U.U.D.M. Project Report 2011:7

Examensarbete i matematik, 30 hpHandledare och examinator: Maciej Klimek

Maj 2011

Department of MathematicsUppsala University

The VIX Volatility Index

Mao Xin

i

Abstract.

VIX plays a very important role in the in financial derivatives pricing, trading, risk control

strategy. It could be said it would not be a finacial market without the financial market

volatility. If it is the lack of risk management tools and the market volatility is too large, the

investors may be worried about the risk and give up trading, then the market is less attractive.

This is why I discuss about it. In this paper, I want to research more about its derivation and

calculation process, and to know how the VIX volatility index changes, in according to S&P

500 Index option prices.

ii

Table of Contents

Abstract. ..................................................................................................................................... i

Table of Contents ..................................................................................................................... ii

Chapter 1. Introduction ........................................................................................................... 1 1.1 The Origin of VIX ............................................................................................................................. 1 1.2 The Development of VIX .................................................................................................................. 1

Chapter 2. The General Information About VIX and Options Swaps ............................... 3 2.1 The Types of Volatility ............................................................................................................... 3

2.1.1 Realized volatility and The historical volatility ............................................................. 3 2.1.2 The implied volatility ...................................................................................................... 4 2.1.3 The volatility index ......................................................................................................... 4 2.1.4 intraday volatility ............................................................................................................ 5

2.2 Principles of Volatility Index .................................................................................................... 5 2.3 Variance Swaps .......................................................................................................................... 6

Chapter 3. EstimatingVolatility From Option Data[2][14] ...................................................... 8

Chapter 4. Calculation of VIX .............................................................................................. 15

4.1—Parameter And Select The Options To Calculate VIX .................................................................. 16 4.1.1 Parameter T: ................................................................................................................ 16 4.1.2 Parameter R: ................................................................................................................ 17 4.1.3 Parameter F: ................................................................................................................ 17 4.1.4 Parameter K0 and Ki. .................................................................................................. 20 4.1.5 Select options ................................................................................................................ 20 4.1.6 Parameter ∆ ............................................................................................................. 22 4.1.7 Parameter .......................................................................................................... 23

4.2—Calculate volatility of both near-term and next-term options ...................................................... 25 4.3—Calculate the 30-day weighted average of both σ1 and σ2 Then take the square root of the value and multiply by 100 to get VIX, according to the formula 4.1 and formula 4.2. ......................... 27

Chapter 5. Conclusion ............................................................................................................ 30

References . ............................................................................................................................. 33

Appendix. ................................................................................................................................ 35 1. Codes ............................................................................................................................................ 35 2. The Calculation Results .............................................................................................................. 39

1

Chapter 1. Introduction

1.1 The Origin of VIX

VIX plays a more and more important role in the in financial derivatives pricing, trading, risk

control strategy. After the global stock market crash in 1987, it is to stabilize the stock market

and protect the investors, the New York Stock Exchange (NYSE) in 1990 introduced a circuit

breaker mechanism (Circuit-breakers). When the stock price changes unusually, it occurs a

temporary suspension of trading, and it is helpful to try to reduce the market volatility in order

to restore the investor confidence on the stock market. However, due to the introduction of

circuit breaker mechanism, there is many new insights for how to measure market volatility,

and it is gradually produced a dynamic display of market volatility requirements. Therefore,

not long after the New York Stock Exchange (NYSE) used Circuit-Breakers to solve the

problem of excessive volatility in the market, the Chicago Board Options Exchange began to

introduce the CBOE market volatility index (VIX) in 1993,which is used to measure market

volatility implied by at the money S&P 100 Index (OEX) option prices[1].

1.2 The Development of VIX

When the stock option transactions began in April 1973, Chicago Board Options Exchange

(CBOE) has envisaged that the market volatility index can be constructed by the option price ,

which could be shown that the expectation of the future volatility in the option market. Since

then there were gradually many various calculation methods proposed by some scholars,

Whaley (1993) proposed a calculation approach which is the preparation of market volatility

index as a measure of future stock market price volatility. In the same year, Chicago Board

Options Exchange (CBOE) started to research the compilation of the CBOE Volatility Index

(VIX), which is based on the implied volatility of the S&P 100 Index options, and at the same

time also calculate implied volatility of call option and put option in order to take into account

the right of traders to buy or sell option under the preferences[8].

It is shown the investors’ expectation of the further stock market volatility by VIX. The

higher the volatility index is, the larger the investors expect the volatility of stock index in the

future is; the lower the VIX index is, the more moderate it shows that the investors believe

2

that the future stock price volatility will be. Since the index (VIX) can reflect the investors’

expectations of the further stock price volatility, and it can be observed the mental

performance of the option participants, also known as "investor sentiment gauge "(The

investor fear gauge). After ten years of development and improvement, the VIX index

gradually was agreed by the stock market, CBOE calculated several other volatility indexes

including, in 2001 NASDAQ 100 index as the underlying volatility index (NASDAQ

Volatility Index, VXN), in 2003 the VIX index based on the S&P 500 Index, which is much

closer to the actual stock market than the S&P 100 Index, CBOE DJIA volatility Index

(VXD), CBOE Russell 2000 Volatility Index (RVX),in 2004 the first volatility futures

(Volatility Index Futures) VIX Futures, and at the same year a second volatility

commercialization futures, that is the variance futures (Variance Futures), subject to three-

month the S&P 500 Index of realized Variance (Realized Variance). In 2006, the VIX Index

options began to trade in the Chicago Board Options Exchange. In 2008, CBOE pioneered the

used of the VIX methodology to estimate the expected volatility of some commodities and

foreign currencies[1]. There are many developments. For example, in India, VIX was launched

in April, 2008 by National stock exchange (NSE). The VIX index of India is based on the

Nifty 50 Index Option prices. The methodology of calculating the VIX index is same as that

for CBOE VIX index. The current focus on the VIX is due to its inherent property of negative

correlation with the underlying price index, and its usefulness for predicting the direction of

the price index[9]. And in HongKong, Hong Kong got its own volatility index for financial

products that allow investors to hedge against excessive market movements. Hang Seng

Indexes Company, the company which owns and manages the benchmark indexes in Hong

Kong including the Hang Seng index (HSI), launched the HSI Volatility index or "VHSI" on

Feb. 21. The index is modeled on the lines of the Chicago Board of Exchanges VIX

index .VIX in that it measures the 30-calendar-day expected volatility of the Hang Seng index

using prices of options traded on the index[10]. In this paper, I plan to discuss more about its

derivation and calculation process in the paper, and to know how the VIX changes.

3

Chapter 2. The General Information About VIX and Options Swaps

2.1 The Types of Volatility

2.1.1 Realized volatility and The historical volatility In order to apply most financial models in practice, it is necessary to be able to use empirical

data to measure the degree of variability of asset prices or market indexes. Suppose that St is

the price of an asset or a market index at time t. The realized volatility of this asset in a period

t , t based on n+1 daily observations S , S , S , S is defined by the formula

∑=−

=n

iirn 12

1252σ

Where

.,1,3,2,1,ln1

nniSS

ri

ii −==

−

K

And 252 is an annualization factor corresponding to the typical number of trading days in a

year[4].

Similarly the historical volatility is defined by a similar formula:

∑=

−−

=n

ii rrn 1

2)(1

252~σ

Where

.,11

returnmeantheisrn

rn

ii∑

=

=

If the returns are supposed to be drawn independently from the same probability distribution,

then r is the sample mean and the historical volatility is simply the annualized sample

standard deviation. The realized volatility is then the annualized sample second moment.

Note that

,)( 21

2

1

2 rnrrrn

ii

n

ii −=− ∑∑

==

and hence

4

2σ

1252~ 22−

−=n

rnσσ

This means that the realized volatility is approximately equal to the historical volatility if the

sample mean is very close to zero. The quantities and are called respectively

the realized variance and the historical variance.

In both cases the factor √252 annualizes the result. In general, if a time interval between two

observations is ∆t (expressed in years), then the annualization factor is 1/√∆t.. In our case

∆t 1/252.

Both the realized volatility and the historical volatility measure variability of existing

financial data and in the financial context in many cases give similar results. Both types of

volatility can be used as predictors of future volatility. Also it is sometimes important to try to

forecast their future values.

2.1.2 The implied volatility The implied volatility is always about market beliefs about the future volatility when the

option investors traded the option, and this awareness has been reflected in the option pricing

process. In theory, it is not difficult to obtain the implied volatility value. In the Black-Scholes

option pricing model there is the five basic parameters ( tS , X, r, tT , and σ) quantitatively

related to the option price, as long as the first four basic parameters and the actual market

option price are known in the option pricing model, the only unknown parameter σ will be

solved, which is the implied volatility. Therefore, the implied volatility also can be regarded

as an expectation of the actual market volatility[4]

2.1.3 The volatility index The volatility index is a weighted average of implied volatilities for options on a particular

index. As we can calculate a stock's volatility or the implied volatility based on its options, we

aslo can calculate the volatility for an index such as the S&P 500. This concept can be taken

one step further. A volatility index has been originated and is usually quoted in the financial

media for many indices,.The following is three most common volatility index:

S&P 100 Volatility Index (VXO)

2σ=V

5

S&P 500 Volatility Index (VIX_)

Nasdaq 100 Volatility Index (VXN)

The above volatility indexes are a weighted average of the implied volatilities for several

series of puts and calls options. Many market participants and observers will look these

indexes as an ascertainment of market sentiment. There are many interesting and important

information about the VIX, other volatility indexes and related products on the CBOE Web

site[11].

