Time-Weighted Vega and the Limits of Compressing Volatility Risk

# Time Weighted Vega of Options


# Time Weighted Vega of Options












What is the correct way to calculate Time weighted vega for options?

From searching in Google, saw this reference: http://www.topquants.nl/wordpress/wp-content/uploads/2015/01/Van-Gulik-Risk-management-at-Optiver.pdf

I also saw from a Risk System, the calculation of time-weighted vega as:

Vega_weighted = (((1-exp(3t))/3)* Vega

On the above two ways, I have few questions:

- Which of the above method is more accurate?

- Why not just divide the Vega by sqrt(T) to annualise the vega?

- Why is there a factor of 3 in both equations?

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/82448

Too long for comment.

In general, trying too hard to have fewer risk measures can obscure material risks. (Think of lossy compression - Mona Lisa collapsed into a single pixel.)

Implied volatility has multiple dimensions, depending on the underlyings, but at least:

- the time remaining to the option's expiry,aka "maturity" (confusing usage because the underlying too may have maturity), aka "DTE", "days to expiry".

- the moneyness, i.e. how far is the option's strike from the underlying's price - is the option in the money or out of the money.

The cited slide doesn't mention the moneyness dimension at all, but only discusses the time to expiry dimension, which makes me suspect that if some portfolio had some large vega at some moneyness and expiry, and the same vega amount with the opposite sign, close expiry, but very different moneyness then the outlined methodology would consider the portfolio's vega to be flat.

Moneyness aside, focusing just on the time to expiry - the goal of the methodology is to compress the sensitivities to the implied volatilities at different expiries into a single number, losing some details - sort of what VaR/ES does. If you collect time series of a large number of implied volatility surfaces, and calculate historical principal components, volatilities, and correlations, you're likely to observe that:

- much (but not all!) of the historical variance of the volatility surface is explained by the first principal component, which is approximately a parallel shift; and yet

- the implied volatilities at longer time to expiry have higher historical volatilities

The outlined methodology assumes relationship $c(t_1,t_2)$ of the historical volatilities of two implied volatilities having different times to expiry $t_1$ and $t_2$, other attributes being the same. I see no fundamental / mathematical reason why this ratio should be this much, or even behave the same for different underlyings. I can hope that this formula is based on the empirical analysis of historical data. It is helpful as a back of envelope estimate, but assuming that options at different expiries can perfectly hedge each other doesn't fully describe the reality.

If some portfolio had some large vega at some expiry $t_1$, and vega amount $\times c(t_1,t_2)$ with the opposite sign at expiry $t_2$, then the outlined methodology would consider the portfolio's vega to be flat. It may be good enough for a "first glance", but some additional ways to monitor the vega exposures at greater detail would be prudent.

## Answer by Scott Howard (score 0)

https://quant.stackexchange.com/a/82445

Vega is defined as the partial derivative of BSM solution for price with respect to IV. It means something on it's own without the scaling. It is correct on its own for that strike and DTE. You don't need those corrections. They are just trying to address specific cases.

Regarding the first one. We often think of IV changes in units of the 30 day IV or VIX. Those equations are saying "if the 30 day IV changes by 1 pct point, how much does this 15 DTE change?" That first equation says that the 15 day IV should change by 1.4 (sqrt of 2). So the weighted vega would be the same as the normal vega times 1.4 if our units of change of IV are in the 30 day VIX or 30 day IV.

The first equation comes from the stochastic volatility view. The 3 in the first equation is just there to handle short DTE going up to infinite weighted vol. I bet they avoid going to shorter than 1/9th whatever there vol scale is so they don't hit that limit often.

The second equation comes from considering a local vol BSM model. The BSM solution uses a local IV at each strike. If you say the local IV is a function of time, your Taylor expansion of price contains a $ \nu \frac{\partial \sigma}{\partial t}\Delta t \Delta \sigma$ term combined with the original vega term, you get

$$ \nu (1+\frac{\partial \sigma}{\partial t}\Delta t) \Delta \sigma = \nu_{weighted}\Delta \sigma$$

They empirically found the weighted vega to be for to the second equation. The 3 is just a coincidence.

Either can be more accurate, depending on how you use them. They akso can be inappropriate and don't solve for the same thing.

Edit: without a clear reference for the second equation, I'm not sure how they got that or their limits. Weighed vega decreases as DTE decreases in the second equation while it increases in the first. It might be for a specific asset? Whatever t is doesn't make sense, is t always negative maybe, as current time minus expiration perhaps?