## Conergence of Monte Carlo algorithms for pricing American options (2002)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Egloff02conergenceof,

author = {Daniel Egloff and Maung Min-oo},

title = {Conergence of Monte Carlo algorithms for pricing American options },

year = {2002}

}

### OpenURL

### Abstract

In this paper we study the convergence of the Longstaff-Schwartz algorithm for the valuation of American options. Our approach is based on empirical risk minimization initiated by Vapnik and Chervonenkis in the early 1970’s and empirical processes techniques. This allows us to prove convergence, derive error estimates and a Central Limit Theorem for the sample estimators. It also opens up a variety of extensions and generalizations.

### Citations

1020 | A Probabilistic Theory of Pattern Recognition - Devroye, Gyorfi, et al. - 1996 |

959 |
On the uniform convergence of relative frequencies of events to their probabilities
- Vapnik, Chervonenkis
- 1971
(Show Context)
Citation Context ...space. The sample error, which is our main concern in this paper, is analyzed in the framework of empirical risk minimization, which has been promoted by Vapnik and Chervonenkis in a series of papers =-=[44, 45, 46]-=- since the early 1970’s. Let X be a random variable defined on a probability space (Ω, P ), taking values in an arbitrary set S, and let G be a class of functions defined on S. Consider a risk functio... |

824 |
Estimation of Dependencies Based on Empirical Data
- Vapnik
- 1982
(Show Context)
Citation Context ...space. The sample error, which is our main concern in this paper, is analyzed in the framework of empirical risk minimization, which has been promoted by Vapnik and Chervonenkis in a series of papers =-=[44, 45, 46]-=- since the early 1970’s. Let X be a random variable defined on a probability space (Ω, P ), taking values in an arbitrary set S, and let G be a class of functions defined on S. Consider a risk functio... |

607 | Option pricing: a simplified approach
- Cox
- 1979
(Show Context)
Citation Context ...res has been considered as an important problem in Computational Finance. Several methods have been developed to numerically solve the related optimal stopping problem. They range from binomial trees =-=[9]-=-, Markov chain approximations [25], to semi-analytical approximations [2], direct integral equation methods, and PDE methods on the basis of variational inequalities, [3, 20], the linear complementary... |

593 | Convergence of Stochastic Processes - Pollard - 1984 |

543 |
Local polynomial modelling and its applications
- Fan, Gijbels
- 1996
(Show Context)
Citation Context ...neralization are local approximation schemes, such as knearest neighbor and other kernel estimators like the Nadaraya-Watson estimator or more generally local polynomial kernel regression estimators, =-=[15]-=-. These types of architectures are appealing because their asymptotic optimality. Stone showed in [38] that if the regression function m(x) = E[Y | X = x] is p-smooth, X, Y are bounded and X has a den... |

387 |
Stochastic Approximation Algorithms and Applications
- Kushner, Yin
- 1997
(Show Context)
Citation Context ...pendently proposed a simpler algorithm for infinite horizon discrete time optimal stopping problem using also parametric approximations. Their approach is based on stochastic approximation techniques =-=[23, 4, 24]-=- and the parameters are calculated by temporal difference updates. Date: 21st August 2002. 2000 Mathematics Subject Classification. Primary 91B28, 60G40, 93E20; Secondary 65C05, 93E24, 62G05. Key word... |

355 |
Adaptive algorithms and Stochastic Approximations
- Benveniste, Métivier, et al.
- 1987
(Show Context)
Citation Context ...pendently proposed a simpler algorithm for infinite horizon discrete time optimal stopping problem using also parametric approximations. Their approach is based on stochastic approximation techniques =-=[23, 4, 24]-=- and the parameters are calculated by temporal difference updates. Date: 21st August 2002. 2000 Mathematics Subject Classification. Primary 91B28, 60G40, 93E20; Secondary 65C05, 93E24, 62G05. Key word... |

315 | Valuing American options by simulation: a simple least-squares approach
- Longstaff, Schwartz
- 2001
(Show Context)
Citation Context ... papers in this direction are [6, 5, 40, 7]. The state of development as of 1998 is described in the overview paper [8]. Recently, in 1999, Longstaff and Scwartz introduced a new Monte Carlo approach =-=[28]-=-. Their method is based on a parametric approximation scheme for Bermudan options in discrete time. They showed how to calculate the parameters algorithmically by solving a sequence of least square pr... |

