Hedging vega risk with varswaps

Question

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?

Answer 1

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.

For (1), I think the article is referring to the Black Scholes vega profile (i.e. BSVega across different spot values).

When you have a portfolio of exotics for various strikes and expiries, delta hedging alone is not enough. Vega hedging is required to offset movements in implied volatility. On top of that, the vega profile of your hedge instruments should closely match the vega profile of exotics. Let say if both your hedge instruments and exotics have a total vega of $10M today, but they have 2 different vega profiles against the spot. Indeed, a one vol increase will cause zero vega P&L today. However, when the spot drifts away from the current spot, the vegas of your hedges and exotics will evolve into 2 different numbers. The mismatch between vega profiles will be reflected in P&L.

For (2), Variance swaps admit a model independent replication, you can re-express a var swap as an weighted strip of OTM options (see Peter Carr's paper). Indeed, var swaps pays you realized variance in the spot. However, since a var swap is essentially a long portfolio of options, var swaps has a positive BSVega.

Note that the dollar gamma and BSVega of a var swap is constant across spot values, unlike vanilla options which exhibit a bell shaped curve around its strike. It is a key advantage of using var swaps for vega hedging versus using vanilla options.