2.1.4 intraday volatility The intraday volatility is the price change in a stock or index on or during a defined trading

day. We can also say that it shows the market swings is the most noticeable and readily

available definition of volatility during the life of a trading day. Intraday volatility is the

Justice Potter Stewart type of volatility, as it is difficult to define but you know it is the

intraday volatility when you see it. A general mistake is people think the intraday volatility is

equated with the implied volatility index. Both of these types of volatility are not

interchangeable, but do carry their own particular importance on a certain extent in measuring

investor sentiment and expectations[11].

2.2 Principles of Volatility Index

The implied volatility is a core data for calculating volatility index (VIX) which is calculated

through the latest deals price in the options market. It is reflected the investors’ expectations

of future market prices. The concept is similar to bond yield to maturity (Yield To Maturity):

As the market price changes, through the appropriate interest rate, the bond principal and

coupon interest discount to the present value, then when the present value of bond is equal to

the discount rate of the market price, it is the bonds’ yield to maturity, that is, the implied

return rate of bonds. In the calculation process based on the bond evaluation model, the yield

to maturity can be obtained according to the market price, which is the implied yield to

maturity.

There are many ways to estimate the implied volatility. Firstly, you must determine the

options evaluation model, the required other parameters values and the present option price,

when computing the implied volatility of the options. For example, in Black-Scholes option

6

pricing model (1973), we can get the option theoretical price, as long as theunderlying price,

strike price, risk-free interest rate, time to maturity, the volatility of stock returns and other

data are put into the option price model formula. If the underlying assets and the option

market are efficient, the option theoretical price has fully reflected its true option value, and at

the same time the option price model is also correct, then we can get the implied volatility,

with taking the option market price into the Black-Scholes option model based on the concept

of the inverse function. Since it shows investors’ expectations of changes in future market

prices, so it is called implied volatility.

CBOE launched the first VIX index (VXO) in 1993, which is based on the option model

which is proposed by Black, Scholes and Merton. In this model, except the volatility, we also

require other many parameters such as the current stock price, the option price, the strike price,

duration, risk-free interest rate, time to expected payment of cash dividends and amount of

expected payment of cash dividends. However, the S&P 100 options of CBOE are American

options and are related to the cash dividends of the underlying stocks, so when CBOE

calculated the VIX index, they used the implied option volatility of the binomial model

proposed by Cox, Ross and Rubinstein (1797)[8].

In 2003, CBOE launched another new VIX index, compared to the old index (VXO), the new

VIX index is more accurate. However, the old VIX index is still sustained released, so in

order to distinguishing easily between the old and new VIX index, the old VIX index was

renamed as VXO index.

2.3 Variance Swaps

Among many volatility products, the variance swaps have become widely used. The variance

swaps, compared to the volatility swaps, have more convenient mathematical characteristics.

In short, a variance swap is a forward contract whose payoff is based on the difference

between the realized variance of the underlying asset during the lifetime of the contract and

the value of Kvar, which is the variance delivery price in the contract. The payment to maturity

can be expressed as

MKpayoff R ×−= )( var2σ

7

Where M is the notional amount of the swap in dollars per annualized volatility point

squared[3], and 2Rσ is the actual variance, that is the square of realized volatility

[4].

∑= −

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

N

i i

iR S

SN 1

2

1

2 ln252σ

Through a series of brief introductions of variances products and the above explanation about

the variance swaps, it is not difficult to find many variance products are more similar with

special futures contracts, which means the “goods” (variance) must be exchanged in

according to the variance delivery price. Because these products we are discussing have the

similar properties with futures options and forward contracts, compared to the value of the

contract itself, we pay more attention to the variance delivery price Kvar in the contract.

However, it is directly related to the volatility arbitrage or the hedging effect of this contract

product on expiration. So at present many research reports and literatures are involved it.

In the early period of volatility trading market, a normal way, which is the statistical arbitrage

method, is used to compute the difference between the actual variance and the the variance

delivery price. As the development of the variance research, a standardized method gradually

appeared which is building a mathematical model to determine a more reasonable and

accurate exchange value of Kvar.

Now, there are two methods to solve it: one is only through bonds and a combination of a

variety of standard European options to simulate a portfolio of variance investment products,

not through a accurate stochastic model, which is based on the paper named as “Towards a

Theory of Volatility Trading” published by Peter Carr and Dilip Madan in 1998[13]. The other

way is to calculate the value based on a definite stochastic volatility model, directly regarding

the volatility as the trade standard, through the GARCH price model.

8

Chapter 3. EstimatingVolatility From Option Data[2][14]

We can assume that the stock price satisfies the following stochastic differential equation

)1.3()( dzdtqrS

dS⋅+−= σ

Where r, q, σ are constant denoting the risk free rate, the continuous divided yield and

volatility respectively, and z is a Wiener process with respect to a risk-neutral probability

measure.

At the same time, from the Ito's lemma

)2.3()2

(ln2

dzdtqrSd ⋅+−−= σσ

Then from the equations (3.1)-(3.2), we can obtain

)3.3(ln2

2)

2()(ln

2

22

SdS

dSdt

dtdzdtqrdzdtqrSdS

dS

−=

=⋅−−−−⋅+−=−

σ

σσσσ

By integrating from time 0 to time T, it can be got

)ln(ln21

ln2

00

2

000

2

SSS

dST

SdS

dSdt

T

T

TTT

−−=

−=

∫

∫∫∫

σ

σ

And so

00

2 ln21

SS

SdST T

T−= ∫σ

Because the variance is the square of the volatility

2σ=V , we get

00

ln21

SS

SdSTV T

T−= ∫

or

)4.3(ln22

00 S

STS

dST

V TT

−= ∫

9

Then take the expectations of the equation (3.4) with respect to the risk-neutral probability

measure

)5.3()(ln2)(2)(0

0 SSE

TSdSE

TVE T

T ∧∧∧−= ∫

Because we can take the expectation of the equation (3.1), the following equation can be got

TqrS

dSE

dzEdtqrES

dSE

T

TTT

)()(

)())(()(

0

000

−=

⋅+−=

∫

∫∫∫∧

∧∧∧

σ

So putting the above equation into (3.5), we get

)(ln2)(2)(0S

SET

TqrT

VE T∧∧

−−=

We know that

XSST exp0= and the random

is Gaussian with

Then

( )TqrT eS

XVarXESSE −∧

∧∧

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+= 00 2

][][exp][

Therefore

)6.3()(ln2ln2)(00

0

SSE

TSF

TVE T

∧∧

−=

Where, which is the forward price of the asset for a contract maturing at time T.

Now, firstly we consider

dKSKK

S

T∫∗

−0 2

)0,max(1

Where S is some value of S.

1. If

0)0,max(10 2

=−∫∗

dKSKK

S

T

TXVarTqrXE 22

][ and )2

(][ σσ =−−=∧∧

)( )2

(2

TzTqrX σσ +−−=

)(0 TSEF∧

=

TSS <*

10

2. If

1ln

)(ln

)(1)0,max(1

*

*

20 2

*

−+=

+=

−=− ∫∫∗∗

SS

SS

KSK

dKSKK

dKSKK

T

T

S

S

T

S

S T

S

T

T

T

Secondly, we consider next

dKKSKS T∫

∞

∗)0,-max(12

1. If ,

0)0,-max(1* 2 =∫∞

dKKSKS T

2. If ,

1ln

lnln1

)ln(

-(1)0,-max(1** 22

−+=

++−−=

−−=

=

∗

∗

∗∗

∞

∗

∫∫

SS

SS

SSSS

KKS

dKKSK

dKKSK

T

T

TT

SS

T

S

S TS T

T

T）

Then we can get that

1ln)0 ,-max(1)0 ,max(1 20 2 −+=+− ∗∗∞

∫∫ ∗∗

SS

SSdKKS

KdKSK

KT

TS T

S

T

For all value of S so that

)7.3()0 ,-max(1)0 ,max(11ln 20 2 dKKSKdKSK

KSS

SS

S T

S

TT

T∫∫∞

∗

∗

∗

∗

−−−−=

We can take the expectations with respect to the risk-neutral probability measure in equation

(3.7)

))0 ,-max(1())0 ,max(1()1()(ln 20 2 dKKSKEdKSK

KE

SSE

SSE

S T

S

TT

T∫∫∞∧∧

∗

∧∗∧

∗

∗

−−−−=

TSS >*

TSS >*

TSS <*

11

Note that

[ ] )()0,max( KpeSKE RTT =−∧

[ ] )()0,max( KceKSE RTT =−∧

and

)(0 TSEF∧

= then we get

)8.3()(1)(11)(ln 20 20 dKKce

KdKKpe

KSF

SSE

S

RTS RT

T∫∫∞

∗

∗∧

∗

∗

−−−=

Where c K and p K are the prices of the call and put option with the strike price K, T is the

maturity time and R is the risk-free interest rate for a maturity of T.

