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1

Stock prices have been modeled using the Lognormal distribution, not the Normal distribution. See this paper http://math.ucsd.edu/~msharpe/stockgrowth.pdf for more detailed information. This does not mean that the current price offer depends on the last price offer.


2

As you have already shown above, \begin{align*} 1 = E\Big(e^{-\theta y - \frac{1}{2}\theta^2 \tau_{-y} }1_{\tau_{-y}<\tau_x} \Big)+E\Big(e^{\theta x - \frac{1}{2}\theta^2 \tau_{x} }1_{\tau_x < \tau_{-y}} \Big). \end{align*} Set $\theta = \sqrt{2u}$, we obtain that \begin{align*} 1 = E\Big(e^{-\sqrt{2u}y - u \tau_{-y} }1_{\tau_{-y}<\tau_x} ...



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