Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $j=1,2$. Their price process is supposed to be modeled by an Itô-process, driven by a Brownian Motion $\widehat{W} = (\widehat{W}^1, \widehat{W}^2)$, with the following solutions:
The price process $S_j(t)$ satisfy some SDE which solutions are given with respect to $\mathbb{P}$ by, where $i,j=1,2$:
$\begin{align} S_{j}(t) & = e^{(r-\frac{1}{2} || \sigma_{j} ||^2 + \sigma_{i} \sigma_{j})\, t + \sigma_{j} \widehat{W}_t } \end{align}$
where $r,\sigma_j,\sigma_i$ are constants. Now one can make the following calculations: \begin{align} P \left( S_{1}(s) \leq S_{2}(s) \Big\vert \mathcal{F}_{t}^{W} \right) & =P \left( \frac{S_{1}(s)}{S_{2}(s)} \leq 1 \Big\vert \mathcal{F}_{t}^{W} \right)\\ & = P \left( \ln\left( \frac{S_{1}(s)}{S_{2}(s)}\right) \leq 0 \Big\vert \mathcal{F}_{t}^{W} \right) \end{align} So far so good. Now I'm at the point where i need help to calculate the last term. I got the solutions but I dont understand them. Any help is welcome. Those are the solutions:
Define $X := \ln\left(\frac{S_{1}(t)}{S_{2}(t)} \right)$. Under $\mathbb{P}$,
\begin{align} X =&\ \ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right) + \sum_{j=1}^{2} (\sigma_{1,j} -\sigma_{2,j} ) (\widehat{W}_s^{j}- \widehat{W}_t^{j}) \\ & - \sum_{j=1}^{2} \left( \frac{1}{2} \left[\sigma_{1,j}^2-\sigma_{2,j}^2 \right] - \left[\sigma_{1,j} -\sigma_{2,j} \right] \sigma_{\pi(i),j} \right) (s-t) \end{align}
where $ s > t$ (why?). And $X$ is normal (why?) with mean $m(t)$ and variance $v(t)$, where: \begin{align} m(t) := \ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right) - \sum_{j=1}^{2} \left( \frac{1}{2} \left[\sigma_{1,j}^2-\sigma_{2,j}^2 \right] - \left[\sigma_{1,j} -\sigma_{2,j} \right] \sigma_{\pi(i),j} \right) (s-t) \end{align} and \begin{align} v(t)^2 := \sum_{j=1}^{2} (\sigma_{1,j} -\sigma_{2,j} ) (s-t) \end{align}.
Also a why? Thanks for any explanations, help and tips.
FrakChris
We assume that \begin{align*} d\left(\! \begin{array}{c} S^1(t)\\ S^2(t) \end{array} \!\right) =\textrm{diag}\left(S^1(t), S^2(t)\right)\bigg[\left(\! \begin{array}{c} r\\ r \end{array} \!\right)dt + \left(\! \begin{array}{cc} \sigma_{1,1} &\sigma_{1,2}\\ \sigma_{2,1} &\sigma_{2,2} \end{array} \!\right)d \left(\! \begin{array}{c} W_t^1\\ W_t^2 \end{array} \!\right)\bigg], \end{align*} where $\{W_t^1, t \ge0\}$ and $\{W_t^1, t \ge0\}$ are two independent standard Brownian motions. That is, for $i=1, 2$, \begin{align*} dS^i(t) = S^i(t)\bigg(rdt + \sum_{j=1}^2 \sigma_{i, j} dW_t^j \bigg). \end{align*} Then, for $s>t$, \begin{align*} S^i(s) = S^i(t)e^{\left(r-\frac{1}{2}\sum_{j=1}^2\sigma_{i,j}^2 \right)(s-t) + \sum_{j=1}^2\sigma_{i, j} \big(W_s^j-W_t^j \big)}. \end{align*} Moreover, \begin{align*} \ln \frac{S^1(s)}{S^2(s)} &= \ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t) + \sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)\big(W_s^j-W_t^j \big)\\ &=\ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t) \\ &\qquad+ \sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2 (s-t)}\,\frac{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)\big(W_s^j-W_t^j \big)}{\sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2(s-t)}}\\ &=\ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t) + \sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2(s-t)}\, Z\\ &=m(t)+v(t) \, Z, \end{align*} where $Z$ is a standard normal random variable that is independent of $\mathcal{F}_t$, \begin{align*} m(t) = \ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t), \end{align*} and \begin{align*} v(t) = \sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2(s-t)}. \end{align*} Therefore, \begin{align*} P\left(\ln \frac{S^1(s)}{S^2(s)} \le 0 \mid \mathcal{F}_t \right) &= \Phi\left( -\frac{m(t)}{v(t)}\right), \end{align*} where $\Phi$ is the cumulative distribution function of a standard normal random variable, assuming that $v(t) \neq 0$.