According to

0

0

00

lnln

lnlnlnln

lnlnln

SS

SS

SSSS

SSSS

T

T

TT

∗

∗

∗∗

+=

−+−=

−=

Taking the expectation of the above the equation, we can get

)9.3()(lnln

)(ln)(ln)(ln

0

00

∗

∧∗

∗∧

∗

∧∧

+=

+=

SSE

SS

SSE

SSE

SSE

T

TT

Now combining the equations (3.6), (3.8) and (3.9),

12

1121 and −−=Δ−=Δ nnn KKKKKK

⎥⎦⎤

⎢⎣⎡ ++⎥⎦

⎤⎢⎣⎡ −−=

⎥⎦⎤

⎢⎣⎡ −−−−−=

⎥⎦⎤

⎢⎣⎡ −−−−+−−=

⎥⎦⎤

⎢⎣⎡ −−−−−=

−−=

⎥⎦

⎤⎢⎣

⎡+−=

−=

∫∫

∫∫

∫∫

∫∫

∞

∗∗

∞

∗∗

∞

∗∗

∞

∗

∗

∗

∧∗

∗

∧∗

∧∧

∗

∗

∗

∗

∗

∗

∗

∗

dKKceK

dKKpeKTS

FTS

FT

dKKceK

dKKpeKS

FT

SFT

dKKceK

dKKpeKS

FT

SSSFT

dKKceK

dKKpeKS

FTS

STS

FT

SSE

TSS

TSF

T

SSE

SS

TSF

T

SSE

TSF

TVE

S

RTS RT

S

RTS RT

S

RTS RT

S

RTS RT

T

T

T

)(1)(1212ln2

)(1)(112)ln(ln2

)(1)(112)lnlnln(ln2

)(1)(112ln2ln2

)(ln2ln2ln2

)(lnln2ln2

)(ln2ln2)(

20 200

20 20

0

20 20

000

20 20

00

0

00

0

00

0

00

0

In conclusion, we get the (3.10), the expected value of the average variance from time 0 to

time T ,

)10.3()(1)(1212ln2)( 20 200

⎥⎦⎤

⎢⎣⎡ ++⎥⎦

⎤⎢⎣⎡ −−= ∫∫

∞

∗∗

∧

∗

∗

dKKceK

dKKpeKTS

FTS

FT

VES

RTS RT

where F is the forward price of the options for a contract maturing time T, c K is the prices

of the call option with the strike price K and time to maturity T, p K is the prices of the put

option with the strike price K and time to maturity T , R is the risk-free interest rate for a

maturity of T, and S is some value of S[2].

We can assume that the price of the options with strike prices K (1 i n) are known, and

and we set S equal to the first strike price below F , then

approximate the integrals as

)11.3()()(1)(1 2n

1i20 2 i

rT

i

iS

RTS RT KQeKKdKKce

KdKKpe

KΔ

=+ ∑∫∫=

∞

∗

∗

Where

1ni2 and 2

11 −≤≤−

=Δ −+ iiiKKK

in particular,

the function Q(K ) is the price of a put option with strike price K . If ,

the function Q(K ) is the price of a call option with strike price K . If ,

nn KKKKK

13

TVE )(∧

If , the function Q(K ) is the average of the price of a call and a put option with strike

price K

Putting the equation (3.11) into equation (3.10), we can get

⎥⎦

⎤⎢⎣

⎡ Δ+⎥⎦

⎤⎢⎣⎡ −−= ∑

=∗∗

∧

)(212ln2)( 2n

1i

00i

rT

i

i KQeKK

TSF

TSF

TVE

or

)12.3()(212ln2)( 2n

1i

00⎥⎦

⎤⎢⎣

⎡ Δ+⎥⎦

⎤⎢⎣⎡ −−= ∑

=∗∗

∧

irT

i

i KQeKK

SF

SFTVE

According to ln N and its the following Maclaurin polynomial

))1(()1(21)1(ln 22 −+−−−= NNNN ο

We can get

))1(()1(21)1(ln 202000 −+−−−= ∗∗∗∗ S

FSF

SF

SF

ο

In the function it can be approximated as follow

2000 )1(21)1(ln −−−= ∗∗∗ S

FSF

SF

or

)13.3( )1(21)1(ln 2000 −−=−− ∗∗∗ S

FSF

SF

We can use equation (3.13) to replace the part of equation (3.12)

(3.14) )(2)1(

)(212ln2)(

2

n

1i

20

2

n

1i

00

⎥⎦

⎤⎢⎣

⎡ Δ+−−=

⎥⎦

⎤⎢⎣

⎡ Δ+⎥⎦

⎤⎢⎣⎡ −−=

∑

∑

=∗

=∗∗

∧

irT

i

i

irT

i

i

KQeKK

SF

KQeKK

SF

SFTVE

The process used on any given day is to calculate for options that trade in the

market and have maturities immediately above and below 30 days . The 30-day risk-neutral

expected cumulative variance is calculated from these two numbers using interpolation. This

is then multiplied by 365/30 and the index is set equal to the square root of the result.

However, we can get the formula used in the above calculation is

14

( )

( )∑

∑

Δ+⎥

⎦

⎤⎢⎣

⎡−−=

Δ+⎥

⎦

⎤⎢⎣

⎡−−=

ii

RT

i

i

ii

RT

i

i

KQeKK

TKF

T

KQeKK

KFT

)15.3(211

21

2

2

0

2

2

2

0

2

σ

σ

15

Chapter 4. Calculation of VIX

In September 22, 2003, CBOE launched the new VIX volatility index, which is calculated on

the basis to the S&P 500 options, while there are also many improvements in the algorithm,

and the index is much closer to the actual market situation. VIX is a volatility index of options

rather than stocks, and each option price shows the market’s expectation of the future

volatility[1]. Similar to the conventional indexes, VIX also chooses the component option and

a formula to compute index values.

The new VIX used the variance and volatility swaps (variance & volatility swaps) method to

update the calculation formula, which reflects better the overall market dynamics, and its

formula is as follows:

)1.4(100 σ×=VIX

Where[1]

T Time to expiration

F Forward index level derived from index option prices

K First strike below the forward index level, F

K

Strike price of thi out-of the-money option;

a call if 0KKi > and a put if 0KKi < l

both put and call 0KKi =

ΔK

Interval between strike prices – half the difference

between the strike on either side of K :

211 −+ −=Δ iii

KKK

(Note: ∆K for the lower strike is simply the difference

between the lowest strike and the next higher strike.

Likewise, ∆K for the highest strike is the difference

between the highest strike and the next lower strike.

R Risk-free interest rate to expiration

16

Q K The midpoint of the bid-ask spread for each option with

strike K

4.1—Parameter And Select The Options To Calculate VIX

4.1.1 Parameter T: VIX is used to measure the 30 days’ expected volatility of the S&P 500 Index, which is

divided to near- and next term put and call option, regularly in the first and second S&P 500

Index (SPX) contract months.

Calculation of the time to maturity is based on minutes, T, in calendar days and separates

every day into minutes for replicating the precision, which is used by option and volatility

traders.

Time to expiration is calculated as follow:

(4.2) yearainminutes

day settlement on the hrs 30:08 tomecurrent ti thefrom minutes=T

Here, we assume today is Tuesday 5, April 2011, in the hypothetical, the near-term and next-

term options have 11 days and 46 days to expiration respectively. (Near-term options expiring

April 16, 2011 and next-term options expiring May 21, 2011 ) and reflect prices observed at

the open time of trading, 8:30 a.m. Chicago time (the source of data from CBOE).

Then we can get the T for the near-term and next-term options,T1 and T2 respectively, is:

{ } 0.030136996024365

6024111 =∗∗

∗∗=T

{ } 0.12602746024365602446

2 =∗∗∗∗

=T

(Notes: there must be more than one week for “Near-term” options to expiration. This is

because there are pricing anomalies when it is very closed to expiration. We make a rule for

the “Near-term” options in order to reduce these anomalies. If the near-term options would

17

expire after less than one week, the VIX will roll to the second and third SPX contract month.

For example, for 11, June, 2010(the second Friday in June), the near-term option and next-

term option will expire in June and July. But if it would change to 14, June, 2010(the second

Monday in June), the near-term option and next-term option will expire in July and August[1].

4.1.2 Parameter R: R, the risk-free interest rate, is the U.S. Treasury yields which has the same maturing as the

relevant SPX option. We also can say, with different maturities, the risk-free interest rates of

near-term options and next-term options are different. However, SPX options are of the

European type with no dividends so this works. The put-call parity formula can be solved for

R as follows:

(4.3) ln1 ⎥⎦

⎤⎢⎣

⎡−+−

−=ttt CPS

KtT

R

Where

K The strike price

tS The underlying S&P500 price

tP Put option price

tC Call option price

Here, we can get the risk-free interest rate of the near-term and the next-term respectively(data from CBOE)

as follow,

0.03385023 11.5-10.3+1332.63

1330ln252

81

1 =⎥⎦⎤

⎢⎣⎡−=R

0.02367618 25.7-27.2+1332.63

1330ln252

331

2 =⎥⎦⎤

⎢⎣⎡−=R

0.0287632 )/2+(=R 21 =RR

4.1.3 Parameter F: We can use the following formula to calculate F:

( ) ）（ 4.4PricePutPriceCallPriceStrike −×+= RTeF

In the formula, the “Call Price minus Put Price” means, the absolute smallest difference

between the call and put prices for the same strike price, then “Strike Price” is the above

18

strike price with the absolute smallest difference between the call and put prices.

