CN104809647A - Volatility index compiling method - Google Patents

Abstract

The invention discloses a volatility index compiling method. The method sequentially comprises the steps of selecting option contract months, calculating residual maturity of the selected months respectively, calculating risk-free interest rates of the current month and the next month respectively, determining option contract price, calculating forward prices of the current month and the next month respectively, screening contracts, calculating volatility of the two months respectively and calculating volatility index. According to the method, theoretical prices of no-deal contracts are calculated by introducing a volatility curved surface method, so that a volatility index calculation formula is allowed to have enough numbers of samples in an assuring manner , and compiling error of the volatility index is reduced to a maximum degree, and needs of options markets in China are met.

Description

A kind of volatility index preparation method

Technical field

The present invention relates to option trade algorithm, particularly relate to a kind of volatility index preparation method.

Background technology

Volatility index (VIX index) preparation method of Chicago Board Options Exchange (CBOE) works out the main algorithm of volatility index at present, the method is applied academicly variance and is exchanged Pricing Principle to calculate volatility index, certain deviation is there is in the algorithm of CBOE with theoretical, mainly truncation error and discretization error, in reality, truncation error and discretization error all cannot be avoided, error size depends on design and the market situation of options products on the one hand, depends on the method for sampling on the other hand.The core of volatility index preparation method, the truth exactly for market designs suitable sampling method, to reduce the error of calculation as far as possible.

The major defect of prior art: the VIX computing method of existing CBOE are when calculating volatility index, adopt strict data screening standard, cause computable sample very few, increase the deviation of result of calculation, especially the method directly applies to the relatively underdeveloped Chinese market in Options market, error will be larger, causes the inaccurate of volatility index result of calculation.

Calculating due to volatility index needs a large amount of option contracts as calculating basis, and contract quantity is more, contract spacing is closeer, and the error of calculation is less.In Real Options market, trade in the market quotation may not be there is in larger a part of contract certain period, the algorithm of CBOE directly gives up these contracts, and give up the contract after " two continuous regular " simultaneously, and only application exists the calculating of the contract participation VIX of bid and offered quotation.Under this algorithm, if a large amount of contract in market is not offered, the option premium data that so can be used for calculating volatility index are very few, can cause larger truncation error, discretization error, cause the deviation of result of calculation to actual value excessive.

Summary of the invention

The technical problem to be solved in the present invention is, for the problem that the CBOE algorithm error of calculation of the prior art is excessive, a kind of volatility index preparation method is provided, and utilizes this preparation method to improve the precision of volatility index calculating, reduce the establishment error of volatility index.

For solving the problems of the technologies described above, the present invention adopts following technical scheme.

A kind of volatility index preparation method, it comprises the steps: step S1, select option contract month, judge of that month contract distance expires whether be greater than for 3 day of trade, if of that month contract distance expires be more than or equal to for 3 day of trade, then choosing of that month and time near moon contract for calculating, if of that month contract distance expires be less than for 3 day of trade, then choosing lower two month contract for calculating; Step S2, calculate the residue time limit in selected month respectively, the computing formula in described residue time limit is:

Wherein, i=1,2, T ₁and T ₂be respectively of that month and the residue time limit of time nearly moon, unit is year, M _{1, i}for the residue the number of minutes at current distance midnight on the same day, M _{2, i}for from the number of minutes between proxima luce (prox. luc) midnight midnight on the same day to date of expiry, M _{3, i}for the number of minutes expired between midnight the previous day to expiration time, described the number of minutes is integer, and rounds downwards, step S3, calculate of that month and time month risk free rate respectively, risk free rate chooses Repo1 days, 7 days price fixing interest rates, and one term Repo exchange the Nian Huahou of interest rate as benchmark interest rate, described exchange reference rate is 7 days repurchase price fixing interest rates, linear interpolation is utilized to obtain the risk free rate in option contract corresponding time limit, step S4, determine option contract price, described option contract price is: if closed within first 15 minutes have transaction, then get same day closing price as the price of this contract, the no deal within first 15 minutes if closed, then get the theoretical prince of this contract, described theoretical prince is the implied volatility being calculated each contract by exchange by the method building stability bandwidth curved surface, and the theoretical prince of this contract is calculated by Black76 Option Pricing Method, the construction method of described implied volatility curved surface is: A, choosing of stock index option contract, in this step, according to the design conditions of Shanghai and Shenzhen 300 stock index option contract month and exercise price sequence, intending chosen distance expires in 7 days-180 days, the implied volatility of option contract estimated by dummy values contract within level values and upper and lower five grades, the middle implied volatility of described at-the-money is the average of the implied volatility of call option and put option, B, the calculating of implied volatility sample data and screening, in this step, corresponding implied volatility is pushed away by the price of Black76 future option pricing model and selected contract is counter, the implied volatility average of the option contract that employing is selected within last 15 minutes of day of trade Mo is as the sample data estimating implied volatility curved surface, the sample data obtained is screened, remaining sample data is used for calculate and generates implied volatility curved surface, C, described implied volatility curved surface is by implied volatility, the curved surface that option contract expiration time and exercise price are formed, utilize this implied volatility curved surface, and adopt Black76 pricing formula to obtain the theoretical prince of each contract, step S5, calculate the forward price of of that month and secondary month respectively, computing formula is:

$F_i=K_i+e^{r_{T_i}\×T_i}\×(C_{K_i}-R_{K_i})$

Wherein, i=1,2, F ₁and F ₂be respectively of that month and the forward price of time nearly moon, K _ifor exercise price, be appointed as the exercise price that call option under this month identical exercise price level, the absolute value minimum value of difference that puts price are corresponding, with represent the call option of corresponding exercise price and the price of put option respectively, step S6, screening contract, in this step, according to the forward price F that step S5 calculates _i, the at-the-money contract exercise price K of the of that month and secondary nearly moon _{0, i}be defined as closest to and lower than F _iexercise price level, determine K _{0, i}after, first select exercise price to equal K _{0, i}level values call option and put option, and get the mean value that is expected to rise and puts price as K _{0, i}corresponding option premium, selects all exercise prices to be greater than K afterwards _{0, i}out-of-the-money call option and exercise price be less than K _{0, i}out-of-the-money put option as calculating; Step S7, calculate the stability bandwidth in two months respectively, computing formula is:

${{\mathrm{\σ}}_i}^2=\frac{2}{T_i}\underset{j}{\mathrm{\Σ}}\frac{{\mathrm{\ΔK}}_j}{K_j^2}{e}^{r_{T_i}\×T_i}Q\left(K_j\right)-\frac{1}{T_i}{[\frac{F_i}{K_{0,i}}-1]}^2$

Wherein, i=1,2, σ ₁ ²and σ ₂ ²represent of that month respectively and time month contract stability bandwidth square, K _jrepresent the 6th step screen the exercise price of contract, Q (K _j) be K _jcorresponding option premium, K _j=K ₀time be the mean value of call option and put option, K _j>K ₀time Q (K _j) be the price of corresponding call option, K _j<K ₀time Q (K _j) be the price of corresponding put option; Step S8, calculate volatility index, calculating 30 daily fluctuation rate exponential formulas is:

Wherein, N ₃₀represent the number of minutes of 30 days, N ₃₆₅represent the number of minutes of 365 days, N is integer, rounds downwards.

In volatility index preparation method disclosed by the invention, carry out selecting the residue time limit in option contract month, respectively month selected by calculating successively, calculate this month and secondary month risk free rate respectively, determine option contract price, calculate the forward price in this month and secondary month respectively, screen contract, calculate the stability bandwidth in two months and calculating volatility index step respectively.The present invention's beneficial effect is compared to existing technologies, because the method introducing stability bandwidth curved surface calculates the theoretical prince without conclusion of the business contract, ensure that in volatility index computing formula and have enough sample sizes, thus reduce the establishment error of volatility index largely, meet the Options market demand of China.

