Variance Swaps
Suppose you made a bet with someone that variance on daily returns for a market index was going to be over 400 this year (meaning a standard deviation of 20%). One month in, the realized standard deviation of returns was just 7% so far, for a variance of 49.
Are you winning?
You're right!
Actually, you are!
Annualized variance is just realized variance scaled to 252 trading days for the year. If you have realized variance of 49 after a month (21 trading days), then your annualized variance is:
> $$\displaystyle \frac{252}{21} \times 49 = 588$$.
If this keeps up, you'll win the bet.
What kind of agreement about the payment from your friend to you for this bet would make sense?
Not quite. Variance is unitless, so you haven't identified anything to pay. You'll need some sort of notional amount.
No, your friend would never agree to this; you would always get paid.
Yes.
Over 400, and you get paid. Under 400, you pay your friend. Maybe you agree on a notional amount of USD 10, so that if you settled today, you'd get that much for each of the 188 points of extra variance for a USD 1,880 payoff. You have just set up something close to a **variance swap**.
How does this mainly differ from other swaps?
Not really; this is similar. An interest rate swap is based on interest rates, and a variance swap is based on an established variance calculation.
Counterparty risk is still there. You may win the bet with your friend, but you only *really* win if you actually get paid.
Right.
There are counterparties and a clear reference, but there's no exchange of notional at the start and no cash flows along the way, just that payment at the end.
In a real variance swap, things aren't as simple as the bet with your friend. It's not based truly on variance, but volatility, measured with natural log of each price relative instead of traditional returns. So a 2% return would enter into the volatility calculation as:
> $$ ln(1.02) \approx 0.0198$$.
What would a -5% return be considered in the volatility calculation?
You got it!
Not quite.
A -5% return comes from a price relative of 95/100 or something like that, and the natural log of 0.95 is about -0.0513. So the Volatility Strike Price X was 20 with your friend, and the variance strike was 400. With variance swaps, you want to talk in terms of volatility strike prices. So what is the volatility strike, really?
No, the expected variance is 400. But 20 is the expected standard deviation.
Precisely.
Then there's the notional. You and your friend agreed on a notional of USD 10. In a variance swap, the trade size is in terms of **vega notional**, which is quite different. To really get the **variance notional** of USD 10, your vega notional would have to be a lot bigger. Here's why:
> $$\displaystyle \text{Variance notional} = \frac{\text{Vega notional}}{2 \times \text{Volatility strike}}$$.
What vega notional would lead to a variance notional of USD 10, given that your bet is about a 20% standard deviation?
Perfect!
That's not it.
> $$\displaystyle \text{Variance notional} = \frac{\text{Vega notional}}{2 \times \text{Volatility strike}} = 10 = \frac{400}{2 \times 20} $$
This can make vega notional look like a useless figure, but suppose the year ended up with a 21% standard deviation. You won. Your payoff would be based on volatility of 441 less the 400 variance strike, giving you a USD 410 payoff. Just about the vega notional!
How might you restate the meaning of vega notional?
Exactly.
You'll see vega notional quoted, along with the volatility strike. Then you can use these to find the volatility notional for the payout at the end. Again, if the standard deviation ends up at 21% for the year, then you'll get:
> $$ N_{Variance} (\sigma^2 - X^2) = 10 (21^2 - 20^2) = \text{ USD } 410$$.
Not really. As you and your friend agreed on USD 10, that is the variance notional.
No, this refers to variance notional.
Some interesting things happen along the way. Suppose that after the first month, you and your friend decided to value the swap. You'll need more information: a risk-free rate for a present value factor, and the implied volatility at this point, which probably changed. Given the way your bet was made, what would your friend hope for?
Right again!
Actually, no. Your friend would really want to see lower implied volatility. That would mean that volatility is expected to be lower, and your friend will have to pay you less money. You would obviously hope for greater implied volatility. Logically, then, how would implied volatility be related to your swap value?
Absolutely! That would mean that volatility is expected to be lower, and your friend will have to pay you less money. You would obviously hope for greater implied volatility. Logically, then, how would implied volatility be related to your swap value?
There you go!
That's not right. Your swap value is based on volatility, so there's a positive relationship.
Assume a 2% risk-free rate, and suppose the implied volatility rose from 20 to 21 in this first month. The value of the swap is calculated as follows.
> $$ VarSwap_t = N_{Variance} \times PV_t(T) \times \left[ \frac{t}{T} \times {RealizedVol(0,t)}^2 + \frac{T - t}{T} \times {ImpliedVol(t,T)}^2 - X^2 \right]$$
In your case, letting $$t$$ be months, this would be:
> $$\displaystyle VarSwap_1 = 10 \times \frac{1}{1 + 0.02 \times \frac{11}{12}} \times \left[ \frac{1}{12} \times 588 + \frac{11}{12} \times 441 - 400 \right] = 522.91$$.
That looks like quite a calculation, but there's your variance notional, and a little PV factor, and then you're just looking for the difference of the actual volatility and the strike, as before.
What does the "actual volatility" look like in this swap valuation?
Actually, it is. You have the realized, the implied, and the time that each represents.
Exactly.
It's a weighted average of what has happened and what's expected to happen, times the notional. It gets pretty straightforward after you look at it for a while. But in the end, if volatility ends up at the implied 21, you'll get your USD 410.
If you play around with other calculations, linearly increasing the realized volatility for that first month, you'll notice that that the swap value grows faster:
| Realized Volatility | Payoff | Difference |
|-----|-----|-----|
| 21 | 410 | |
| 22 | 840 | 430 |
| 23 | 1,290 | 450 |
| 24 | 1,760 | 470 |
| 25 | 2,250 | 490 |
It's convex.
Not really. The variances are already calculated and sitting there with timings attached.
To summarize:
[[summary]]
Yes
No
Payment is variance times notional
Payment is variance minus 400 times notional
Payment is variance minus 400
There's no reference rate
There are no cash flows until the end
There's no counterparty risk
Other responses
-0.051
Expected variance
Expected standard deviation
Other responses
400
It's basically the traditional meaning of "notional" from other swaps
It estimates the change in the value of the variance swap
It is the multiplier in determining the payoff at the end of a variance swap
Lower implied volatility
Higher implied volatility
Positively
Negatively
Positively
Negatively
A weighted average
A variance calculation
Nothing intuitive
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