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53
Numerical Valuation of High Dimensional Multivariate American Securities
, 1994
"... We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted ..."
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Cited by 130 (0 self)
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We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process. Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the
Robust Numerical Methods for PDE Models of Asian Options
- Journal of Computational Finance
, 1998
"... We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this p ..."
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Cited by 58 (15 self)
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We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. We adapt flux limiting techniques originally developed in the field of computational fluid dynamics in order to rapidly obtain accurate solutions. We show that flux limiting methods are total variation diminishing (and hence free of spurious oscillations) for non-conservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit schemes. We also modify the van Leer flux limiter so that the second-order total variation diminishing property is preserved for non-uniform grid spacing. 1 Introduction Asian options are securities with payoffs...
Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance
- NONLINEAR PROBLEMS IN MATHEMATICAL PHYSICS AND RELATED TOPICS VOL. II IN HONOR OF PROFESSOR O.A. LADYZHENSKAYA”. INTERNATIONAL MATHEMATICAL SERIES
, 2002
"... This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorov-type operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, ar ..."
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Cited by 24 (15 self)
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This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorov-type operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, are also announced.
Competitive Monte Carlo methods for the Pricing of Asian Options
- Journal of Computational Finance
, 2000
"... We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithm for the computation of the price of Asian options. We give evidence of the eciency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods. K ..."
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Cited by 23 (2 self)
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We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithm for the computation of the price of Asian options. We give evidence of the eciency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods. Key Words: Asian option, Monte Carlo methods, Numerical methods, Diusion process. 1 Introduction Monte Carlo methods are known to be useful when the state dimension is large. This is widely true but we will give here an example of a small dimension problem coming from nance where a Monte Carlo (helped by a variance reduction technique) can be more ecient than other known methods. This example is based on the price of an Asian option (see subsection 2.1). This problem is known to be computationally hard and a lot of literature deals with this problem: using either analytic methods ([10], [9]), numerical methods based on the partial dierential equation associated ([4], [7], [12], [16]) or M...
A Semi-Lagrangian approach for American Asian options under jump diffusion
- SIAM Journal on Scientific Computing
, 2003
"... version 1.7 A semi-Lagrangian 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 semi-Lagrangian timestepping. Crank-Nicolson ..."
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Cited by 22 (7 self)
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version 1.7 A semi-Lagrangian 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 semi-Lagrangian timestepping. Crank-Nicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. With low volatility values, it is observed that the non-smoothness at the strike in the payoff affects the convergence rate; sub-quadratic convergence rate is observed.
Double-exponential fast Gauss transform algorithms for pricing discrete lookback options
, 2005
"... This paper presents fast and accurate algorithms for computing the prices of discretely sampled lookback options. Under the Black-Scholes framework, the pricing of a discrete lookback option can be reduced to a series of convolutions of a function with the Gaussian distribution. Using this fact, an ..."
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Cited by 20 (0 self)
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This paper presents fast and accurate algorithms for computing the prices of discretely sampled lookback options. Under the Black-Scholes framework, the pricing of a discrete lookback option can be reduced to a series of convolutions of a function with the Gaussian distribution. Using this fact, an efficient algorithm, which computes these convolutions by a combination of the double-exponential integration formula and the fast Gauss transform, has been proposed recently. We extend this algorithm to lookback options under Merton’s jump-diffusion model and American lookback options. Numerical experiments show that our method is much faster and more accurate than conventional methods for lookback options under Merton’s model. For American lookback options, our method outperforms conventional methods when required accuracy is relatively high. A lookback option is the right to sell an asset at the end of a time period at the highest price the asset took during the period (lookback put option),
Free boundary and optimal stopping problems for American Asian options
- Finance and Stochastics
"... Abstract We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent p ..."
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Cited by 19 (9 self)
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Abstract We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.