Embodiment

Below by embodiment, the present invention is described in more detail.

The invention discloses a kind of volatility index preparation method, the method its comprise the steps:

Step S1, selects option contract month

Judging of that month contract distance expires whether be greater than for 3 day of trade, if of that month contract distance expires be more than or equal to for 3 day of trade, then choosing of that month and time near moon contract for calculating; If of that month contract distance expires be less than for 3 day of trade, then choose lower two month contract for calculating.

Step S2, calculates the residue time limit in selected month respectively

Residue time limit (by the calendar day but not the day of trade calculate) computing formula is:

Wherein, i=1,2, T ₁and T ₂be respectively of that month and the residue time limit of time nearly moon, unit is year.M _{1, i}for the residue the number of minutes at current distance midnight on the same day.M _{2, i}for from the number of minutes between proxima luce (prox. luc) midnight midnight on the same day to date of expiry.M _{3, i}for the number of minutes expired between midnight the previous day to expiration time.The number of minutes is integer, and rounds downwards.

Step S3, calculates the risk free rate of of that month and secondary month respectively

Risk free rate chooses the best Repo1 of mobility days, 7 days price fixing interest rates, and one term Repo exchange the Nian Huahou of interest rate (exchanging reference rate is 7 days repurchase price fixing interest rates) as benchmark interest rate, utilize linear interpolation to obtain the risk free rate in option contract corresponding time limit.

Repo1 days, 7 days price fixing interest rate r ₀₀₁and r ₀₀₇at 11 at noon every day is issued by foreign exchange trading center, can directly apply after acquisition.RepoSwap interest rate is issued every day continuously, chooses the data of 11 at noon every day.Due to the announcement of RepoSwap interest rate is season interest rate, and raw data needs after obtaining first to carry out yearization conversion:

$r_{\mathrm{SWAP}}={(1+\frac{r_S}{4})}^4-1$

Wherein, r _sfor a term Repo of original acquisition exchanges interest rate, r _sWAPfor a term Repo in year exchanges interest rate.Finally use r ₀₀₁, r ₀₀₇and r _sWAPthese three interest rates carry out interpolation calculation option residue time limit T _icorresponding interest rate:

If T _i> T ₀₀₇, computing formula is:

$r_{T_i}=\frac{T_{\mathrm{SWAP}}-T_i}{T_{\mathrm{SWAP}}-T_{007}}{r}_{007}+\frac{T_i-T_{007}}{T_{\mathrm{SWAP}}-T_{007}}{r}_{\mathrm{SWAP}}$

If T _i< T ₀₀₇, computing formula is:

$r_{T_i}=\frac{T_{007}-T_i}{T_{007}-T_{001}}{r}_{001}+\frac{T_i-T_{001}}{T_{007}-T_{001}}{r}_{007}$

Wherein, T ₀₀₁, T ₀₀₇and T _sWAPthe time limit in year of expression 1 day, 7 days and 1 year correspondence respectively, $T_{001}=\frac{1}{365},T_{001}=\frac{7}{365},T_{\mathrm{SWAp}}=1.$

Step S4, determines option contract price

This step is the main difference place that the present invention is different from prior art.This method determines the option contract price calculating volatility index in the following manner:

There is transaction if closed within first 15 minutes, then get same day closing price as the price of this contract.

If closed, no deal within first 15 minutes, then get the theoretical prince of this contract.

