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Dynamics of FX rate

Here is a way to demonstrate the logic of standard FX rate models. The key takeaway is that these models assume that uncovered interest parity (UIP) holds but still imply that a US rate hike leads to ...
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Dynamics of FX rate

I am answering now instead of commenting. The rate of change in FX is naturally forward looking in this case. What you confuse is what happened to Spot due to changes in interest rate environments ...
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SDE linear combination of stock price

Let's just write $p'(t)$ for $\frac{\partial p}{\partial t}$. Then your last expression for $V$ should rather be $$ V(t)=V(0)\exp\Bigg\{\int_0^t\frac{p'(s)}{p(s)}\,ds-\frac{\sigma^2}{2}\color{red}{t}+\...
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