269 |
Stochastic approximation methods for constrained and unconstrained systems
- Kushner, Clark
- 1978
(Show Context)
Citation Context ...pendently proposed a simpler algorithm for infinite horizon discrete time optimal stopping problem using also parametric approximations. Their approach is based on stochastic approximation techniques =-=[23, 4, 24]-=- and the parameters are calculated by temporal difference updates. Date: 21st August 2002. 2000 Mathematics Subject Classification. Primary 91B28, 60G40, 93E20; Secondary 65C05, 93E24, 62G05. Key word... |

252 |
Weak Convergence and Empirical Processes With Applications to Statistics
- Vaart, Wellner
- 1996
(Show Context)
Citation Context ...sults on empirical process theory which are needed later on. A thorough development of the topic can be found in the books of Pollard [35, chapter II], [36], Dudley [13], or van der Vaart and Wellner =-=[42]-=-. Consider a sequence of iid random variables X1, . . . , Xn, . . . on a probability space (Ω, P, F). Introduce the empirical measure Pnf = 1 n� n� 1 δXif = f(Xi) (5.1) n n i=1 where δx is the Dirac m... |

168 | Efficient analytic approximation of American option values
- Barone-Adesi, Whaley
- 1987
(Show Context)
Citation Context ... Several methods have been developed to numerically solve the related optimal stopping problem. They range from binomial trees [9], Markov chain approximations [25], to semi-analytical approximations =-=[2]-=-, direct integral equation methods, and PDE methods on the basis of variational inequalities, [3, 20], the linear complementary problem [19, 10], or the free boundary value problem [43]. An increase i... |

168 |
Options: a Monte Carlo Approach
- Boyle
- 1977
(Show Context)
Citation Context ...development of Monte Carlo methods to evade the curse of dimensionality caused by these new products and models. Various approaches have been proposed. The first landmark papers in this direction are =-=[6, 5, 40, 7]-=-. The state of development as of 1998 is described in the overview paper [8]. Recently, in 1999, Longstaff and Scwartz introduced a new Monte Carlo approach [28]. Their method is based on a parametric... |

143 |
bounds for Gaussian and empirical processes
- Talagrand, Sharper
- 1994
(Show Context)
Citation Context ...Similar uniform deviation inequalities have been obtained earlier by Vapnik and Chervonenkis for VC classes of sets and VC-major classes. This estimate has been substantially improved by Talagrand in =-=[39]-=-. He considered functions with values in [0, 1]. Let G be a class of measurable functions with range [0, K] of pseudo-dimension d. Then by (5.26) N(ε, G/K, d2,Pn ) = N(εK, G, d2,Pn ) ≤ e(d + 1) �√ 8e ... |

140 |
Central Limit Theorems for Empirical Measures
- Dudley
- 1978
(Show Context)
Citation Context ...omial rate. Let (Ω, P, F) be a probability space and introduce on F the pseudo-metric dP (A, B) = P (A∆B), where ∆ is the symmetric difference. Haussler improved the L1 packing number bound of Dudley =-=[12]-=- by a logarithmic factor. Theorem 5.4 (Haussler [18]). Let X be a set. For any probability measure P on X , any class C of P -measurable sets of dimV C = d < ∞ and any ε > 0, � �d 2e N(ε, C, dP ) ≤ D(... |

135 | Methods of Mathematical Finance - Karatzas, Shreve - 1998 |

113 |
Pricing American-style securities using simulation
- Broadie, Glasserman
- 1997
(Show Context)
Citation Context ...development of Monte Carlo methods to evade the curse of dimensionality caused by these new products and models. Various approaches have been proposed. The first landmark papers in this direction are =-=[6, 5, 40, 7]-=-. The state of development as of 1998 is described in the overview paper [8]. Recently, in 1999, Longstaff and Scwartz introduced a new Monte Carlo approach [28]. Their method is based on a parametric... |

