# Using Taylor formula with logarithmic returns

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 $$r_t-r_{t-1}$$

• Are you computing this with respect to prices? Because as the formula reads it is change in returns not change in prices. – rubikscube09 Oct 8 '20 at 18:14
• 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. – Chris Oct 9 '20 at 6:49
• @Chris It is not a long FX position. It is a portfolio of mostly linear instruments with underlying FX risk factors. – user92234 Oct 9 '20 at 8:15

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
$$PnL\equiv f(X_0e^r)-f(X_0) \approx \left.\frac{\partial f }{\partial r}dr + \frac{1}{2}\frac{\partial^2f }{\partial r^2}(dr)^2\right|_{r=r_0=0}$$
$$PnL \approx \frac{\partial f}{\partial X_0}X_0dr + \frac{1}{2}\left(\frac{\partial^2 f}{\partial X_0^2}X_0^2+\frac{\partial f}{\partial X_0}X_0\right)(dr)^2$$
In this formulation, your $$\Delta r$$ can indeed be specified using log returns.But it is important to use the 'correct' sensitivities, though... HTH?