Similarities and Differences of the Volatility Smiles of Euro- Bund and 10-year T-Note Futures Options

Transcription

1 Similarities and Differences of the Volatility Smiles of Euro- Bund and 10-year T-Note Futures Options Shengxiong Wu Kent State University and Dr. Axel Vischer Eurex This version: November, 2009

2 Abstract This paper compares the behavior of the volatility series implied by options using daily settlement prices of Euro-Bund and 10-year T-Note futures options from January 2 nd, 2002 to October 3 rd, We examine the statistical similarities and differences of the volatility smiles of the two products in terms of several descriptive observables, such as 30-day ATM (at the money) implied volatility, 30-day ATM implied volatility skewness and kurtosis. We find that in general the volatilities of the two products move together over time in terms of level, skewness, and kurtosis. However, the 30-day ATM implied volatility of Euro-Bund futures options is on average lower than that of 10-year T-Note futures options. We also find that the difference of the two volatility series is negatively correlated to the difference of short-term interest rates in the U.S. and Euro-Zone, but positively correlated to the difference of term structure spreads in the U.S. and Germany over time. This is consistent with the notion that from an investor s point of view price changes of bonds with different maturities and coupons depend not only on the expected change in future interest rate but also on the uncertainty of expected movements. 1

3 I Introduction The Black-Scholes (1973) option pricing model for futures assumes constant volatility of the underlying asset up to option maturity. However, this is not what is observed in the option markets. It is widely agreed today that there is volatility smile/skewness across strikes and term structure across time to maturity. Thus, the knowledge of implied volatility has become indispensable to option market participants. Especially, understanding the statistical characteristics of volatility smiles/skewness over time is very important to all option market participants because these characteristics can be used to enhance risk management in either option or underlying asset markets. This paper uses the daily settlement price data to compare the volatility smiles of two similar financial derivatives products, Euro-Bund and 10-year T-Note futures options in terms of several major descriptive observables, such as 30-day ATM implied volatility, 30-day ATM implied volatility skewness, and 30-day ATM implied volatility kurtosis. The comparison is very important for those traders who are not familiar with the Euro-Bund futures options market, but are actively trading in 10- year T-Note futures options. If the two products are similar in terms of these descriptive statistics, traders could in principle use the results to trade the two option classes in one book. If the two products are different, traders could use the findings to adjust their model to accommodate for the differences in statistical behaviors. In general, the paper observes the statistical similarities and differences of these two similar interest rate derivatives products. Basically, we find that 30-day ATM implied volatility of Euro-Bund and 10-year T-Note futures options move together over time. The correlation between them is about 83% during the sample period. 30-day ATM implied volatility skewness and kurtosis also exhibit similar patterns over time. Moreover, the skewness and kurtosis of 30-day ATM implied volatility of the two products on average behave very similarly as a function of days to expiration. Economically speaking, the 30-day ATM implied volatility skewness moves as a function of one over the squared root of time to expiration while the 30-day ATM implied volatility kurtosis behaves as a function of one over time to expiration. However, there are differences. First, on average the 30-day ATM implied volatility of Euro-Bund futures options is lower than that of 10-year T-Notes futures options. Second, the 30-day ATM implied volatility of Euro-Bund futures options is less 2

4 skewed than that of 10-year T-Note futures options. Third, the Euro-Bund futures options have less curvature or kurtosis than 10-year T-Note futures options. The differences are further explored. We define the implied volatility spread as 30-day ATM implied volatility of 10-year T-Note futures option minus that of Euro-Bund futures option, and interest rate spread as the effective Fed Funds Rate minus the European Overnight Index Average (EONIA). We also define the term structure spread as the difference between the yields of 10-year and 3-month constant maturity government securities, and the difference of the term structure spread as the difference between the term structure spread in the U.S. and in Germany. We find that there is a negative relationship between implied volatility spreads and interest rate spreads, while there is a positive relationship between implied volatility spread and the difference between the two term structure spreads. The paper is organized as follows. The first section illustrates the contract specifications of the two interest rate derivatives products. In addition we explore whether embedded options in interest rate futures will affect futures option values through the delivery process. The second section discusses the models used in interpolating the time series of implied volatility and implied volatility estimates. The third section presents the empirical results with regard to statistical similarities and differences in volatility smiles of the two products and explores the economic reasons underlying the differences. II. Characteristics of Futures and Futures Options Contracts In this section, we first review the contract specifications of Euro-Bund and 10-year T-Note futures and futures options. Afterwards we compare the delivery process for the underlying futures of the two options. In general, we can observe that options on Euro-Bund and 10-year T-Note futures are very similar. Any embedded delivery options, such as quality and timing options, do not significantly affect the implied volatility of two interest rate futures options Futures and Futures Options Contract Specifications Table 1 illustrates the contract specifications of the two futures analyzed in this paper. One Euro-Bund futures contract is for the delivery of 100,000 of German 3