Now, we use my computer program based on the Yahoo financial data to calculate the

smallest difference, then we get the absolute difference of near-term option and next-term

option in the following table,

Near-Term Option

Strike price Call Put

Absolute Difference Bid Ask Bid Ask

... ... ... ... ... ...

1295 37.8 40.6 2.7 3.6 36.050

1300 33.5 36.1 3.6 4.1 30.950

1305 29.2 31.8 4 5.4 25.800

1310 25.1 27.8 4.7 6.2 21.000

1315 20.6 23.6 5.7 7.5 15.500

1320 17 19.8 7 8.8 10.500

1325 13.8 16.2 9.1 9.9 5.500

1330 11.5 13 10.3 12 1.100

1335 8.2 9.5 12.4 14.2 4.450

1340 5.7 7.6 14.9 16.9 9.250

1345 3.9 5.2 17.8 20 14.350

1350 2.9 3.3 21.2 23 19.000

1355 1.6 2.3 24.9 27.3 24.150

1360 1.25 1.35 29.1 31.5 29.000

1365 0.5 0.8 33.8 36.8 34.650

... ... ... ... ... ...

Table 4.1：The Difference of Near-Term Options

From Table 4.1, we can see 1.100 (the strike price is 1330) is the smallest difference of the

near-term options, so 1330 is the related strike price, then we get

19

( ) ( ) ( )

( )

1331.101100.11330

2123.10

21311.51330

0.030136990.02841502

0.030136990.028415021

=

×+=

⎟⎠⎞⎜

⎝⎛ +−+×+=

∗

∗

e

eF

Next-Term Option

Strike price Call Put

Absolute Difference Bid Ask Bid Ask

... ... ... ... ... ...

1305 41.7 45.3 18.8 21.2 23.500

1310 38.4 41.9 20.2 23.2 18.450

1315 35 38.4 21.8 24.3 13.650

1320 31.6 34.9 23.5 26.1 8.450

1325 28.6 31.8 25.4 28.3 3.350

1330 25.7 28.7 27.2 29.6 1.200

1335 22.8 25.7 29.3 31.8 6.300

1340 20.2 23 31.5 34.2 11.250

1345 17.6 20.5 34 36.7 16.300

1350 15.4 17.8 36.5 39.6 21.450

1355 13.3 14.5 39.3 42.4 26.950

1360 11.3 12.4 42.2 45.4 31.950

1365 9.7 11.3 45.3 48.5 36.400

1370 8 9.9 48.5 52 41.300

1375 6.8 8.4 52 55.5 46.150

1380 5.3 7.1 55.7 59.2 51.250

... ... ... ... ... ...

Table 4.2：The Difference of Next-Term Options

From Table 4.2 we can see 1.200 is the smallest difference of the next-term options, so 1330

is the related strike price, then we get

20

( ) ( ) ( )

( )

1331.204200.11330

26.292.27

27.287.251330

0.12602740.02841502

0.12602740.028415022

=

×+=

⎟⎠⎞

⎜⎝⎛ +−

+×+=

∗

∗

e

eF

4.1.4 Parameter K0 and Ki. If there are N options with different strike prices in the near-term S&P 500 options (including

N call option and N put option, a total of 2N),with the order of the N option by strike price, in

fact, the strike price which is first lower than the F value is defined as the K , the rest of the

option with the order from small strike price to large strike price are defined as

K , K , K , , KN , KN . while we defined the set φ=S .

From the above calculation, we know 1331.204 , 1331.101 21 == FF , then

3301 , 3301 2,01,0 == FK .

4.1.5 Select options It is shown VIX index is compiled, according to certain criteria of all the near-term options

and next-term options,we can filter these options into a standard option set S. Then the VIX

index is calculated by the strike prices and call/put prices of the options of set S. However,

now in the following select process, I will use the near-term options as the example to filter

the suitable options into set S.

If 0KKi < , we choose the strike price iK of put options, set iKSS +=

In fact, selecting set S starts with the strike price of put options lower than 0K and

move to the following lower strike prices, at the same time, reject any option of

which a put bid price is equal to zero. If it occurs that the two consecutive strike

prices have zero bid prices, the lower strike are not considered.

Near-Term Option

Strike price Put

Include? Bid Ask

990 0 0.05Not Considered

995 0 0.05

21

1000 0 0.05Not Considered

1005 0 0.05

1010 0 0.05 NO

1015 0 0.1 NO

1020 0.05 0.1 YES

1025 0.05 0.1 YES

1030 0.05 0.15 YES

1035 0.05 0.15 YES

1040 0.05 0.1 YES

1045 0.05 0.15 YES

1050 0.05 0.15 YES

1055 0.05 0.15 YES

1060 0.05 0.15 YES

... ... ... ...

Table 4.3：Selection of Near-Term Put Options

If ii KK 0> , we choose the strike price iK of call options, set iKSS +=

Selecting out-of- the-money call options is the same as the above method, there is

only one difference which is based on call options. Start with the call option strike

price higher than 0K and move to the following higher strike prices, at the same time,

we must reject any option of which a call bid price is equal to zero.However , if the

two consecutive strike prices which both have zero bid prices it occur, the higher

strike price are not considered. We will stop to select the options. Then the options

with higher strike price are not useful.

Near-Term Option

Strike price Call

Include? Bid Ask

... ... ... ...

1400 0.1 0.15 YES

1405 0.1 0.15 YES

1410 0.05 0.1 YES

1415 0.05 0.1 YES

22

1420 0.05 0.1 YES

1425 0 0.1 YES

1430 0.05 0.1 YES

1435 0 0.1 NO

1440 0 0.1 NO

1445 0 0.05

Not Considered

1450 0 0.05

1455 0 0.05

1460 0 0.05

... ... ..

Table 4.4：Selection of Near-Term Call Options

If 0KKi = , we put both call option and put option of the strike price 0K into set S .

In conclusion, from the above filter process we can see, when 0KKi < , only choose the put

option with strike price iK ; when ii KK 0> , only choose the call option with strike price iK ;

when 0KKi = , choose both put and call option with strike price 0K .

4.1.6 Parameter ∆

If SKi ∈ , we set

211 −+ −=Δ iii

KKK

Note: If dK is the lowest strike price of the put options in set S ,

If uK is the highest strike price of the call options in set S,

For example, from the above Table 4.3, we can see the strike price 1020 is the lowest

price of set S, and the strike price 1025 is the neighbour of the strike price 1020 in the

set S, so

5102010251020 =−=Δ =pricestrikeK

ddd KKK −=Δ +1

1−−=Δ uuu KKK

23

4.1.7 Parameter

is the average of quoted bid and ask option prices, or mid-quote prices.

2 )( priceAskpriceBidKQ i

+=

From our data, we can get the following tables:

Near-Term Option

Strike price Option Type Bid Ask Mid-quote

Price

... ... ... ... ...

1290 Put 2.2 3 2.600

1295 Put 2.7 3.6 3.150

1300 Put 3.6 4.1 3.850

1305 Put 4 5.4 4.700

1310 Put 4.7 6.2 5.450

1315 Put 5.7 7.5 6.600

1320 Put 7 8.8 7.900

1325 Put 9.1 9.9 9.500

1330 Put/Call Average 12.25 11.15 11.700

1335 Call 8.2 9.5 8.850

1340 Call 5.7 7.6 6.650

1345 Call 3.9 5.2 4.550

1350 Call 2.9 3.3 3.100

1355 Call 1.6 2.3 1.950

1360 Call 1.25 1.35 1.300

1365 Call 0.5 0.8 0.650

1370 Call 0.3 0.65 0.475

1375 Call 0.3 0.4 0.350

1380 Call 0.15 0.3 0.225

1385 Call 0.15 0.3 0.225

... ... ... ... ...

Table 4.5： Mid-quote Price of Near-Term Options

24

Next-Term Option

Strike price Option Type Bid Ask Mid-quote

Price

... ... ... ... ...

1300 Put 17.3 19.7 18.500

1305 Put 18.8 21.2 20.000

1310 Put 20.2 23.2 21.700

1315 Put 21.8 24.3 23.050

1320 Put 23.5 26.1 24.800

1325 Put 25.4 28.3 26.850

1330 Put/Call Average 27.2 28.4 27.800

1335 Call 22.8 25.7 24.250

1340 Call 20.2 23 21.600

1345 Call 17.6 20.5 19.050

1350 Call 15.4 17.8 16.600

1355 Call 13.3 14.5 13.900

1360 Call 11.3 12.4 11.850

1365 Call 9.7 11.3 10.500

1370 Call 8 9.9 8.950

1375 Call 6.8 8.4 7.600

1380 Call 5.3 7.1 6.200

1385 Call 4.3 5.7 5.000

1390 Call 3.3 4.8 4.050

1395 Call 2.6 3.6 3.100

1400 Call 2.5 2.65 2.575

1405 Call 1.75 2.55 2.150

1410 Call 1.3 2.05 1.675

... ... ... ... ...

