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## Monday, June 4, 2012

### var swap daily mark-to-market

Imagine we buy a week-long var swap with variance notional \$400K. Since there are 4 price relatives in a week and hence 4 daily realized var (DRVar) numbers, we divide the notional into 4 equal slices. Each day we record the DRVar and compute our "spread" over the strike level (say 30% annualized vol), before computing our gain or loss realized on that day.

If today's DRVol annualized is -64.8029% our spread would be 0.648*0.648 - 0.09 = 0.33. We actually earn 0.33 * \$100k on this day. It's a realized gain. By the way, the minus sign is ignored !

If today's DRVol annualized is 12.7507% our spread would be 0.1275*0.1275 - 0.09 = 0.0737. We actually lose 0.0737*\$100k. This is realized loss.

This is like a cabby earning a daily wage. Each day's earning is realized by end of the day. Next day starts on a clean slate. He could also make a daily loss due to rent, fuel, parking, or servicing.

Now imagine 400 slices. It's possible to have a lot of small losses in the early phase, then some big profit, perhaps due to dividend anouncement. A shrewd market observer may have some insider/insight on these volatility swings -- the volatility of volatility.

It's wrong to project those early losses (or profits) into the remaining "slices". This important principle deserves a special illustration. Supppose I strike a deal to sell you all the eggs my hen produces -- one per day for 100 days, at a total price of \$100. If you initially get a lot of small eggs (losses) you may complain and want to cancel at a negotiated price. However, the initial price is actually a reasonable price, because market participants expect bigger eggs later. After 20 days, the 80 upcoming eggs now have a fair market price of \$88. First 20 eggs were perhaps priced from the onset at around \$11.50-\$12.70, though people couldn't have predicted the exact value of first 20 eggs. Now if you do terminate, you get \$88 + 20 eggs. Fair. Your 20 eggs' value may exceed \$12.

( Now imagine a long var swap position with 100 "slices" with 20 realized.....)   The eggs is an analogy of var swap daily market-to-market. It's similar (slight messier than) to futures mark-to-market. Here "Market" refers to the market expectation of the remaining 80 daily price relatives (all remaining eggs). Remember each of the 80 slices would result in a daily realized PnL contribution. The market participants have some idea about the SUM[2] of these 80 forthcoming DRVariances -- 80 eggs.

[2] variance is additive; volatility isn't.