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I would like to calculate PnL scenarios for an FX portfolio using Taylor series approximation:

$$ \begin{align} \text{PnL} \approx \delta \Delta r + \frac{1}{2} (\Delta r)^2 \Gamma \end{align} $$

I have the $\delta$ the $\Gamma$ and the $\Delta r$s which are given as log returns: $ln(\frac{r_t}{r_{t-1}})$. Is it ok to just plug in the log returns into the Taylor formula or must $\Delta r$ equal $r_t-r_{t-1}$

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    $\begingroup$ Are you computing this with respect to prices? Because as the formula reads it is change in returns not change in prices. $\endgroup$ – rubikscube09 Oct 8 '20 at 18:14
  • $\begingroup$ Your question doesn't really make sense (or at a minimum you're missing details). It isn't at all clear why you'd use a Taylor expansion to approximate the return (or PnL) of a long currency position when you can just calculate it directly. $\endgroup$ – Chris Oct 9 '20 at 6:49
  • $\begingroup$ @Chris It is not a long FX position. It is a portfolio of mostly linear instruments with underlying FX risk factors. $\endgroup$ – user92234 Oct 9 '20 at 8:15
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Assuming your are modeling a product that is not linear in the underlying risk factor (not the FX rate per se), and assuming you are using logarithmic FX returns, you may arrive at the following: Let $f$ denote the value of your FX-product, $X_0$ denote today's exchange rate, and $r$ denote the log return (change) of your FX rate. Linearising around $r=0$ yields

$$ PnL\equiv f(X_0e^r)-f(X_0) \approx \left.\frac{\partial f }{\partial r}dr + \frac{1}{2}\frac{\partial^2f }{\partial r^2}(dr)^2\right|_{r=r_0=0} $$

yielding

$$ PnL \approx \frac{\partial f}{\partial X_0}X_0dr + \frac{1}{2}\left(\frac{\partial^2 f}{\partial X_0^2}X_0^2+\frac{\partial f}{\partial X_0}X_0\right)(dr)^2$$

In this formulation, your $\Delta r$ can indeed be specified using log returns.But it is important to use the 'correct' sensitivities, though... HTH?

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