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?