Table 4.6： Mid-quote Price of Next-Term Options

25

4.2—Calculate volatility of both near-term and next-term options

From the formula 3.15, we can get the volatility formula of near-term and next-term option

with time to expiration of 1T and 2T respectively.

( )

( )2

20

2

22

2

22

2

10

1

12

1

21

112

112

2

1

⎥⎦

⎤⎢⎣

⎡−−

Δ=

⎥⎦

⎤⎢⎣

⎡−−

Δ=

∑

∑

，

，

KF

TKQe

KK

T

KF

TKQe

KK

T

ii

RT

i

i

ii

RT

i

i

σ

σ

Now, we want to compute the contribution values of both near-term and next-next options.

For example, the contribution of the near-term 1020 put option is given by:

( ) ( ) 36075000000.00.0751020

5 0201 03013699002841502022 1020

1020 1 =×=Δ ∗ ..RT

Put

Put ePutQeKK

The contribution of the next-term 1450 call option is given by:

( ) ( ) 77570000000.0.32501450

5Call 4501 0.12602740.0284150222 1450

1450 2 =×=Δ ∗eQeKK RT

Call

Call

A similar calculation is used for every option, then we get the following table:

Near-Term Option Next-Term Option

Strike

price

Option

Type

Mid-

quote

Price

Contribution

by Strike

Strike

price

Option

Type

Mid-

quote

Price

Contribution

by Strike

1220 Put 0.075 3.607509e-07 880 Put 0.175 9.072083e-06

1225 Put 0.075 3.572399e-07 890 Put 0.175 2.217340e-06

1230 Put 0.100 4.717067e-07 900 Put 0.175 1.626255e-06

1235 Put 0.100 4.671601e-07 905 Put 0.125 7.658737e-07

1240 Put 0.075 3.470092e-07 910 Put 0.225 1.363465e-06

... ... ... ... ... ... ... ...

1310 Put 5.450 1.589280e-05 1310 Put 21.700 6.345436e-05

1315 Put 6.600 1.910024e-05 1315 Put 23.050 6.689039e-05

26

1320 Put 7.900 2.268954e-05 1320 Put 24.800 7.142465e-05

1325 Put 9.500 2.707935e-05 1325 Put 26.850 7.674619e-05

1330

Put/Call

Average 11.700 3.310008e-05

1330

Put/Call

Average27.800 7.886527e-05

1335 Call 8.850 2.485005e-05 1335 Call 24.250 6.828000e-05

1340 Call 6.650 1.853355e-05 1340 Call 21.600 6.036545e-05

1345 Call 4.550 1.258674e-05 1345 Call 19.050 5.284388e-05

1350 Call 3.100 8.512177e-06 1350 Call 16.600 4.570722e-05

1355 Call 1.950 5.314990e-06 1355 Call 13.900 3.799098e-05

1360 Call 1.300 3.517321e-06 1360 Call 11.850 3.215029e-05

1365 Call 0.650 1.745800e-06 1365 Call 10.500 2.827928e-05

1370 Call 0.475 1.266482e-06 1370 Call 8.950 2.392909e-05

1375 Call 0.350 9.264225e-07 1375 Call 7.600 2.017216e-05

... ... ... ... ... ... ... ...

1405 Call 0.125 3.168866e-07 1430 Call 0.600 1.472392e-06

1410 Call 0.0750 1.887859e-07 1435 Call 0.525 1.279381e-06

1415 Call 0.0750 1.874541e-07 1440 Call 0.475 1.149510e-06

1420 Call 0.0750 2.792044e-07 1445 Call 0.350 8.411562e-07

1430 Call 0.0750 3.670842e-07 1450 Call 0.325 7.756962e-07

( )∑ Δi

iRT

i

i KQeKK

T1

21

2

0.02206404 ( )∑ Δ

ii

RT

i

i KQeKK

T2

22

2

0.03004726

Table 4.7： Contribution of Near-Term and Next-Term Options

Next, we calculate of both near-term and next-term options:

Now put the values of the parameters into the formula 3.15. we get

0000065026.011330

1331.2040.1260274

111

000022740.011330

1331.101030136990

111

22

2,0

2

2

22

1,0

1

1

=⎥⎦⎤

⎢⎣⎡ −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

=⎥⎦⎤

⎢⎣⎡ −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

KF

T

.KF

T

2

0

11 ⎥⎦

⎤⎢⎣

⎡−

KF

T

27

( )

( )

0.030040750000065026.0-0.03004726

112

0.0220413000022740.0-0.02206404

112

2

20

2

22

2

22

2

10

1

12

1

21

2

1

=

=

⎥⎦

⎤⎢⎣

⎡−−

Δ=

=

=

⎥⎦

⎤⎢⎣

⎡−−

Δ=

∑

∑

，

，

KF

TKQe

KK

T

KF

TKQe

KK

T

ii

RT

i

i

ii

RT

i

i

σ

σ

4.3—Calculate the 30-day weighted average of both σ1 and σ2 Then take the square root

of the value and multiply by 100 to get VIX, according to the formula 4.1 and formula 4.2.

30

365

12

130222

12

302211100 N

NNTNTNTNT

NTNTNNTTVIX ×

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−−

×= σσ

Where[1],

1NT Number of minutes to settlement of the near-term options

2NT Number of minutes to settlement of the next-term options

30N Number of minutes in 30 days (30*24*60=43200)

365N Number of minutes in 365-day year (365*24*60=525600)

In the above formula, if the near-term options have less than 30 days to expiration and the

next-term options have more than 30 days to expiration, from the value of VIX we know it

have been shown an interpolation of 21σ and

22σ .On the other hand, if both the near-term and

next-term options have more than 30 days to expiration, the VIX would “roll”. The same

above formula will be used to compute the 30-day weighted average, but this result is an

extrapolation of 21σ and

22σ .

In conclusion, we get the value of VIX

43200525600

158406624015840432000.030040750.1260274

158406624043200662400.0220413030136990100

10030

365

12

130222

12

302211

×⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡

−−

××+⎥⎦⎤

⎢⎣⎡

−−

×××=

×⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−−

×=

.

NN

NTNTNTNT

NTNTNNTTVIX σσ

28

So we get the final value of VIX of the April SPX options,

94.1616.941041694104.0100 ≈=×=VIX

As the same method, we can get the VIX values of other month SPX options, which is shown

in the following table.

Table 4.8： VIX of April and May

Table 4.9： VIX of May and June

Table 4.10： VIX of June and July

Current

day

Option

VIX Settlement day

Days Trading

days R F 0K 2σ

Apr 5 April 16 11 8 0.03385023 1331.101 1330 0.022041

16.941May 21 46 33 0.02367618 1331.204 1330 0.030017

Current

day

Option

VIX Settlement day

Days Trading

days R F 0K 2σ

Apr 5 May 21 46 33 0.02367618 1331.203 1330 0.030017

15.056June 18 74 53 0.02123212 1327.310 1325 0.035231

Current

day

Option

VIX Settlement day

Days Trading

days R F 0K 2σ

Apr 5 June 18 74 53 0.02123212 1327.309 1325 0.035214

14.471July 16 102 73 0.01878529 1325.754 1325 0.035231

Then ac

ccording to

1313.514

14.5

15

15.5

16

16.5

17

M

Table 4.8, ,

May

Table 4.9, a

Figure

J

29

and Table 4

4.1 The VIX

June

4.10 we get

IX Index

v

the followin

July

vix

ng figure

30

Chapter 5. Conclusion

From the above description, we know that the original intention of involving VIX index

reflects investors’ expectations for the future of the market. However, with the deepening of

the study, people gradually found that, except the mentioned earlier point, the VIX index also

has a wider range of applications. For example, after years of empirical testing, it was

discovered that the VIX index often has a negative correlation with returns of the stock.

However, this is what prompted people to think about that the VIX index may be used to

hedge the equity portfolio risk, so CBOE launched the first VIX index of futures contracts in

2004, two years later, it launched VIX index options contracts. The futures and options

contracts, are a powerful tool to prevent the risk, and helpful to get stable returns for

investors[8].

Now we can see the following figure is the comparison between VIX, S&P500, and 30-day

Historical Volatility from March, 2005 to December 2010.

Figure 5.1 VIX, S&P500, and 30-day Historical Volatility

From the Figure 5.1, we found that VHN and VIX index over the same period showed a high

degree of negative correlation with S&P500; when the VIX and VXN index is relatively high,

that is the higher volatility, there is a big change on the S&P500. In particular, there is a rapid

31

increase on VIX and VHX index in October 2008, at the same time, S&P500 decreased

swiftly. It is the famous event that Lehman Brothers went bankrupt in October 2008.

In short, if the stock declines, usually VIX would continue to rise. It is shown the investors

expect the volatility index will increase in future. When the VIX index hits the high point, the

investors because of increasing panic, would buy a lot of put options without consider, which

is a huge effect for market to quickly reach the bottom; when the index rises, VIX is down,

then investors are blind optimism, without any hedging action, and the stock prices will tend

to reverse. we believe that the calculation of implied volatility index, compared to the other

estimated methods, contains most information, and has a good increasing ability to predict the

future actual volatility.

