The VIX Volatility Index

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

Examensarbete i matematik, 30 hp

Handledare och examinator: Maciej Klimek Maj 2011

Department of Mathematics

Uppsala University

The VIX Volatility Index

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

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

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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 i i r n 1 2 1 252 σ Where . , 1 , 3 , 2 , 1 , ln 1 n n i S S r i i i = = − − 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 i i r r n 1 2 ) ( 1 252 ~ σ Where . , 1 1 return mean the is r n r n i i

∑

= =

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 , ) ( 2 1 2 1 2 r n r r r n i i n i i − =

∑

−

∑

= = and hence

4 2 σ 1 252 ~2 2 − − = n r n σ σ

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 (S_t, X, r, T_t, 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:

¾ S&P 100 Volatility Index (VXO)

2

σ

=

5

¾ S&P 500 Volatility Index (VIX_)

¾ 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

M K

7

Where M is the notional amount of the swap in dollars per annualized volatility point squared[3], and 2

R

σ is the actual variance, that is the square of realized volatility[4].

∑

= − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = N i i i R S S N 1 2 1 2 252 _ln σ

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 ( ) (r q dt dz S 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 ( ln 2 dz dt q r S d = − −σ +σ ⋅ Then from the equations (3.1)-(3.2), we can obtain

) 3 . 3 ( ln 2 2 ) 2 ( ) ( ln 2 2 2 S d S dS dt dt dz dt q r dz dt q r S d S dS − = = ⋅ − − − − ⋅ + − = − σ σ σ σ σ

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

) ln (ln 2 1 ln 2 0 0 2 0 0 0 2 S S S dS T S d S dS dt T T T T T − − = − =

∫

∫

∫

∫

σ σ And so 0 0 2 _ln 2 1 S S S dS T =

∫

T − T σ

Because the variance is the square of the volatility 2 σ = V _, we get 0 0 ln 2 1 S S S dS T V =

∫

T − T or ) 4 . 3 ( ln 2 2 0 0 _S S T S dS T V =

∫

T − T

9

Then take the expectations of the equation (3.4) with respect to the risk-neutral probability measure ) 5 . 3 ( ) (ln 2 ) ( 2 ) ( 0 0 _S S E T S dS E T V E∧ = ∧ T − ∧ T

∫

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

T q r S dS E dz E dt q r E S dS E T T T T ) ( ) ( ) ( ) ) ( ( ) ( 0 0 0 0 − = ⋅ + − =

∫

∫

∫

∫

∧ ∧ ∧ ∧ σ

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

) (ln 2 ) ( 2 ) ( 0 S S E T T q r T V E∧ = − − ∧ T We know that X S S_T = ₀exp and the random

is Gaussian with Then (r q)T T S e X Var X E S S E − ∧ ∧ ∧ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + = _{0 0} 2 ] [ ] [ exp ] [ Therefore ) 6 . 3 ( ) (ln 2 ln 2 ) ( 0 0 0 S S E T S F T V E T ∧ ∧ − =

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

dK S K K S T

∫

∗ − 0 2 max( ,0) 1

Where S is some value of S. 1. If 0 ) 0 , max( 1 0 2 − =

∫

∗ K S dK K S T T X Var T q r X E 2 2 ] [ and ) 2 ( ] [ ∧ = − −σ ∧ =σ ) ( ) 2 ( 2 T z T q r X = − −σ +σ ) ( 0 E ST F ∧ = T S S* <

10 2. If 1 ln ) (ln ) ( 1 ) 0 , max( 1 * * 2 0 2 * − + = + = − = −

∫

∫

∗ ∗ S S S S K S K dK S K K dK S K K T T S S T S S T S T T T

Secondly, we consider next

dK K S K S T

∫

∞∗ max( - ,0) 1 2 1. If , 0 ) 0 , -max( 1 * 2 =

∫

∞ S K dK K S T 2. If , 1 ln ln ln 1 ) ln ( -( 1 ) 0 , -max( 1 * * 2 2 − + = + + − − = − − = = ∗ ∗ ∗ ∗ ∞ ∗

