Several months ago Xena Exchange launched XBTVAR — futures on the volatility of Bitcoin (to USD). This is the first exchange-traded volatility contract on the market.
Now let us take a look at the physical and economic sense of this menagerie.
1. For simplicity’s sake, we can assume that the variance is just the sum of the squares of changes in a random variable. For example, if the SV for 5 tests took on the following values: 1, -1, -2, 1, 0, then the variance will be (-1 - 1) 2 + (-2 - -1) 2 + (1 - -2) 2 + ( 0 - 1) 2. Variance is additive as the sum of N random variables with the same distribution (with mean μ and variance σ2) is spread normally with expectation Nμ and variance Nσ2 (central limit theorem). Volatility (which is essentially the standard deviation σ or the root of the variance) does not possess this property, so we cannot simply make a contract for it.
2. We are interested in the variance in annual terms. By using additivity, we can write the following formula: Variance on an annualized basis = Number of days in a year / Number of measurements* Variance over the measurement period (i.e., we can simply approximate it to a year linearly).
3. Why in annual terms? In order not to depend on the life of the contract and to be able to compare different contracts with each other, etc.
4. The contract itself implies that people are essentially betting on the variance at the end of the contract. For example, I bought variance at 100, and if at the end of the life of the contract it turned out to be 110, then I earned 10, and my counterparty lost the same amount.
5. Historically, the terms (root) for volatility are used rather than variance. Therefore, all the prices are expressed in terms of annual volatility, i.e. not 100, but 102. Later on in the text, I will use variance and volatility as synonyms, the reader just needs to remember that volatility is the root of the summation of two volatilities – in fact, their squaring, addition and root extraction.
6. The price at which we entered the trade is an implied variance – the way we estimate the variance that will be accumulated in a month.
7. Everything is really simple in the example from paragraph 5, because there are only two participants who enter the contract at the very beginning and exit it at the very end, and there is only one calculation according to the final formula.
8. Variance is the green area on the middle chart. If we only took it into account in the contract, such a contract would be impossible to trade (because it grows linearly). Why not make a prediction for a year from the current number of measurements ? I don’t know, probably, because of the very high volatility at the very beginning of the contract’s existence (a sharp peak and then a decline on the chart):
9. Accordingly, we mix in a linearly decreasing implied variance (the orange zone on the chart). This graph assumes that IV == 50 (in percent per annum) throughout the life of the contract (while, according to our formulas, its contribution decreases linearly with time). The sum of the contribution of the realized variance and the implied variance gives us a value (k2) that fluctuates around the same value. This value can be traded. The linear decrease is done so that the prices in the order book do not depend on the time to expiration:
a) Let us assume the market thinks that at the beginning of the life of the contract variance 152 will accumulate in a month. The key word here is ‘month’. In the middle of the life of the contract (when, for example, 100 units of variance have already accumulated), prices should reflect that only 11.22 (sqrt (152 - 100)) will have accumulated by the end of the contract. While market expectations have not changed, the price has changed. To prevent this from happening, prices are always set in terms of the annual wall.
10. Until now, we have only talked about measurement results once a day. From the point of view of the final calculation, we are not interested in what happened within the day (because the contract is for the variance of daily prices. It is clear that if we calculate the variance not daily, but, for example, by hour bars, it will turn out to be a completely different value).
11. On the other hand, we need to look at the current price within the day. For example, if the intraday price rallies (increasing the variance), the likelihood increases that the price will be around this level at the close of the day. Therefore, it is necessary to calculate the current risk (and is it time to liquidate some entity) in real time. The question of why not take into account the accumulated variance at the current moment within the day at each clearing has no simple answer (perhaps, in our case, it would be worth trying).
a) It is rather clear why this is done in the original CBOE contract. The variance is hedged by an options package, and the options are likely to be cleared on a CBOE only once a day. Accordingly, any fluctuations in the realized wall within the day should not be taken into account, so as not to violate the “futures + options package” strategy. In addition, there are also variance swaps (OTC instrument) on the traditional market, in which the variance is calculated by day. Futures must be settled in the same way as swaps (in which case they can be hedged against each other) for the market to function properly. The contract is designed so that if you trade intraday, you are only trading the implied wall.
12. In general, it turns out that, on the one hand, we cannot use the realized variance within the day (apparently, due to distortions introduced into trading), and on the other hand, we must use it (so as not to get liquidation at clearing when a new piece of the realized variance is added)... This problem can be circumvented by forcing implied volatility to trade around the current (real-time) realized variance. If the realized wall grows sharply within the day, the implied wall will grow along with it (at least due to the price corridor) and the revaluation of the contract will change accordingly.
13. The width of the corridor (as for our other contracts) depends on the leverage. The width guarantees that no liquidations will occur even when the price changes to the maximum within the corridor. Meaning that even if the contract’s value equaled the upper border a second before the transition to a new day, and the daily value became equal to the lower border, liquidations will not happen as a result.
14. And now let us examine why the formula for calculating the bars differs by a factor (365 / n instead of 365 / Basis), and why you need to take into account the minutes within the day:
a) As shown earlier, the 365 / Basis formula gives an annualized variance at the time of contract expiration. Meaning that if 1000 is accumulated over 30 days of the contract, we say that the annual variance is 365/30 * 1000. This is not the case until the contract expires (for example, in the middle of the same contract, the accumulated variance was about 500, 365/30 * 1000 which gives half the annual variance). The implied wall (prices in the order book) is always set in annual terms. To compare these two parameters, we need to convert the accumulated variance into annual.
b) Why take minutes into account? Let us assume that, on average, we have 10 units of variance per day. This means that an average of 10 / (24 * 60) is accumulated per minute. We had some value of the accumulated variance at the beginning of the day, for example 100 (the contract lasted 10 days). By the end of the day, the value should be (100 + 10) / (10 + 1). The question is when should we add this unit to the days. If we consider the beginning of the day, the planks fall sharply in the morning (the value of the variation will be (100 + 10 / (24 * 60) for the first minute) / (10 + 1). If we consider the end of the day, the planks fall sharply in the evening by following the same principle (we would calculate them as (100 + 10 - 10 / (24 * 60) a second before clearing) / 10. It is the same at clearing, but divided by 11.) Therefore, we add linearities to our approximation to determine the boundaries.