# Hedging vega risk with varswaps

I have encountered a statement that in summary reads like this:

Varswaps became popular after the LTCM meltdown due to high levels of implied volatility the market was seeing at the time. Hedge funds took advantage of this by selling the realized variance (i.e. being short varswaps which are a proxies for the quadratic variation of the log-spot process). Dealers wanted to buy vega at these levels because they were structurally short vega.

The vega mentioned here is probably the bucketed vega, i.e. the partial derivative of the portfolio PV with respect to different $$\sigma_{imp}(K,T)$$.

Questions:

1. There is not just one vega (there are as many as $$(K,T)$$ pairs we use to construct the volatility surface). Does the statement above refer to vega as a whole (i.e. the surface as a whole became more expensive)?
2. How can they neutralize vega by buying what is essentially a linear product paying the quadratic variation of the log-spot? Why adding varswap positions can be equated to long vega?
• This is a very timely question in view of the fact that recently Allianz Global investors had to shut down its hedge fund Allianz Structured Alpha 1000 because of lossses connected with Point 2 raised in your question. From what I understand the fund was long S&P Puts and short Varswaps. Essentially long vol and short vol squared. – noob2 Apr 19 '20 at 21:05

There are many forms of vega. For var swaps, you can directly differentiate its strike to get a vega that is the sensitivity against the var strike. In LV, you can get a vega by a parallel bump in the entire volatility surface (beware of arbitrage though). People seldom calculate vega as a partial derivative against the implied vol of a certain strike and expiry. In Black Scholes, vega is the sensitivity of the option against $$\sigma$$, which can be thought of a parallel bump in a flat volatility surface.