94 | Sphere packing numbers for subsets of the Boolean n-cube with bounded VapnikChervonenkis dimension
- Haussler
- 1995
(Show Context)
Citation Context ... introduce on F the pseudo-metric dP (A, B) = P (A∆B), where ∆ is the symmetric difference. Haussler improved the L1 packing number bound of Dudley [12] by a logarithmic factor. Theorem 5.4 (Haussler =-=[18]-=-). Let X be a set. For any probability measure P on X , any class C of P -measurable sets of dimV C = d < ∞ and any ε > 0, � �d 2e N(ε, C, dP ) ≤ D(ε, C, dP ) ≤ e(d + 1) . (5.22) ε It is worthwhile to... |

82 | Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing highdimensional financial derivatives
- Tsitsiklis, Roy
- 1999
(Show Context)
Citation Context ...rem is then discussed in the work of Clément, Lamberton and Protter [37]. However, their approach is different from the methods used in this paper. It is also worth noting that Tsitsiklis and Van Roy =-=[41]-=- independently proposed a simpler algorithm for infinite horizon discrete time optimal stopping problem using also parametric approximations. Their approach is based on stochastic approximation techni... |

71 |
Valuing American options in a path simulation model
- Tilley
- 1983
(Show Context)
Citation Context ...development of Monte Carlo methods to evade the curse of dimensionality caused by these new products and models. Various approaches have been proposed. The first landmark papers in this direction are =-=[6, 5, 40, 7]-=-. The state of development as of 1998 is described in the overview paper [8]. Recently, in 1999, Longstaff and Scwartz introduced a new Monte Carlo approach [28]. Their method is based on a parametric... |

57 | 1990]: \Variational Inequalities and the pricing of the American options
- Jaillet, Lamberton, et al.
(Show Context)
Citation Context ...hey range from binomial trees [9], Markov chain approximations [25], to semi-analytical approximations [2], direct integral equation methods, and PDE methods on the basis of variational inequalities, =-=[3, 20]-=-, the linear complementary problem [19, 10], or the free boundary value problem [43]. An increase in product sophistication and model complexity during the last decade has called for new methods which... |

55 |
On the pricing of American options
- Karatzas
(Show Context)
Citation Context ...fficient condition such that the expectation in (2.1) makes sense for all stopping times is that the discounted payoff process exp(− � t 0 r(s)ds) ft is of class D. For additional details we refer to =-=[22, 34]-=- and [21, Appendix D]. 2.2. Bermudan Approximations. The first approximation is to replace the continuous time problem (2.1) by a discrete time approximation. Restricting exercise and trading dates to... |

30 |
The pricing of the American option
- Myneni
(Show Context)
Citation Context ...fficient condition such that the expectation in (2.1) makes sense for all stopping times is that the discounted payoff process exp(− � t 0 r(s)ds) ft is of class D. For additional details we refer to =-=[22, 34]-=- and [21, Appendix D]. 2.2. Bermudan Approximations. The first approximation is to replace the continuous time problem (2.1) by a discrete time approximation. Restricting exercise and trading dates to... |

24 |
Optimal rate of convergence for nonparametric regression
- Stone
- 1982
(Show Context)
Citation Context ...ke the Nadaraya-Watson estimator or more generally local polynomial kernel regression estimators, [15]. These types of architectures are appealing because their asymptotic optimality. Stone showed in =-=[38]-=- that if the regression function m(x) = E[Y | X = x] is p-smooth, X, Y are bounded and X has a density, then the L2-error of a local polynomial kernel estimator converge to zero at a rate of n 2p/(2p+... |

22 |
Applications of variational inequalities in stochastic control, volume 12
- BENSOUSSAN, LIONS
- 1982
(Show Context)
Citation Context ...hey range from binomial trees [9], Markov chain approximations [25], to semi-analytical approximations [2], direct integral equation methods, and PDE methods on the basis of variational inequalities, =-=[3, 20]-=-, the linear complementary problem [19, 10], or the free boundary value problem [43]. An increase in product sophistication and model complexity during the last decade has called for new methods which... |

19 | Option Pricing and Linear Complementarity
- Huang, Pang
- 1997
(Show Context)
Citation Context ...chain approximations [25], to semi-analytical approximations [2], direct integral equation methods, and PDE methods on the basis of variational inequalities, [3, 20], the linear complementary problem =-=[19, 10]-=-, or the free boundary value problem [43]. An increase in product sophistication and model complexity during the last decade has called for new methods which can value American-style options with comp... |

