Fast Evaluation of the Asian Basket Option by Singular Value Decomposition
We investigate the use of singular value decomposition of the noise term in the Asian basket option problem. By performing this decomposition the problem can be formulated as an integral. We find a critérium for deciding the effective dimension of the int
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Department of Mathematics, Norwegian University of Science and Technology, N-7491 Tr ondheim , Norway and Sto rebrand In vestment s, PO Box 1380, N- 01l4 Oslo, Norway (corres po ndence address) Em ail: lar s.oswald.dahl @storebrand.com Uri: ht tp :// www .m ath .ntnu .n o/~I arsosw Dep ar tment of Mathem ati cs, Univers ity of Oslo, P.O . Box 1053, Blindern , N- 0316 Oslo, Nor way and MaPhySt o - Centre for Mathem ati cal Physics and St och astics, University of Aarhus, Ny Munkegad e, DK -8000 Arhus, Denmark Email: [email protected] o Uri : http ://www .m ath .uio .no/~ fredb
Abstract We investigate t he use of singular value decompositi on of the noise t erm in t he Asian basket op t ion pr obl em. By performing this decomposition t he problem can be formulat ed as an int egral. We find a crite rium for deciding t he effect ive dimen sion of t he integrand in t he fram ework of t he singular value decomposition. The resul t ing int egrat ion problem is calculated by a suite d quas i Monte Carlo method . The sim ulation resul t s show t hat t he pr op osed crite rium works well, and t hat t he com puting t ime can b e reduced significantly com pared to t he full problem .
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Introduction
It is well known t hat many finan ce problems hold certain properties t hat can be exploited to increase t he convergence rat e when calculating t heir values with qu asi Monte Carlo (QMC) method s. This is don e by combining the QMC method with variance reducti on techniques. Singular valu e decomp osition (SVD) of the noise term in t he problem is suggest ed . In the pricing of t he European-style Asian arit hmet ic average basket option (hereaft er Asian basket option) , the correlation structure of the noise is a mix of the corre lat ion structure of the Brownian paths and the correlation structure of the assets in the basket. We have in [1] develop ed a method to decompose the full problem into ort hogona l factors. For the single asset Asian option t he correlation st ruct ure is static, and t he effective dimensi on only dep ends on the chosen time discretiz ati on. This problem is ofte n referred to in the QMC literature, see e.g. [10]. The Asian basket opti on , however , has a more complex , non-st ati c, correlation structure - dep ending also on K.‒T. Fang et al. (eds.)., Monte Carlo and Quasi-Monte Carlo Methods 2000 © Springer-Verlag Berlin Heidelberg 2002
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the basket. The effective dimension will therefore vary among different baskets. The goal of this paper is to show how the decomposition of the full Asian basket option problem is performed, and to quantify the effective dimension for the single asset Asian option and for different Asian basket options. We discuss the link of our approach to the notion of the ANOVA decomposition discussed in [8] and [9]. The outline of the article is as follows: We give some required properties of the Brownian motion in sec. 2. In sec. 3 we formulate the Asian basket option pricing problem as a multi-dimensional integration problem. In sec. 4 we implement the SVD solutio
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