Now, it has been applied to many countries such as Indian, China, and many European

countries. India has an India VIX, Germany has the VDAX, China Hongkong has the HSI

Volatility Index (VHSI) and some other countries (in particular largely in Europe) have their

own country volatility indices. But Outside of North America and Europe, the volatility index

of the other countries pickings are slim. In short, fear and anxiety may be rising slowly in U.S.

markets, but in the critical economies of Brazil, Russia, India, and China, signs of panic are

much more widespread. In indian, NSE has said that there is no intention to introduce

tradeable products based on the India VIX in the immediate future, as it is important that the

market investors get used to understanding and tracking the value of the India VIX and what

it signifies. The exchange said in a release. "Once market participants are comfortable, India

VIX futures and options contracts can be introduced in the Indian markets, based on

regulatory approvals, to enable investors to buy and sell volatility and take positions based on

the movement of India VIX."[15]

In China Hongkong, we can say that the VHSI is a measure of market perceived volatility in

direction. Hence VHSI readings mean investors expect that the market will move significantly,

regardless of direction. Real-time display of HSI Volatility Index (VHSI) is now available on

HKEx website homepage for public reference. VHSI measures expectations of volatility or

fluctuations in price of the Hang Seng Index and is published by Hang Seng Indexes

Company Limited (HSIL)[16].

32

VHSI shows the market’s expectations of stock market volatility in the next 30-day period. It

is calculated real-time using prices of the Hang Seng Index Options which is listed on HKEx.

The VHSI is always quoted in percentage points. If the VHSI is higher value, it shows that

investors expect the value of the HSI to change sharply - up, down, or both - in the next 30

days[16].

In conclusion, VIX is known as 'investor fear gauge' and 'fear index', reflects investors' best

prediction of near-term market volatility. If emerging markets are the buffer the continued

growth of which is supposed to buttress developed markets with slowdown of this economic,

then emerging markets need to find their own firm ground and balance anxious investors

before they can be expected to lubricate the wheels of the global economy.

33

References .

[1]. The CBOE Volatility Index-VIX, White paper, CBOE, 2003.

[2]. John Hull, Options, Futures, and Other Derivatives, 7th Edition, Pearson-Prentice

Hall 2009.

[3]. Kresimir Demeterfi, Emanuel Derman, Michael Kamal, and Joseph Zou, More Than

You Ever Wanted To Know About Volatility Swaps, Quantitative Strategies Research

Notes, March, 1999.

[4]. Sebastien Bossu, Eva Strasser and Regis Guichard, Just Want You Need To Know

About Variance Swaps, Equity Derivatives Investor, Quantitative Research &

Development, JPMorgan, February, 2005.

[5]. Grant V. Farnsworth, Econometrics in R,October 26, 2008.

[6]. Allen and Harris, VolatilityVehicles, JPMorgan Equity Derivatives Strategy Product

Note.

[7]. Peter Carrand Liuren Wu, A Tale of Two Indices, The Journal Of Derivative, Spring,

2006.

[8]. Brenner, M., and M. Subrahmanyam, A Simple Formula to Compute theImplied

Standard Deviation,Financial AnalystsJournal, 44.

[9]. 'National Stock Exchange of India', http://www.nseindia.com, Last accessed on 14

October, 2009.

[10]. Hang Seng indexes to launch Hong Kong's "VIX", http://www.reuters.com/article, Mon

Jan 31, 2011 4:56am EST

[11] Understanding the Four Measures of Volatility, http://www.thestreet.com/story, 3

December,2007.

[12] Alireza Javaheri, Paul Wilmott and Espen G. Haug, GARCH and Volatility Swaps,

Published on wilmott.com, January 2002.

[13] Peter Carr and Dilip Madan, Towards a Theory of Volatility Trading, New Estimation

Techniques for Pricing Derivatives, London:Risk Publications,1998.

[14] J.Hull, Technical Note 22.

http://www.rotman.utoronto.ca/hull/TechnicalNotes/TechnicalNote22.pdf.

34

[15]. How to Create Your Own Portable VXV,

http://vixandmore.blogspot.com/search/label/India, Wednesday, April 22, 2009.

[16]. HSI Volatility Index (VHSI), HKEx, April,2011.

35

Appendix.

1. Codes ## Data

neardata

36

##Step 2

##Ordering

##Find k0

k0=0

for(i in 1:numo)

{if(nedata[i,1]=k0&&nedata[j1,2]!=0)

{j2=j2+1}

if(j1==numo||(nedata[j1+1,2]==0&&nedata[j1+2,2]==0))

{break}

j1=j1+1

}

c=k0&&nedata[j2,2]!=0)

{

for(i in 1:5)

{

c[j1,i]=nedata[j2,i]

}

j1=j1+1

}

if((nedata[j2,2]==0&&nedata[j2+1,2]==0)||(j2==dim(nedata)[1]))

37

{break}

j2=j2+1

}

## Check calls

c

## Puts

j1=1

j2=0

repeat

{

if(nedata[j1,4]!=0)

{j2=j2+1}

if(nedata[j1,1]>=k0)

{break}

j1=j1+1

}

p

38

{break}

j2=j2+1

}

##Check that if puts is full element

p

##Final data

fdata

39

s=0

for(i in 1:dim(fdata)[1])

{s=s+dk[i]*qk[i]/(fdata[i,1]^2)

}

cont r2

[1] 0.02367618

> r=(r1+r2)/2

40

> r

[1] 0.0287632

> cvix(neardata,r,11/365)

$Use.Data

[,1] [,2] [,3] [,4] [,5]

