# VaR using normal vol vS lognormal

We are using a vendor's software to calculate the Parametric VaR (using RiskMetrics approach) that take as input the volatility figure of the risk factors. The volatility used so far was the lognormal. But, due to negative rates (ie EUR swap), we have to switch to normal ones.

Is it OK to just replace the vol figures from the lognormal calc with the ones from the normal?

## 1 Answer

My answer is to a certain extent a question: "Switching from lognormal to normal vol" reminds me pretty much of implied volatility. Could it be that you plug some implied volatility into you parametric VaR?

If this is the case then I would recommend you to change this and to look at the volatility in the physical and not the risk-neutral measure. This means that if your model of the return is: $$R= -D * \Delta r$$ where $$D$$ is some duration and $$\Delta r$$ is the delta of your rate then we can (under certain assumptions) assume that $$R \sim N(0,D \sigma)$$ where $$\sigma$$ is the volatility of $$\Delta r$$. Then you could estimate $$\sigma$$ from a timeseries $$(\Delta r_t)_{t=1}^n$$ using basic or more sophisticated methods. For estimating $$\sigma$$ it does not matter whether $$r_t$$ is positive or negative.

Furthermore, if you work in the phyiscal (real world) measure, then you can take correlations to other risk factors into account. In the risk-neutral (implied) setting you would need something like basket products that have all you risk factors as underlying and would only getsomething like a global implied correlation [see e.g. https://quant.stackexchange.com/questions/8689/average-correlation-of-index-portfolio].

But for risk measuring and management most probably the implied/risk neutral world is the wrong view.