Canonical Valuation of Options in the Presence of Stochastic Volatility

CANONICAL VALUATION OF OPTIONS IN THE PRESENCE OF STOCHASTIC VOLATILITY

PHILIP GRAY* SCOTT NEWMAN

Proposed by M. Stutzer (1996), canonical valuation is a new method for valuingderivative securities under the risk-neutral framework. It is nonparametric, simple to apply, and, unlike many alternative approaches, does not require any option data. Although canonical valuation has great potential, its applicability in realistic scenarios has not yet been widely tested. This article documents the ability of canonical valuation to price derivatives in a number of settings. In a constant-volatility world, canonical estimates of option prices struggle to match a Black-Scholes estimate based on historical volatility. However, in a more realistic stochastic-volatility setting, canonical valuation outperforms the Black-Scholes model. As the volatility generating process becomes further removed from the constant-volatility world, the relative performance edge of canonical valuation is more evident. In general, the results are encouraging that canonical valuation is a useful technique for valuingderivatives. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:1–19, 2005

The authors are grateful for the comments and suggestions of Jamie Alcock, Stephen Gray, Philip Hoang, Egon Kalotay, an anonymous referee, participants at the 16th Australian Finance and Banking conference, and funding from a UQ Business School summer research grant. *Correspondence author, UQ Business School, The University of Queensland, St. Lucia 4072, Australia; e-mail: [email protected] Received November 2003; Accepted March 2004

I Philip Gray is an Associate Professor at UQ Business School at the University of Queensland in Brisbane, Australia. I Scott Newman is with the UQ Business School at the University of Queensland in Brisbane, Australia.

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INTRODUCTION Thirty years after the seminal work of Black and Scholes (1973) and Merton (1973), the benchmark in option pricingcontinues to be the widely applied Black-Scholes formula. However, while the strongpara- metric assumptions underlyingthe Black-Scholes model allow a simple, closed-form solution to the price of a European call option, empirical tests suggest that the assumptions are violated in practice. For example, rather than beingconstant, implied volatilities from observed option prices are systematically related to moneyness and maturity (see Derman & Kani, 1994; MacBeth & Merville, 1979; Rubinstein, 1985). There is also considerable evidence that stock returns are not normally distributed (see Jackwerth & Rubinstein, 1996; Kon, 1994). In light of these problems, alternative methods have been developed to price options. One such example is canonical valuation. Developed by Stutzer (1996), canonical valuation is a nonparametric technique for valuing derivatives. Unlike the Black-Scholes model, it makes no restrictive assumptions about the underlying asset’s return generating process; rather, the historical distribution of returns on the underlying asset is used to predict the distribution of future stock prices. A maximum- entropy principle is employed to transform this real-world distribution into its risk-neutral counterpart, from which option prices follow easily using the standard risk-neutral approach. In addition to being relatively simple to implement, a major advantage of canonical valuation is that option price data are not required as input.1 Despite its potential, canonical valuation has only been examined in a handful of papers to date. Usinga simulated Black-Scholes world, Stutzer (1996) reports that the accuracy of canonical estimates of option prices is comparable to Black-Scholes estimates usinghistorical volatility. Foster and Whiteman (1999) modify canonical valuation to incorporate a more sophisticated Bayesian predictive model. In an application to the soybean futures options market, the modiﬁed model performs well with reference to both the simple canonical valuation model usinghistorical returns and Black-Scholes. Finally, Stutzer and Chowdhury (1999) apply canonical valuation to bond futures options, with results also suggesting that the method performs well. This article explores the potential usefulness of canonical valuation in two directions. First, the analysis of Stutzer (1996) is extended to document the accuracy of canonical valuation across various levels of

1Although no option prices are strictly required, such data are easily incorporated into canonical valuation if desired.