Wherein, theoretical prince is the implied volatility being calculated each contract by exchange by the method building stability bandwidth curved surface, and is calculated the theoretical prince of this contract by Black76 Option Pricing Method.The building method of implied volatility curved surface is as follows:

The construction method of implied volatility curved surface mainly comprises stock index option contract and chooses, generation three steps of the calculating of implied volatility sample data and screening and implied volatility curved surface, and concrete condition is as follows:

Choosing of A, stock index option contract

In order to ensure the Pricing Efficiency of the option contract for estimating implied volatility curved surface, choosing and there is the necessarily good option contract of representational, mobility.According to the design conditions of current Shanghai and Shenzhen 300 stock index option contract month and exercise price sequence, intend that chosen distance expires in 7 days-180 days, dummy values contract within level values and upper and lower five grades to be to estimate the implied volatility of option contract.Wherein, the middle implied volatility of at-the-money is the average of the implied volatility of call option and put option.

The calculating of B, implied volatility sample data and screening

Corresponding implied volatility is pushed away by the price of Black76 future option pricing model and selected contract is counter.The implied volatility average of the option contract that employing is selected within last 15 minutes of day of trade Mo is as the sample data estimating implied volatility curved surface.

The sample data obtained is screened, rejects very high or low-down abnormal implied volatility.3 times that are greater than Historic Volatility or the sample data being less than Historic Volatility 1/3 will be by, remaining sample data are used for calculating generating implied volatility curved surface.Historic Volatility is calculated by exponential weighted moving average model(EWMA) (EWMA) model.

The generation of C, implied volatility curved surface

Implied volatility curved surface is the curved surface be made up of implied volatility, option contract expiration time and exercise price.By carrying out least square regression to 2 order polynomials of stability bandwidth curved surface, obtain corresponding regression coefficient.

Stability bandwidth curved surface is at middle implied volatility Z, option time expiry x, and the curved surface in exercise price y tri-coordinate axis.Implied volatility Z can be counted as the function of time expiry and exercise price, as: Z=Z (x, y).Suppose that stability bandwidth curved surface is provided by 2 order polynomial curved surfaces:

Z(x，y)＝c ₀+c ₁x+x ₂x ²+c ₄y+c ₅y ²+c ₆xy

Because above formula has 6 coefficients, therefore at least need 6 groups of sample datas to return and obtain coefficient c _k.Least square regression is utilized to obtain coefficient c _k, then use c _k, time expiry and exercise price calculate the implied volatility of any point on stability bandwidth curved surface.

Regression coefficient is used to estimate its implied volatility for not having the option contract of the market price in sample data.Thus, the implied volatility curved surface comprising each option contract implied volatility on market can be generated.Application implied volatility curved surface, and adopt Black76 pricing formula just can obtain the theoretical prince of each contract.

Step S5, calculate the forward price of of that month and secondary month respectively, computing formula is:

$F_i=K_i+e^{r_{T_i}\&times;T_i}\&times;(C_{K_i}-R_{K_i})$

Wherein, i=1,2, F ₁and F ₂be respectively of that month and the forward price of time nearly moon.K _ifor exercise price, be appointed as the exercise price that call option under this month identical exercise price level, the absolute value minimum value of difference that puts price are corresponding, with represent the call option of corresponding exercise price and the price of put option respectively.

Step S6, screening contract

According to the forward price F that step S5 calculates _i, the at-the-money contract exercise price K of the of that month and secondary nearly moon _{0, i}be defined as closest to and lower than F _iexercise price level.

Determine K _{0, i}after, first, select exercise price to equal K _{0, i}level values call option and put option, and get the mean value that is expected to rise and puts price as K _{0, i}corresponding option premium.The second, select all exercise prices to be greater than K _{0, i}out-of-the-money call option and exercise price be less than K _{0, i}out-of-the-money put option as calculating.

Step S7, calculates the stability bandwidth in two months respectively

Computing formula is:

${{\mathrm{\&sigma;}}_i}^2=\frac{2}{T_i}\underset{j}{\mathrm{\&Sigma;}}\frac{{\mathrm{\&Delta;K}}_j}{K_j^2}{e}^{r_{T_i}\&times;T_i}Q\left(K_j\right)-\frac{1}{T_i}{[\frac{F_i}{K_{0,i}}-1]}^2$

Wherein, i=1,2, σ ₁ ²and σ ₂ ²represent of that month respectively and time month contract stability bandwidth square.K _jrepresent the 6th step screen the exercise price of contract, Q (K _j) be K _jcorresponding option premium, K _j=K ₀time be the mean value of call option and put option, K _j>K ₀time Q (K _j) be the price of corresponding call option, K _j<K ₀time Q (K _j) be the price of corresponding put option.Δ K _jfor exercise price spacing, be taken as 50 points at present.

