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ColumnVector b = sm.calcBetas(); // compute coef cients return b; } /********************************************************************************** calcLogDerives : computes derivative of the log likelihood function [in]: ColumnVector theta : matrix column (3x1) of parameters [out]: ColumnVector LLD: matrix column with log derivatives with respect to parameters **********************************************************************************/ ColumnVector StatModel::calcLogDerives(ColumnVector theta) { int i; h.clear(); // initial h[0] double sum = accumulate(e2.begin(), e2.end(),0.0); double ave = sum/T; // compute sum of squared residuals // get average of sum of squared residuals

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If we compare the similarity of this to (t ) = f (0, t ) + 2 2a2 (1 e at )2

But the truth, even more, is that life is perpetually weaving fresh threads which link one individual and one event to another, and that these threads are crossed and recrossed, doubled and redoubled to thicken the web, so that between any slightest point of our past and all the others a rich network of memories gives us an almost infinite variety of communicating paths to choose from.4

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time t = 2 is shown in Table 14.2. Note that at each time step only those paths in the regression that are in-the-money are included; that is, path 8 is excluded. After discounting the cash ows (Table 14.3), the American price is $3.39. For a more detailed example with all the intermediate regressions calculations, see Longstaff and Schwartz (2001). Longstaff and Schwartz also show that the LSM method can be applied to many types of exotic path-dependent derivatives including an American-Bermudan-Asian call option on the average of the price of a stock over some time horizon, where the call option can be exercised after some initial lockout period. As basis functions in the regression, they use a constant, the rst two Laguerre polynomials evaluated at both the stock and average stock price, as well as the cross-product of these Laguerre polynomials up to third-order terms. Their results are compared with those using a nite-difference approximation using an alternating direction implicit (ADI) algorithm. The results of the LSM and nitedifference prices are quite similar and the differences in early exercise values are quite small, within bid-ask spread cost bounds.

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Multiplexing is the reversible operation of combining several information-bearing signals to form a single, more complex signal The signals that are combined in a multiplexer usually come from independent sources, such as subscribers in a telephone network Prior to multiplexing, each signal travels over a separate electrical path, such as a pair of wires or a cable, whereas the multiplexed signal can be transmitted over a single communication medium of sufficient capacity The reversibility of the multiplexing operation permits recovery of the original signals which often have different final destinations at the receiving end of the transmission link This inverse operation whereby the original signals are recovered is called demultiplexing Multiple access may be considered as a special kind of multiplexing in which two or more signals, each from a different location, are sent by radio to a single transponder.

This work so far has been based on visual comparisons. Statistical analysis appears to confirm what the charts are showing, namely that there is a strong negative correlation between the CRB Index and bond prices. A study prepared by Powers Research, Inc. (Jersey City, NJ 07302), entitled The CRB Index White Paper: An Investigation into Non-Traditional Trading Applications for CRB Index Futures (March, 1988), reported the results of correlation analysis over several time periods between the CRB Index and the other financial sectors. The results showed that over the 10 years from 1978 to 1987, the CRB Index had an 82 percent positive correlation with 10-year Treasury yields with a lead time of four months. In the five years from 1982 to 1987, the correlation was an even more impressive +92 percent. Besides providing statistical evidence supporting the linkage between

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