TLFeBOOK Canonical Valuation of Options 3 maturity and moneyness. Working in a constant-volatility Black-Scholes world, stock prices are simulated under a geometric Brownian motion so that true option prices are known. Prices are then estimated using both canonical (CAN) and historical-volatility-based Black-Scholes (HBS) methods, and the properties of pricing errors are documented. We also examine the potential to reduce pricing errors by incorporating a token amount of option data. The canonical estimator is modified such that the risk-neutral density is estimated subject to the constraint that a single at-the-money option is correctly priced. The performance of the constrained canonical estimator (CON) is compared to CAN and HBS estimators. The sensitivity of all findings to the number of returns used to estimate the risk-neutral density is also examined. Because there is considerable evidence that volatility is noncon- stant, Stutzer (1996) foreshadows that the usefulness of canonical valuation will be most apparent when we move beyond the Black-Scholes world. This issue has not previously been examined. The second contribution of this article, therefore, is to evaluate the performance of canonical valuation in the presence of stochastic volatility. Heston’s (1993) model provides an ideal environment to test this conjecture as it admits a closed-form solution to the price of a call option under stochastic volatility. Stock prices are simulated under Heston’s stochastic volatility model, and the performance of CAN and HBS estimates is assessed relative to the true price. Simulation results are reported for a range of moneyness and time to maturity. Finally, the sensitivity of results to key parameters in the stochastic volatility model is examined. There are several key findings in this article. Not surprisingly, the Black-Scholes model outperforms canonical valuation in a constant- volatility world. HBS estimates are less biased than standard canonical estimates, and this performance edge persists even as sample size increases. The practice of incorporatingminimal option data into the estimation of the risk-neutral density under canonical valuation produces significant improvements in pricing performance. The constrained canonical estimator arguably outperforms HBS estimates, particularly for deep out-of-the-money options which can be difficult to price. Moving to simulations assuming stochastic volatility, the magnitude of pricing errors under both CAN and HBS estimators rises markedly highlighting the difficulty in pricing real-world options. The out-of-the- money superiority of the HBS estimator over CAN documented under constant volatility disappears. Plain-vanilla canonical estimation produces less biased estimates for out-of-the-money options, and the

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constrained canonical estimator performs admirably regardless of moneyness. Sensitivity analysis identifies the volatility of the volatility dynamics as the key parameter impacting on the success of alternative pricing methods. The remainder of the article is structured as follows. The second section, An Overview of Canonical Valuation, reviews other popular nonparametric approaches to pricing derivatives, outlines the potential advantages of canonical valuation, and provides a brief overview of the canonical valuation approach. The next two sections, Canonical Valuation in a Black-Scholes World and Canonical Valuation in a Stochastic Volatility World, conduct simulation experiments to document the properties of alternative pricing methods in constant-volatility and stochastic-volatility worlds, respectively. The last section is the Conclusion.

AN OVERVIEW OF CANONICAL VALUATION A risk-neutral approach is often adopted to price derivatives. The primary task is to estimate the risk-neutral probability distribution of the underlying asset, from which the expected payoff to the derivative can be calculated. There are, however, different ways to estimate the required risk-neutral density. Black and Scholes (1973) typify the parametric approach by specifying the dynamics of the underlying asset from which the risk-neutral density (lognormal in this case) is derived. In contrast, Rubinstein (1994), Hutchinson, Lo, and Poggio (1994), Jackwerth and Rubinstein (1996), and Aït-Sahalia and Lo (1998) propose nonparametric methods of estimating the risk-neutral density. Although they make fewer restrictive assumptions over the data- generating process, these nonparametric methods require as input large quantities of market option prices across a range of strike prices. An advantage of canonical valuation is that option prices are unnecessary; the method can be implemented merely using a time-series of data for the underlying asset. Note also that the ability to price options without using observed market prices classiﬁes canonical valuation as an option pricing theory. The nonparametric methods just cited are best viewed as interpolation and extrapolation algorithms that predict some option prices from the observed prices of other options.2 Consider valuing a European call option on a stock expiring at time T (i.e., T years forward). Obviously the possible option payoffs depend on

2We are grateful to an anonymous referee for elucidating this point.

TLFeBOOK Canonical Valuation of Options 5 the distribution of the underlying stock price at time T. Stutzer (1996) begins with the stock’s historical distribution of T-year returns

ϭ p Ri, i 1, ,n, where returns are expressed as price relatives. An advantage of the historical distribution is that it is more likely to capture stylized features of the data (such as skewness and leptokurtosis) than parametric models. From the returns, the distribution of possible prices,

Pi, for the underlying asset T-years forward is constructed:

ϭ ϭ p Pi P0 Ri, i 1, ,n (1) where P0 is the current price of the underlyingasset. Each possible future price computed in (1) is assigned an equal prior real-world proba- ϭ 1 bilitypˆi, such thatpˆ i n . These prior probabilities are transformed into their risk-neutral counterparts subject to the constraint that the expected return on the stock is the riskless rate; equivalently, the discounted expected return is unity:

n R 1 ϭ pˆ*a i b a i ϩ T (2) iϭ1 (1 r) where pˆ *i denotes the risk-neutral probability of return Ri, and r is the risk-free rate of interest. Employing the maximum entropy principle of information theory, Stutzer (1996) shows that the risk-neutral probabilities, pˆ *,i are given by:

Ri exp ag* b (1 ϩ r)T ˆ ϭ p*i n (3) Ri exp ag* b a ϩ T iϭ1 (1 r) where g* is the Lagrange multiplier, given by the following minimization problem:3

n Ri (4) g* ϭ arg min a exp c ga Ϫ 1 bd . g ϩ T iϭ1 (1 r)

The final step in canonical valuation is to compute the expected discounted payoff to the derivative security usingthe risk-neutral probabilities

3In this article, g* is calculated using a single variable optimization routine in Matlab. Alternatively, the optimization is equally simple using Solver in Microsoft Excel.

