Mexican Volatility Index

Mexican Volatility Index

Definition of VIMEX The VIMEX is an index that measures the short term (90 days calendar) expected volatility for the Mexican stock market through the IPC Futures Options listed on MexDer. The VIMEX construction was based on the methodology described on the paper from Fleming, Ostdiek & Whaley: Predicting stock market volatility: a new measure, this paper was published on the Journal of Futures Markets, vol.15 (3), pp. 265-302, on 1995. VIMEX sample The VIMEX index uses a sample of 8 Options on IPC Futures for its calculation: A Call and a Put with a strike price above the current IPC level and with the nearest quarterly maturity date. A Call and a Put with a strike price below the current IPC level and with the nearest quarterly maturity date. A Call and a Put with a strike price above the current IPC level and with the second nearest quarterly maturity date. A Call and a Put with a strike price below the current IPC level and with the second nearest quarterly maturity date. The VIMEX sample shall always observe these properties; it is subject to the movement of the IPC level, in such a way that an important decrement or increment during the trading session could cause a modification on the VIMEX sample.

Calculation methodology of the VIMEX 1. A simple average of the implied volatilities is calculated from the pairs of Options that are above and below the theoretical ATM (at-the-money) exercise price. Then, 4 different indexes are obtained. Let i, j, k be the implied volatility where: i = j = k = C Call P Put 1 Nearest quarterly maturity date 2 Second nearest quarterly maturity date a Strike price above the IPC level ( K > S ) b Strike price below the IPC level ( K < S ) The simple average of the implied volatilities for the nearest quarterly maturity date is calculated as: 1, a = 1, b = In the same way, the simple average of the implied volatilities is calculated for the second nearest quarterly maturity dates: 2, a = 2, b =

2. The implied volatilities of the ATM exercise price for the nearest quarterly and second nearest quarterly maturities are obtained by interpolating the previous volatilities in the next expression: 1 = ( ) ( ) 2 = ( ) ( ) Where: = Exercise price above the IPC s level at the moment of calculation = Exercise price below the IPC s level at the moment of calculation = IPC s level at the moment of calculation 3. Finally, the implied volatilities previously obtained are weighted in order to create a constant period of 90 days calendar for every quarter, approximately, that the series listed in MexDer have. The formula for the VIMEX final calculation is given on the next expression: VIMEX = ( ) ( ) Where: = Calendar days remaining for the Option with the nearest quarterly maturity date: [(Maturity date Current date)] = Calendar days remaining for the Option with the second nearest quarterly maturity date: [(Maturity date Current date)]

Important considerations for the VIMEX calculation To calculate the VIMEX is required to have one of the eight implied volatilities that form the sample. Such volatility will be calculated every time that one of the following scenarios happens: 1. A bid and an ask quote are registered. 2. When the best bid-ask quote change in relation to the quotes used in the previous implied volatility calculation. 3. A trade is registered. First calculation of the VIMEX during the MexDer trading session It happens when a bid and an ask quote of the sample are registered. The Option contracts composing the sample must have quarterly cycles. The remaining seven Options will take the implied volatility value registered at the market close of the previous day. Inputs to obtain the first implied volatility I. Price of the Call or Put (c or p) It will be the crossed volume weighted average of the best bid-ask quotes of the Calls and Puts that compose the sample. Crossed volume weighted average= [(bid quote*ask volume + ask quote*bid volume) / (ask volume + bid volume)] II. Underlying value F a. It will be the value of the last trade in the trading session for the IPC Future. In case that there were no trades, next subsection will be applied. b. The crossed volume weighted average of the best bid-ask quotes of the IPC Future. In case it doesn t exist, the next subsection will be applied. c. It will be the IPC Future settlement price from the previous session plus the percentage change of the IPC accumulated from the previous day until the moment of calculation.

III. IV. Exercise price K It is the exercise price of the Option s IPC Future. Time to maturity t It s the time to maturity in years (360 days calendar) and is held constant during the session and is calculated as: [(Maturity date Calculation date)/360] V. Risk-free interest rate r It is the risk-free interest rate FRASWAP provided by the price vendor VALMER, during the session, for the Option s time to maturity. VALMER provides this interest rate three times per day. At the moment of calculation, the available FRASWAP rate will be taken. Subsequent calculations of the VIMEX in a MexDer trading session The following VIMEX calculation happens when the best bid-ask quotes change in one of the eight Options that compose the sample. The remaining seven Options will take the implied volatilities previously calculated during the session, in case there weren t any, they will take the value of the market close from the previous day. Calculation is also done when bid and ask quotes appear in the elements of the sample that didn t have any quotes before. Finally, trades in the sample mentioned before are the latest criteria to give way to a VIMEX calculation. Inputs for the subsequent implied volatilities I. Call or Put value (c or p) It will be the crossed volume weighted average of the best bid-ask quotes of the Calls and Puts that compose the sample, provided these quotes have changed in relation to the quotes that induced the first calculation. II. Underlying value F a. It will be the last trade of the IPC Future at the moment of calculation, provided it has not exceeded five minutes since its register. Otherwise, next subsection will be applied.

