Fallacy of the Log Normal Approximation to Optimal Portfolio Decision Making Ove
Fallacy of the Log Normal Approximation to Optimal Portfolio Decision Making Ove
Robert C Merton
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(T, T;h) ], then from (7. 5), we have that Y. (T;h) S Z X. (k, h) where N = T/h, (7. 17) 3 k=l ^ and from the independence and identical distribution of the X. ( ;h), the moment-generating function of Y. Will satisfy 0. (X;h, T) = E{exp[XY. (T;h)]} (7. 18) Taking logs of both sides of (7. 18) and using Taylor series, we have that log[0^(X;h, T)] = (T/h)log[4^. (X;h)] (7. 19) 00 = (T/h)log[ I t|^/^^(0;h)x'^/k!] k=0 ^ = (T/h)log[ Z m. (k;h)A^/k!] k=0 ^ = (T/h)log[l + m. (l;h)X + l/2m. (2;h)X^ + o...(h) ] = T[X(m. (l;h)/h) + ^(m. (2;h)/h) + 0(h)]. 2 2 2 Substituting y (h)h for m. (l;h) and O. (h)h + y, (h)h for m. (2;h) and taking the limit as h^o in (7. 19), we have that log[0. (X;O, T)] = lim log[0(X;h, T) ] (7. 20) J h->0 = Xy. T + l/2a.^TX^, and therefore, 0. (X;O, T) is the moment-generating function for a normally- 2 distributed random variable with mean y. T and variance O. T. 35 Thus, in what is essentially a valid application of the Central- Limit theorem, we have shown that the limit distribution for Y.
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