### Transcription of A Tale of Two Indices - Baruch College

1 **a tale of two** **Indices** T. A. M. PETER CARR AND LIUREN WU. R. FO. Y. N. A. PETER CARR In 1993, the Chicago Board of Options Exchange based on historical option prices. The new def- IN. is the director of the (CBOE) introduced the CBOE Volatility Index. inition uses the S&P 500 index (SPX) to replace Quantitative Finance This index has become the de facto benchmark for the OEX as the underlying stock index. Fur- LE. Research group at stock market volatility. On September 22, 2003, thermore, the new index measures a weighted Bloomberg LP and the C. director of the Masters in the CBOE revamped the definition and calculation average of option prices across all strikes at two TI. Mathematical Finance of the volatility index and back-calculated the new nearby maturities. On March 26, 2004, the index to 1990 based on historical option prices. On CBOE launched a new exchange, the Chicago R. program at the Courant Institute of New York March 26, 2004, the CBOE launched a new Futures Exchange (CFE), to start trading futures University, NY.

2 Exchange, the Chicago Futures Exchange, and A on the new VIX. At the time of writing, IS. started trading futures on the new volatility index. options on the VIX are also planned. TH. LIUREN WU is an Options on the new volatility index are also planned. In this article, we describe the major dif- associate professor of This article describes the major differences between ferences in the definitions and calculations of E. economics and finance at the old and the new volatility indexes, derives the the old and the new volatility indexes. We C. the Zicklin School of theoretical underpinnings for the two indexes, and derive the theoretical underpinnings of the U. Business, **Baruch** **College** , discusses the practical motivations behind the recent two indexes and discuss the practical motiva- D. City University of New switch. It also looks at the historical behavior of the tions for the switch from the old to the new O. York, NY.

3 New volatility index and discusses the pricing of VIX VIX. We also study the historical behavior of R. futures and options. the new volatility index and analyze how it EP. interacts with stock index returns and realized n 1993, the Chicago Board of Options volatilities. Finally, we discuss how to use I. R. Exchange (CBOE) introduced the options on the underlying S&P 500 index to TO. CBOE Volatility Index (VIX). This index define valuation bounds on VIX futures and has become the de factor benchmark for how to exploit information in the underlying L. stock market volatility. The original construc- options market and VIX futures to price A. tion of this volatility index uses options data options on the new VIX. G. on the S&P 100 index (OEX) to compute an LE. average of the Black and Scholes [1973] option DEFINITIONS AND. IL. implied volatility with strike prices close to the CALCULATIONS. current spot index level and maturities inter- IS.

4 Polated at about one month. The market often The Old VXO. regards this implied volatility measure as a fore- IT. cast of subsequent realized volatility and also The CBOE renamed the old VIX the as an indicator of market stress (Whaley [2000]). VXO, and it continues to provide quotes on On September 22, 2003, the CBOE this index. The VXO is based on options on revamped the definition and calculation of the the OEX. It is an average of the Black-Scholes VIX and back-calculated the new VIX to 1990 implied volatilities on eight near-the-money SPRING 2006 THE JOURNAL OF DERIVATIVES 13. Copyright 2006. options at the two nearest maturities. When the time to it no longer comparable to annualized realized volatilities the nearest maturity is within eight calendar days, the computed from index returns. Thus, the VXO compu- next two nearest maturities are used instead. tation methodology has drawn criticism from both acad- At each maturity, the CBOE chooses two call and emia and industry for its artificially induced upward bias.

5 Two put options at the two strike prices that straddle the spot level and are nearest to it. The CBOE first averages The New VIX. the two implied volatilities from the put and call at each strike price and then linearly interpolates between the two In contrast to the old VXO, which is based on near- average implied volatilities at the two strike prices to obtain the-money Black-Scholes implied volatilities of OEX. the at-the-money spot implied volatility. The interpolated options, the CBOE calculates the new volatility index at-the-money implied volatilities at the two maturities are VIX using market prices instead of implied volatilities. It further interpolated along the maturity dimension to create also uses SPX options instead of OEX options. The gen- a 22-trading-day volatility, which constitutes the VXO. eral formula for the new VIX calculation at time t is The Black-Scholes implied volatility is the annual- 2.

