Correlation variance swap.

[SP_SK]

hi, please explain related to
Paying fixed in a variance swap on an index and receiving fixed on individual
what does the following statement mean:
If correlation increases, so will the variance. As a consequence, the present value for the variance swap buyer, the
fixed variance swap payer, will increase. This increase is expected to outperform
the potential losses from the short variance swap positions on the individual
components.

 

[David Harper CFA FRM]

HI @SP_SK

To me, it's a complicated trade but the idea is a pair trade of sorts:

• Pay fixed in a single variance swap on an index (aka, portfolio), plus
• Receive fixed in multiple variance swaps on the components of the index

Abstractly, imagine the index is only two components, Stock A and Stock B, equally weighted. Say each stock has volatility of 10.0%, or variance of 0.10^2. As you know, the variance of the index is given by .5^2*0.10^2 +.5^2*0.10^2 + 2*.5*.5*.10*.1*ρ, where ρ is the correlation between the components.

Say the component volatilities do not change, for illustration purposes of the simple case (ie, both A and B hold at 10%), but their correlation, ρ, increases. With respect to the multiple variance swaps on individual components, there will be roughly no gain/loss: we are receiving the fixed at 0.10^2 and paying the realized variance at 0.10^2. However, if the ρ increases, then the variance of the index increases (per the formula: an increase in correlation implies an increase in portfolio variance!), so we may pay on a fixed variance at the initial (eg) ρ = 0 such that index variance = .5^2*0.10^2 +.5^2*0.10^2 + 0, but we will receive on the higher realized variance .5^2*0.10^2 +.5^2*0.10^2 + X, even as component volatilities are unchanged. The same "pair trade" type logic applies when we generalize to cases where the component volatilities do change: if component volatilities decrease, there is a gain on the individual component swaps but that is offset (hedged) by a loss on the index swap, and vice-versa. The net position's gain/loss will hinge on the correlation. I hope that's helpful!

 

[SP_SK]

Thanks David I think I am understanding this better. To confirm there are 2 pairs trade that is happening here. Related to the index while the buyer of the variance swap is paying fixed on the variance of the index determined at the start of the transaction he is also receiving the realised variance of the index variance. This part was not clear since as per the definition he was receiving fixed the variance of the variance of the underlying stocks of the index hence I got confused that if the index correlation increased how would the position be profitable to variance buyer. Thanks.

 

[Linette Joseph]

hi David,

In reference to Meissner's chapter 'Some Correlation Basics' - Example of Var for a 2-asset portfolio, the portfolio volatility is taken as the product of (horizontal beta vector, covariance matrix, vertical beta vector); I understand this is how the example calculates it as the covariance matrix has been provided, but I tried calculating the portfolio volatility in the usual way and the answer does not match -

.67^2*.02^2 + .33^2*.01^2 + 2*.67*.33*.02*.01 and then the sq root of this

This approach was used in the previous example of the chapter for portfolio volatility.

Could you please help me understand why it wouldn't work or why we should only use the cov matrix formula as stated in Meissner's example?

 

[David Harper CFA FRM]

Hi @Linette Joseph Here is the XLS (see screenshot below) of Meissner's calculation in Chapter 1 https://www.dropbox.com/s/jgd2v6suq9mk42h/0224-meissner-port-sigma.xlsx?dl=0 .... I don't understand where you are getting your weights (67% and 33%) and standard deviations (2.0% and 1.0%) and your formula omits the correlation, or implicitly assumes the ρ = 1.0.

The 2-asset matrix version is the exact same as the explicated formula; the formula is a straightforward post-then-pre multiplication of the vector-matrix-vector. I think the confusion might be due to the fact that covariance(X,Y) = correlation(X,Y)*StdDev(X)*StdDev(Y), so there are two equivalent versions:

• σ^2(P) = w(x)^2*σ(x)^2 +w(y)^2*σ(y)^2 + 2*w(x)*w(y)*covariance(x,y); and because covariance(x,y) = σ(x)*σ(y)*ρ(x,y), also:
• σ^2(P) = w(x)^2*σ(x)^2 +w(y)^2*σ(y)^2 + 2*w(x)*w(y)*σ(x)*σ(y)*ρ(x,y)

When Meissner solves for portfolio volatility of 16.66% (green cell below) of the equally-weighted (50% and 50%), he's using the covariance directly from the pair series. He doesn't show the correlation, which you can see is -0.7403 such that:

• σ^2(P) = 0.50^2*0.4451^2 +0.50^2*0.4758^2 + 2*0.50*0.50*-0.1568 = 0.16655^2; or,
• σ^2(P) = 0.50^2*0.4451^2 +0.50^2*0.4758^2 + 2*0.50*0.50*0.4451*0.4758*-0.7403 = 0.16655^2

In this way, also, the formula you are showing is correct for assumptions (i.e., weights = 67%/33% and volatilities = 1%/2%) if the correlation is 1.0, otherwise it omits the correlation, ρ. I hope that explains it!