Variance swap

[S2000magician]

@Stardust99

Good morning! Thanks for your reply Pyng. I get the scaling factor (10,000) for the realized variance.
But when they calculate variance notional using Vega notional, it was divided by 2K(it’s not the same scaling factor here). See the attached.

Using 20%, Variance notional = 250,000.

So the result would be: [(0.045175 x 10,000) - (0.04 x 10,000)] x $250,000 = $1,293.75
It’s a different outcome, isn’t it?

Yes, it’s a different result.

The people who created variance swaps decided that they would write “20” to mean 20% (= 0.2) in their formula.

If they had decided to write “0.2” to mean 20%, then their formula would have been different (e.g., variance\ notional = \frac{vega\ notional}{0.02×strike\ price}), so that the amount comes out to be the same.

The point is that for the variance swap described in that question, the answer is supposed to be $129,375. You can get that number by changing how you write percentages and how you compute the variance notional, as long as you change them together.

In a way, it’s similar to how convexity is computed and used for bond pricing. They used to have a “2” in the denominator of the formula for calculating convexity, but not in the formula for using that calculated number to price a bond. More recently, they decided to drop the “2” from the formula for calculating convexity, but include a factor of “½” in the formula to use that convexity. The upshot is that the mount of convexity adjustment is the same either way, there are simply two routes one can use to determine it.

[Stardust99]

Thanks! I think i might finally got it!!! Your magic works wonders

[Dan_Thorn]

possibly a helpful way to break down the equation:
Variance Swap Payoff

Vega notional is a value that represents the potential profit or loss of a variance swap if volatility changes by 1% from the strike price. It’s a dollar value per volatility point.

• If a variance swap has a vega notional of $10,000, then a 1% change in volatility would result in a profit or loss of about $10,000.

We must bear in mind that this is an approximation because the variance swap payoff is convex and the profit and loss is not linear for changes in the realized volatility.

This just means the price change is linear and the change in volatility is a curve (think bond convexity vs duration)

So how do we calculate the ‘convexity’?: variance notional

Variance notional considers the convex nature of the swap’s payoff. It is similar to the relationship between duration and convexity in fixed income. Duration approximates small shifts from the current state, while convexity accounts for the exact measurement of changes in bond present value.

NOTE: variance notional is both linear estimation and curve adjustment – whereas convexity in bonds is just the curve adjustment and you need to add straight-line duration estimate to the curve adjustment of convexity to get the total change when doing this for bonds.

So Three simple steps:

Step 1: Find Variance Notional: you have vega notional – they HAVE to give you this, its part of the contract and it can be whatever amount you need it to be to hedge your position, its just the agreed upon payoff for each percentage point difference between realized volatility, and the strike volatility. – but it is a straightline estimate of a curve

Variance Notional equals: Vega Notional / (2 x Strike)

Just memorize, and remember its adjusting vega for the curve

Now you need to make two more adjustments: Step 2: Find Present Value (PV) and Step 2: find weighted average of realized and remaining implied volatility (When calculating the variance of a swap during its lifetime)

Step 2: PV is just the PV Factor: 1 / (1+r^(t/T) – this just adjusts r for the actual time left and finds PV

Step 3: The weighted average volatility – this just weights the realized volatility so far, with the remaining period implied volatility.

So all together

Step 1: variance notional (ie the combined linear and curve change in value) Vega Notional / (2 x Strike)

Times
Step 2: PV Factor (ie the payoff value is realized in the future but you want the value today) 1 / (1+r^(t/T)

Times

Step 3: The difference in realized and implied vol minus strike vol [t/T×[RealizedVol(0,t)]^2+(T−(t/T))×[ImpliedVol(t,T)]2] –Strike^2

NOTE: you don’t know actual realized value for the full period so you take a weighted average of the (volatility realized so far + implied volatility for remaining period) – strike volatility

In Summary:

Payoffs follow a curve so you need to adjust for payoff by converting vega notional to variance notional

then you need to find the present value of the payoff

then you need to calculate the payoff which is the strike minus realized volatility, but of course we don’t know actual realized volatility so we get a weighted average of what we know so far, plus the remaining implied volatility.

Value of Swap = Variance notional × PVt(T) × { w of [RealizedVol(0,t)]^2+w ×[ImpliedVol(t,T)]^2–Strike**^2**}

Note vol is quoted in standard deviation terms; so square all the terms

Final Note:

Payoff at maturity:

It’s the same calculation however, obviously no PV or weighted average:

Vega Notional * (( Realized Variance – Strike Variance)/( 2 * Strike Price))

You see it more clearly if you rewrite as
Variance Notional * (Realized Variance – Strike Variance)

And remember that variance notional is just: Vega Notional/( 2 * Strike Price)