Using Taylor formula with logarithmic returns

Question

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 $r_t-r_{t-1}$

Answer 1

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?