13 | Symmetrization approach to concentration inequalities for empirical processes
- PANCHENKO
- 2003
(Show Context)
Citation Context ...ows that the convergence rate of the expectation is � � 1 P sup |Png − P g| ≤ O √n , (5.34) g∈G whereas (5.31) would result in a rate of O �� log n/n � . For other new deviation inequalities see also =-=[14]-=-. 5.5. Uniform Central Limit Theorems. Consider a class G of functions endowed with a semi-metric dG. The class G is said to be asymptotically equicontinuous if for every ε > 0, lim lim sup δ→0 m→∞ P ... |

12 |
On optimal stopping and free boundary problems, Archive for Rational Mechanics and Analysis 60 (1975/76), no. 2, 101–148. Ghada Alobaidi: Department of Mathematics and Statistics
- vanMoerbeke
(Show Context)
Citation Context ... approximations [2], direct integral equation methods, and PDE methods on the basis of variational inequalities, [3, 20], the linear complementary problem [19, 10], or the free boundary value problem =-=[43]-=-. An increase in product sophistication and model complexity during the last decade has called for new methods which can value American-style options with complicated exercise structures and depending... |

11 |
Pricing American Stock Options by Linear Programming
- Dempster, Hutton
- 1999
(Show Context)
Citation Context ...chain approximations [25], to semi-analytical approximations [2], direct integral equation methods, and PDE methods on the basis of variational inequalities, [3, 20], the linear complementary problem =-=[19, 10]-=-, or the free boundary value problem [43]. An increase in product sophistication and model complexity during the last decade has called for new methods which can value American-style options with comp... |

11 |
P.: An analysis of the Longstaff-Schwartz algorithm for American option pricing
- Clément, Lamberton, et al.
- 2002
(Show Context)
Citation Context ... proof for convergence of the algorithm is then outlined. A more detailed analysis of the convergence proof and a Central Limit Theorem is then discussed in the work of Clément, Lamberton and Protter =-=[37]-=-. However, their approach is different from the methods used in this paper. It is also worth noting that Tsitsiklis and Van Roy [41] independently proposed a simpler algorithm for infinite horizon dis... |

10 |
1990]: \Sur l'approximation des reduites
- Lamberton, Pages
(Show Context)
Citation Context .... However, convergence of discrete time approximations to the continuous time limit is rather subtle and requires additional conditions, primarily on the filtration. We refer the interested reader to =-=[26, 33]-=-. 2.3. Discrete Time Optimal Stopping Problems. Henceforth we fixed a suitable time discretization (2.2) and consider the related discrete time problem. Let ft, t = 0, . . . , T be a discrete time pay... |

9 |
Functional Convergence of Snell Envelopes� Applications to American Options Approximations
- Mulinacci, Pratelli
- 1996
(Show Context)
Citation Context .... However, convergence of discrete time approximations to the continuous time limit is rather subtle and requires additional conditions, primarily on the filtration. We refer the interested reader to =-=[26, 33]-=-. 2.3. Discrete Time Optimal Stopping Problems. Henceforth we fixed a suitable time discretization (2.2) and consider the related discrete time problem. Let ft, t = 0, . . . , T be a discrete time pay... |

4 |
On polynomial approximation with respect to general weights, Lecture
- FREUD
(Show Context)
Citation Context ...et W k p,w(R d ) be the weighted Sobolev space with weak derivatives in Lp,w(R d ) up to order k. The following theorem is a multivariate extension of the univariate weighted approximation results of =-=[16]-=- for special tensor product weights of Freud type. A survey of univariate weighted polynomial approximation can be found in [30], an extensive presentation of the whole theory is available in [31]. Th... |

4 |
Fundmentals of approximation theory
- Mhaskar, Pai
- 2000
(Show Context)
Citation Context ...d be a smooth bounded domain and denote by W k p (O) the usual Sobolev space with weak derivatives in Lp(O) up to order k. The following is a classical result in approximation theory, see for example =-=[29]-=-. Theorem 4.2. Let 1 ≤ p ≤ ∞. Then, for any f ∈ W k p (O) R d inf �f − p�Lp(O) ≤ C m p∈Pm −k �f�W k p (O) ≤ C dim(Pm) −k/d �f�W k p (O), (4.4) where the constant C is independent of f and m. Polynomia... |

