Interpreting FRTB Vega Sensitivity as a Relative Volatility Shock

# Basel FRTB Vega Sensitivity for Market Risk Capital Standardised Approach


# Basel FRTB Vega Sensitivity for Market Risk Capital Standardised Approach












Sorry if this is too simple/obvious a question, but I'm a bit lost looking at the FRTB definition of vega sensitivity for Standardised Approach. Per section 21.25:

```
s_k = vega x implied volatility
```

I see a trading book with long 1,000,000 Black-Sholes vega per 1 pct additive and implied vol trading around 30 pct (0.30). That means vega sensitivity is 300,000?? Seems low!

I find one example calculation online where that writer uses 8 instead of 0.08 to as an example FX vol. So then should my sensitivity be $30,000,000?!

And I'm not grasping if or how this multiplication is supposed to represent a normalization. If the vega were insensitive to volatility, then the multiplication kind of represents extrinsic value. So the idea is loosely extrinsic always going to zero for longs (seems too conservative) and doubling for shorts (seems under-conservative maybe)? All the delta definitions in 21.19 to 21.24 are so nice and clear...

## Answer by ir7 (score 2, accepted)

https://quant.stackexchange.com/a/80540

Basel must have had in mind an explicit forward finite difference FRTB Vega definition that uses a relative shock (just like for the FRTB Equity Delta), rather than an absolute shock (see FRTB IR Delta):

$$ \frac{V(\sigma + 0.01 \sigma )- V(\sigma)}{0.01} \stackrel{Taylor}{\approx} \frac{\partial V}{\partial \sigma}(\sigma)\cdot \sigma. $$

So, unless the local regulator re-specifies the definition, either of the two numbers above are usable.

Note: Basel's statement in para 21.28(1) might refer to two pricing functions $V_{LN}$ (Black) and $V_N$ (Bachelier}, hitting the same market price, using respective market implied vols $\sigma_{LN}$ and $\sigma_N$. See this for how the two implied vols relate. At least at ATM and for short TTE, it turns out that: $$\sigma_N \approx \sigma_{LN}F.$$ It follows that: $$ V_{N}(\sigma_{LN} F)\approx V_{LN}(\sigma_{LN}).$$ Taking derivatives wrt $\sigma_{LN}$ on both sides, we get: $$ \frac{\partial V_{N}}{\partial \sigma_N } F\approx \frac{\partial V_{LN}}{\partial \sigma_{LN} }. $$ So, if the risk manager reports "Vega X Vol" in both cases, we do have: $$ \frac{\partial V_{N}}{\partial \sigma_N }\sigma_N \approx \frac{\partial V_{LN}}{\partial \sigma_{LN} }\sigma_{LN}. $$