Guide to the CBOE / CBOT 10 Year Treasury Note Volatility Index (TYVIX SM Index) Part I: Introduction to the TYVIX Index

Transcription

1 Guide to the CBOE / CBOT 1 Year Treasury Note Volatility Index ( SM Index) Part I: Introduction to the Index

2 1 Table of Contents I. Introduction to the Index II. III. IV. Index Calculation Conversion to TYVOL, a Yield Volatility Salient Features 1. Descriptive Statistics of and VIX 2. Empirical Properties of Time Series

3 1 Year Trea sury Yiel d 1/2/3 1/2/4 1/2/5 1/2/6 1/2/7 1/2/8 1/2/9 1/2/1 1/2/11 1/2/12 1/2/13 1/2/14 2 I. Introduction to the SM Index This is the first module of CBOE s Handbook to futures. It describes the calculation and salient features of the CBOE/CBOT 1-Year U.S. Treasury Note Volatility Index ( SM ). The second module, called Futures Primer, describes futures and their basic empirical properties, and the third module, called Compendium of Empirical Findings presents additional empirical facts about the index and the futures. The Index in a Nutshell The is the first exchange-traded volatility benchmark for U.S. Treasuries. Similar to the CBOE Volatility Index (VIX Index), measures expected percentage changes in its underlying over a one-month period. The underlying is CBOT futures on 1-year Treasury Notes (ticker TY). TY futures are the most actively traded U.S. Treasury futures, and their volatility is aligned with the volatility of a variety of fixed income assets, such as spot Treasuries, interest rate swaps, mortgage-backed securities, and corporate bonds. is calculated every 15 seconds from 7: am to 3:15 pm Central time and is disseminated to data vendors under the ticker. Figure 1. and the 1-Year Treasury Yield, Year Treasury Yield 6% 5% 4% 3% 2% 1% % Source: CBOE, Federal Reserve Board

4 3 The historical series of from 23 to 214 (Figure 1) shows that the index reacts strongly to sudden and extreme variations in the 1-year constant maturity Treasury yield. II. Index Calculation is an estimate of the expected 3-day volatility of CBOT Ten-Year Treasury futures (TY), and, by extension, of the volatility of Ten-Year Treasury Notes. Like the VIX, is model independent. It applies the VIX methodology to the market prices of OZN options on TY futures 1. The times series of daily values as well as the term structure may be found at The VIX formula which forms the basis of the calculation is: 365 K rτ Ki = 1 * [2e { Pr icek + i 3 K K K i F } ( K j Pr ice 2 2 K j i K j j ATM 1) 2 ] summarized as 365 = 1 * VAR, the square root of annualized variance 3 expressed in percentage points. This formula has its roots in theoretical results of Neuberger (199) and Carr and Madan (22) that show that the N-day variance of an asset (VAR) is approximated by the forward value of a portfolio of at- and out-of the-money N-day options on the asset. 1 The formula is consistent with a random spot rate of interest. Jamshidian (1989) has shown that when the spot rate of interest is random, the prices of related bond futures and options can be interpreted as the discounted expected value of their final payoff, with the nuance that the expectation for each expiration is now taken under a different probability measure called the forward measure for that expiration. The forward measure is a transformation of the risk-neutral measure. Following Jamshidian, Mele and Obayashi (212) have expressed the price of bond variance in terms of bond option prices. This is the basis of the formula. There are two other nuances that don t affect the essential information conveyed by XTYN. First, the formula strictly holds for European-style options and CBOT Treasury options have American-style exercise. The effect of ignoring the early exercise premium on volatility is likely to be small because the Treasury options used to calculate the index are out-of-themoney and short-dated. Second, is interpolated from adjacent futures to arrive at a 3 day measure, and each futures expiration is associated with its own forward measure.

