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Topic: an integral question
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Vijay Fafat

Posts: 3
Registered: 12/12/04
an integral question
Posted: Jun 14, 1996 6:39 PM
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I have the following questions:

Let N(x) be the normal cumulative distribution function and f(x) be
the standard normal distribution kernel. Cox and Rubinstein, in their
book, "Options Markets", have the following result:

M(a,b) = integral from -inf to inf of f(x)N(a+bx) = N(a/sqrt(1+b^2)).

This does not seem to be right since for a=0, the
integral seems to be independent of b. The questions are:

1) How do you go about evaluating the above integral and what's the
correct answer?

2) In general, what kinds of arguments for N(.) will give you such closed
form expressions?

Now for the paradox: Consider M(0,b). Differentiate both
sides above w.r.t. b. This
generates an integral of an odd integrand from -inf to inf (if I've
applied Leibnitz correctly). Thus, M(0,b) is insensitive to b. What gives?


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