5 government securities with a remaining maturity between 8.5 and 10.5 years. The prices of Euro-Bund futures are quoted in percent of the nominal value of the notional bond to three decimal places. The minimum price movement is 0.01 percent, a tick or one basis point. The corresponding value of a tick is 10 for Euro-Bund futures. The Euro-Bund futures contract months listed are always the three nearest quarterly months of the March, June, September, and December cycle. And the regular trading hours for Euro-Bund futures are between 8:00 and 22:00 Central European Time (CET). The daily settlement price for the front-month futures is derived from the volume-weighted average of the prices of all transactions during the minute before 17:15 CET, provided that more than five trades are transacted within this period. The daily settlement price for a back-month contract is determined based on the average bid-ask spread of the combination order book. Similarly, 10-year T-Note futures are deliverable into $100,000 of any U.S. Treasury Notes with a remaining maturity in the range of 6.5 to 10 years. The prices are quoted in one half of 1/32 nd of a point. Thus, the minimum price increment is 1/64 of a point, equivalent to a dollar value of $ The 10-year T-Note futures contract months listed are the five nearest quarterly months of the March, June, September, and December cycle. The regular trading hours are from 5:30 p.m. on day t-1 to 4:00 p.m. on day t Eastern Standard Time (EST). The daily futures settlement price for the front-month on day t is derived from the volume-weighted average of the prices of all trades on Globex during the period between 14:59:30 and 15:00:00 EST. For the remaining maturities daily settlement prices are determined based on the trades or the quoted bid-ask spread in the book. Table 1 Contract Specifications of Interest Rate Futures 10-year T-Note Futures Euro-Bund Futures Remaining Term of 6.5 to to 10.5 Years Coupon Percent 6 6 Regular Trading Hours 17:30-16:00 EST 8:00-22:00 CET Price Quotation halves of 1/32 of a point In percent of par with three decimal places Tick (%) 1/ Tick (Euro) $ Contract Months The five nearest quarterly months of the March, June, The three nearest quarterly months of the March, June, September, and December 4

6 September, and cycle. December cycle. Size of Contract $100, ,000 Last Trading Day Seventh business day preceding the delivery day of the delivery month Last Delivery day Last business day of the delivery month Daily Settlement Time 17:15 CET 15:00 EST Two business days preceding the delivery day of the delivery month The tenth calendar day of the respective quarterly month, if is an exchange day; otherwise, the succeeding exchange day. Table 2 presents contact specifications of Euro-Bund and 10-year T-Note futures options. Both futures options are American-style. Buying a futures option contract entitles the option holder the right, but not the obligation, to either buy or sell one contract of the underlying futures at a predetermined price on any exchange trading day throughout the lifetime of the option. For example, if the buyer of a contract of the Euro-Bund futures call option exercises her right, she will buy the Euro-Bund futures with specific maturity at the exercise price of the option, resulting in the opening of a long position of the underlying Euro-Bund futures. Both futures option contracts also have the same tick size as the underlying futures contract as well as the same increment for the exercise price. The margining process for both contracts shows some essential differences. Euro- Bund futures options are margined futures style. The full premium is paid by option holders when the option is exercised. A buyer of an option would pay the premium at exercise or at the expiration date of the option. Any change in contract price during the lifetime of the option is marked-to-market through a variation margin, just as with futures. In contrast, the margin required for the purchase of one 10-year T-Note futures option contract is equity-style, i.e. the option buyer has to pay the option premium upfront. Going forward, we will find that the difference in margining is crucial to understand the different shapes of the early exercise boundaries. Regular trading hours for 10-year T-Note futures options are aligned with the corresponding futures contracts starting from 5:30 p.m. on day t-1 to 4:00 p.m. on day t EST. In contrast, Euro-Bund options close earlier than their underlying futures, 5

7 opening at 8:00 a.m. CET and closing at 7:00 p.m. CET instead of the 10 pm CET close of the futures. Eurex offers the first three consecutive calendar months for a given trading day plus one additional month within the March, June, September and December quarterly cycle for its Euro-Bund futures options. Typically there are four maturities available. The CME also offers the first three consecutive contract months for a given trading day, but in addition the following four months within the March, June, September, and December quarterly cycle for 10-year T-Note futures options are also listed. A total of seven expirations are typically listed. There are at least nine exercise prices available for Euro-Bund futures options for each term, four in-the-money, four out-of-the-money options and one at-the-money option. For 10-year T-Note futures options there are about 100 exercise prices available, fifty in-the-money and fifty out-of-the-money options. For both products, the actively traded options series in terms of exercise price are concentrated in the ATM region, i.e. about 3% up and down from the underlying futures price. The daily settlement prices of Euro-Bund futures options are derived using both quoted and traded prices at the end of the trading day. The implied volatility smile is considered in order to set out-of-the-money settlement price. Similarly the daily settlement prices of 10-year T-Note futures options are derived from the mid-point of bid and ask prices on Globex. The skewness of the implied volatilities is also included to settle out-of-the-money options The Delivery Process of the Underlying Futures: Does it Matter for Implied Volatility? It is widely known that the delivery options embedded in 10-Year T-note futures affect the futures prices. These delivery options are driven by the structure of the delivery process at the CME. Basically, the whole delivery process requires three consecutive business days: notice day, tender day, and delivery day. On notice day the short gives the official notice that she will make a delivery by 9:00 p.m. EST. On tender day the short will precisely indicate which security she would like to deliver by 3:00 p.m. EST. On delivery day, the short has to deliver the specified security to the long s bank by 11:00 p.m EST. 6