3 |
Simulation estimators of optimal early exercise. Working paper, Carnegie-Mellon Univ
- Boessarts
- 1989
(Show Context)
Citation Context |

3 |
Carlo methods for pricing high-dimensional American options: An overview, Monte Carlo
- Monte
- 1998
(Show Context)
Citation Context ...e new products and models. Various approaches have been proposed. The first landmark papers in this direction are [6, 5, 40, 7]. The state of development as of 1998 is described in the overview paper =-=[8]-=-. Recently, in 1999, Longstaff and Scwartz introduced a new Monte Carlo approach [28]. Their method is based on a parametric approximation scheme for Bermudan options in discrete time. They showed how... |

3 |
Monte Carlo valuing of American options
- Rogers
(Show Context)
Citation Context ...ithms we referred to so far, approximate the value function or the early exercise rule in some way, hence provide a lower bound to the true option value. In contrast to this, a recent paper of Rogers =-=[17]-=- focuses on the dual problem to calculate upper bounds. Finally, a recent comparative study of various Monte Carlo approaches can be found in [27]. 1.2. Our Approach and Contributions. In this paper, ... |

3 |
Pricing American options: A comparision of Monte Carlo simulation approaches
- Laprise, Su, et al.
(Show Context)
Citation Context ... value. In contrast to this, a recent paper of Rogers [17] focuses on the dual problem to calculate upper bounds. Finally, a recent comparative study of various Monte Carlo approaches can be found in =-=[27]-=-. 1.2. Our Approach and Contributions. In this paper, we analyze the convergence and error estimates of the Longstaff-Schwartz algorithm, which is based on a linear approximation of the so called q-fu... |

3 |
Weighted polynomial approximation
- Mhaskar
- 1996
(Show Context)
Citation Context ... nature and its convergence is controlled by functional analytic properties of the approximating architecture and and the degree of smoothness of the q-function. Results of G. Freud and H. N. Mhaskar =-=[30]-=- on weighted polynomial approximation can be applied to prove convergence without restricting ourselves to compact domains of the state space. The sample error, which is our main concern in this paper... |

2 |
Numerical methods for stochastic control in finance, Mathematics of Derivative Securities
- Kushner
- 1997
(Show Context)
Citation Context ...ortant problem in Computational Finance. Several methods have been developed to numerically solve the related optimal stopping problem. They range from binomial trees [9], Markov chain approximations =-=[25]-=-, to semi-analytical approximations [2], direct integral equation methods, and PDE methods on the basis of variational inequalities, [3, 20], the linear complementary problem [19, 10], or the free bou... |

1 |
Arcones, A remark on approximate m-estimators
- A
- 1998
(Show Context)
Citation Context ...r the sample minimizers, which we establish in Section 7, under the assumption of a bounded payoff function and L∞ approximation architecture. Our setup also makes a connection to the work of Arcones =-=[1]-=-, who investigated limit behavior of approximate M-estimators. Another advantage of the machinery of empirical risk minimization and empirical process theory is that it provides a coherent framework f... |

1 |
central limit theorems, Cambridge Studies in advanced mathematics
- Uniform
- 1999
(Show Context)
Citation Context ...or we give an excerpt of the key results on empirical process theory which are needed later on. A thorough development of the topic can be found in the books of Pollard [35, chapter II], [36], Dudley =-=[13]-=-, or van der Vaart and Wellner [42]. Consider a sequence of iid random variables X1, . . . , Xn, . . . on a probability space (Ω, P, F). Introduce the empirical measure Pnf = 1 n� n� 1 δXif = f(Xi) (5... |

1 |
processes: Theory and applications
- Empirical
- 1990
(Show Context)
Citation Context ...he sample error we give an excerpt of the key results on empirical process theory which are needed later on. A thorough development of the topic can be found in the books of Pollard [35, chapter II], =-=[36]-=-, Dudley [13], or van der Vaart and Wellner [42]. Consider a sequence of iid random variables X1, . . . , Xn, . . . on a probability space (Ω, P, F). Introduce the empirical measure Pnf = 1 n� n� 1 δX... |