5 4 In Equation (1): The variable Price Ki denotes mid-quotes of out-of-the-money puts and calls with strikes K i and K j respectively,and the average of the mid-quotes of the at-the-money put and call. K 2 K 2, where K is the strike interval associated with K, is the number of options at strike K included in the portfolio. e rτ is the interest rate factor that converts the price of each option to a forward value. F 2 The term ε = ( 1) under the square root compensates for the error K ATM introduced by the substitution of K ATM, the strike deemed at-the-money, for the forward price. The substitution is necessary because there is usually no listed strike equal to the forward price. 365 The factor converts the 3-day variance into an annualized variance, 3 and is defined as the square root of this annualized variance multiplied by 1 to express it in percentage points. Looking at the formula, an immediate question arises: what should one do if there are no OZN options expiring in 3 days? This question is answered in the following numerical example which describes how the formula is implemented step by step. Numerical Example of Calculation Step1: Find OZN Contract Months At 3:15 p.m. CT on 11/1/214, the closest to 3-day options listed were CME OZN options expiring at 4: p.m. on 11/21/214 and on 12/26/214. Note that the CME calls these December 214 and January 215 options. They settle to December 214 TY and March 215 TY futures respectively. A 3-day variance will be interpolated from at-and-out-of-the money OZN December 214 and January 215 options.

6 5 Step 2. Exclusion of Illiquid OZN Strikes Next, download bid and ask quotes of OZN Dec 214 and Jan 215 options from Bloomberg. On Bloomberg, the quotes are sorted in ascending strike order as shown in Table 1: Table 1.3:15 p.m. 11/1/214, Dec 214 and Jan 215 OZN options Dec 214 OZN Options Jan 215 OZN Options Strike Price call bid call ask put bid put ask strike call bid call ask put bid put ask n.a. n.a n.a. n.a n.a. n.a n.a. n.a n.a. n.a n.a. n.a n.a. n.a n.a. n.a n.a. n.a. Sources: Bloomberg, CBOE Options that are zero-bid are deemed illiquid and are excluded from the calculation. Strikes with bids of.1 correspond to cabinet trades and are also excluded. Excluded strikes are grayed out in Table 1, leaving a range of to 129 for December 214 OZN options and 12.5 to for January 215 OZN options. Note that as volatility rises and falls, the strike price range of options with positive bids tends to expand and contract. As a result, the number of options

7 6 used in the calculation may vary from month-to-month, day-to-day and possibly, even minute-to-minute. Step 3. Determination of at- and out-of-the-money strikes. Table 2 shows mid-quotes for all December 214 and January 215 OZN options that were not screened out in Table 1. Table 2. Midquotes of OZN options Dec 214 OZN Jan 215 OZN Strike Price Call Put Difference Strike Call Put Difference n.a n.a n.a n.a n.a n.a n.a n.a n.a. Sources: Bloomberg, CBOE The theoretical at-the-money strike is the forward price, and the difference between call and put prices at this strike is equal to zero. In Table 2, we see there is no strike at which this difference is zero. The forward price must therefore be estimated by applying put call parity: F = Strike Price + e rtau * (Call Price - Put Price)

8 7 Put call parity holds at any strike but a more robust estimate of the forward price results if one uses the strike where the absolute difference between the call and put price is smallest. The Dec 214 strike with the closest put and call is (difference =-.2344), highlighted in yellow, and for Jan 215 options it is (.5469), highlighted in orange. To apply put-call parity, find r, the rate of interest to the expiration of the option, and tau, the time to expiration in year units. The rate of interest r is found by cubic spline interpolation of Treasury bill rates published at On 11/1/214, the rates applied to Dec 214 and Nov 215 OZN options were.444 and.35 respectively. The number of minutes in a year is 525,6. Hence tau, or time to expiration in year units was: December 214 OZN: = 15,885 / 525,6 January 215 OZN: = 66,285 / 525,6 Substitute these values in the put-call parity formula to obtain: F1 = exp(.444 x.32263) *( ) = F2 = exp(.35 x ) x ( ) = The resulting OZN option strikes to use in the calculation are OTM Puts ATM Call & Put OTM Calls Dec 214 OZN Jan 215 OZN All puts with strikes smaller than the at-the-money strike are included in the calculation, as well as all calls with strikes greater than the at-the-money strike. Both at-the-money put and call are included.

