Brownian Motion for Price of a Stock

Question

Suppose that the current ( t = 0 ) price of a stock is 1, the drift µ = 1 and the volatility σ = 0.5. I am willing to sell you the option to buy from me at a price 2 at time t = 1. What would be the fair price to charge for this option? your reasoning for determining the price?

Answer 1

$S_t = S_0 + \mu t + \sigma^2 t^2\\ S_1 = 2 + Z$

Z is a normally distributed random variable with standard deviation $\sigma = 0.5$

The option is in the money if $Z>0$ and out of the money of $Z\le 0$

The expected value $= \int_0^{\infty} x \frac {1}{\sigma\sqrt{2\pi}}e^{-\frac {x^2}{2\sigma^2}} dx$

$= \int_0^{\infty} \frac {2x}{\sqrt{2\pi}}e^{-2x^2} dx = \frac {1}{2\sqrt{2\pi}}$

I suppose there should be a NPV adjustment, as I pay today to exercise at $t=1$

$e^{-rt}\frac {1}{2\sqrt{2\pi}}$