Suppose your risk analytics does not cover Variance Swaps and you decide to proxy it for example using a future on the related volatility index (e.g., VIX future with same maturity as the variance swap on VIX). How can you gauge the basis risk?
PS I understand that you can replicate the variance swap by a basket of options, but suppose you do not want to follow this path as it will be more operationally intensive. Thanks
Calculating Var for a Variance swap by proxying
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 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: Calculating Var for a Variance swap by proxying
Why not just use the VIX index itself as a proxy for the S\&P 500 variance swap rate (vsr)?
Then use data on s\&P 500 and VIX to compute variance risk premium (vrp) = difference between realised variance and vsr of same maturity as the swap.
If you are an issuing bank (paying realised) it is vrp that you need to use measure VaR, otherwise just compute VaR from this vrp series. The vrp is typically small and negative, so issuing bank who pays realised usually makes a small profit. But vrp shoots upwards during volatile periods, and these banks can make enormous losses.
Using historical VaR can give very misleading picture because the risks on vrp are so highly sensitive to the market conditions. For issuing bank it is especially important to ensure your data includes banking crisis  measure VaR using data from 1 Jan 2000 to 31 Dec 2009 and then use Jan 2010  now for backtesting, rolling the 10 year sample daily and computing 1day VaR then doing usual coverage tests as described in Vol IV. Stress tests should always include banking crisis period, even if vrp has been small and neagtive for many years  it can shoot up at any time without warning.
Hope this helps, Carol
PS Digression: There is a considerable difference between VIX and its futures. There is no trading on VIX, so noarb doesn't apply and the `basis' VIX  VIX futures prices is very high and extremely variable. To measure variance of the variance swap rate, you'd need to use the S\&P indices (which are constant maturity VIX futures trackers representing investable returns) for the futures. They include a small roll cost which is almost always negative (when the futures term structure is in contango) but swings to be very high and positive at the onset of a crisis (when the term structure has a brief period of backwardation)
We have rtn VIX (y) = rtn S\&P Index (x) + rtn basis (e) where all rtns are daily log returns ln(P_t/P_{t1}) for some price P
Then Variance operator V > V(y) = V(x) + V(e) + 2Cov(x,e)
So if you use Variance of S&P indices of the same maturity as the variance swap on S\&P 500, ie using only V(x) instead of V(y), then you also need to add V(e)  variance of returns on basis  and Cov(x,e) which is definitely nonzero and probably high and positive (since 'basis' increases when volatility is high)
Then use data on s\&P 500 and VIX to compute variance risk premium (vrp) = difference between realised variance and vsr of same maturity as the swap.
If you are an issuing bank (paying realised) it is vrp that you need to use measure VaR, otherwise just compute VaR from this vrp series. The vrp is typically small and negative, so issuing bank who pays realised usually makes a small profit. But vrp shoots upwards during volatile periods, and these banks can make enormous losses.
Using historical VaR can give very misleading picture because the risks on vrp are so highly sensitive to the market conditions. For issuing bank it is especially important to ensure your data includes banking crisis  measure VaR using data from 1 Jan 2000 to 31 Dec 2009 and then use Jan 2010  now for backtesting, rolling the 10 year sample daily and computing 1day VaR then doing usual coverage tests as described in Vol IV. Stress tests should always include banking crisis period, even if vrp has been small and neagtive for many years  it can shoot up at any time without warning.
Hope this helps, Carol
PS Digression: There is a considerable difference between VIX and its futures. There is no trading on VIX, so noarb doesn't apply and the `basis' VIX  VIX futures prices is very high and extremely variable. To measure variance of the variance swap rate, you'd need to use the S\&P indices (which are constant maturity VIX futures trackers representing investable returns) for the futures. They include a small roll cost which is almost always negative (when the futures term structure is in contango) but swings to be very high and positive at the onset of a crisis (when the term structure has a brief period of backwardation)
We have rtn VIX (y) = rtn S\&P Index (x) + rtn basis (e) where all rtns are daily log returns ln(P_t/P_{t1}) for some price P
Then Variance operator V > V(y) = V(x) + V(e) + 2Cov(x,e)
So if you use Variance of S&P indices of the same maturity as the variance swap on S\&P 500, ie using only V(x) instead of V(y), then you also need to add V(e)  variance of returns on basis  and Cov(x,e) which is definitely nonzero and probably high and positive (since 'basis' increases when volatility is high)
Re: Calculating Var for a Variance swap by proxying
Thank you very much for your reply. I have another question. Does anyone know how to map a vega notional of a Variance Swap to 1) the notional on VIX Future with same maturity; 2) notional exposure of VIX?

 Posts: 815
 Joined: Sun Sep 28, 2008 10:30 pm
Re: Calculating Var for a Variance swap by proxying
The variance swap rate is based on VIXtype formula, and there is a very significant difference between statistical properties of VIX and VIX futures  see my discussion paper on Diversification with VIX futures with Korovilas on ssrn or my webpage. So for this reason, no mapping is possible.
For mapping to VIX index, again I cannot see myself how a mapping would work, since vega exposure on variance swap is to the variance risk premium, i.e. difference between realised variance and implied variance not to the implied leg only (and VIX is just the implied leg).
Cheers, Carol
For mapping to VIX index, again I cannot see myself how a mapping would work, since vega exposure on variance swap is to the variance risk premium, i.e. difference between realised variance and implied variance not to the implied leg only (and VIX is just the implied leg).
Cheers, Carol
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