Convexity and Variance Swaps

CFA Level III Portfolio Management
#1

Guys, I read about convexity in variance swaps, but understood nothing. Do you know any video lectures maybe?

This is extract from Schweser

" Convexity Because the payoffs on a variance swap are based on variance, while the strike price is expressed in terms of volatility, the payoffs for a variance swap are convex with respect to volatility. Compared to the payoffs on a volatility derivative with payoffs that are linear with respect to volatility, (1) when realized volatility is below the strike, the losses on the variance swap are smaller than the losses on the volatility derivative, and (2) when realized volatility is above the strike, the gains on the variance swap are greater than the gains on the volatility derivative. This convexity is similar in nature to the convexity of bond prices with respect to yield. With the variance swap, payoffs are increasing at an increasing rate when volatility rises and decreasing at a decreasing rate when volatility falls. Convexity is an attractive feature to those who use variance swaps to hedge tail risk. When equity values fall sharply and volatility increases dramatically, decreases in portfolio value are offset by profits on a variance swap that increase at an accelerating rate."

#2

Or maybe you have any other ways to recommend how to understand this

#3

That explanation is pretty confusing. It is no wonder you didn’t get it. If I only had this info to go by, i would be lost too. The text is comparing the payoffs from a volatility swap and variance swap. If you recall interest rate swaps, these are pretty much the same conceptually. One ends pays a notional amount N times a fixed rate for the volatility, while the other end pays the notional M times the realized volatility. The variance swap is similar, except the notional amount differs. to make things more concrete, let

Nvol be the notional amount for the volatility swap

Nvar be the notional amount for the variance swap

σk be the strike volatility (the fixed rate for the volatility)

σrealized be the realized volatility

then

payoff from variance swap = Nvar x ((σrealized)2 - σk2)

payoff from a volatility swap = Nvol x (σrealized - σk)

it turns out (according to my book anyhow) that Nvar = 1/(2σk) Nvol,

and so the payoff for variance can be rewritten as

payoff from variance swap = Nvol / (2σk) x ((σrealized)2 - σk2)

now, consider what happens when realized volatility is lower than the strike volatility.

that is, σrealized < σk. if we let σrealized = σk - ε , where ε is any positive constant,

and substitute it into the payoff formula, we see that

payoff from variance swap = payoff from volatility swap + a positive constant

this means that, when volatility increases, the payoff for a variance swap is greater than the payoff for a volatility swap. in other words, you lose less money with a variance swap compared to a volatility swap. this is point #1.

next, consider what happens when σrealized > σk.

this time, let σrealized = σk + ε, where once again ε is a positive constant.

if you substitute this into the payoff formula, you’ll find that

payoff from variance swap is also greater than volatility swap. this is point #2.

edit: as for convexity:

A single-variable function that can be differentiated twice is a convex if and only if

the second derivative is non-negative.

i.e. f(x) is convex iff f’’(x) >= 0

otherwise, if f’’(x) <= 0, then it is a concave function.

https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/cv1/t

here, the book says that the payoff is a convex function of the volatility, and is similar to the convexity of bond prices. in this case, we are treating the payoff as a single variable of the volatility (this is the ‘x’)

as a reminder, the convexity for a bond price is given by

[1/P(i)] x [(d2/di2) P(i)]

the payoff for a variance swap is given by

Payoff = notional x ((σrealized)2 - σk2).

differentiating twice with respect to σrealized results in a positive constant, so

[d2/d(σrealized)2] (Payoff of variance swap) > 0

because the second derivative of the payoff function is positive, by definition it is a convex function.

3 Likes
#4

I learned so much from this post. Thanks so much!

#5

Hi,

I understand the need for adjustment to go from Vega notional to Variance notional but I don’t understand why we multiply Variance notional by 2K to get the vega notional.

Thank you.

#6

I still dint understand why payoff for Variance Swap is convex

#7

Vega notional= Variance notional/ (2K) where K is the volatility strike price set at the initiation of the variance swap.

Vega notional is the monetary value of a 1% change in volatility.

The payoff of the variance swap is what we call the settlement amount=
Variance notional *(realized variance-strike variance)

Where realized variance = (realized volatility) ^2
strike variance= (volatility strike) ^2

Suppose a variance swap with a volatility strike price of 25% and a notional of $1500.

The Vega notional is = 1500* (2*25)=75 000

It means that a 1% increase in volatility (up to 26%) is equivalent to $75 000.

If you use variance and not volatility the monetary value associated with a 1% increase in volatility will be equivalent to:

1500*(26^2-25^2)=1500*51=$76 500

You can notice that $76 500>$75 000

The payoff of the variance swap is 2% higher than the payoff using volatility.

It means that for a 1% increase in volatility the payoff you get using VARIANCE (aka (VOLATILITY)^2) vs using VOLATILITY will be greater.

Now imagine that the volatility rises by 2% up to 27%. The Vega notional is still $75 000 because the volatility strike remains the same and the monetary value of a 2% increase will be 2* 75 000 because the Vega notional like I said is equivalent to a 1% increase in volatility.

Now let’s calculate the payoff we get using variance:

Settlement amount (variance swap payoff) =1500* (27^2-25^2)=1500*104=$156 000

Payoff using volatility = $150 000

Payoff using variance = $156 000

The payoff of the variance swap is now 4% higher than the payoff using volatility.

The payoff increases at a geometric pace using variance while the payoff increases at an arithmetic pace using volatility.

There is:

1- no linear relation between the 2 payoffs,
2- a greater payoff using VARIANCE vs VOLATILITY,
3- a payoff using VARIANCE vs VOLATILITY increasing at an accelerating pace,
AND SO:
4-a CONVEX payoff of the VARIANCE SWAP

#8

thanks a lot for the detailed revert…I understood the point with the math of the payoffs…

1 Like
#9

My pleasure

#10

Can you suggest me something…I have been writing mocks but my score is not improving…I am not able to articulate my thoughts properly even when I know the asnwer…How should I approach an AM question? Any framework to apply ?

#11

What are your average score on the am section?

I have not passed the level 3 yet. I will sit for the exam in early September. So I am not sure I am legit but you can DM me if you want.