financial maths - payoff options

Question

Consider the payoff of a call option $C=\max\{0,S(1)-K\}$, where $S(1)=S(0)(1+\mu+\sigma_X)$, X has standard normal distribution. Take $S(0)=80$, $\mu=0.3$, $\sigma=0.4$, $K=100$ (strike price). Assume that the option price is $C(0)=10$. An investor borrowed 10 at the rate 10% (annual compounding) and bought one option. Let $Y$ be the payoff for the investor at time 1 (after one year). The question is as follows: Find the probability $P(Y>0)$? Thanks.

Answer 1

$Y>0$ if $S(1)>K+10 \times(1+10\%)$. That is, if $80(1+0.3+0.4X)>111$. That is, if $X>0.21875$. That probability is 0.4134 from normal distribution tables.