how to solve E[exp(exp(x))]

ychen · January 13th, 2013, 2:21 am

x~N(0,1)How to solve the expectation of exp(exp(x))? Thanks!

bearish · January 13th, 2013, 12:50 pm

It explodes.

Antonio · January 13th, 2013, 2:17 pm

The moment generating function of the lognormal distribution is only defined on the negative real axis, but not on the positive half-line.One way to see it is: take the characteristic function of the lognormal, expand it by Taylor series. And then you see that this series diverge on the positive half-line.Did you actually try in the first place?The point of the forums is to show that you tried something, it gives more incentive to people to help you.Best,

quantiquequant · January 14th, 2013, 5:56 pm

Does anyone has an idea why this converges for trees methods,for example when using log-normal short rates models (BDT or Dothan) the cash expectation Eexp(int _0^t r_s ds) converge where even the expectation is not defined. Eexp(exp(N)) ??

QuantEquity · January 15th, 2013, 8:21 am

see attachement

ZhuLiAn · January 15th, 2013, 11:32 am

I came accross this problem. First to see the divergence just write which shows that the chi-2 diverges as well. In practise you need to discretize your model and assume not continuous compounding. A funny thing is that when scaled by the volatiltily the divergence can happens quite far and some numerical integration schemes can miss out.

mj · January 24th, 2013, 8:21 pm

QuoteOriginally posted by: quantiquequantDoes anyone has an idea why this converges for trees methods,for example when using log-normal short rates models (BDT or Dothan) the cash expectation Eexp(int _0^t r_s ds) converge where even the expectation is not defined. Eexp(exp(N)) ??i don't think it does converge in the Dothan model.BDT get round the problem by using a binomial tree. So whilst the limit process has the issue, the approximating tree does not.