8 Table 2 Contract Specifications of Interest Rate Futures Options 10-year T-Note Futures Options Euro-Bund Futures Options Size of Contract One interest rate futures contract One interest rate futures contract Regular Trading 17:30-16:00 EST 8:00-19:00 CET Hours Underlying 10-year T-Note futures Euro-Bund futures Tick (%) 1/ Tick (Euro) $ Contract Months Last Trading Day The three consecutive contract months as well as the following four months within the March, June, September, and December quarterly cycle The last Friday preceding by at least two business days the last business day of the month preceding the option month. The three successive calendar months, as well as the following month within the March, June, September and December quarterly cycle. Six exchange days prior to the first calendar day of the option expiration month. 1 Style American option American option Incremental Exercise Price Number of Exercise Prices Option Premium Daily Settlement Time The minimum strike price range will include the atthe-money strike price closest to the current futures price plus the next fifty consecutive higher and the next fifty consecutive lower strike prices. At least nine exercise prices available for each term for each call and put, such that four in-themoney and four out-ofmoney. Equity-style posting Future-style posting method method 17:15 CET 15:00 EST We can see that the first notice day for the 10-year T-note futures is two business days before the start of futures contract month providing a timing option for the short. In addition the short can also choose any day within the futures contact month to make a 1 On May 25 th, 2009 the last trading date of Euro-Bund futures options has changed to the exchange day prior to the first calendar day of the option expiration month. 7

9 delivery. Moreover, after the futures stop trading on the seventh business day prior the last business day of the delivery month, the short still can make his delivery. This is the so called end-of-month timing option. On days when both cash and futures market are open, there are in addition wild card options and accrue interest timing option. Figure 1 shows a comparison of the delivery schedule of Euro-Bund and 10-year T- note futures in the futures contract month. The delivery day for Euro-Bund futures is fixed on the tenth calendar day of the futures contract month, if is an exchange day; otherwise, the succeeding exchange day. While delivery at the CME can take place on any day of the futures expiration month and the actual delivery is arranged directly between the clearing members, the delivery process of the underlying for Euro-Bund futures at Eurex is automatically carried out in the central clearing system of Eurex Clearing AG only on the delivery day. Having the delivery process automated through the central clearing system at a particular point in time eliminates all delivery options for Euro-Bund futures. In Table 2 we show the last trading day of Euro-Bund and10-year T-Note futures options. The last trading day for Euro-Bund Futures Options is six business days prior to the first calendar day of the option expiration month during the time period of our investigation. Since May 25 th, 2009 the last trading date of Euro-Bund futures options is the exchange day prior to the first calendar day of the option expiration month. The last trading day for 10-year T-Note futures options is the last Friday preceding by at least two business days the last business day of the month preceding the option expiration month. Comparing the delivery process of futures with the respective futures options, we can see that interest rate futures options at both exchanges always expire earlier than the start of the delivery process of the underlying futures. Therefore the futures delivery process, in particular the various embedded options of the 10-year Treasury futures delivery process, do not influence option pricing or volatilities. After we examine the contract specifications of futures and futures options, we now proceed to look at the overall market structure and trading environment. Figure 1 shows monthly average daily volumes of traded contracts since July Both markets reached their peaks in Volume dropped off slightly ahead of the financial crisis and decreased dramatically after the onset of the financial crisis in 8

10 September The average daily volume of the 10-year T-Note futures options decreased more significantly than that of the Euro-Bund futures options. The difference of average daily volumes of the two products shrank after August 2008 and has stabilized since then. Trading outside the electronic order book dominates the markets of both products. In contrast to CME, Eurex does not offer floor trading. Trades at Eurex can be arranged by telephone and directly routed into Eurex Clearing via the Eurex block trading facility. Figure 2 breaks down the traded volume in 2008 into two parts, electronically traded or order book volume on the one hand and OTC block trade or pit volume on the other hand. The portion of electronic order book trading for the two products is higher at Eurex than at the CME. Overall 31.6% of all trades happened in Eurex order book while only 24.28% of trades took place in the Globex system at CME. 400,000 Figure 1 Average Daily Volume Since , , ,000 Volume 200,000 CME Eurex 150, ,000 50,000 0 Jul-07 Nov-07 Mar-08 Jul-08 Nov-08 Mar-09 Jul-09 Month 9

