Question: The stock price starts at 100\$. At any given time, there is 50% probability that stock price increases further by 1 and 50% probability that stock price goes back to 100\$. You are paying 1\$ to trade. What price should you trade this stock if it starts moving upwards?

I tried modeling this problem as markov chain but was soon lost due to lack of experience. Any help will be appreciated.

Answer: 
To answer you question (1): Of course it is possible: since your original $X$ has a continuous distribution function $F$, you can use the $Y=G^{-1}(F(X))$ trick to generate a random variable $Y$ with any desired distribution function $G$. In your case this $g(t)=G^{-1}(F(t))$ function will be continuous and monotone increasing.
The random variable $Y$ is obtained from $X \sim \mathcal{N}\left( \mu,\sigma^{2}\right)$ by random sampling (with probability in your case) from the normal distribution :
\begin{equation\*}
Y=\left\{ 
\begin{array}{c}
X \text{ with probability }1/2 \\ 
-X \text{ with probability }1/2
\end{array}
\right.
\end{equation\*}

This fully describes the relationship between $X$ and $Y$ and as $g$ is not a one-to-one transformation, I am not convinced that the "change-of-variable technique" is useful here. The "distribution function technique", instead directly gives use the cdf and pdf of $Y$ (which fully describes this random variable):
\begin{eqnarray\*}
F\left( y;\mu,\sigma \right) &\equiv & \Pr \left[ Y\leq y\right] \\
&=& 1/2 \Pr \left[ X\leq y\right] + 1/2 \Pr \left[ -X\leq y\right] \\
&=& 1/2 \int\_{-\infty }^{y}\phi \left( u;\mu,\sigma^{2}\right) du+
1/2 \int\_{-y }^{\infty 
}\phi \left( u;\mu,\sigma^{2}\right) du,
\end{eqnarray\*}
\begin{eqnarray\*}
f\left( y;\mu,\sigma^{2} \right)
&=&1/2 \phi \left( y;\mu,\sigma^{2}\right) + 1/2 \phi \left( -y;\mu,\sigma^2\right) \\
&=&\frac{1/2}{\sqrt{2\pi }\sigma}\exp \left[ -\frac{1}{2\sigma^{2}
}\left( y-\mu\right) ^{2}\right] +\frac{1/2 }{\sqrt{
2\pi }\sigma}\exp \left[ -\frac{1}{2\sigma^{2}}\left( -y-\mu
\right) ^{2}\right].
\end{eqnarray\*}