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$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-martingale with mean 1?

The questions doesn't specify anything about the filtration beside that we are in the standard Black-SCholes model. My concern here is how to treat $S_0$. Will $S_0$ be a process or simply a constant?

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  • $\begingroup$ This is a question given to me by a professor at Stockholm University. It is related to Bjork chapter 24 and 26. The level is for graduate students. I have solved many of his questions but this is by far the most difficult one $\endgroup$ – user32098 Mar 4 '18 at 13:39
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Assuming standard BS dynamics for $S_t$, you have

$\frac{Z_t}{Z_0}\equiv(\frac{S_t}{S_0})^p = \exp{((rt-\frac{\sigma^2}{2}t+\sigma W_t)(1-\frac{2r}{\sigma^2}))}$

Now, this is a lognormally distributed r.v. and the Gaussian inside the exponential has mean and variance

$m=2rt - \frac{\sigma^2}{2}t -\frac{2r^2}{\sigma^2}t$

$v = (\sigma-\frac{2r}{\sigma})^2 t$

Applying the standard formula for the expectation of a lognormal r.v. $E(e^X)=e^{m_X+\frac{v_X}{2}}$, you will find after a little arithmetic that

$E(\frac{Z_t}{Z_0}) = 1$

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As a starting point: for price dynamics $dS(t) = rS(t)dt + \sigma S(t) dW^\mathbb{Q}(t)$, to show that $Z(t)/Z(0)$ is a positive mean 1 Q-martingale, use Itô's formula to get:

$dZ(t)=p \sigma Z(t)dW^\mathbb{Q}(t)$

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  • $\begingroup$ How do I go from there? Because I can't seem to wrap my head around $Z_0$. Can I simply let it be a constant? If not How do define the process $dZ_0$ when I need to apply Ito to $Z_t/Z_0$ $\endgroup$ – user32098 Mar 4 '18 at 14:44

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