9 8 Step 4. Calculation of Variances from OZN Options For each OZN expiration, Table 3 stacks the puts over the calls centered around the ATM strikes which are highlighted in green. Table 3. Variances from OZN Dec 214 and Jan 215 Options Dec 214 OZN Options Strike Price Number Options Option Price Forward value Portfolio N=2* Delta K/ K^2 P = Mid-Quote e rtau *N * P Weight E E % E % E E % E E % E E % E E % E E-6 6.3% E E % E E % E E % Sum E-5 1.% Epsilon E-6 Variance E-5 Jan 215 OZN Options Strike PriceNumber Options Option Price Forward value oportfolio N=2* Delta K/ K^2 Mid-Quote e rtau *N * P Weight E E-7.48% E E-7.48% E E-6.79% E E-6 1.9% E E % E E % E E % E E-6 5.6% E E % E % E E % E E % E E % E E % E E % E E % E E % E E % E E % E E % E E-6.96% E E-6.68% E E-6.67% E E-7.4% E E-7.4% E E-7.39% E E-7.39% Sum % Epsilon E-7 Variance

10 9 From left to right, Table 3 columns contain: 1. The strike of the put and call K 2. 2 K 2, the number of options included at each strike, where the strike interval K is set as half the distance between adjacent strikes, or the distance from the next strike for end strikes. The strike interval is usually equal to.5 at all OZN strikes. The number of options at the strike is 2*.5/ = E-5 OZN Dec 14 OZN puts 3.The mid-quotes at each strike. The mid-quote at the ATM strike is the average of the put and call mid-quotes. Thus, for Dec 214 OZN options:.4625 = average (.54688, ) 4.The forward value of the option at each strike. For example, the forward value of the price of the OZN Dec 214 call is E-6 = exp(.444 x.32263)* E-5* The portfolio weight of the forward value of the option at each strike. For each OZN expiration,the sum of the forward values across strikes is in the bottom row labeled sum. The variance is found by substracting the adjustment factor epsilon = (F/K ATM - 1) 2 from this sum: Epsilon 1 = ( /126-1) 2 = E-6 Epsilon 2 = ( / )2 =1.8991E-7 Variance from December 214 OZN Options VAR 1 = E E-6 = E-5 Variance from January 215 OZN Options VAR 2 = E-7=

11 1 Step 5: Interpolation of a 3-Day Variance and Determination of The last step to is to interpolate a 3-day variance from December 214 and January 215 OZN variances, annualize the variance, take the square root and multiply by 1 to express it in percentage points: 365 = 1 * ( wvar1 + (1 w) VAR2 ) 3 VAR1 is the variance of December 214 futures from 11/1/214 to 11/21/214 and VAR2 is the variance of January 215 TY futures from 11/1/214 to 12/26/214. The weights are w and 1-w. For the purpose of the calculation, time is recorded in minutes. Thirty days = 3*24 *6 mn = 43,2 mn variance. The December 214 OZN options have 15,885 mn to expiration ( , = 15,885) and the January 214 OZN options have 66,285 mn to expiration ( ,8=66,285). Hence, 66,285 43,2 w = = ,285 15,885 1 w =.542 Substitute the weights and variances in the formula : 365 T YVIX = 1* (.458* E * ) 3 T YVIX = 5.11 Conversion of to TYVOL, a Yield Volatility In the fixed income market, there are two conventions for calculating and quoting volatility. The first expresses volatility as the variation of a rate of interest or yield and quotes it in basis points. The second expresses volatility as the variation of

12 12/19/ 26 5/19/2 7 11/19/ 29 4/19/2 1 9/19/2 1 2/19/2 11 7/19/ /19/ 211 5/19/2 12 1/19/ 212 3/19/2 13 8/19/2 13 1/19/2 14 6/19/ /19/ 214 1/19/ 212 3/19/2 13 8/19/2 13 1/19/2 14 6/19/ /19/ the rate of return of the asset and quotes it in percentage points. VIX belongs to this second category, as does. To facilitate the comparison between and yield volatilities, CBOE converts the Index to a yield volatility index called Ten-Year Treasury Yield Volatility Index (TYVOL SM ). Figure 2 shows TYVOL, and the Merrill Option Volatility Expectations Index (MOVE index). The MOVE index is a measure of the average yield volatility of 2, 5, 1, and 2 year Treasury bonds. The derivation of TYVOL is described in a separate note titled Conversion of the Index SM to the Ten-Year Treasury Yield Volatility Index (TYVOL SM Index). Figure 2 TYVOL, MOVE, and TYVOL MOVE TYVOL /19/ 27 3/19/2 8 8/19/2 8 1/19/2 9 6/19/2 9 12/19/ 26 5/19/2 7 1/19/ 27 3/19/2 8 8/19/2 8 1/19/2 9 6/19/2 9 11/19/ 29 4/19/2 1 9/19/2 1 2/19/2 11 7/19/ /19/ 211 5/19/2 12 Source: CBOE, Bloomberg IV. Salient Features of 1. Descriptive Statistics, and VIX Treasuries are reputed to be less volatile than stocks. To confirm, the frequency distributions and standard statistics for and VIX are reported in Figure 4. In the histogram on the left, the distribution of is located to the left of the distribution of VIX. It is also more compact, less skewed and less kurtotic than VIX. Standard descriptive statistics in the top right panel of the table tell the same story in numbers.