11 Figure 2 Euro-Bund Futures Options 10-year T-Note Futures Options 31.62% 68.38% 24.28% 75.72% OTC Block Trade Orderbook Volume Pit Volume Electronic Volume III. Data and Models for Implied Volatility Estimation The sample data used in this paper cover the period of January 2 nd, 2002 to October 3 rd, Daily settlement price data for Euro-Bund Futures and Futures Options were provided directly from Eurex. Daily settlement price data for 10-year T-Note Futures and Futures Options are from Bloomberg. The data for the effective Fed Funds Rate, the yield of 10-year constant maturity T-Note and 3-month constant maturity T-Bill were taken from the website of the Federal Reserve Bank, St. Louis. The data for EONIA, and the yield of 10-year constant maturity German government security and 3-month constant maturity German government bill were drawn from Bloomberg. There are two reasons why we use settlement prices of interest rate futures and futures options. First, trades in interest rate futures and futures options markets heavily concentrate on near-to-money and slightly out-of-money calls and puts of the frontmonth contract. As a result, reliable traded prices for all back-month contracts are difficult to obtain. Second, daily settlement prices are drawn from the market at the same time period every day. As a result, the problem of non-synchronous prices is avoided. The daily closing prices of the two products are not determined at the same point in time. Euro-Bund Futures Options close about 4 hours earlier then 10-year T- Note future options at the CME, i.e. the two market closing prices reflect different amounts of information. In this paper we do not attempt to analyze which market digests external shocks and information first. Rather we try to compare statistical observables between the two markets. The strong correlations we find between some 10

12 of the observables of the two markets would be potentially even larger if the two markets would determine their settlement prices at the same point in time. The model used to derive the implied volatility of Euro-Bund futures options is the interest rate futures option model with future-style margining derived by Lieu (1990). c = FN( d 1 p = KN ( d ) KN( d 2 2 ) ) FN ( d ) 1 where d 1 2 = [ln( F / K) + 0.5σ t]/ σ t and d2 = d1 σ t, c and p are the prices of a European futures call and put options, F is the price of Euro-Bund futures contract underlying the option contract, K is the exercise price, σ is the annualized standard deviation of the futures contract, t is the time to the option s expiration date measured in years, and N(.) is the cumulative normal distribution function. The futures option valuation model with futures-style margining is basically a variation of Black s model. The main difference between the standard Black model and the futures options model with futures-style margining is that there is no discount factor in the later. Lieu s results are subject to the criticism that expected cash flows are discounted by fixed interest rate, whereas security futures prices are assumed to be stochastic. However, Chen and Scott (1993) show that the results in Lieu hold in a general equilibrium model with stochastic interest rates. They argue that futures options with futures-style margining should not be exercised early because their prices are always greater than the intrinsic value. Thus, American style options on futures with futures style margining will have the same prices as comparable European style options on futures. As a result, we can simply price American style options on futures with futures style margining with a European pricing model. American Euro-Bund futures options have a futures-style option margining, so it is appropriate to use the interest rate futures option valuation model with futures-style margining to derive implied volatilities. Options on the 10-year T-Note futures are also American style but follow equity option style margining. The cost of carry of 10-year T-Note futures options is zero. Thus, as long as the risk-free rate is positive there does exist an early exercise boundary according to Barone-Adesi and Whaley (1987). Recent market data showed that there are early exercises of both deep in the money calls and puts. There are two 11

13 possible reasons. One is that traders exercise deep in the money calls and puts for cash in order to improve cash position. The other is that the difference between the borrowing and lending rate is so large for some traders that the early exercise of deep in the money options can reduce their cost of borrowing. To account for the early exercise boundary of 10-year Treasury Note futures options we evaluate implied volatilities through an American-style binomial pricing model by using 70 binomial steps. In summary it is important to note here that Euro-Bund futures options do not get exercised early, while early exercises in 10-year T-Note futures options do take place. While an early exercise in Euro-Bund Futures Options would most likely trigger a investigation into an erroneous order entry, early exercises of far out-of-the-money 10-year T-Note futures options do happen regularly. To account for speed and accuracy we also examined the quality of a European style model for ATM options of 10-year T-Note futures options. The model used to test this assumption and to derive the implied volatility of 10-year T-Note futures option is Black (1976) European futures option model c = e -rt p = e [ FN ( d -rt 1 [ KN( d ) KN ( d 2 2 )] ) FN ( d where r is the interest rate used to set settlement price, which is the average of the broker loan rate and the fed fund effective rate 2. The rest of parameters are defined in the same way as those in previous equations. There is a vast of majority of literature showing that Black s model is a fairly good approximation for pricing American-style T-Bond futures option and 10-year T-Note futures option. Cakici, Chatterjee, and Wolf (1993) test Black and Barone-Adesi and Whaley (1987) option valuation models for pricing American options on T-Note and T-Bond futures. They show that the option prices from the Black model are indistinguishable from the quadratic approximation in Barnone-Adesi and Whaley. And they find no mispricing with regard to out of and at the money options when they 1 )] 2 This is drawn from the statement of CME group daily settlement procedures online, 12

