Quantitative Finance

Appendix 2: Proof of DeMoivre–Laplace Theorem

To prove the DeMoivre–Laplace theorem, a preliminary result, called in math language a “lemma,” makes things much easier.

Lemma A2.1

Suppose X is a random variable with moment-generating function MX(t). Further, suppose that, for fixed constants a and b, a ≠ 0, Y is another random variable such that Y = aX + b. Then

$M_Y(t)=e^{bt}{M}_X(at)$

Proof

Assume X is discrete. Then $P_Y(k)=P_X((k-b)\text{/}a).$ Thus

$M_Y(t)=E[e^{tY}]={\displaystyle \sum _{\text{all}k}{e}^{kt}}{P}_Y(k)$

Now, in the argument of the exponential, write k = a[(k − b)/a + b], to obtain

$M_Y(t)={\displaystyle \sum _{\text{all}k}{e}^{a[((k-b)/a)+b]t}}{P}_Y(k)$

Now, use $P_Y(k)=P_X((k-b)\text{/}a)$ to rewrite the expression as

$M_Y(t)={\displaystyle \sum _{\text{all}k}{e}^{a[((k-b)/a)+b]t}}{P}_X((k-b)\text{/}a)$

Now, write j = (k − b)/a and note that the sum must now be over all j, to ...