[1,] 1020 309.10 311.90 0.05 0.10

[2,] 1025 304.10 306.90 0.05 0.10

[3,] 1030 299.10 301.90 0.05 0.15

[4,] 1035 294.10 296.90 0.05 0.15

[5,] 1040 289.10 291.90 0.05 0.10

[6,] 1045 284.10 286.90 0.05 0.15

[7,] 1050 279.10 281.90 0.05 0.15

[8,] 1055 274.10 276.90 0.05 0.15

[9,] 1060 269.10 271.90 0.05 0.15

[10,] 1065 264.10 266.90 0.05 0.15

[11,] 1070 259.10 261.90 0.05 0.15

[12,] 1075 254.20 257.00 0.10 0.15

[13,] 1080 249.20 252.00 0.10 0.15

[14,] 1085 244.20 247.00 0.05 0.15

[15,] 1090 239.20 242.00 0.05 0.30

[16,] 1095 234.20 237.00 0.05 0.20

[17,] 1100 229.00 232.00 0.15 0.20

[18,] 1105 224.30 227.00 0.15 0.20

[19,] 1110 219.30 222.10 0.05 0.20

[20,] 1115 214.30 217.10 0.10 0.25

[21,] 1120 209.30 212.10 0.15 0.30

[22,] 1125 204.30 207.10 0.15 0.30

[23,] 1130 199.30 202.10 0.15 0.35

[24,] 1135 194.30 197.10 0.10 0.35

[25,] 1140 189.30 192.10 0.15 0.30

[26,] 1145 184.30 187.10 0.15 0.30

[27,] 1150 179.30 182.10 0.20 0.35

[28,] 1155 174.30 177.20 0.15 0.45

[29,] 1160 169.30 172.20 0.25 0.40

41

[30,] 1165 164.30 167.20 0.25 0.40

[31,] 1170 159.40 162.30 0.25 0.40

[32,] 1175 154.40 157.30 0.30 0.40

[33,] 1180 149.40 152.30 0.30 0.55

[34,] 1185 144.40 147.40 0.30 0.65

[35,] 1190 139.50 142.40 0.35 0.50

[36,] 1195 134.50 137.50 0.40 0.50

[37,] 1200 129.50 132.60 0.35 0.55

[38,] 1205 124.50 127.60 0.50 0.60

[39,] 1210 119.60 122.70 0.50 0.60

[40,] 1215 114.60 117.80 0.50 0.65

[41,] 1220 109.60 112.80 0.60 0.70

[42,] 1225 104.60 107.90 0.60 0.80

[43,] 1230 99.70 102.90 0.60 0.90

[44,] 1235 94.70 98.00 0.60 1.00

[45,] 1240 89.40 93.00 0.60 1.00

[46,] 1245 84.50 88.10 0.65 1.30

[47,] 1250 80.60 83.20 0.95 1.40

[48,] 1255 75.70 78.40 0.90 1.25

[49,] 1260 70.80 73.50 1.00 1.50

[50,] 1265 65.70 68.70 1.20 1.95

[51,] 1270 61.10 63.90 1.10 1.80

[52,] 1275 56.10 59.10 1.60 2.00

[53,] 1280 51.60 54.40 1.70 2.65

[54,] 1285 46.90 49.70 2.10 2.60

[55,] 1290 42.40 45.10 2.20 3.00

[56,] 1295 37.80 40.60 2.70 3.60

[57,] 1300 33.50 36.10 3.60 4.10

[58,] 1305 29.20 31.80 4.00 5.40

[59,] 1310 25.10 27.80 4.70 6.20

[60,] 1315 20.60 23.60 5.70 7.50

[61,] 1320 17.00 19.80 7.00 8.80

[62,] 1325 13.80 16.20 9.10 9.90

[63,] 1330 11.50 13.00 10.30 12.00

42

[64,] 1335 8.20 9.50 12.40 14.20

[65,] 1340 5.70 7.60 14.90 16.90

[66,] 1345 3.90 5.20 17.80 20.00

[67,] 1350 2.90 3.30 21.20 23.00

[68,] 1355 1.60 2.30 24.90 27.30

[69,] 1360 1.25 1.35 29.10 31.50

[70,] 1365 0.50 0.80 33.80 36.80

[71,] 1370 0.30 0.65 38.50 41.40

[72,] 1375 0.30 0.40 43.30 46.20

[73,] 1380 0.15 0.30 48.20 51.10

[74,] 1385 0.15 0.30 53.20 56.00

[75,] 1390 0.15 0.20 58.20 61.00

[76,] 1395 0.10 0.15 63.20 66.00

[77,] 1400 0.10 0.15 68.10 70.90

[78,] 1405 0.10 0.15 73.10 75.90

[79,] 1410 0.05 0.10 78.10 80.90

[80,] 1415 0.05 0.10 83.10 85.90

[81,] 1420 0.05 0.10 88.10 90.90

[82,] 1430 0.05 0.10 98.10 100.90

$Z

[1] 830.375 730.375 680.375 630.475 580.475 570.475 560.475 550.475 540.475

[10] 530.475 520.475 510.475 505.475 500.475 490.475 480.475 470.475 460.475

[19] 455.475 450.475 440.475 430.475 425.475 420.475 415.475 410.475 405.475

[28] 400.475 395.475 390.475 385.475 380.475 375.475 370.475 365.475 360.475

[37] 355.475 350.475 345.475 340.475 335.475 330.475 325.475 320.475 315.450

[46] 310.425 305.425 300.400 295.400 290.425 285.400 280.400 275.400 270.400

[55] 265.400 260.400 255.475 250.475 245.500 240.425 235.475 230.325 225.475

[64] 220.575 215.525 210.475 205.475 200.450 195.475 190.475 185.475 180.425

[73] 175.450 170.425 165.425 160.525 155.500 150.425 145.425 140.525 135.550

[82] 130.600 125.500 120.600 115.625 110.550 105.550 100.550 95.550 90.400

[91] 85.325 80.725 75.975 70.900 65.625 61.050 55.800 50.825 45.950

[100] 41.150 36.050 30.950 25.800 21.000 15.500 10.500 5.500 1.100

[109] 4.450 9.250 14.350 19.000 24.150 29.000 34.650 39.475 44.400

43

[118] 49.425 54.375 59.425 64.475 69.375 74.375 79.425 84.425 89.425

[127] 94.450 99.425 104.450 109.450 114.475 119.475 124.475 129.475 134.475

[136] 144.475 149.425 154.425 159.425 164.425 169.425 179.425 184.475 189.475

[145] 194.475 219.425 269.375 319.375 369.375 419.375 469.375

$Strike.Price

[1] 1330

$k0

[1] 1330

$F

[1] 1331.101

$Q.k0

[1] 0.075 0.075 0.100 0.100 0.075 0.100 0.100 0.100 0.100 0.100

[11] 0.100 0.125 0.125 0.100 0.175 0.125 0.175 0.175 0.125 0.175

[21] 0.225 0.225 0.250 0.225 0.225 0.225 0.275 0.300 0.325 0.325

[31] 0.325 0.350 0.425 0.475 0.425 0.450 0.450 0.550 0.550 0.575

[41] 0.650 0.700 0.750 0.800 0.800 0.975 1.175 1.075 1.250 1.575

[51] 1.450 1.800 2.175 2.350 2.600 3.150 3.850 4.700 5.450 6.600

[61] 7.900 9.500 11.700 8.850 6.650 4.550 3.100 1.950 1.300 0.650

[71] 0.475 0.350 0.225 0.225 0.175 0.125 0.125 0.125 0.075 0.075

[81] 0.075 0.075

$Contribution

[1] 3.607509e-07 3.572399e-07 4.717067e-07 4.671601e-07 3.470092e-07

[6] 4.582620e-07 4.539080e-07 4.496158e-07 4.453841e-07 4.412119e-07

[11] 4.370981e-07 5.413019e-07 5.363014e-07 4.250960e-07 7.371087e-07

[16] 5.217089e-07 7.237676e-07 7.172325e-07 5.077039e-07 7.044250e-07

[21] 8.976209e-07 8.896597e-07 9.797823e-07 8.740520e-07 8.664017e-07

[26] 8.588514e-07 1.040599e-06 1.125392e-06 1.208687e-06 1.198334e-06

[31] 1.188114e-06 1.268641e-06 1.527465e-06 1.692791e-06 1.501902e-06

[36] 1.576969e-06 1.563855e-06 1.895549e-06 1.879916e-06 1.949224e-06

44

[41] 2.185446e-06 2.334384e-06 2.480833e-06 2.624838e-06 2.603713e-06

[46] 3.147838e-06 3.763261e-06 3.415604e-06 3.940174e-06 4.925451e-06

[51] 4.498907e-06 5.541133e-06 6.643329e-06 7.122100e-06 7.818805e-06

[56] 9.399776e-06 1.140041e-05 1.381094e-05 1.589280e-05 1.910024e-05

[61] 2.268954e-05 2.707935e-05 3.310008e-05 2.485005e-05 1.853355e-05

[66] 1.258674e-05 8.512177e-06 5.314990e-06 3.517321e-06 1.745800e-06

[71] 1.266482e-06 9.264225e-07 5.912495e-07 5.869883e-07 4.532678e-07

[76] 3.214460e-07 3.191541e-07 3.168866e-07 1.887859e-07 1.874541e-07

[81] 2.792044e-07 3.670842e-07

$Sum

[1] 0.02206404

$sigma2

[1] 0.0220413

> cvix(nextdata,r,46/365)

$Use.Data

[,1] [,2] [,3] [,4] [,5]

