Pricing and Hedging Asian Basket Options with Quasi-Monte ... - Infn

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Methodol Comput Appl Probab1 Introduction and MotivationIn a few recent papers Dahl and Benth (2002) and Wang (2009) have investigatedthe efficiency and the computational cost of the Principal Component Analysis(PCA) used in the Quasi-Monte Carlo (QMC) simulations for the pricing of highdimensionalAsian basket options in a multi-dimensional Black–Scholes (BS) modelwith constant volatilities. In particular they have shown the essential role of theKronecker product both for a fast implementation, and for identifying the effectivedimension in the analysis of variance (ANOVA, see below). Since the convergencerate of the QMC method is O ( N −1 log d N ) —here N is the number of simulationtrials, and d the nominal dimension of the problem—the theoretically higher asymptoticconvergence rate of QMC could not be practically achieved in high dimensions.On the other hand, particular applications in finance (see Paskov and Traub 1995)have shown that the QMC provides an accuracy higher than the standard MonteCarlo (MC) even for high dimensions.To explain the success of the QMC in high dimensions Caflisch et al. (1997)have introduced two notions of effective dimensions based on the ANOVA ofthe integrand function. Consider an integrand function f for a MC problem withnominal dimension d,andletA ={1,...,d} denote the labels of the input variablesof the function f : the effective dimension of f in the superposition sense is thesmallest integer d S such that ∑ |u|≤d Sσ 2 ( f u ) ≥ pσ 2 ( f ), where f u is a function withvariables in the set u ⊆ A , σ 2 (·) denotes the variance of the given function, |u| is thecardinality of the set and 0 ≤ p ≤ 1 (for instance p = 0.99). On the other hand theeffective dimension of f in the truncation sense is the smallest integer d T such that∑u⊆{1,2,...,d T } σ 2 ( f u ) = pσ 2 ( f ). In other words the truncation dimension indicates thenumber of variables essential to capture the given function f , while the superpositiondimension takes into account that, for some f ’s, the inputs might influence theoutcome through their joint action within smaller groups.Different techniques have been proposed for a dimension reduction: the PCAdecomposition and the Brownian bridge (BB) however achieve this result independentlyfrom the particular payoff of a European option. Imai and Tan (2006) haveinstead proposed a general dimension reduction construction, the Linear Transformation(LT), that depends on the payoff function, and that minimizes the effectivedimension in the truncation sense. Several studies have investigated the efficiency ofthe dimension reduction produced by these approaches. Wang (2009), for example,has shown that the accuracy of the QMC simulations depends on both the dimensionreduction technique, and the quasi-random points. He also proved that the PCAdecomposition is always outperforming the BB as a result of the different groupingstrategies developed (see the cited article for more details). Moreover, Papageorgiou(2002) has demonstrated that the accuracy of the QMC method used for thepricing of certain specific derivative contracts is not substantially improved by aBB construction. Finally Imai and Tan (2006) have shown that the LT approachis more accurate than the standard PCA and BB, but has a higher computationalcost. In a previous paper one of the authors (Sabino 2011) has described how toefficiently implement this technique and, even with a slower computer, has obtainedcomputational times that are about 30 times shorter than those originally presentedby Imai and Tan (2006).In the present paper we address the problem of the time-dependent volatilities,and since we can no longer rely on the properties of the Kronecker product, our task
Methodol Comput Appl Probabwill be computationally harder. In order to reduce this computational complexity,we first introduce a fast Cholesky (CH) decomposition algorithm tailored for blockmatrices: this will already severely lower the computational cost. Then we presenta new path-generation technique based on the Kronecker Product Approximation(KPA) of the correlation matrix of a multi-dimensional Brownian path: this returnsa suboptimal ANOVA decomposition with a remarkable computational advance.In the case of time-dependent volatilities, the BB procedure also requires a slightlydifferent algorithm (see Sabino 2009 for details), but on the basis of the previousobservations we have decided not to include it in this study.Our numerical simulations consist first in calculating the Randomized QMC(RQMC) estimation of both the prices and the deltas of high-dimensional Asianbasket options in a BS market with time-dependent volatilities. In order to computethe deltas we extend to a dependent multi-assets model the procedure employedby Kohatsu-Higa and Montero (2003) in a single-asset setting. To do that we takeadvantage of the Malliavin Calculus, and we allow a certain flexibility to enhance thelocalization techniques introduced by Fournié et al. (1999). As far as the computationof the Asian options prices is concerned, the KPA and LT approaches are checkedboth in terms of accuracy and computational cost. We show that the LT procedure ismore efficient than the PCA—even from a computational point of view—providedthat we adopt our CH algorithm and the approach described in Sabino (2011). TheKPA and the PCA constructions perform equally well in terms of accuracy, butwith the former requiring a considerably shorter computational time. In the samevein, the KPA and the LT display comparable accuracies in the computation of thedeltas. Finally we contrast our simulations—also using the standard CH and the PCAdecomposition methods—with both pseudo-random and Latin Hypercube Sampling(LHS) generators.We finally remark that all the methods described here can accommodate a marketwith stochastic volatility where the evolution of the risky securities is modeled by amixture of multi-dimensional dynamics as in the papers by Brigo et al. (2004a, b). Itis noteworthy to say instead that none of these procedures can be applied to theHeston-like multi-dimensional stochastic volatility models. In principle we mightstill use the LT for the Euler discretization of the Heston model, but this couldbe no longer applicable within more realistic frameworks involving discrete randomvariables as proposed for instance by Alfonsi (2005).The paper is organized as follows: the Section 2 describes Asian options, while theSection 3 discusses some path-generation techniques and in particular, presents thefast CH algorithm and the KPA construction. Section 4 shows then the numericalsimulations for the Asian option pricing, and Section 5 explains how to represent thedeltas of Asian basket options as expected values with the aid of Malliavin Calculus,and shows their estimated values by RQMC. Section 6 finally summarizes the mostimportant results and concludes the paper.2 Asian Basket OptionsAssume a multi-dimensional BS market with M risky securities and one riskfreeasset. Denote B (t) = (B 1 (t) ,...,B M (t)) an M-dimensional Brownian motion(BM) with correlated components and (F t ) t≥0 the filtration generated by this BM.
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