How-to: What is Russian Volatility Index and how it is calculated

Note: This text is published as part of an experiment - we have already covered a fairly large number of introductory theoretical aspects of the stock market in our blog. Today we will try to “go to the next level” and talk about a deeper and more complex topic - volatility indices, in particular, a similar index for the Russian market.

On April 16, 2014, the Moscow Exchange launched the calculation and publication of a new volatility index of the Russian market - the RVI index.
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In a press release of the exchange on the occasion of the launch of the RVI index, it is indicated that the new index allows to assess the level of volatility of the Russian market, and also expands the financial capabilities of option traders, hedgers and institutional investors.

The RVI index is calculated according to five basic principles:

1. The index is calculated to obtain the values ​​of thirty-day volatility;
2. The calculation is based on two series of options on futures on the RTS Index, namely: the options for the next and next series included in the quarterly or monthly series, but not included in the weekly series, the term before the expiration date of which is inclusive more than 7 days;
3. The index calculation also includes futures quotes, which are the underlying asset of the next series option and the next series option.
4. In the absence of quotations and transactions, it is possible to calculate the RVI Index at the theoretical price of the option, determined on the basis of the futures quote, which is the underlying asset of such an option, and the volatility curve at the time of the calculation;
5. The index is calculated every 15 seconds during the main and evening trading sessions on the Derivatives Market (from 10:00 to 18:45 and from 19:00 to 23:50 Moscow time).

According to the Methodology for Calculating the Volatility Index of the Russian Market, approved by the decision of the Board of the Moscow Exchange OJSC (Minutes No. 24 dated April 11, 2014), the RVI index is calculated using the formula:


Where

T ₃₆₅ - 365 days in shares of the calendar year (year = 365 days);
T ₃₀ - 30 days in shares of the calendar year (year = 365 days);
T ₁ - time before the expiration date of the next option series inclusive in shares of the calendar year (year = 365 days);
T ₂ - time before the expiration date of the next series of options, inclusive, in shares of the calendar year (year = 365 days);
σ ₁ is the implied volatility of the next option series;
σ ₂ is the implied volatility of the next series of options.

The volatilities of the next and next series of options are determined by the equations:


Where

ΔK _i - strike step (for the purpose of calculating the Index, the main strikes are used, intermediate strikes are not used);
T ₁ is the time before the expiration date of the near-series option, inclusive, in shares of the calendar year (year = 365 days). Changes every 15 seconds;
T ₂ - time before the expiration date of the option of the far series, inclusive, in shares of the calendar year (year = 365 days). Changes every 15 seconds;
K _i - i-th strike. At the same time (for the purpose of calculating the Index, the main strikes are used, intermediate strikes are not used);
F ₁ , F ₂ - quotes of futures contracts, which are the underlying asset of the next series option and the next series option, respectively.

The futures contract quotation is either the price of the last transaction, or the price of the best active sell order that is lower than the price of the last transaction, or the price of the best active buy order that is higher than the price of the last transaction at the moment. If there were no transactions in the current session until the moment when the futures contract was calculated, the arithmetic average between the prices of the best active buy order and the best active sell order is used. If, at the time of settlement, there are no active purchase orders and no active sales orders, the settlement price determined based on the results of the nearest previous settlement period is used.
Pr (K _i ) is the value of the option for the i-th strike, determined according to a certain algorithm [1].

Details on the method of calculating the RVI index can be found on the Moscow Exchange website in the "Indices / Volatility Index" section. Unfortunately there you will not find links to resources explaining how the above formulas are obtained, and what is their economic meaning. To search for primary sources, we turn to the method of calculating the volatility index of the Chicago Exchange CBOE - VIX.

The CBOE Volatility Index - VIX

In 1993, the Chicago Board Options Exchange (CBOE) began to calculate and publish the values ​​of the CBOE Volatility Index (VIX). This volatility index was created to estimate market expectations regarding 30-day volatility and was calculated using at-the-money market prices of options on the S & P 100 Index (OEX). After 10 years in 2003, CBOE, together with Goldman Sachs, updated the VIX calculation methodology. The new VIX is based on the S & P 500 Index (SPX) and estimates the expected volatility by averaging the prices of the options on the SPX, selected on a broad list of strikes with specific weights. By allowing the volatility indicator to be expressed through the SPX options portfolio, the new methodology has transformed the VIX from an abstract concept into a practical standard for trading and hedging volatility.



In March 2004, the CBOE listed the first VIX futures contract on the stock exchange. Two years later in February 2006, CBOE launches VIX options, the most successful product in the history of the exchange [2].

Let me remind you that the Moscow Exchange, represented by Roman Sulzhik in April of this year, announced that it plans this summer to launch a futures contract on the volatility index RVI. Let's hope that this will happen in the stated time frame, and that the new product will be in demand by the market.