Step S8, calculate volatility index, calculating 30 daily fluctuation rate exponential formulas is:

Wherein, N ₃₀represent the number of minutes of 30 days, N ₃₆₅represent the number of minutes of 365 days, N is integer, rounds downwards.

In volatility index preparation method disclosed by the invention, carry out selection option contract month successively, calculate the residue time limit in selected month respectively, calculate of that month and time month risk free rate respectively, determine option contract price, calculate the forward price of of that month and secondary month respectively, screening contract, calculate the stability bandwidth in two months respectively and calculate volatility index step, in said process, because the method introducing stability bandwidth curved surface calculates the theoretical prince without conclusion of the business contract, ensure that in volatility index computing formula and have enough sample sizes, thus reduce the establishment error of volatility index largely, meet the Options market demand of China.

The above is preferred embodiment of the present invention, is not limited to the present invention, all make in technical scope of the present invention amendment, equivalent to replace or improvement etc., all should be included in scope that the present invention protects.

Claims (2)

1. a volatility index preparation method, is characterized in that, comprises the steps:

Step S1, select option contract month, judge of that month contract distance expires whether be greater than for 3 day of trade, if of that month contract distance expires be more than or equal to for 3 day of trade, then choose of that month and time near moon contract for calculating, if of that month contract distance expires be less than for 3 day of trade, then choose lower two month contract for calculating;

Step S2, calculate the residue time limit in selected month respectively, the computing formula in described residue time limit is:

Wherein, i=1,2;

T ₁and T ₂be respectively of that month and the residue time limit of time nearly moon, unit is year;

M _{1, i}for the residue the number of minutes at current distance midnight on the same day;

M _{2, i}for from the number of minutes between proxima luce (prox. luc) midnight midnight on the same day to date of expiry;

M _{3, i}for the number of minutes expired between midnight the previous day to expiration time;

Described the number of minutes is integer, and rounds downwards;

Step S3, calculate of that month and time month risk free rate respectively, risk free rate chooses Repo1 days, 7 days price fixing interest rates, and one term Repo exchange the Nian Huahou of interest rate as benchmark interest rate, described exchange reference rate is 7 days repurchase price fixing interest rates, utilizes linear interpolation to obtain the risk free rate in option contract corresponding time limit;

Step S4, determines option contract price, and described option contract price is:

There is transaction if closed within first 15 minutes, then get same day closing price as the price of this contract;

The no deal within first 15 minutes if closed, then get the theoretical prince of this contract, described theoretical prince is the implied volatility being calculated each contract by exchange by the method building stability bandwidth curved surface, and the theoretical prince of this contract is calculated by Black76 Option Pricing Method, the construction method of described implied volatility curved surface is:

Choosing of A, stock index option contract, in this step, according to the design conditions of Shanghai and Shenzhen 300 stock index option contract month and exercise price sequence, intend that chosen distance expires in 7 days-180 days, dummy values contract within level values and upper and lower five grades to estimate the implied volatility of option contract, the middle implied volatility of described at-the-money is the average of the implied volatility of call option and put option;

The calculating of B, implied volatility sample data and screening, in this step, corresponding implied volatility is pushed away by the price of Black76 future option pricing model and selected contract is counter, the implied volatility average of the option contract that employing is selected within last 15 minutes of day of trade Mo is as the sample data estimating implied volatility curved surface, the sample data obtained is screened, remaining sample data is used for calculate generation implied volatility curved surface;

