An alternative Monte Carlo portfolio estimator has been developed byCvitanic et al.
(2003). Their method relies on an approximation of the covariation of the optimal wealth process with the Brownian innovation,
Xt t0
t D lim
!0
1
X; W
tC
X; W
t
D lim
!0
1
Z
t Xss0sds:
(25.66) AsXtDEtŒTI .yT/, it follows that
X; W
tC
X; W
t DEtŒTI .yT/ .WtCWt/ : (25.67) A Monte Carlo estimator of the portfolio, for a given Lagrange multipliery, is
1
XttM;N D t01
EMt
TNI
yTN WtCWt
; (25.68)
where is some selected time interval.
The Monte Carlo portfolio estimator (25.68) is easy to calculate. But it is based on a formula which approximates the optimal portfolio (and not the exact portfolio rule). As shown byDetemple et al.(2005c), this introduces an additional second-order bias term and reduces the speed of convergence to M1=3. The numerical efficiency studies in Detemple et al. (2005b,c) also show that the efficiency gains of estimators based on exact portfolio formulas (such as MCMD) can be considerable.
Similar results apply to Monte Carlo finite difference estimators (MCFD). These estimators use Monte Carlo simulation, but approximate the derivatives of the value function by finite difference perturbations of the relevant arguments. These finite difference perturbations replace the correct expressions given by expected values
of functionals containing Malliavin derivatives. Again, the corresponding portfolio estimator relies on a formula that holds just in the limit, introducing an additional approximation error and reducing the convergence speed. The numerical studies in Detemple et al.(2005b,c) also illustrate the superior performance of MCMD relative to MCFD estimators.
The Monte Carlo methods based on Malliavin calculus rely only on forward sim- ulations. In contrast, the Bellman equation naturally suggests backward simulation algorithms. In order to obtain adapted portfolio policies using backward algorithms, conditional expectations have to be calculated repeatedly. This is computationally costly, especially for high-dimensional problems. Brandt et al. (2005) consider polynomial regressions on basis functions to reduce the computational burden in backward calculations of the value function. These methods have proven useful for optimal stopping problems (Tsitsiklis and Van Roy 2001;Longstaff and Schwartz 2001; Gobet et al. 2005) where the control is binary. The optimal policy in a portfolio problem is more complex and the numerical precision depends directly on the estimates of the derivatives of the value function. A general convergence proof for these methods is unavailable at this stage (Partial results are reported in Cl´ement et al. (2002) and Glasserman and Yu (2004)). The numerical efficiency study inDetemple et al. (2005b) shows that this approach is dominated by other MC methods.
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Low-Discrepancy Simulation
Harald Niederreiter
Abstract This article presents a survey of low-discrepancy sequences and their applications to quasi-Monte Carlo methods for multidimensional numerical inte- gration. Quasi-Monte Carlo methods are deterministic versions of Monte Carlo methods which outperform Monte Carlo methods for many types of integrals and have thus been found enormously useful in computational finance. First a general background on quasi-Monte Carlo methods is given. Then we describe principles for the construction of low-discrepancy sequences, with a special emphasis on the currently most powerful constructions based on the digital method and the theory of.T; s/-sequences. Next, the important concepts of effective dimension and tractability are discussed. A synopsis of randomized quasi-Monte Carlo methods and their applications to computational finance is presented. A numerical example concludes the article.