∫

∫

S S S S S S S S K K S dK K S K dK K S K T T T T S S T S S T S T T T ）

Then we can get that

1 ln ) 0 , -max( 1 ) 0 , max( 1 2 0 2 − + = + ∗ − ∗ ∞

∫

∫

∗ ∗ S S S S dK K S K dK S K K T T S T S T

For all value of S so that

) 7 . 3 ( ) 0 , -max( 1 ) 0 , max( 1 1 ln ₂ 0 _K2 K S dK _K S K dK S S S S S T S T T 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 ₂ 0 _K2 K S dK E _K S K dK E S S E S S E S T S T T T

∫

∫

∞ ∧ ∧ ∗ ∧ ∗ ∧ ∗ ∗ − − − − = T S S* > T S S* > T S S* <

11 Note that

[

max(K S ,0)

]

e p(K) E RT T = − ∧

[

max(S K,0)

]

e c(K) E T − = RT ∧ and ) ( 0 E ST F ∧ = then we get ) 8 . 3 ( ) ( 1 ) ( 1 1 ) (ln ₂ 0 2 0 _{e c K dK} K dK K p e K S F S S E S RT S 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 0 0 ln ln ln ln ln ln ln ln ln S S S S S S S S S S S S T T T T ∗ ∗ ∗ ∗ + = − + − = − =

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

) 9 . 3 ( ) (ln ln ) (ln ) (ln ) (ln 0 0 0 ∗ ∧ ∗ ∗ ∧ ∗ ∧ ∧ + = + = S S E S S S S E S S E S S E T T T

12 1 1 2 1 = − and Δ = − − ΔK K K K_n K_n K_n ⎥⎦ ⎤ ⎢⎣ ⎡ ₊ + ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ − = ⎥⎦ ⎤ ⎢⎣ ⎡ _{− − −} − − = ⎥⎦ ⎤ ⎢⎣ ⎡ _{− − −} − + − − = ⎥⎦ ⎤ ⎢⎣ ⎡ _{− − −} − − = − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = − =

∫

∫

∫

∫

∫

∫

∫

∫

∞ ∗ ∗ ∞ ∗ ∗ ∞ ∗ ∗ ∞ ∗ ∗ ∗ ∧ ∗ ∗ ∧ ∗ ∧ ∧ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ dK K c e K dK K p e K T S F T S F T dK K c e K dK K p e K S F T S F T dK K c e K dK K p e K S F T S S S F T dK K c e K dK K p e K S F T S S T S F T S S E T S S T S F T S S E S S T S F T S S E T S F T V E S RT S _RT S RT S RT S RT S RT S RT S _RT T T T ) ( 1 ) ( 1 2 1 2 ln 2 ) ( 1 ) ( 1 1 2 ) ln (ln 2 ) ( 1 ) ( 1 1 2 ) ln ln ln (ln 2 ) ( 1 ) ( 1 1 2 ln 2 ln 2 ) (ln 2 ln 2 ln 2 ) (ln ln 2 ln 2 ) (ln 2 ln 2 ) ( 2 0 2 0 0 2 0 2 0 0 2 0 2 0 0 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0

In conclusion, we get the (3.10), the expected value of the average variance from time 0 to time T , ) 10 . 3 ( ) ( 1 ) ( 1 2 1 2 ln 2 ) ( ₂ 0 2 0 0 ⎥⎦ ⎤ ⎢⎣ ⎡ ₊ + ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ − = _{∗ ∗}

_∫

_∫

∞ ∧ ∗ ∗ dK K c e K dK K p e K T S F T S F T V E S RT S _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 2 n 1 i 2 0 2 i rT i i S RT S RT K Q e K K dK K c e K dK K p e K Δ = +

∫

∑

∫

= ∞ ∗ ∗ Where 1 n i 2 and 2 1 1− _{≤ ≤ −} = Δ i+ i− i K K K in particular,

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

z If ,

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

z If , n n K K K K K₁ < ₂ < ₃ <_L< ₋₁ < * S Ki < * S Ki > * S Ki =

13

T V E( )

∧

z 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 + ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ − =

∑

= ∗ ∗ ∧ ) ( 2 1 2 ln 2 ) ( n ₂ 1 i 0 0 i rT i i K Q e K K T S F T S F T V E or ) 12 . 3 ( ) ( 2 1 2 ln 2 ) ( n ₂ 1 i 0 0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ + ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ − =

∑

= ∗ ∗ ∧ i rT i i_{e Q K} K K S F S F T V E

According to ln N and its the following Maclaurin polynomial

) ) 1 (( ) 1 ( 2 1 ) 1 ( ln _{= − − −} 2 _{+ −} 2 N N N N ο We can get ) ) 1 (( ) 1 ( 2 1 ) 1 ( ln 0 ₌ 0 _{− −} 0 ₋ 2 ₊ 0 ₋ 2 ∗ ∗ ∗ ∗ S F S F S F S F _ο

In the function it can be approximated as follow 2 0 0 0 _{( 1)} 2 1 ) 1 ( ln _∗ = _∗ − − _∗ − S F S F S F or ) 13 . 3 ( ) 1 ( 2 1 ) 1 ( ln 0 ₋ 0 _{− =−} 0 ₋ 2 ∗ ∗ ∗ S F S F S F

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

(3.14) ) ( 2 ) 1 ( ) ( 2 1 2 ln 2 ) ( 2 n 1 i 2 0 2 n 1 i 0 0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ + − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ + ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ − =

∑

∑

= ∗ = ∗ ∗ ∧ i rT i i i rT i i K Q e K K S F K Q e K K S F S F T V E

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 + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = i i RT i i i i RT i i K Q e K K T K F T K Q e K K K F T ) 15 . 3 ( 2 1 1 2 1 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 th

i out-of the-money option;

a call if K_i >K₀ and a put if K_i <K₀l both put and call K_i =K₀

ΔK Interval between strike prices – half the difference between the strike on either side of K :

2 1 1 − + − = Δ i i i K K K

(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) year a in minutes day settlement on the hrs 30 : 08 to me current ti the from 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-next-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:

{

}

_0.03013699 60 24 365 60 24 11 1 _{∗ ∗} = ∗ ∗ = T

{

}

_0.1260274 60 24 365 60 24 46 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) ln 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − − = t t t P C S K t T R Where

K The strike price

t

S The underlying S&P500 price

t

P Put option price

t

C 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 1330 ln 252 8 1 1 ⎥⎦= ⎤ ⎢⎣ ⎡ − = R 0.02367618 25.7 -27.2 + 1332.63 1330 ln 252 33 1 2 ⎥⎦= ⎤ ⎢⎣ ⎡ − = R 0.0287632 )/2 + ( = R R₁ R₂ = 4.1.3 Parameter F:

We can use the following formula to calculate F:

(

Call Price Put Price

)

（4.4） Price

Strike + × −

= RT

e F

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 ( )

(

)

(

)

( ) 1331.101 100 . 1 1330 2 12 3 . 10 2 13 11.5 1330 0.03013699 0.02841502 0.03013699 0.02841502 1 = × + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ₋ + × + = ∗ ∗ e e F 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 ( )

(

) (

)

( ) 1331.204 200 . 1 1330 2 6 . 29 2 . 27 2 7 . 28 7 . 25 1330 0.1260274 0.02841502 0.1260274 0.02841502 2 = × + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − + × + = ∗ ∗ e e F 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 , , K_N , K_{N .} while we defined the set S =φ.

From the above calculation, we know F₁ =1331.101, F₂ =1331.204 , then

330 1 , 330 1 0,2 1 , 0 = F = K . 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.

¾ If K_i <K₀, we choose the strike price K_i of put options, set S=S+K_i

In fact, selecting set S starts with the strike price of put options lower than K₀ 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 0.05

Not Considered

21 1000 0 0.05 Not 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

¾ If K_i > K_0i, we choose the strike price Ki of call options, set S=S+Ki

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 K₀ 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

¾ If K_i =K₀, we put both call option and put option of the strike price K₀ into set S. In conclusion, from the above filter process we can see, when Ki <K₀, only choose the put option with strike price K_i; when Ki > K0_i, only choose the call option with strike price Ki;

when Ki =K₀, choose both put and call option with strike price K0. 4.1.6 Parameter ∆ If K_i∈S, we set 2 1 1 − + − = Δ i i i K K K

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

If K_u 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 5 1020 1025 1020 = − = ΔKstrikeprice= d d d K K K = − Δ ₊₁ 1 − − = ΔK_u K_u K_u

23 4.1.7 Parameter

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

2 )

(K Bid price Ask price

Q 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 ... ... ... ... ...