14 compare model prices to actual prices in the market. Hull and White (1987) show that the Black-Scholes (1973) model s prediction for implied volatility is consistent with that from a valuation model in which volatility follows a stochastic process, and the prediction is especially good for near-to-the-money options. Brooks and Oozeer (2002) show that it is valid to use Black s European option pricing model to determine implied volatility of at the money Long Gilt futures options. Therefore, it is reasonable to use Black s model to calculate implied volatility in a close to ATM region. Simon (1997) also argues that the ATM call options are optimal to exercise early when they are about 20% in the money. Comparing our binomial tree model with the European style model we find that the differences are negligible. There were almost no early exercises during the sample period for slightly out-of-the-money and near-tothe money 10-year T-Note futures options, both in our calculation and in the data. After the selection of models for the two types of interest rate derivatives, implied volatilities can be estimated by inverting the option price model for the near-the money and out-of-money call and put. We estimate implied volatilities across moneyness, where moneyness is defined as K/F. The selected range of moneyness we use to interpolate the implied volatility of at-the-money option is from 0.97 to This will guarantee that there are at least four strike series of options in each option maturity. Next we estimate the 30-day ATM implied volatility, skewness (slope) and kurtosis (curvature). In particular Following Shimko (1993), we use the quadratic functional form to interpolate across the implied volatilities of options for different moneyness, but for a given maturity. We then calculate the ATM implied volatility by setting moneyness equal to one in our functional form. We also calculate the skeweness (slope) and kurtosis (curvature) of ATM implied volatility smile by taking the first and second order derivatives of our functional form and evaluating it at moneyness equal to one. We repeat the process for futures options with different maturities. 13

15 We then use the interpolated ATM implied volatility, ATM implied volatility skewness, and ATM implied volatility kurtosis across different maturities to generate 30-day ATM implied volatility, 30-day ATM implied volatility skewness and kurtosis by using the same quadratic interpolation method between maturities. As a further robust check of our procedure we introduce the time-adjusted moneyness, 1 T K ln( ), and perform the same procedures illustrated above. With this adjusted F 1 variable skewness and kurtosis scales like and 1/T respectively. We will show T below that the data exhibit this behavior very accurately. 14

16 Figure 1 The Comparison of Delivery Processes of 10-year T-Note and Euro-Bund Futures Options Eurex The start of the futures month Last Trading Day; two business day before the delivery day Last Delivery Day; the 10 the business day t Option expiration date; 6 th business day CME First Notice Day; 2 nd last business day First Tender Day; last business day Last Delivery Day; the end of the month First Delivery Day; the start of the month Last Trading Day; 7th business day before the end of the futures contract month t Option expiration date; last Friday preceding by at least two business days 15

17 IV. Statistical Similarities and Differences of 30-day ATM Implied Volatility 4.1. Statistical Similarities of 30-day ATM Implied Volatility First we would like to address the question of to what extent implied volatilities represent forward market expectations, i.e. forward historical volatilities 3. We define historical volatility as the forward 30-day realized volatility rolling over every 30-day window in the sample period. Figure 2.1 shows that the forward historical volatilities of Euro-Bund Futures and 10-year T-Note Futures move in parallel. The two historical volatility series have a high correlation of 75% (see Table 3). Figure 2.2 shows that the historical volatility of 10-year T-Note futures moves pretty much in step with 30-day ATM implied volatility of 10-year T-Note futures option. Both time series show a correlation of 78%. Similar results hold for Euro-Bund Futures and Futures Options in Figure 2.3, with a correlation of 75% in Table 3. The 30-day ATM implied volatility of Euro-Bund futures options and 10-year T-Note futures options show in Figure 2.4 an even stronger correlation of 83%. Visually, the two series move closer together than the historical one. However, we can observe that on average the 30-day ATM implied volatility of Euro-Bund futures options is lower than that of 10- year T-Note futures options in the sample period. We investigate the reason for this difference in next section. Next we look at the behavior of the 30-day ATM implied volatility skewness and kurtosis as a function of calendar day. Figure 5.1 compares the two time series of the ATM implied volatility skewness over time. One can see in the graph that the two time series are very closely related in the long term. But it seems that there are more oscillations in the 30-day ATM implied volatility skewness of 10-year T-Note futures options. To resolve these oscillations we apply a 30-day moving average. The results are shown in Figure 7.1 emphasizing both the similarities between the two time series as well as the oscillations of the Treasury options. 3 From a trader s perspective one would want to compare a historical volatility based on the last 30 trading days to the current implied volatility, selling implied volatility if it is larger then historical volatility and vice versa. We are interested in evaluating how good a forecast of the current implied volatility is. Thus, we compare instead a 30-day forward looking historical volatility with the current implied volatility. 16

