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?
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$S_t = S_0 + \mu t + \sigma^2 t^2\\ S_1 = 2 + Z$
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 $t=1$
$e^{-rt}\frac {1}{2\sqrt{2\pi}}$
Doug M
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@ Doug M Would you elaborate your reasoning please? I need to further model the price of stock as Geometric Brownian Motion. – user427820 May 22 '17 at 20:17
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Geometric brownian motion.... $S_1/S_0 = Z$ with mean $\mu$ and standard deviation $\sigma$. $\log S_1 -\log S_0 = \log (Z)$ and the expectation is a log-normal distribution, And then you find the mean of the tail to the right of the strike price to find the value of the option. – Doug M May 22 '17 at 20:24
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So i should find the expected value of stock at time t=1 and see if the expected value is greater than 2 and how much( lets say it's 3.5). If that's the case, a reasonable price for the option could be 0.5 ( because the price of stock at t = 1 will still be greater than our total investment i.e 2 + 0.5 = 2.5 ) Am I going the right way? – user427820 May 23 '17 at 15:48