13 1/2/2 3 1/2/2 4 1/2/2 5 1/2/2 6 1/2/2 7 1/2/2 8 1/2/2 9 1/2/2 1 1/2/2 11 1/2/2 12 1/2/2 13 1/2/2 14 VIX 12 Figure 4. Historical Range & Behavior of and VIX Values Source: CBOE Futures traders are especially interested in the volatility of the underlying asset. The bottom of the right panel of Figure 4 shows statistics for the absolute daily log-return of the index. By this measure, the volatility of is comparable to that of VIX. Figure 5. and VIX VIX Source: CBOE Continuing with our comparison, estimates the volatility of 1-year Treasury Notes, while VIX estimates the volatility of the S&P 5 Index. Since these two asset classes are impacted by different factors, one would not expect their volatilities to behave in the same way. And as shown in Figure 5, they do not. There are marked differences even during the financial crisis of 28. and VIX both peak during the third quarter, but starts its ascent

14 13 earlier and retreats more gradually. Out of global crisis mode, the paths of and VIX diverge even more. 2. Empirical Properties of Time Series a. Mean Reversion of A hallmark of volatility is that it is range-bound. The upper and lower bounds of the range act as reflecting barriers which lend the index the appearance of reverting to a mean. To explore the mean-reverting tendencies of, we follow the six-month path of conditional on its initial value. Each panel of Figure 6 shows future ranges of from one week to six months out, starting from a different range of initial values. For example, in the first panel, initial values range from 4 to 7. The blue lines represent the initial and future ranges of, from one week to 6 months later. The black rectangles mark the average and median values of from one week to six months later. Figure 6. Six-Month Conditional Volatility Cones of, Initial Range Initial Range Initial Range 11 to 14 2 Initial Range 14 to Spot 1w 2w 1m 2m 3m 4m 5m 6m Spot 1w 2w 1m 2m 3m 4m 5m 6m Spot 1w 2w 1m 2m 3m 4m 5m 6m Spot 1w 2w 1m 2m 3m 4m 5m 6m Source: CBOE Based on these historical six-month patterns, the average and median values of persist for as long as six months when the current value of is at or below its long-term average of 7, and they gradually decrease when is above its long-term average. On average, and this is also borne out by the evolution of the range of ; mean-reversion appears stronger when is at a historically high level and less so when is at a historically low level. b. Volatility Premium A key attraction of volatility as an asset is its explosive behavior during tail events that impact the rate of return of the underlying asset. Investors pay a risk

15 1/19/ 7 6/19/ 7 11/19/ 7 4/19/ 8 9/19/ 8 2/19/ 9 7/19/ 9 12/19/ 9 5/19/1 1/19/ 1 3/19/1 1 8/19/1 1 1/19/1 2 6/19/1 2 11/19/ 12 4/19/1 3 9/19/1 3 2/19/ premium for the natural hedge that volatility provides against adverse returns. The risk premium is embedded in the spread between expected and realized volatility, but is sometimes overshadowed by large forecasting errors. If forecasting errors are unbiased, the average spread should be positive. In Figure 7, we see that indeed tends to be greater than the realized volatility of futures on 1-year Treasury notes. Exceptions occur more frequently in the period surrounding the 28 credit crisis when the market significantly underestimated future volatility. Figure 7. Implied vs. Realized Risk Premium 2 15 Realized Risk Premium Source: CBOE c. What Moves? is highly sensitive to announced changes in monetary policy and to macroeconomic events that affect interest rates. The broad impact of monetary policy is captured in Figure 8, where is overlaid on the Federal Fund Target rate. steadily decreases during the slow ramp up of the Federal Fund Target rate from 24 to 26, but it increases more rapidly when the Federal Fund Target rate starts to cascade down in late 27. Once the rate hits bottom, reacts to different events and news such as talk about tapering that may lift the rate.