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

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 T₁ and T₂ respectively.

( )

( )

2 2 0 2 2 2 2 2 2 2 1 0 1 1 2 1 2 1 1 1 2 1 1 2 2 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − Δ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − Δ =

∑

∑

， ， K F T K Q e K K T K F T K Q e K K T i i RT i i i i 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:

(

)

( ) _{0.075 0.00000036075} 1020 5 020 1 002841502 003013699 2 2 1020 1020 1 = × = Δ RT . ∗ . Put Put e Put Q e K K

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

(

)

( ) _{0.325 0.00000077570} 1450 5 Call 450 1 0.028415020.1260274 2 2 1450 1450 2 = × = Δ _∗ e Q e K K 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

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

Average 27.800 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

( )

∑

Δ i i RT i i _{e Q K} K K T 1 2 1 2 0.02206404

∑

Δ

( )

i i RT i i _{e Q K} K K T 2 2 2 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 . 0 1 1330 1331.204 0.1260274 1 1 1 000022740 . 0 1 1330 1331.101 03013699 0 1 1 1 2 2 2 , 0 2 2 2 2 1 , 0 1 1 = ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − K F T . K F T 2 0 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − K F T

27

( )

( )

0.03004075 0000065026 . 0 -0.03004726 1 1 2 0.0220413 000022740 . 0 -0.02206404 1 1 2 2 2 0 2 2 2 2 2 2 2 1 0 1 1 2 1 2 1 2 1 = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − Δ = = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − Δ =

∑

∑

， ， K F T K Q e K K T K F T K Q e K K T i i RT i i i i 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 1 2 1 30 2 2 2 1 2 30 2 2 1 1 100 N N NT NT NT N T NT NT N NT T VIX × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − × = σ σ Where[1], 1

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

NT Number of minutes to settlement of the next-term options 30

N Number of minutes in 30 days (30*24*60=43200) 365

N 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 2

1

σ and 2 2

σ .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 2

1

σ and 2 2

σ .

In conclusion, we get the value of VIX

43200 525600 15840 66240 15840 43200 0.03004075 0.1260274 15840 66240 43200 66240 0.0220413 03013699 0 100 100 30 365 1 2 1 30 2 2 2 1 2 30 2 2 1 1 × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥⎦ ⎤ ⎢⎣ ⎡ − − × × + ⎥⎦ ⎤ ⎢⎣ ⎡ − − × × × = × ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − × = . N N NT NT NT N T NT NT N NT T VIX σ σ

28

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

94

.

16

16.94104

1694104

.

0

100

×

=

≈

=

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 K0 2 σ Apr 5 April 16 11 8 0.03385023 1331.101 1330 0.022041 16.941 May 21 46 33 0.02367618 1331.204 1330 0.030017 Current day Option VIX Settlement day Days Trading days R F K0 2 σ Apr 5 May 21 46 33 0.02367618 1331.203 1330 0.030017 15.056 June 18 74 53 0.02123212 1327.310 1325 0.035231 Current day Option VIX Settlement day Days Trading days R F K0 2 σ Apr 5 June 18 74 53 0.02123212 1327.309 1325 0.035214 14.471 July 16 102 73 0.01878529 1325.754 1325 0.035231

Then acccording to 13 13.5 14 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.

