Expansion of a function of a GBM

Question

I'm a bit stuck on the following question: Suppose $X$ is a Geometric Brownian Motion with a drift $\mu$ and volatility $\sigma$. How can you expand, or approximate, the function $f(X)=(a + bX)^c$ in terms of powers of $X$, where $a$, $b$, and $c$ are real numbers? Can anyone solve it?

Answer 1

I suggest that you use Taylor's expansion with binomial series:

$$f(X)=a^c\left(1+\frac{b}{a}X\right)^c = a^c \cdot \sum_{n=0}^{\infty} \left(\pmatrix{c \\n} b^{n}a^{-n}X^n \right)$$ where $$\pmatrix{c \\n}=\frac{c(c-1)...(c-n+1)}{n!}$$