18 In Figure 5.2 we take the difference between the skewnesses of the two markets. To determine if this difference is significant we perform one-sample t statistics test with the null hypothesis that mean of difference is equal to zero. The test shows that it is statistically significant. Thus we can reject the null hypothesis. However, if we look at the mean value of the difference of skewness in the sample it is Therefore, the difference is very small. Figure 6.1 shows the time series of the 30-day ATM implied volatility kurtosis as a function of calendar days. It is hard to observe any discernible patterns. We have reached the highest derivative we can analyze in this model and the finest resolution we can apply. In figure 7.2 we apply a 30-day moving average to the kurtosis time series. In general the two series behave very similar. But any typical characteristics are harder to identify than for the 30-day ATM implied volatility and its skewness. We do see that the 30-day ATM implied volatility kurtosis of Euro-Bund futures options is lower than that of 10-year T-Note futures options. Figure 6.2 shows the difference between two kurtosis time series. One can see that the difference between the two is significant in terms of statistical and economic magnitude. The mean of the difference is 4.28, and the one-sample t-test is statistically significant. The previous analysis was based on calendar days, i.e. tracking the behavior of the two products over time. We proceed now to plot the data as a function of days to expiration instead of calendar days. This will allow us to investigate the temporal structure of the volatility surface, i.e. the term structure of skewness and kurtosis. Basically, we run a simple regression of the logarithm of skewness or kurtosis against the logarithm of days to expiration. We limit the sample data with days to expiration greater or equal to 10. Figures 3.1, 3.2, 3.3, and 3.4 show the resulting regression plots and Table 4 contains the regression parameters. From Figures 3.1 and 3.2 we can see that the slopes of the regression of the logarithm of skewness against the logarithm of days to expiration of the Euro-Bund and 10-year T-Note futures options are close to negative one half. Both are statistically and economically significantly different from zero. In other words the skewnesses of implied volatility of both products are changing with the days to expiration approximately as a function of one over the squared root of days to expiration. We also find in Figures 3.3 and Figure 3.4 that the slopes of the regression of the kurtosis of 10-year T-Note and Euro-Bund futures options are very close to minus one. Also this result is statistically and economically 17

19 significant as shown by the coefficients in Table 4. Thus, we can say that the two time series of Kurtosis on average behave approximately as a function of one over days to expiration. 1 K In Figure 4.1, 4.2, 4.3, and 4.4 we use instead time-adjusted moneyness, ln( ) as T F the independent variable to construct the Smiles and to examine the behavior of skewness and kurtosis as a function of days to expiration. This variable is often preferred by traders when designing their volatility surface models since it automatically introduces a scaling behavior for skewness and kurtosis as one over square-root of time to expiration and one over time to expiration respectively. Using time-adjusted moneyness the functional behavior with regard to time to expiration is automatically taken care off. Repeating the regression analysis from the previous paragraph should lead now to graphs without any discernible dependence on time to expiration. This result is well demonstrated in Figures 4.1 to 4.4. Table 4 provides the resulting summary statistics. Most of the coefficients for the slope are statistically significant. But they are also very close to zero. Therefore, from an economic sense, they confirm the findings in Figure 3.1, 3.2, 3.3 and 3.4. In summary skewness and kurtosis show a scaling behavior in terms of days to expiration. Skewness scales approximately like 1 and kurtosis like 1/T. T 4.2. What are the Reasons Underlying the Differences of 30-day ATM Implied Volatilities? The similarities of 10-year T-Note and Euro-Bund Futures Options are both strengthened and potentially caused by top investment institutions monitoring both products within the same screen. They use the same model to fit both interest rate derivatives products, often modeling the differences to turn the smile of one product into the one of the other product. However, we not only observe strong similarities, but also see important differences in the statistical characteristics of the two interest rate derivatives. Thus, the more interesting question is what drives the differences? This is the question we want to investigate in this section. Veronesi (1999) and David and Veronesi (2004) suggest that asset market volatility increases with the degree of economic fundamental uncertainty. Market participants 18

20 make predictions on asset prices based on observation of economic fundamentals even if the average growth rates of fundamentals are time-varying and not directly observable. When uncertainty is high, news will cause asset prices to move more than when it is low. As a result, the asset market volatility will be high. Moreover, according to the Taylor rule, monetary policy is linked to expectations regarding inflation and economic activity through the central bank s reaction function. Cook and Hahn (1989) find that changes in the Fed Fund target rate have large impact on short-term interest rates, a moderate impact on intermediate-term rates, and a small impact on long-term rates during the 1970s. Kutter (2001) measures monetary policy surprise as the difference between actual target rate and the expected rate embedded in the Fed Funds Futures. He finds that interest rates across different maturities react strongly to the monetary policy surprises. Gurkaynak, Sack and Swanson (2005) find that monetary policy surprise has strong effect on long-term forward interest rates, leading to a claim that market participants change their expectations in response to change in economic fundamentals and monetary policy. Litterman, Scheinkman, and Weiss (1991) find that there is a strong correlation between the level of interest rate volatility and the shape of the yield curve as measured by the yields of zero-coupon 1-month, 3-year and 10-year bonds. Wu (2001) empirically studied the relationship between the movement of the slope factor of the yield curve and the Fed s monetary policy surprises in the U.S. after He identifies that there is strong correlation between monetary policy surprises and the movement of the slope factor overtime. He found that the Fed s monetary policy action explains 80% - 90% of the movement of the slope factor, but the effect is short-lived within one to two month in the sample period. Therefore, we project that the difference of 30-day ATM implied volatility is mainly driven by the difference between the inter-bank borrowing rates and the difference between term structure spreads as measured by the difference between yields of 10- year Treasury Notes and 3-month Treasury Bill between the U.S. and Germany. Figure 8.1 plots the movement of the implied volatility spread as measured by the difference between 30-day ATM implied volatility of 10-year T-Note futures options and that of Euro-Bund Futures Options and the interest rate spread as measured by the difference of EONIA minus effective Fed Funds Rate. It clearly shows that the two 19