6 K i rt (T t ) Ft . VS(t ,T ) = Ot (K i ,T ) . 2 1. ized volatility that equates the Black-Scholes formula value to the options market quote. The annualization is based T t . i K i2. e T t . K. 0. 1 .. (4). on an actual/365 day-counting convention. Instead of using this implied volatility directly, the CBOE intro- where T is the common expiry date for all of the options duced an artificial trading-day conversion into the cal- involved in this calculation, Ft is the time-t forward index culation of the VXO. Specifically, let ATMV (t, T ) denote level derived from coterminal index option prices, Ki is the time-t Black-Scholes at-the-money implied volatility the strike price of the i-th out-of-the-money option in the as an annualized percentage with expiry date T. The calculation, Ot(Ki,T ) denotes the time-t midquote price CBOE converts this percentage to trading-day volatility of the out-of-the-money option at strike Ki, K0 is the first TV (t, T ) as strike below the forward index level Ft, rt denotes the time- t risk-free rate with maturity T, and DKi denotes the interval TV (t ,T ) = ATMV (t ,T ) NC / NT (1) between strike prices, defined as DKi 5 (Ki11 2 Ki)/2.

7 For notational clarity, we suppress the dependence of rt and Ft where NC and NT are the number of actual calendar days on the maturity date T as no confusion will result. and the number of trading days between time t and the Equation (4) uses only out-of-the-money options option expiry date T, respectively. The CBOE converts except at K0, where Ot(K0,T ) represents the average of the the number of calendar days into the number of trading call and put option prices at this strike. Since K0 # Ft, days according to the following formula: the average at K0 implies that the CBOE uses one unit of the in-the-money call at K0. The last term in Equation NT 5 NC 2 2 3 int(NC/7) (2) (4) represents the adjustment needed to convert this in- the-money call into an out-of-the-money put using put- The VXO represents an interpolated trading-day volatility call parity. at 22 trading days based on the two trading-day volatili- The calculation involves all available call options at ties at the two nearest maturities (TV (t,T1) and TV (t,T2)): strikes greater than Ft and all put options at strikes lower than Ft.

8 The bids of these options must be strictly posi- NT 2 22 22 NT1. VXOt = TV (t ,T1 ) + TV (t ,T 2 ) (3) tive to be included. At the extreme strikes of the avail- NT 2 NT1 NT 2 NT1 able options, the definition for the interval DK is modified as follows: DK for the lowest strike is the difference where NT1 and NT2 denote the number of trading days between the lowest strike and the next lowest strike. Like- between time t and the two option expiry dates T1 and wise, DK for the highest strike is the difference between T2, respectively. the highest strike and the next highest strike. Since each month has around 22 trading days, the To determine the forward index level Ft, the CBOE. VXO represents a one-month at-the-money implied chooses a pair of put and call options with prices that are volatility estimate. Nevertheless, the trading-day conver- closest to each other. Then, the forward price is derived sion in Equation (1) raises the level of the VXO and makes via the put-call parity relation: 14 **a tale of two** **Indices** SPRING 2006.

9 Copyright 2006. Ft 5 ert(T2t)(Ct(K,T) 2 Pt(K,T)) 1 K (5) model. They show under general market settings that the time-t at-the-money implied volatility with expiry at time T. The CBOE uses Equation (4) to calculate VS(t,T ) represents an accurate approximation of the conditional at two of the nearest maturities of the available options, risk-neutral expectation of the return volatility during T1 and T2. Then, the CBOE interpolates between VS(t,T1) the time period [t, T ]: and VS(t,T2)to obtain a VS(t,T) estimate at 30 days to ATMV(t,T) EtQ [RV olt,T] (7). maturity. The VIX represents an annualized volatility per- centage of this 30-day VS, using an actual/365 day- where EtQ [.] denotes the expectation operator under the counting convention: risk-neutral measure Q conditional on time-t filtration Ft 365 NC 2 30 30 NC 1 . , and RVolt,T denotes the realized return volatility VIX t = 100 (T1 t )VS (t ,T1 ) + (T 2 t )VS (t ,T 2 ) (6) in annualized percentages over the time horizon [t, T ].

10 30 NC 2 NC 1 NC 2 NC 1 . Appendix A details the underlying assumptions and deriva- where NC1 and NC2 denote the number of actual days to tions for this approximation. expiration for the two maturities. When the nearest time The result in Equation (7) assigns new economic to maturity is 8 days or fewer, the CBOE switches to the meanings to the VXO, which approximates the volatility next-nearest maturity in order to avoid microstructure swap rate with a one-month maturity, if we readjust the effects. The annualization in Equation (6) follows the upward bias induced by the trading-day conversion. actual/365 day-counting convention and does not suffer Volatility swap contracts are traded actively over the from the artificial upward bias incurred in the VXO counter on major currencies and some equity indexes. calculation. At maturity, the long side of the volatility swap contract receives the realized return volatility and pays a fixed ECONOMIC AND THEORETICAL volatility rate, which is the volatility swap rate.