Consider a continuous time framework where stock prices follow a GBM process. You observe a Stock X that currently trades at $$45$ per share, does not pay dividends, and has a volatility of $20\%.$ Suppose the riskless rate is $2\%$ per annum $($continuously compounded$)$

$($i$)$ What is the probability under the risk$-$neutral measure that this Stock X is worth at least $$50/$share one year from today? $(30$ points$)$

$($ii$)$ What is the probability under the physical measure? $(30$ points$)$