C, described implied volatility curved surface are the curved surfaces be made up of implied volatility, option contract expiration time and exercise price, utilize this implied volatility curved surface, and adopt Black76 pricing formula to obtain the theoretical prince of each contract;

Step S5, calculate the forward price of of that month and secondary month respectively, computing formula is:

$F_i=K_i+e^{r_{T_i}\&times;T_i}\&times;(C_{K_i}-R_{K_i})$

Wherein, i=1,2;

F ₁and F ₂be respectively of that month and the forward price of time nearly moon;

K _ifor exercise price, be appointed as the exercise price that call option under this month identical exercise price level, the absolute value minimum value of difference that puts price are corresponding, with represent the call option of corresponding exercise price and the price of put option respectively;

Step S6, screening contract, in this step, according to the forward price F that step S5 calculates _i, the at-the-money contract exercise price K of the of that month and secondary nearly moon _{0, i}be defined as closest to and lower than F _iexercise price level, determine K _{0, i}after, first select exercise price to equal K _{0, i}level values call option and put option, and get the mean value that is expected to rise and puts price as K _{0, i}corresponding option premium, selects all exercise prices to be greater than K afterwards _{0, i}out-of-the-money call option and exercise price be less than K _{0, i}out-of-the-money put option as calculating;

Step S7, calculate the stability bandwidth in two months respectively, computing formula is:

${{\mathrm{\&sigma;}}_i}^2=\frac{2}{T_i}\underset{j}{\mathrm{\&Sigma;}}\frac{{\mathrm{\&Delta;K}}_j}{K_j^2}{e}^{r_{T_i}\&times;T_i}Q\left(K_j\right)-\frac{1}{T_i}{[\frac{F_i}{K_{0,i}}-1]}^2$

Wherein, i=1,2;

σ ₁ ²and σ ₂ ²represent of that month respectively and time month contract stability bandwidth square;

K _jrepresent the 6th step screen the exercise price of contract;

Q (K _j) be K _jcorresponding option premium, K _j=K ₀time be the mean value of call option and put option, K _j>K ₀time Q (K _j) be the price of corresponding call option, K _j<K ₀time Q (K _j) be the price of corresponding put option;

Step S8, calculate volatility index, calculating 30 daily fluctuation rate exponential formulas is:

Wherein, N ₃₀represent the number of minutes of 30 days, N ₃₆₅represent the number of minutes of 365 days, N is integer, rounds downwards.

2. volatility index preparation method as claimed in claim 1, is characterized in that, in described step S3, utilize linear interpolation to obtain in the risk free rate in option contract corresponding time limit, raw data needs after obtaining first to carry out yearization conversion:

$r_{\mathrm{SWAP}}={(1+\frac{r_S}{4})}^4-1$

Wherein, r _sfor a term Repo of original acquisition exchanges interest rate, r _sWAPfor a term Repo in year exchanges interest rate, use r afterwards ₀₀₁, r ₀₀₇and r _sWAPthese three interest rates carry out interpolation calculation option residue time limit T _icorresponding interest rate:

If T _i> T ₀₀₇, computing formula is:

$r_{T_i}=\frac{T_{\mathrm{SWAP}}-T_i}{T_{\mathrm{SWAP}}-T_{007}}{r}_{007}+\frac{T_i-T_{007}}{T_{\mathrm{SWAP}}-T_{007}}{r}_{\mathrm{SWAP}}$

If T _i< T ₀₀₇, computing formula is:

$r_{T_i}=\frac{T_{007}-T_i}{T_{007}-T_{001}}{r}_{001}+\frac{T_i-T_{001}}{T_{007}-T_{001}}{r}_{007}$

Wherein, T ₀₀₁, T ₀₀₇and T _sWAPthe time limit in year of expression 1 day, 7 days and 1 year correspondence respectively, $T_{001}=\frac{1}{365},T_{001}=\frac{7}{365},T_{\mathrm{SWAp}}=1.$