b. The value will be the crossed volume weighted average of the best bidask quotes of the IPC Future. In case it doesn t exist, the next subsection will be applied. c. It will be the IPC Future settlement price from the previous session plus the percentage change of the IPC accumulated from the previous day until the moment of calculation. VI. The Option s exercise price, the time to maturity and the interest rate will have the same value of the first calculation. VIMEX calculation at market close The final VIMEX calculation is done with smoothed implied volatilities produced by MexDer with settlement prices adjusted with the Heston model. The VIMEX level at market close will be the opening level on the next trading day. Rollover of the VIMEX sample Options The rollover is the substitution of one of the Options on the VIMEX sample for another with greater time to maturity. In the VIMEX this is done with Options that have 10 days to maturity. The reason for this change is because implied volatility increases meaningfully as maturity gets closer creating distortions on the calculation. The substitute Option shall comply with the characteristics required for the 8 VIMEX sample Options. During the rollover period, the VIMEX calculation has a weighting resulting from an extrapolation of and ; the with closest maturity has a weighting greater than 1 and the furthest has a negative compensatory weighting, in such a way that the sum of both weightings equals one. (e.g. 1.25 and -0.25). This extrapolation also happens in the quarters that last more than 90 days, in the period where both Options have time to maturity greater than 90 days. From the VIMEX calculation formula and the time to maturity for the new sample this statement can be demonstrated: VIMEX = ( ) ( )

Particularities on the VIMEX calculation Indetermination or zero value of the implied volatility If the convergence to zero is not achieved after 20 iterations of the Newton Raphson model, the implied volatility gets indeterminate. Such convergence to zero refers to the difference between the volatility obtained from the model and a seed volatility value (an arbitrarily set volatility). This indetermination also occurs when such difference fluctuates in size (from smaller to larger or just when it grows larger). Indetermination arises with disproportioned quotes, far below or far above fair prices. In case of indetermination or that the volatility takes zero value, the previously calculated volatility on the same exercise price/series will be taken. In case of its absence, it will take the value from the previous market close. Triggers of the implied volatility calculation The only trigger of the implied volatility calculation, and therefore, the VIMEX calculation, are the quotes and trades of the IPC Options from the VIMEX sample. Nor the changes in quotes and trades on the underlying Future and the risk-free rate activate the implied volatility calculation. The reason for this is that the VIMEX is an implied volatility model arising from Option trading. Time to maturity treatment Discrete periods of days are considered in the time to maturity.

Implied volatility calculation The Black76 model is used for implied volatility calculation; the Option s theoretical price comes from these variables: the exercise price of the Option K, the price of the underlying Future F, the premium of the Option in the market C or P, the time to maturity T-t, the risk-free rate r and the volatility of the underlying asset. For the call Option: ( ) ( ) ( ) ( ) ( ) [ ( ) ( )] For the Put Option: ( ) [ ( ) ( )] Where: ( ) ( ) ( ) ( ) To obtain the implied volatility, the Black76 formula has to be inverted. The dependent variable will be the volatility and the Option s premium will be the independent variable. This resulting equation is solved through the numerical method Newton Raphson which needs the following information: a) The theoretical price for the Call or Put calculated with Black76. b) The parameter lambda ( ) which is the change rate in the Option value in respect to the change in the volatility of the underlying Future. c) The Option s Premium in the market. d) A seed volatility value for the volatility to start the iterative search. The required accuracy level determines the end of the iterative search:

Where: = Required confidence level. Theoretical price obtained from Black76 with volatility. Last registered trade price. Once the level of accuracy is established, the theoretical price is calculated using the Black76 model. The inputs are the Future price of the underlying asset F, the exercise price K, the time to maturity T-t, the risk-free rate r and an arbitrary seed value for the volatility. The iterative process is done with the Newton Raphson search algorithm: Where: ( ) ( ) Volatility value of the i-th iteration. Vega or lambda of the i-th iteration. The iterative process ends when the volatility. The Mexican Volatility Index VIMEX has been designed and is calculated by MexDer with value reliable techniques, however, MexDer, Mercado Mexicano de Derivados, S.A. de C.V. is not liable for possible mistakes, neither for the interpretation or decisions third parts could make regarding this Index.It does not involve in any way purchasing or selling advices of securities, methodology, or any information based on this index. VIMEX is a copyright registered by MexDer, Mercado Mexicano de Derivados, S.A. de C.V. It must be mentioned that MexDer is the source when it appears in the media. IPC is a copyright registered by Bolsa Mexicana de Valores, S.A.B. de C.V.