Using Log Returns in Delta-Gamma PnL Approximation

Summary

The document asks whether log returns can serve as the risk-factor change in a second-order Taylor approximation of portfolio PnL, or whether the change must instead be the arithmetic difference in the FX rate. The answer treats the product value as a function of the exchange rate and expresses the rate as its initial value multiplied by the exponential of the log return. Expanding around a zero log return gives a delta-gamma approximation in log-return space.

This means log returns can be used when the sensitivities are defined with respect to that coordinate. The answer also gives the conversion from sensitivities with respect to the exchange rate to the corresponding first- and second-order terms for log returns. This qualification matters: directly combining log returns with sensitivities defined for arithmetic rate changes would mismatch units and derivatives. The treatment is local, based on a Taylor approximation, and the document supplies no accuracy assessment for large moves or a worked portfolio example.

Key ideas

• A Taylor PnL expansion must use risk changes and sensitivities defined in compatible coordinates.
• Log FX returns can be used when the derivatives are transformed to log-return space.
• The log-return gamma term includes a contribution from the original delta sensitivity.
• The approximation is local around the current exchange rate and its accuracy for large moves is not assessed.

Tags

# Using Taylor formula with logarithmic returns


# 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 $r_t-r_{t-1}$

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/58567

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.