21 series move together over time. 4 That is, the higher the Fed Funds effective rate relative to EONIA in Europe, the smaller the 30-day ATM implied volatility of the 10-year T-Note futures options versus Euro-Bund Futures Options. Table 5 confirms this assessment with a correlation coefficient of 79% between the two spreads. Figure 8.2 plots the movement of the implied volatility spread and the difference between the term structure spreads as measured by the difference between yields of 10-year Treasury Notes and 3-month Treasury Bill between the U.S. and Germany. Also here we find that two series move closely during the sample period. The steeper the terms structure spread in the U.S. relative to Germany, the more uncertainty of interest rate movement in the U.S. relative to Germany, the higher the 30-day ATM implied volatility of 10-year T-Note futures options relative to Euro-Bund Futures Options. The correlation coefficient in Table 5 is very high, with a value of 83%. The observations show that the relative monetary policy in the two regions affect the difference of 30-day ATM implied volatility of Euro-Bund and 10-year T-Note futures options. During the sample period there are, in addition, more changes of the Fed Funds target rate than refinancing rate in European central bank, 26 vs. 12. The more changes in the target Fed Funds rate, the larger the uncertainty in the sample period, the higher the expected volatility in the U.S. An additional source for expected volatility of the underlying futures is futures price changes in response to a change in the cheapest to deliver. The cheapest to delivery (CTD) changes over time, usually because the level of yield or the slope of the yield curve changes. Occasionally, due to a new bond being added to the eligible basket of CTDs through the Treasury s funding of the budget deficit, the supply and the demand of CTDs is changed, which lowers and raises the implied volatility respectively. The set of available CTD securities in the U.S. is far larger than that in Germany. As a result, when there is uncertainty in the market, traders in short positions in futures have more options to choose the CTD. This will impact futures price strongly, leading to higher volatility. 4 We see a break-down of this relationship in 2008 during the financial crisis. A similar breakdown is observable in 2003 after the Irak war and at point in time the Federal Reserve lowered its Fed Fund target rate aggressively creating deflationary fears. Both instances are characterized by 10-year Treasury yields dropping to 3.5% and below. 20

22 V. Conclusions This paper examines the statistical similarities and differences of the 30-day ATM implied volatility smile of Euro-Bund and 10-year T-Note futures options. On average we find that the 30-day ATM implied volatility, 30-day ATM implied volatility skewness and kurtosis of Euro-Bund and 10-year T-Note futures options show very similar behavior and are strongly correlated. However, there are some differences. In general, 30-day ATM implied volatility of Euro-Bund Futures Options is lower than that of 10-year T-Note futures options. Also, the Euro-Bund Futures Options exhibit less 30-day ATM implied volatility skewness and kurtosis than 10- year T-Note futures options. The skewness and kurtosis of both Euro-Bund Futures Options and the 10-year T-Note futures options behave approximately like one over square-root of time to expiration and one over time to expiration respectively. We also find that there is a negative correlation between the difference of the two 30-day ATM implied volatilities and the difference of short-term interest rates in the U.S. and Euro zone, but positive correlation between the volatility spread and the difference of term structure spreads in the U.S. and Germany over time. This is consistent with long investors understanding that price change of bonds with different maturities and coupons not only affect the expected change in future interest rate but also on the uncertainty of expected market movements. 21

23 References Barone-Adesi, G., and Whaley, R. E., (1987), Efficient Analysis Approximation of American Option Values, Journal of Finance, Vol. 42, pp Black, F. (1976), The Pricing of Commodity Contracts, Journal of Financial Economics, Vol. 3, pp Black, F. and Scholes, M. (1973), The Pricing of Options and Corporate Liabilities, Journal of Political Economy, Vol. 8, pp Brooks, C. and C. Oozeer (2002), Modelling the Implied Volatility of Options on Long Gilt Futures, Journal of Business Finance and Accounting, Vol. 29, pp Cakici, N., S. Chatterjee and A. Wolf (1993), Empirical Tests of Valuation Models for Options on T-Note and T-Bond Futures, Journal of Futures Markets, Vol. 13, pp Chen, R. and Scott, L. (1993), Pricing Interest Rate Futures Options with Futures- Style Margining, Journal of Futures Markets, Vol. 13, pp Clare, A. D. and S. H. Thomas (1992), International Evidence for the Predictability of Stock and Bond Returns, Economic Letters, Vol. 40, pp Connolly, R. J., C. Stivers, and L. Sun (2005), Stock Market Uncertainty and the Stock-Bond Return Relation, Journal of Financial and Quantitative Analysis, Forthcoming. Cook, T. and T. Hahn (1989), The Effect of Changes in the Federal Funds Rate Target on Market Interest Rate in the 1970s, Journal of Monetary Economics, Vol. 24, PP Cox, J. C., S. A. Ross, and M. Rubinstein (1979), Option Pricing: A Simplified Approach, Journal of Financial Economics, Vol. 7, pp David, A. and P. Veronesi (2004), Inflation and Earnings Uncertainty and Volatility Forecasts, working paper, University of Calgary Haskyane School of Business. Gurkaynak, R.S., B. Sack and E. Swanson (2005), The Sensitivity of Long-Term Interest Rates to Economic News: Evidence and Implications for Macroeconomic Models, American Economic Review, Vol. 95, pp Hull, J. and A. White (1987), The Pricing of Options on Assets with Stochastic Volatilities, Journal of Finance, Vol. 42, pp Kuttner, K. N. (2001), Monetary Policy Surprises and Interest Rates: Evidence from the Fed Funds Futures Market, Journal of Monetary Economics, Vol. 47, pp

