VIX, the CBOE Volatility Index

Transcription

1 VIX, the CBOE Volatility Index Ser-Huang Poon September 5, 008 The volatility index compiled by the CBOE (Chicago Board of Option Exchange) has been shown to capture nancial turmoil and produce good volatility forecast of S&P 00 volatility (Fleming, Ostdiek and Whaley (995), Ederington and Guan (000), Blair, Poon and Taylor (00), Hol and Koopman (00)). De nition of V IX According to CBOE, V IX is calculated as the aggregate value of a weighted strip of options using the formula below: vix = T X i K i Ki e rt Q (K i ) T F () K 0 F = K 0 + e rt (c 0 p 0 ) () K i = K i+ K i (3) where r is the continuously compounded risk free interest rate to expiration, T is the time to expiration, F is the forward price of the index calculated using put-call parity in (), K 0 is the rst strike just below F, K i is the strike price of ith out-of-the-money options (i.e. call if K i > F and put if

2 K i < F ), Q (K i ) is the option price at strike price K i, K i in (3) is the interval between strike prices. K i = K i+ K i (or K i = K i K i ). If i is the lowest (or highest ) strike, then Equation () is applied to two sets of options contracts for the nearest term T and the next near term T to derive a constant 30-day volatility index V IX V IX = 00 s T NT N 30 N T N T + T N30 N T N T N T N 365 N 30 (4) where N is the number of minutes (N 30 = 30 ; 400 = 43; 00 and N 365 = 365 ; 400 = 55; 600), T and T are the time to expiration for the two near term contracts. measured in minutes. In the formula used by CBOE (4), T and T are V IX calculated according to (4) is an annualised volatility assuming there are 365 days in a year. When (conditional or realised) volatility is calculated from historical time series data, it is estimated for trading days only where there are price observations. There are typically 5 trading p days in a year. Hence, V IX is about. times (i.e. 365=5) greater than historical annual volatility computed using trading day data. this reason, we need to make adjustment to time series estimate before comparing it with V IX. Assuming that d;t is the estimated daily variance (conditional or realised) for day t, then v u t 365 X d;t = annual (5) t= annual is now in a scale comparable to V IX. For the case of daily volatility, = and we simply multiply the daily estimate by p 365 = 9:05. Note that the option price is a call price if K i > F and a put price if K i < F. The actual option price used in the calculation is the midpoint of the bid and ask quotes. For

3 Note that if d;t is calculated from squared percentage returns (i.e. r t = 00ln (P t /P t )) then annual in (5) is comparable to V IX in (4). If d;t is calculated from squared returns (i.e. r t = ln (P t /P t ) without multiplying by 00) then annual in (5) has to be multiplied by 00 before it can be compared with the V IX in (4) and published by the CBOE. What is the V XO? Prior to September 003, V IX is calculated di erent. di erence between V IX and V XO: There are important (i) V IX uses information from out-of-the-money call and put options of a wide range of strike prices whereas V XO uses eight at- and near-the money options. (ii) V IX is model free whereas V XO is a weighted average of Black- Scholes implied volatility. (iii) V IX is compiled on a real time basis aiming to re ect the volatility over the next 30 calendar days, whereas V XO has a constant 8 calendar days to expiry. (The constant maturity is achieved by combining the rst nearby and second nearby options around the targeted days to maturity). (iv) V IX is based on S&P 500 index options whereas V XO is based on S&P 00 index options. At the time of writing, the old version of V IX, now renamed as V XO, continued to be calculated and released on CBOE web site ( 3

4 V XO, the predecessor of V IX, was released in 993. V XO is an implied volatility composite compiled from eight options written on the S&P 00. Eight option prices are used, including four calls and four puts, to reduce any pricing bias and measurement errors caused by staleness in the recorded index level. Since options written on S&P 00 are American style, a cashdividend adjusted binomial model was used to capture the e ect of early exercise. The mid bid-ask option price is used instead of traded price because transaction prices are subject to bid-ask bounce. (See Whaley (993) and Fleming, Ostdiek and Whaley (995) for further details.) 3 Reason for the change There are many reasons for the change if anything else the new volatility index is hedgeable and the old one is not. V IX can be replicated with a static portfolio of S&P 500 options. Hence, it allows hedging and, more importantly, the corrective arbitraging of V IX derivatives if prices are not correct. The CBOE claims that V IX re ect information in a broader range of options rather than just the few at-the-money options. More importantly, V IX is aiming to capture the information in the volatility skew. It is linked in now to the broader based S&P 500 index instead of the S&P 00 index. The S&P 500 is the primary index for many portfolio benchmarking so derivative products that are closer linked to S&P 500 will facilitate risk management. Although the two volatility indices are compiled very di erently, their statistical properties are very similar. Figure 3.(a) and 3.(b) show the time series plots of V IX and V XO over the period January 990 to 8 June 004, and Figure 3.(c) provides a scatter plot showing the 4

5 Figure : VIX relationship between the two. than the old V XO. V IX has a smaller mean and is more stable References [] Carr Peter and Liuren Wu (006) A tale of two indices, Journal of Derivatives, 3, 3, 3-9. [] CBOE (003) The new CBOE volatility index - VIX, CBOE white paper ( [3] Corrado Charles J. and T.W. Miller (005) The forecast quality of CBOE implied volatility indexes, Journal of Futures Markets, 5,