[1,] 800 526.50 529.90 0.05 0.10

[2,] 810 516.50 519.90 0.05 0.20

[3,] 880 446.60 450.10 0.05 0.30

[4,] 890 436.60 440.10 0.05 0.30

[5,] 900 426.60 430.10 0.15 0.20

[6,] 905 421.60 425.10 0.05 0.20

[7,] 910 416.60 420.10 0.05 0.40

[8,] 915 411.70 415.20 0.05 0.40

[9,] 920 406.70 410.20 0.05 0.25

[10,] 925 401.70 405.20 0.05 0.40

[11,] 930 396.50 400.20 0.05 0.45

[12,] 935 391.70 395.20 0.05 0.45

[13,] 940 386.60 390.50 0.10 0.50

[14,] 945 381.60 385.50 0.10 0.50

[15,] 950 376.80 380.30 0.15 0.60

45

[16,] 955 371.80 375.30 0.15 0.65

[17,] 960 366.80 370.30 0.20 0.65

[18,] 965 361.80 365.40 0.20 0.70

[19,] 970 356.90 360.40 0.25 0.70

[20,] 975 351.90 355.40 0.25 0.75

[21,] 980 346.90 350.50 0.30 0.80

[22,] 985 341.90 345.50 0.50 0.80

[23,] 990 337.00 340.50 0.55 0.85

[24,] 995 332.00 335.50 0.40 0.85

[25,] 1000 327.00 330.60 0.55 0.65

[26,] 1005 322.00 325.60 0.45 0.90

[27,] 1010 317.00 320.60 0.45 0.95

[28,] 1015 312.10 315.70 0.50 1.00

[29,] 1020 307.10 310.70 0.55 1.05

[30,] 1025 302.60 306.20 0.55 1.05

[31,] 1030 297.10 300.80 0.55 1.15

[32,] 1035 292.20 295.80 0.60 1.20

[33,] 1040 287.70 291.20 0.80 1.25

[34,] 1045 282.00 285.90 0.85 1.30

[35,] 1050 277.80 281.30 0.80 0.95

[36,] 1055 272.30 276.10 0.70 1.50

[37,] 1060 267.40 271.10 0.75 1.55

[38,] 1065 262.40 266.20 0.80 1.60

[39,] 1070 257.40 261.20 0.85 1.65

[40,] 1075 252.60 256.50 0.85 1.75

[41,] 1080 247.70 251.60 0.90 1.80

[42,] 1085 242.60 246.40 0.95 1.90

[43,] 1090 237.80 241.70 1.00 1.95

[44,] 1095 232.90 236.80 1.05 2.00

[45,] 1100 228.00 231.90 1.35 1.65

[46,] 1105 223.00 226.90 1.20 2.15

[47,] 1110 218.10 222.00 1.30 2.25

[48,] 1115 213.20 217.10 1.40 2.30

[49,] 1120 208.30 212.20 1.50 2.45

46

[50,] 1125 203.40 207.30 1.50 2.10

[51,] 1130 198.50 202.40 1.75 2.70

[52,] 1135 194.10 198.00 1.85 2.80

[53,] 1140 189.20 193.10 1.95 2.80

[54,] 1145 184.40 188.30 2.00 3.40

[55,] 1150 179.50 183.40 2.10 2.85

[56,] 1155 174.60 178.50 2.10 3.50

[57,] 1160 169.80 173.70 2.55 3.50

[58,] 1165 165.00 168.90 2.45 3.80

[59,] 1170 160.20 164.10 2.60 4.00

[60,] 1175 155.40 159.30 2.75 4.00

[61,] 1180 150.60 154.50 3.00 4.50

[62,] 1185 145.80 149.70 3.20 4.70

[63,] 1190 141.00 144.90 3.50 4.50

[64,] 1195 136.30 140.20 3.70 5.20

[65,] 1200 131.60 135.50 4.00 5.50

[66,] 1205 126.90 130.80 4.40 5.90

[67,] 1210 122.20 126.10 4.70 6.20

[68,] 1215 117.60 121.50 5.00 7.00

[69,] 1220 113.00 116.90 5.30 7.10

[70,] 1225 108.40 112.30 5.70 7.50

[71,] 1230 103.80 107.70 6.20 8.00

[72,] 1235 99.30 103.20 6.60 8.30

[73,] 1240 94.80 98.70 7.30 9.20

[74,] 1245 90.40 94.30 7.70 9.40

[75,] 1250 86.00 89.90 8.20 9.40

[76,] 1255 81.60 85.50 8.80 10.70

[77,] 1260 77.40 81.30 10.00 11.50

[78,] 1265 73.20 77.10 10.30 12.50

[79,] 1270 69.00 72.90 11.00 13.10

[80,] 1275 64.80 68.70 12.40 14.00

[81,] 1280 60.80 64.70 12.80 15.10

[82,] 1285 56.80 60.70 13.80 16.10

[83,] 1290 52.90 56.80 15.00 17.30

47

[84,] 1295 49.10 52.70 16.20 18.60

[85,] 1300 45.30 49.20 17.30 19.70

[86,] 1305 41.70 45.30 18.80 21.20

[87,] 1310 38.40 41.90 20.20 23.20

[88,] 1315 35.00 38.40 21.80 24.30

[89,] 1320 31.60 34.90 23.50 26.10

[90,] 1325 28.60 31.80 25.40 28.30

[91,] 1330 25.70 28.70 27.20 29.60

[92,] 1335 22.80 25.70 29.30 31.80

[93,] 1340 20.20 23.00 31.50 34.20

[94,] 1345 17.60 20.50 34.00 36.70

[95,] 1350 15.40 17.80 36.50 39.60

[96,] 1355 13.30 14.50 39.30 42.40

[97,] 1360 11.30 12.40 42.20 45.40

[98,] 1365 9.70 11.30 45.30 48.50

[99,] 1370 8.00 9.90 48.50 52.00

[100,] 1375 6.80 8.40 52.00 55.50

[101,] 1380 5.30 7.10 55.70 59.20

[102,] 1385 4.30 5.70 59.40 63.00

[103,] 1390 3.30 4.80 63.60 67.10

[104,] 1395 2.60 3.60 67.90 71.20

[105,] 1400 2.50 2.65 72.10 75.60

[106,] 1405 1.75 2.55 76.70 80.00

[107,] 1410 1.30 2.05 81.00 84.60

[108,] 1415 1.00 1.60 85.90 89.00

[109,] 1420 0.70 1.30 90.70 93.70

[110,] 1425 0.55 1.05 95.50 98.50

[111,] 1430 0.40 0.80 100.40 103.30

[112,] 1435 0.30 0.75 105.30 108.20

[113,] 1440 0.25 0.70 110.20 113.10

[114,] 1445 0.10 0.60 114.90 118.00

[115,] 1450 0.25 0.40 120.10 122.90

$Z

48

[1] 827.975 728.025 678.025 628.025 578.100 552.975 528.125 518.075 508.150

[10] 503.200 498.100 488.200 478.175 468.100 458.150 453.275 448.175 438.175

[19] 428.175 423.225 418.125 413.225 408.300 403.225 398.100 393.200 388.250

[28] 383.250 378.175 373.150 368.125 363.150 358.175 353.150 348.150 343.050

[37] 338.050 333.125 328.200 323.125 318.100 313.150 308.100 303.600 298.100

[46] 293.100 288.425 282.875 278.675 273.100 268.100 263.100 258.050 253.250

[55] 248.300 243.075 238.275 233.325 228.450 223.275 218.275 213.300 208.275

[64] 203.550 198.225 193.725 188.775 183.650 178.975 173.750 168.725 163.825

[73] 158.850 153.975 148.800 143.800 138.950 133.800 128.800 123.700 118.700

[82] 113.550 108.750 103.750 98.650 93.800 88.500 83.800 79.150 73.800

[91] 68.600 63.750 58.900 53.550 48.800 43.800 38.700 33.500 28.750

[100] 23.500 18.450 13.650 8.450 3.350 1.200 6.300 11.250 16.300

[109] 21.450 26.950 31.950 36.400 41.300 46.150 51.250 56.200 61.300

[118] 66.450 71.275 76.200 81.125 86.150 91.200 96.200 101.250 106.225

[127] 111.175 116.100 121.175 131.700 141.150 146.275 151.175 161.625 171.750

[136] 196.100 221.475 246.475 271.450 321.400 371.400 471.400

$Strike.Price

[1] 1330

$k0

[1] 1330

$F

[1] 1331.204

$Q.k0

[1] 0.075 0.125 0.175 0.175 0.175 0.125 0.225 0.225 0.150 0.225

[11] 0.250 0.250 0.300 0.300 0.375 0.400 0.425 0.450 0.475 0.500

[21] 0.550 0.650 0.700 0.625 0.600 0.675 0.700 0.750 0.800 0.800

[31] 0.850 0.900 1.025 1.075 0.875 1.100 1.150 1.200 1.250 1.300

[41] 1.350 1.425 1.475 1.525 1.500 1.675 1.775 1.850 1.975 1.800

[51] 2.225 2.325 2.375 2.700 2.475 2.800 3.025 3.125 3.300 3.375

[61] 3.750 3.950 4.000 4.450 4.750 5.150 5.450 6.000 6.200 6.600

49

[71] 7.100 7.450 8.250 8.550 8.800 9.750 10.750 11.400 12.050 13.200

[81] 13.950 14.950 16.150 17.400 18.500 20.000 21.700 23.050 24.800 26.850

[91] 27.800 24.250 21.600 19.050 16.600 13.900 11.850 10.500 8.950 7.600

[101] 6.200 5.000 4.050 3.100 2.575 2.150 1.675 1.300 1.000 0.800

[111] 0.600 0.525 0.475 0.350 0.325

$Contribution

[1] 1.176131e-06 7.648465e-06 9.072083e-06 2.217340e-06 1.626255e-06

[6] 7.658737e-07 1.363465e-06 1.348605e-06 8.893238e-07 1.319603e-06

[11] 1.450502e-06 1.435030e-06 1.703766e-06 1.685784e-06 2.085107e-06

[16] 2.200886e-06 2.314146e-06 2.424947e-06 2.533346e-06 2.639399e-06

[21] 2.873789e-06 3.361903e-06 3.584033e-06 3.167949e-06 3.010895e-06

[26] 3.353636e-06 3.443496e-06 3.653200e-06 3.858637e-06 3.821084e-06

[31] 4.020581e-06 4.216053e-06 4.755558e-06 4.939923e-06 3.982665e-06

[36] 4.959433e-06 5.136064e-06 5.309166e-06 5.478817e-06 5.645088e-06

[41] 5.808053e-06 6.074348e-06 6.229932e-06 6.382428e-06 6.220857e-06

[46] 6.883900e-06 7.229308e-06 7.467346e-06 7.900878e-06 7.136935e-06

[51] 8.744147e-06 9.056816e-06 9.170610e-06 1.033468e-05 9.391259e-06

[56] 1.053267e-05 1.128116e-05 1.155427e-05 1.209725e-05 1.226711e-05

[61] 1.351486e-05 1.411577e-05 1.417459e-05 1.563754e-05 1.655295e-05

[66] 1.779826e-05 1.867971e-05 2.039591e-05 2.090337e-05 2.207070e-05

[71] 2.355008e-05 2.451132e-05 2.692495e-05 2.768036e-05 2.826226e-05

[76] 3.106429e-05 3.397908e-05 3.574935e-05 3.749073e-05 4.074721e-05

[81] 4.272662e-05 4.543382e-05 4.870095e-05 5.206598e-05 5.493249e-05

[86] 5.893228e-05 6.345436e-05 6.689039e-05 7.142465e-05 7.674619e-05

[91] 7.886527e-05 6.828000e-05 6.036545e-05 5.284388e-05 4.570722e-05

[96] 3.799098e-05 3.215029e-05 2.827928e-05 2.392909e-05 2.017216e-05

[101] 1.633721e-05 1.308021e-05 1.051889e-05 7.993879e-06 6.592733e-06

[106] 5.465503e-06 4.227863e-06 3.258178e-06 2.488672e-06 1.976990e-06

[111] 1.472392e-06 1.279381e-06 1.149510e-06 8.411562e-07 7.756962e-07

$Sum

[1] 0.03004726

50

$sigma2

[1] 0.03004075

>

> vix vix(neardata,nextdata,r,11/365,46/365)

[1] 16.94104