## 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.

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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, ... |

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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... |

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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... |

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