16 2-Jan-3 2-Jan-4 2-Jan-5 2-Jan-6 2-Jan-7 2-Jan-8 2-Jan-9 2-Jan-1 2-Jan-11 2-Jan-12 2-Jan-13 2-Jan-14 1/8/2 4 1/8/2 5 1/8/2 6 1/8/2 7 1/8/2 8 1/8/2 9 1/8/2 1 1/8/2 11 1/8/2 12 1/8/2 13 1/8/2 14 Fixed Income Rates 15 Figure 8. Time Series vs. Federal Fund Target Rate Federal Funds Target Rate Z IRP Flash Crash Operation Twist Taper Talk Source: CBOE, Bloomberg Figure 9 illustrates that the joint impact of the Federal Reserve s QE programs, uncertainty about future Federal Reserve policy, and macro-crises, such as Standard & Poor s downgrade of U.S. debt in August 211, have made longer term yields jumpier than in the period preceding the 28 credit crisis. has followed suit and has become more sensitive to rate fluctuations. Figure 9. and Changes in Select Rates of Interest 3-YR Conventional Mortgage Rate 1Y Swap Rate AAA Corp Yield Source: CBOE and Bloomberg

17 16 Futures trading is not suitable for all investors, and involves risk of loss. Options involve risk and are not suitable for all investors. Prior to buying or selling an option, a person must receive a copy of Characteristics and Risks of Standardized Options. Copies are available from your broker, by calling OPTIONS, or from The Options Clearing Corporation at The information in this document is provided solely for general education and information purposes, and is not intended to provide, and should not be relied on for financial or legal advice. No statement within this document should be construed as a recommendation to buy or sell a future or security or to provide investment advice. It is not possible to invest directly in an index. Past performance does not guarantee future results. Supporting documentation for any claims, comparisons, statistics or other technical data in this document is available from CBOE upon request. Visit for more information. The CBOE/CBOT 1-Year U.S. Treasury Note Volatility Index ( SM Index), CBOE Ten-Year Treasury Yield Volatility Index (TYVOL SM index) and CBOE Volatility Index (VIX index) and all other information provided by Chicago Board Options Exchange, Incorporated (CBOE) and its affiliates and their respective directors, officers, employees, agents, representatives and third party providers of information (the Parties ) in connection with the, TYVOL and VIX Indexes (collectively Data ) are presented "as is" and without representations or warranties of any kind. The Parties shall not be liable for loss or damage, direct, indirect or consequential, arising from any use of the Data or action taken in reliance upon the Data. This document contains comparisons regarding the performance of indexes based on backtesting, i.e., calculations of how the indexes might have performed in the past if they had existed. Backtested performance information is purely hypothetical and is provided in this document solely for informational purposes. Backtested performance does not represent actual performance, and should not be interpreted as an indication of actual performance. The VIX index methodology is the property of CBOE. CBOE, CBOE Volatility Index, Execute Success and VIX are registered trademarks and CBOE Ten-Year Treasury Yield Volatility Index, TYVOL and are service marks of CBOE. CBOT is a trademark of CME Group, Inc. (CME). CBOE has, with the permission of CME, used such trademark in the CBOE/CBOT 1 Year U.S. T-Note Volatility Index. CME makes no representation regarding the advisability of investing in any investment product that is based on such index. Standard & Poor's, S&P and S&P 5 are registered trademarks of Standard & Poor's Financial Services, LLC and have been licensed for use by CBOE. Financial products based on S&P indices are not sponsored, endorsed, sold or promoted by Standard & Poor s, and Standard & Poor s makes no representation regarding the advisability of investing in such products. MOVE is a registered trademark of Bank of America Corporation..Redistribution, reproduction and/or photocopying in whole or in part are prohibited without the written permission of CBOE. Copyright 215. All rights reserved.