Vtyjkyuk

Let $X_T$ be a random variable with $$ln(X_T) sim mathcalNleft(ln(x)+fracT-t2,T-tright).$$
What is $mathbbEleft(X_T^4right)$?

Hint

I will let $X_T=X$ and write $ln Xsim mathcal N(mu,sigma^2)$ in place of your expressions, for simplicity.

The pdf of the lognormal distribution is $(xsigmasqrt2pi)^-1exp(frac-(ln x-mu)^22sigma^2)$ for $x>0$. Therefore,
$$
E[X^4]=int_0^infty x^4cdot frac1xsigmasqrt2picdot expleft(frac-(ln x-mu)^22sigma^2right),dx
$$
To compute this, start with the change of variables $z=ln x$, so that $dz=dx/x$. Then
$$
E[X^4]=frac1sigmasqrt2piint_-infty^infty e^4zcdot expleft(frac-(z-mu)^22sigma^2right),dx=frac1sigmasqrt2piint_-infty^infty expleft(4z-frac(z-mu)^22sigma^2right),dx
$$
There is a quadratic function of $z$ inside the $exp()$. The next step is to complete the square so that function becomes $exp(frac-(z-c)2sigma^2)$ for some $c$. Once you have done this, the integrand just becomes a shifted version of the normal cdf, which integrates to one. The result is just the constant you had to multiply by in order to complete the square.

Given:


You are given that
$$
ln(X_T) sim mathcalNleft(ln(x) + dfracT-t2, T-t right) ;.
$$

Simplifications:


To begin with, let us define
$$
Y := ln(X_T) ;,
qquad
mu := ln(x) + dfracT-t2 ;,
qquadtextandqquad
sigma := T - t ;.
$$
Then the given expression is equivalent to the much simpler expression $Y sim mathcalN(mu, sigma)$. For this, we already know that
$$
mathbbE(Y) = colorgreenmu ;,
qquad
mathbbSD(Y) = colorbrownsigma ;,
qquadtextandqquad
mathbbE(X_T) = mathrme^colorgreenmu + frac12colorbrownsigma^2 ;.
$$
To get from $mathbbE(X_T)$ to $mathbbE(X_T^4)$, we need an additional step.

Identities:


Notice that $ln(X_T^c) = cY$ for every $c > 0$. Using the identities
$$
mathbbE(cY) = cmathbbE(Y)
qquadtextandqquad
mathbbSD(cY) = cmathbbSD(Y) ;,
$$
we obtain $ln(X_T^c) = cY sim mathcalN(cmu, csigma)$. From this it follows that
$$
mathbbE(cY) = colorgreencmu ;,
qquad
mathbbSD(cY) = colorbrowncsigma ;,
qquadtextandqquad
mathbbE(X_T^c) = mathrme^colorgreencmu + frac12(colorbrowncsigma)^2 ;.
$$

Final result:


Now we just need to set $c = 4$, to get the final result

$$
mathbbE(X_T^4)
= mathrme^4mu + frac12(4sigma)^2
= underlineunderlinex^4 cdot mathrme^2(T-t) + 8(T-t)^2 ;.
$$