The generalized formula for calculating the VIX is:



Where

σ - VIX / 100;
T - time to expiration (in years);
F is the forward price of the S & P 500 index obtained from the prices of the SPX options;
K ₀ - strike nearest to the price of the forward index;
K _i - the strike of the i-th out-of-the-money call option, if K _i > K ₀ , or put, if K _i <K ₀ , and both put and call, if K _i = K ₀ ;
ΔK _i - the interval between strikes, half the difference between strikes on each side of K _i :



Note: ΔK _i for the lowest strike is equal to the difference between the lowest strike and the next strike. For the topmost strike, the difference between the topmost and the previous one.

R is the risk-free expiration rate;
Q (K _i ) is the average value between the bid and ask prices for an option with strike K _i ; for K ₀ , this is the average bid and ask prices of two put and call options.

The VIX measures the 30-day expected volatility of the S & P 500 index. The VIX components are the call and put options of the near and next series, usually the first and second SPX monthly contracts. The closest series should have a period of at least one week before expiration. For each selected option series, the square of volatility is calculated - sigma ² ₁ and sigma ² ₂ according to the formula (4). Next is their 30-day weighted average by the formula:



Comparing the formulas (1–3) and (4–5) with each other, we conclude that the Moscow Exchange RVI volatility index is an exact copy of the Chicago option exchange index VIX. The difference between RVI and VIX is only in the absence of the growth factor e ^RT , as well as in the way of determining the options involved in the calculations and their prices. The first follows from the fact that options for a futures contract, and not an equity asset like that of VIX, participate in the calculations of the RVI index. The latter is probably dictated by the low liquidity of the Russian option market.

The CBOE document [2], which discloses the VIX calculation method, contains a reference to Goldman Sachs materials [3], which describe how to evaluate the Volatility Swap and Variance Swap. Mathematics VIX is closely intertwined with the mathematics of pricing of volatility swaps. And it is not by chance, because The new VIX (launched in 2003) was developed by CBOE together with the investment bank Goldman Sachs. The latter, in turn, actively promoted the trade in volatility swaps - in 1999, the famous article “Goldman Sachs quantitative analysts from Goldman Sachs described the valuation methods of these swaps” was published. The idea of ​​creating a volatility index tied to a real instrument, the cash flow of which is directly dependent on this volatility, turned out to be very successful - futures and options on the VIX became very popular among investors.

Variance swap

As noted above, the method of calculating the VIX index is closely related to the theory of volatility swaps. The basis of this class of financial instruments is the concept of Variance Swap. Variance Swap (VS) is a forward contract for the annual variance (variance), the square of realized volatility (realized volatility). The expiry payment formula for this swap is described by the formula:



Where

σ ² _R is the realized dispersion of the stock, futures, index, etc., indicated in annual terms, for the period of circulation of the contract;
K _var - contract delivery price;
N - the value of the swap per unit of annual dispersion.

Upon expiration of the contract, the VS owner will receive N dollars for each item for which the realized variance σ ² _R exceeds the price of delivery K _var . Therefore, the fair value of the variance (according to the market) is equal to the delivery price VS, at which the cost of the swap will be zero. The fair value of the variance in this context serves as a good guide to the value of the VIX volatility index. Thus, the VS valuation method can be applied in the VIX calculations.

The VS value is estimated using the swap replication strategy through an option portfolio. This strategy is based on the concept of a log contract - an exotic option per share (index, futures, etc.), which hedging provides for payment of the equivalent dispersion of the price returns of this share. The log contract in turn can be replicated through a portfolio of vanilla options on the same underlying asset. This makes it possible to express the value of VS through option prices.

Let V denote the sensitivity index of the price of the option C _BS to the dispersion of the underlying asset σ ² (let's call it Variance Vega), which determines how much the price of the option will change if the dispersion of its underlying asset changes:



Figure 1 shows the graphs of the variation of the Variance Vega indicator for options with different strikes depending on the price of the underlying asset (left side), as well as the Variance Vega graphs of portfolios consisting of these options (right side).



Fig. 1: Variance Vega of call options portfolios with different strikes as a function of the price of the underlying asset. Each graph on the left shows the contribution of an individual option to the V portfolio. The corresponding graph on the right shows the sum of these contributions, weighted in two ways: the dashed line - with equal weights, the solid line - with weights inversely proportional to the square of the strike. The number of options increases, and the distance between strikes decreases from the top chart to the bottom.

Thus, the Variance Vega portfolio, consisting of options for all strikes, weighted inversely to the square of the strike, does not depend on the price of the underlying asset. This is exactly what you need to trade dispersion. What does a similar option portfolio look like, and how does trading with this portfolio depend on the dispersion of the underlying asset?

