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BullsI
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Joined: August 17th, 2020, 9:25 am

Variance Swap (VarSwap) Volga hedge

September 7th, 2020, 12:02 pm

Hi

I have been looking at this product recently and there's something I'm not sure I'm interpreting correctly.

I understand a long Varswap can be theoretically replicated with a portfolio of sold options so what I was wondering is whether there's any net Volga risk. The ATM option currently having the largest contribution in terms of Vega hedge won't show any volga (as it's ATM) although the other options (both upside and downside) will indeed show some volga.

So my question is, in a portfolio where a variance swap is hedged with a replicating portfolio of options, does the change in Vega in the Varswap generated by, let's say a rapid increase in vol, get offset by the change in Vega driven by the ITM and OTM options in the portfolio?

Or is it the case that I'll have to be rehedging by longing more options (assuming I'm short the Varswap) as vols go up so the Vega exposure is hedged down to 0 again? Conversely, as vols come back off (if they do) I'd be selling the extra options I bought to reduce my Vega hedge as the Varswap Vega decreases.

Is it like that? Or does the Volga in the options portfolio actually offset the one from the Varswap?

Thanks!
 
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Alan
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Re: Variance Swap (VarSwap) Volga hedge

September 7th, 2020, 2:38 pm

Haven't traded it, so just some general observations.

First, the replication is a theoretical one, requiring various assumptions to hold. In an idealized mathematical world where the assumptions held, there would be no Volga or other risk. This requires a diffusion process generates the stock price data and you initially buy a continuum of all positive option strikes. Then, indeed -- no (option) rebalancing required. 

(p.s. Actually, somewhat weaker assumptions might still work theoretically: stock prices continuous but stochastic vol can jump, etc)

Second, in the real world, you might try to answer your question via a back-test replication -- using a good source of varswap prices and historical options db. Don't know about the former, but the CBOE is a good source for US-based historical options. For S&P500 varswaps, you might be able to answer using just the varswap prices and VIX data. An issue might be new (downside) strikes that come into play with a big vol spike -- strikes missing from your original portfolio.  

Finally, another practical issue might be if the (ISDA) contract had caps.
 
BullsI
Topic Author
Posts: 6
Joined: August 17th, 2020, 9:25 am

Re: Variance Swap (VarSwap) Volga hedge

September 7th, 2020, 7:49 pm

Hi Alan
Thanks for your response but I'm not sure I follow though.
I had taken for granted that the Varswap did have Volga and my question was more with regards to whether the hedging portfolio would also offset that Volga.
However, from your response I understand you're saying the Varswap wouldn't have Volga either but that can't be the case, can it? So, considering the payout is based on Variance, rather than vol, the return of 1 vol point from a vol level at 10 (i.e. the P&L after vols go from 10 to 11) must be less than if the vol was at 20 (and went up by 1 vol point to 21). No?
Am I not seeing correctly? Doesn't a Varswap show Volga as per the above?
And if it does, does the replicating option portfolio (i.e. the hedge) also hedge Volga? Or would it require rebalancing as I suspect?
Apologies if you've answered already but I just can't get my head around the Varswap not having Volga.
Thanks
 
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Alan
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Re: Variance Swap (VarSwap) Volga hedge

September 8th, 2020, 3:11 am

Sorry for the confusion. What I should have said is: I'm not even sure what Volga means when volatility is stochastic. Instead. the claim is* 

We have shown in Equation 20 that a variance swap is theoretically equivalent to a dynamically adjusted, constant-dollar exposure to the stock, together with a static long position in a portfolio of options and a forward that together replicate the payoff of a log contract. This portfolio strategy captures variance exactly, provided the portfolio of options contains all strikes between zero and infinity in the appropriate weight to match the log payoff, and provided the stock price evolves continuously. 
(emphasis mine).

*Derman, E., and Kamal, M., 'More Than You Ever Wanted to Know About Volatility Swaps', Goldman Sachs Quantitative Strategies Research Notes, March 8, 1999.

Another way to say it is: whatever you define varswap Volga to mean -- yes, the replicating strategy offsets it (under the assumptions). No option portfolio rebalancing is needed. I hope that's clearer.
 
 
BullsI
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Posts: 6
Joined: August 17th, 2020, 9:25 am

Re: Variance Swap (VarSwap) Volga hedge

September 8th, 2020, 6:16 am

I just went through the paper you referenced and it makes sense now I think.

Thanks for your help!

(The way in which I usually interpret/define Volga is the change in the sensitivity of the price to a 1 vol point parallel shift (Vega) before and after having shifted the current vol by 1 vol point - parallel).
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