates model. It is leads to solution of sequence of onedimensional LCPs which can be solved very efficiently using the Brennan and Schwartz algorithm. The numerical experiments demonstrate the componentwise splitting method to be essentially as accurate as the PSOR method, but order of magnitude faster. Furthermore, pricing under the Bates model is less than twice more expensive computationally than under the Heston model in the experiments. 1
Asymptotics for exponential Lévy processes and their volatility smile: survey and new results
 Int. J. Theor. Appl. Finance
, 2013
"... Abstract. Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfac ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the nonasymptotic regimes, option prices are described by the LewisLipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the implied volatility have yielded several expressions for the shorttime, longtime, and wing asymptotics. In order to study the volatility surface in required detail, in this paper we use the FX conventions and describe the implied volatility as a function of the BlackScholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts regarding the asymptotic behavior of the implied volatility, and present several
Operator splitting methods for pricing American options with stochastic volatility
, 2004
"... Summary. Pricing American options using partial (integro)differential equation based methods leads to linear complementarity problems (LCPs). The numerical solution of these problems resulting from the BlackScholes model, Kou’s jumpdiffusion model, and Heston’s stochastic volatility model are cons ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Summary. Pricing American options using partial (integro)differential equation based methods leads to linear complementarity problems (LCPs). The numerical solution of these problems resulting from the BlackScholes model, Kou’s jumpdiffusion model, and Heston’s stochastic volatility model are considered. The finite difference discretization is described. The solutions of the discrete LCPs are approximated using an operator splitting method which separates the linear problem and the early exercise constraint to separate fractional steps. The numerical experiments demonstrate that the price of options can be computed in a few milliseconds on a PC. 1