Consider a portfolio (S, σ, τ) , consisting of options with all strikes K with the same term before expiration τ , weighted inversely to the square of K In view of the fact that out-of-the-money options are usually more liquid, we will use put options P(S, K, σ, τ) for strikes K taking values ​​from 0 to some estimated price S _* , and call options C(S, K, σ, τ) for strikes from S _* to infinity. The price S _* can be viewed as at-the-money level of a forward asset on the underlying asset (or futures price) with a term τ , indicating the boundary between liquid put and call options.

At the time of expiration, when t = T, it can be shown that the total payment of all options of the above portfolio is:



Similarly, at time t, summing up all the prices of options, the value of the portfolio will be:



Variance Vega of this portfolio:



To get the initial Variance Vega of $ 1 per unit of volatility square, you need to open (2 / T) portfolio units. Take for the cost of a new portfolio:



The first term in the payment on the portfolio from formula (11) describes 1 / S _* forward contracts per share with the price of delivery S _t . This is not an option, it is a linear asset that can be statically hedged once and for the entire period, without any assessment of the volatility of the stock. The second term log (S _* / S _t ) describes a short position in a log contract with an estimated price S _* . Hedging of this contract depends on the volatility of the stock. Thus, the sensitivity of the portfolio to the volatility of the underlying asset fully contains a log contract.

Suppose that the price dynamics of the underlying asset can be described using the equation:



For simplicity, suppose also that stock dividends are not charged.
The formula for the dispersion of a random process is:



The swap valuation procedure is no different from any other derivative valuation procedure. The value of forward contract F for future realized volatility with strike K is equal to the expected payment in current prices under the risk of a neutral measure:



Where

r is the risk-free rate corresponding to time T;
E is the mathematical expectation.

The fair value of the future realized variance is strike K _var , for which the current value of the contract is zero:



Using formulas (13) and (15), we obtain the formula for calculating the fair value of the variance:



Applying the Ito lemma for log S _t , we find:



Subtracting (17) from (12), we get:



Summing (18) from 0 to T, we get:



This identity defines the dispersion replication strategy. The first term in brackets corresponds to the financial result from the continuous rebalancing of the position in the shares in such a way that the value of 1 / S _{t of the} shares at any time is $ 1. The second term is a static short position in the contract, the payment on which for expiration is the logarithm of the stock return for the period.

Formula (19) suggests a method for calculating the fair value of the variance. In this case, the risk-neutral expected value of the right side of the formula represents the cost of replication:



The expected value of the first term in brackets in formula (20) is the cost of rebalancing a portfolio consisting of 1 / S _t . In a risk-neutral world with a constant risk-free rate, the price of hedging such a portfolio will be:



The second term is a log contract. In view of the fact that there is no such contract freely negotiable, to reproduce the payments on it, it is necessary to separate the linear and nonlinear parts from its expiration payment profile and replicate each one separately. The linear part can be duplicated using a forward contract for a share with a term T , the remaining non-linear part can be duplicated using vanilla options with an expiration date T

To determine the boundary between liquid put and call options, we introduce the parameter S _* , which denotes this boundary. Payment under the log contract will be presented in the form:



The second term on the right-hand side of (22) is a constant independent of the final stock price S _T , therefore, only the first term should be replicated. How to repeat it is known from formula (8) - this is a forward contract plus an option portfolio with weights inverse to the square of the strike value:



The fair value of the future variance can be related to the initial fair value of each term in formula (21). Using equations (21) and (23), we get:



where P(K) and C(K) mean, respectively, the current fair prices of the put and call options with strike K. If the current market prices of the options are used, the current market estimate of the future variance is obtained. It is this formula that underlies the calculation of the VIX volatility index.

Formula (4) is a discrete copy of formula (24), from which the first three terms are removed (probably for simplicity). Certain integrals are presented in the form of corresponding sums. The last term in formula (4) is the adjustment element in cases when the price of the forward F does not coincide with the central strike K ₀ .

Conclusion

The basis for calculating the volatility index of the Russian RVI market is the method of calculating the volatility index VIX of the Chicago Stock Exchange. According to this method, the index value is a weighted average of the volatilities of the two option series. This averaging implies that the result is a volatility value corresponding to a 30-day period. The method for calculating volatility for each option series involved in the VIX (and RVI) calculations is based on the theory of volatility swaps, described in detail in [3]. Modeling the fair value of a volatility swap has a number of assumptions, one (and the main one, according to the author) of them is that price dynamics is a geometric Brownian motion, i.e. continuous process without jumps (jumps). This assumption may be the reason for the underestimation of volatility indexes VIX and RVI.

Bibliography

1. Methodology for calculating the Russian Market Volatility Index, Moscow Exchange, 04/11/2014 , fs.moex.com/files/6756
2. The CBOE Volatility Index - VIX, The Chicago Board Options Exchange, September 22, 2003, www.cboe.com/micro/VIX/vixwhite.pdf
3. Demeterfi K., Derman E., Kamal M. and Zou J., Goldman Sachs Quantitative Strategies Research Notes, March 1999

Author: Oleg Mubarakshin, Quant Lab

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