A TALE OF TWO INDICES.pdf
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1、 A TALE OF TWO INDICES Peter Carr Liuren Wu Working Paper Series Working Paper Series CENTER FOR FINANCIAL ECONOMETRICSCENTER FOR FINANCIAL ECONOMETRICS A Tale of Two Indices PETERCARR Bloomberg L.P. and Courant Institute LIURENWU Zicklin School of Business, Baruch College First draft: October 20, 2
2、003 This version: March 19, 2004 We thank Harvey Stein, and Benjamin Wurzburger for inspiring discussions. We welcome comments, including refer- ences we have inadvertently missed. 499 Park Avenue, New York, NY 10022; tel: (212) 893-5056; fax: (917) 369-5629;. One Bernard Baruch Way, Box B10-254, Ne
3、w York, NY 10010; tel: (646) 312-3509; fax:(646) 312-3451; Liuren Wubaruch.cuny.edu;http:/faculty.baruch.cuny.edu/lwu. A Tale of Two Indices ABSTRACT In 1993, the Chicago Board of Options Exchange (CBOE) introduced the COBE Volatility In- dex (VIX). This index has become the de facto benchmark for s
4、tock market volatility. On Septem- ber 22, 2003, the CBOE revamped the defi nition and calculation of the VIX, and back-calculated the new VIX up to 1990 based on historical option prices. The CBOE is also planning to launch futures and options on the new VIX. In this paper, we describe the major di
5、fferences between the old and the new VIXs, derive the theoretical underpinnings for the two indices, and discuss the practical motivation for the recent switch. We also study the historical behaviors of the two indices. A Tale of Two Indices In1993, theChicagoBoardofOptionsExchange(CBOE)introducedt
6、heCOBEVolatilityIndex(VIX). Thisindexhasbecomethedefactorbenchmarkforstockmarketvolatility. Itiswidelyfollowedandhas been cited in hundreds of news articles in the Wall Street Journal, Barrons and other leading fi nancial publications. The volatility index uses options data on S&P 100 index (OEX) an
7、d computes an average of the Black and Scholes (1973) option implied volatility with strike prices close to the current spot index level and maturities interpolated at about one month. The market often regards this implied volatility measure as a forecast of subsequent realized volatility and also a
8、s an indicator on market stress. On September 22, 2003, following suggestions from the industry,1 CBOE revamped the defi nition and calculation of the VIX, and back-calculated the new VIX up to 1990 based on historical option prices. The new defi nition uses the more actively traded S&P 500 index op
9、tions to replace the S&P 100 index as the underlying index. Furthermore, the new index measures a weighted average of option prices across all strikes at two nearby maturities. Currently, the CBOE keeps track of both volatility indexes and rename the old index as VXO. The CBOE has also been planning
10、 to launch futures and options on the new VIX. In this paper, we describe the major differences in the defi nition and calculation of the two volatility indices. We also derive the theoretical underpinnings for the two defi nitions and discuss the practical motivations for the switch from the old to
11、 the new VIX. Finally, we study the historical behavior of the two volatility indexes. 1. Defi nitions and Calculations 1.1. The old VXO The CBOE renames the old VIX now as VXO and continues to provide the quotes on this index. The calculation of the VXO index is based on options on the S&P 100 inde
12、x (OEX). It is an average of the 1See “Developing the New VIX A Practitioners Tale,” by Sandy Rattray at Goldman, Sachs & Co. Black-Scholes implied volatility quotes on eight near-the-money options at the two nearby maturities. At each maturity, the CBOE picks the two call and two put options that a
13、re closest to the money and average their implied volatility quotes to obtain an estimate of the approximately at-the-money implied volatility at that maturity. Then, the CBOE linearly interpolates between the two at-the-money implied volatility estimates to obtain an at-the-money implied volatility
14、 estimate at the one-month maturity level. The interpolation is based on the number of business days. 1.2. The new VIX The CBOE calculates the new volatility index, VIX, using market prices on the S&P 500 index options. The general formula for the new VIX calculation is 2= 2 T i K K2 i erTP(Ki,T) 1
15、T F K0 1 2 ,(1) where T is the common time to maturity for the all the options involved in this calculation, F is the forward index level derived from the index option prices, Kiis the strike price of the i-th out-of-the- money option in the calculation, P(Ki,T) denotes the midquote price of the out
16、-of-the-money option at strike Ki, K0 is the fi rst strike below the forward index level F, r denotes the riskfree rate of maturity T, and Ki denotes the interval between strike prices, defi ned as Ki= Ki+1Ki 2 .(2) The formula in equation (1) only uses out-of-the-money options. Thus, P(Ki,T) repres
17、ents the call option price when Ki F and the put option price when Ki t that marks to market continuously. No arbitrage implies that there exists a risk-neutral probability measure Q defi ned on a probability space (,F,Q) such that the futures price Ftsolves the following stochastic differential equ
18、ation, dFt/Ft= tdWt+ Z R0 (ex1)(dx,dt)t(x)dxdt,t 0,T,(12) starting at some fi xed and known value F0 0. In equation (12), Ftdenotes the futures price at time t just prior to a jump, R0denotes the real line excluding zero, Wtis a Q standard Brownian motion, and the random measure (dx,dt) counts the n
19、umber of jumps of size exin the asset price at time t. The process t(x),x R0,t 0,T compensates the jump process Jt Rt 0 R R0(ex1)(dx,ds), so that 6 the last term in equation (12) is the increment of a Q-pure jump martingale. The process t(x) must have the following properties (see Prokhorov and Shir
20、yaev (1998), 0(x) = 0,t(0) = 0, Z R0 |x|2 1t(x)dx F0and a put option when K F0). The approximation error is zero when the futures dynamics is purely continuous. When the 7 futures dynamics has a discontinuous component, the approximation error is of order O h dF F 3i and is determined by the compens
21、ator of this discontinuous component, = 2EQ 0 Z T 0 Z R0 ex1x x2 2 t(x)dxdt.(15) We refer the interested readers to Appendix A for the details of the proof. Carr and Madan (1998) and Demeterfi , Derman, Kamal, and Zou (1999a,b) have derived similar relations under the assumption of continuous sample
22、 path for the underlying futures. It is important to note that the return quadratic variation can be written as lnFT,lnFT=2 Z F0 0 1 K2 (KST)+dK+ Z F0 1 K2 (STK)+dK +2 Z T 0 1 Fs 1 F0 dFs 2 Z T 0 Z R0 ex1x x2 2 (dx,ds).(16) Thus, we can replicate the return quadratic variation up to time T by the su
23、m of (i) the payoff from a static position in dK K2 European options on the underlying spot at strike K and expiry T (fi rst line), (ii) the payoff from a dynamic trading strategy holding 2er(Ts) h 1 Fs 1 Ft i futures at time s (second line), and (iii) a higher-order error term induced by the discon
24、tinuity in the futures price dynamics (third line). The options are all out-of-the money forward, i.e., call options when Ft K and put options when K Ft. Taking expectations under measure Q on both sides, we obtain the risk-neutral expected value of the quadratic variation on the left hand side. We
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