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Longstaff and Schwartz developed a method to value American- and Bermudanstyle options by computing both the intrinsic value and the expected value of holding the option, waiting for a later exercise date. As discussed earlier, due to complex dynamics, the LIBOR market model doesn t lend itself well to computational approximations with lattice methods, so Monte Carlo simulation must be used. However, a key problem is that the holding value is not immediately facilitated by simulation, in contrast to lattice methods. The Longstaff and Schwartz method (LSM) overcomes this problem by approximating the holding value by assuming that the holding value, considered as an expectation conditional on continuation, is a simple function of state variables observable at the exercise date in question. Using standard regression techniques, whereby ex-post realized payoffs are regressed against the speci ed ex-ante observed values, these functions can be estimated from cross-sectional information obtained in the simulation. This type of regression is referred to as least-squares Monte Carlo (LSMC). Estimation of such functions for each exercise date will provide an approximated functional expression for holding values at all exercise dates, which can be used to value American or Bermudan options directly by Monte Carlo.12 The method is based on the mathematical fact that any twice-differentiable function, such as the conditional expectation of the holding value, can be approximated by a countable set of linear independent basis functions. Except for linear independence, there are no restrictions on the choice of basis functions, except that the chosen basis functions in uence the quality of the approximation, and so those

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U Charash, Reception through Nakagami fading multipath channels with random delays, IEEE Trans Commun, vol COM-27, April 1979, pp 657 670 61 J F Weng and S H Leung, Analysis of DPSK with equal gain combining in Nakagami fading channels, Electron Lett, vol 33, April 1997, pp 654 656 62 F Patenaude, J H Lodge, and J-Y Chouinard, Noncoherent diversity reception over Nakagami-fading channels, IEEE Trans Commun, vol COM-46, August 1998, pp 985 991; see also Proc IEEE Veh Technol Conf (VTC 97), Phoenix, AZ, May 1997, pp 1484 1487 63 L-F Tsaur and D C Lee, Simplifying performance expressions for noncoherent and differentially coherent modulations over generalized fading multiple channels, IEEE Trans Commun, vol COM-49, April 2001, pp 583 584 64 S Stein, Uni ed analysis of certain coherent and noncoherent binary communication systems, IEEE Trans Inform Theory, vol IT-10, January 1964, pp.

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43 51 65 G M Dillard, Recursive computation of the generalized Q-function, IEEE Trans Aerosp Electron Syst, vol AES-9, July 1973, pp 614 615 66 T Eng and L B Milstein, Comparison of hybrid FDMA/CDMA systems in frequency selective Rayleigh fading, IEEE J Select Areas Commun, vol SAC-12, June 1994, pp 938 951 67 G P Efthymoglou, V A Aalo, and H Helmken, Performance analysis of noncoherent binary DS/CDMA systems in a Nakagami multipath channel with arbitrary parameters, Proc IEEE Global Communication Conf (GLOBECOM 96), London, UK, November 1996, pp 1296 1300; full paper published in IEE Proc Communication, vol 144, June 1997, pp 166 172 68 P M Hahn, Theoretical diversity improvement in multiple frequency shift keying, IRE Trans Commun Syst, vol CS-10, June 1962, pp 177 184 69 P J.

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Crepeau, Uncoded and coded performance of MFSK and DPSK in Nakagami fading channels, IEEE Trans Commun, vol COM-40, March 1992, pp 487 493 70 J F Weng and S H Leung, Analysis of M-ary FSK square law combiner under Nakagami fading conditions, Electron Lett, vol 33, September 1997, pp 1671 1673 71 Q Bi, Performance analysis of a CDMA system in the multipath fading environment, Proc IEEE Int Symp Personal, Indoor, and Mobile Communication (PIMRC 92), Boston, MA, October 1992, pp 108 111..

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Figure 11.17 Block diagram of a TDMA system equipped with Lt antenna elements at the transmitter and Lr antenna elements at the receiver.

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