24 Lieu, D. (1990), Option Pricing with Futures-Style Margining, Journal of Futures Markets, Vol. 10, pp Litterman, R., J. Scheinkman, and L. Weiss (1991), Volatility and the Yield Curve, Journal of Fixed Income, Vol. 1, pp Shimko, D. C. (1993), Bounds of Probability, Risk, Vol. 6, pp Simon, D. P. (1997), Implied Volatility Asymmetries in Treasury Bond Futures Options, Journal of Futures Markets, Vol. 17, pp Veronesi, P. (1999), Stock Market Overreaction to Bad News in Good Times: A Rational Expectation Equilibrium Model, Review of Financial Studies, Vol. 12, pp Wu, T. (2001), Monetary Policy and the Slope Factor in Empirical Term Structure,Estimations. FRB San Francisco Working Paper

25 Figure 2.1 Movement of 30-day Forward Realized Volatility at CME and Eurex Figure 2.2 Movement of Implied Volatility and Historical Volatility at CME 20% 18% 16% CMEHistVola30 EurexHistVola30 20% 18% 16% CMEATMVolatility30 CmeHistVola30 14% 14% Volatility 12% 10% 8% Volatility 12% 10% 8% 6% 6% 4% 4% 2% 2% 0% Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date Figure 2.3 Movement of Implied Volatility and Historical Volatility at Eurex 0% Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date Figure 2.4 Movement of Implied Volatility of 10y TN Futures Options and Eurex- Bund Futures Options 14% 12% EurexATMVolatility30 EurexHistVola30 14% 12% CMEATMVolatility30 EurexATMVolatility30 Volatility 10% 8% 6% 4% Implied Volatility 10% 8% 6% 4% 2% 2% 0% Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date 0% Jan-02 Jan-03 Jan-04 Jan-05Date Jan-06 Jan-07 Jan-08 24

26 Figure 3.1 Figure Y TN Futures Options from to Euro-Bund Futures Options from to Figure Y TN Futures Options from to Figure 3.4 Euro-Bund Futures Options from to

27 Figure 4.1 Figure Y TN Futures Options from to Euro-Bund Futures Options from to Figure Y TN Futures Options from to Figure 4.4 Euro-Bund Futures Options from to

28 0.6 Figure 5.1 Movement of 30-day ATM Implied Volatility Skewness at CME and Eurex CMEATMSkewness EurexATMSkewness Skewness Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date Figure 5.2 Movement of 30-day ATM Implied Volatility Skewness Difference 0.6 SkewSpread Skewness Spread Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date 27

29 Figure 6.1 Movement of 30-day ATM implied volatility Kurtosis at CME and Eurex CMEATMKurtosis EurexATMKurtosis Kurtosis Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date Figure 6.2 Movement of 30-day ATM implied volatility Kurtosis Difference 150 Kurtosis spread 100 Kurtosis Spread Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date 28

30 Figure 7.1 Movement of 30-day Moving Average of ATM Implied Volatility Skewness CMEskewness Eurexskewness Skewness Feb-02 Feb-03 Feb-04 Feb-05 Feb-06 Feb-07 Feb-08 Date Figure 7.2 Movement of 30-day Moving Average of 30-day ATM Implied Volatility Kurtosis 35 CMEKurtosis30MA EurexKurtosis30MA Kurtosis Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date 29

31 Figure 8.1 Movement of Interest Rate Spread and Implied Volatility Spread 6% 5% EFSpread VolatilitySpread 4% 3% Spread 2% 1% 0% -1% -2% -3% Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Date Figure8.2 Movement of Yield Curve Slope Spread and Implied Volatility Spread 6% 5% YCSlopeSpread VolatilitySpread 4% 3% 2% 1% 0% -1% -2% Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 30

32 Table 2 Correlation of Volatility Measures CmeVola30 EurexVola30 CmeHistVola30 EurexHistVola30 CmeVola NA EurexVola NA 0.75 CmeHistVola NA EurexHistVola30 NA Table 3 Test for the slope of Skewness and Kurtosis Moneyness = K/F Moneyness = 1/sqrt(T)*ln(K/F) Eurex Skewness <0 Skewness <0 Estimate s.e. p Estimate s.e. p Intercept Intercept Slope Slope Kurtosis Kurtosis Intercept Intercept Slope Slope CME Skewness <0 Skewness <0 Estimate s.e. p Estimate s.e. p Intercept Intercept Slope Slope Kurtosis Kurtosis Intercept Intercept Slope Slope Table 4 Correlation between implied volatility spread and term structure spread as well as short-term interest rate spread SlopeSpread VolaSpread EFSpread SlopeSpread VolaSpread EFSpread