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 <- read.csv("C:/Documents and Settings/Owner/桌 面/毕 设/新 数 据

/4.csv",header = FALSE)

nextdata <- read.csv("C:/Documents and Settings/Owner/桌 面/毕 设/新 数 据

/5.csv",header = FALSE)

#neardata <- read.csv("c:/data.csv",header = FALSE) #nextdata <- read.csv("c:/data2.csv",header = FALSE) cvix<-function(nedata,r,t) { ## Step 1 ## Find Z numo=dim(nedata)[1] Z<-numeric(numo) for(i in 1:numo) {Z[i]=abs((nedata[i,2]+nedata[i,3]-nedata[i,4]-nedata[i,5])/2) } zmin<-min(Z) S=0 #Strike Price

sn=0 #number of Strike Price for(i in 1:numo)

{if(Z[i]==zmin) {S=nedata[i,1] sn<-i}

}

## Give values to r and t #r=0.0038

#t=14/365 ##Find F

F<-S+exp(r*t)*zmin F

36 ##Step 2 ##Ordering ##Find k0 k0=0 for(i in 1:numo) {if(nedata[i,1]<F) k0=nedata[i,1] } k0 ##find call j1=1 j2=0 repeat { if(nedata[j1,1]>=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<-matrix(0,j2,5) j1=1 j2=1 repeat { if(nedata[j2,1]>=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<-matrix(0,j2,5) j1=1 j2=1 repeat { if(nedata[j2,1]<=k0&&nedata[j2,4]!=0) { for(i in 1:5) { p[j1,i]=nedata[j2,i] } j1=j1+1 } if(nedata[j2,1]>k0)

38 {break}

j2=j2+1 }

##Check that if puts is full element p ##Final data fdata<-matrix(0,dim(p)[1]+dim(c)[1]-1,5) for(i in 1:dim(p)[1]) { fdata[i,]<-p[i,] } for(i in 1:dim(c)[1]-1) { fdata[i+dim(p)[1],]<-c[i+1,] } ##Calculate sigma2 ##1.Calculate delta k dk=numeric(dim(fdata)[1]) dk[1]<-(fdata[2,1]-fdata[1,1]) dk[dim(fdata)[1]]<-(fdata[dim(fdata)[1],1]-fdata[dim(fdata)[1]-1,1]) for(i in 2:(dim(fdata)[1]-1)) {dk[i]=(fdata[i+1,1]-fdata[i-1,1])/2 } ##2.Calculate Q(k) qk=numeric(dim(fdata)[1]) for(i in 1:(dim(p)[1]-1)) {qk[i]=(fdata[i,4]+fdata[i,5])/2 } qk[dim(p)[1]]=(fdata[dim(p)[1],2]+fdata[dim(p)[1],3]+fdata[dim(p)[1],4]+fdata[dim( p)[1],5])/4 for(i in (dim(p)[1]+1):dim(fdata)[1]) {qk[i]=(fdata[i,2]+fdata[i,3])/2 }

39 s=0 for(i in 1:dim(fdata)[1]) {s=s+dk[i]*qk[i]/(fdata[i,1]^2) } cont<-numeric(dim(fdata)[1]) for(i in 1:dim(fdata)[1]) {cont[i]=exp(r*t)*dk[i]*qk[i]/(fdata[i,1]^2) } ##4.Sigma2 sigma2<-2*exp(r*t)*s/t-(F/k0-1)^2/t SE<-2*exp(r*t)*s/t list(Use.Data=fdata,Z=Z,Strike.Price=S,k0=k0,F=F,Q.k0=qk,Contribution=cont,Sum =SE,sigma2=sigma2) } cvix(neardata,r,11/365) cvix(nextdata,r,46/365) vix<-function(neardata,nextdata,r,t1,t2) {vix=100*sqrt(((t1*cvix(neardata,r,t1)$sigma2*(t2*365*24*60- 30*24*60)/(t2*365*24*60-t1*365*24*60))+(t2*cvix(nextdata,r,t2)$sigma2*(30*24*60-t1*365*24*60)/(t2*365*24*60-t1*365*24*60)))*(365/30)) vix} vix(neardata,nextdata,r,11/365,46/365) 2. The Calculation Results

> r1=-(1/(8/252))*log(1330/(1332.63+10.3-11.5)) > r1 [1] 0.03385023 > r2=-(1/(33/252))*log(1330/(1332.63+27.2-25.7)) > 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