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calculated in (3). The price of a European call option expiringat T with an exercisepriceofXissimply:

n Ϫ max(P0 Ri X,0) (5) C ϭ a bpˆ *. a ϩ T i iϭ1 (1 r)

To summarize, canonical valuation uses the historical time-series of prices on the underlying asset to estimate the future distribution of the asset price. A maximum-entropic technique transforms the distribution into the required risk-neutral density. Most importantly, no option price

data is required. If desired, the risk-neutral probabilities pˆ *i can be estimated subject to an additional constraint that they correctly price one or more traded options (see Stutzer, 1996, Equations 11 and 12). This article also explores the usefulness of this marginally more complex procedure.

CANONICAL VALUATION IN A BLACK-SCHOLES WORLD To investigate its applicability in the most basic setting, canonical valuation is applied to price European call options across a range of moneyness and maturities in a simulated Black-Scholes world. Under these conditions, stock price follows a geometric Brownian motion:

ϭ ϩ dSt mSt dt sSt dzt (6)

where St is the stock price at time t, m is the average stock return, s is the constant instantaneous volatility of the process and dzt is the increment in a standard Wiener process. Under this model, the distribution of continuously compounded T-year returns is normal:

1 2 2 (7) ෂ Ϫ ln(Ri) N((m 2 s )T, s T).

For each time to maturity T, 200 returns are drawn from the normal distribution (7), and the distribution of possible future stock prices is constructed as per Equation (1). Risk-neutral probabilities andg* are estimated from Equations (3) and (4), respectively. Canonical option prices follow from Equation (5). The simulation employs a drift m of 10% and annual volatility s of 20%. The riskless rate of interest is assumed to be a constant 5% continuously compounded.4 These values are consistent

4To ensure comparability, the discrete compounding equivalent of this continuously compounded rate is employed in Equations (2)–(5) for canonical estimates.

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TABLE I MPE of Canonical and Black-Scholes Estimates in a Black-Scholes World

Time to expiration (years) Moneyness (spot/strike) 1͞52 1͞13 1͞41͞23͞41

Deep out-of-the-money n/a 0.0155 Ϫ0.0020 Ϫ0.0028 Ϫ0.0013 Ϫ0.0022 (0.90) n/a Ϫ0.0272 Ϫ0.0245 Ϫ0.0292 Ϫ0.0321 Ϫ0.0372 n/a Ϫ0.0212 Ϫ0.0057 Ϫ0.0037 Ϫ0.0030 Ϫ0.0024 Out-of-the-money 0.0003 Ϫ0.0021 Ϫ0.0022 Ϫ0.0020 Ϫ0.0009 Ϫ0.0015 (0.97) Ϫ0.0098 Ϫ0.0077 Ϫ0.0111 Ϫ0.0158 Ϫ0.0191 Ϫ0.0234 Ϫ0.0052 Ϫ0.0008 Ϫ0.0001 Ϫ0.0001 Ϫ0.0002 Ϫ0.0001 At-the-money Ϫ0.0012 Ϫ0.0014 Ϫ0.0015 Ϫ0.0007 Ϫ0.0009 Ϫ0.0011 (1.00) Ϫ0.0032 Ϫ0.0042 Ϫ0.0078 Ϫ0.0120 Ϫ0.0150 Ϫ0.0191 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 In-the-money 0.0000 Ϫ0.0005 Ϫ0.0009 Ϫ0.0010 Ϫ0.0005 Ϫ0.0009 (1.03) Ϫ0.0005 Ϫ0.0020 Ϫ0.0052 Ϫ0.0091 Ϫ0.0118 Ϫ0.0155 Ϫ0.0003 Ϫ0.0004 Ϫ0.0004 Ϫ0.0004 Ϫ0.0003 Ϫ0.0003 Deep in-the-money 0.0000 0.0001 0.0000 Ϫ0.0001 0.0000 Ϫ0.0002 (1.125) 0.0000 Ϫ0.0001 Ϫ0.0012 Ϫ0.0033 Ϫ0.0052 Ϫ0.0077 Ϫ0.0001 Ϫ0.0001 Ϫ0.0007 Ϫ0.0009 Ϫ0.0010 Ϫ0.0011

Note. Canonical and HBS estimates are compared to the true Black-Scholes call price, where stock prices are simulated by a geometric Brownian motion with m ϭ 0.1 and s ϭ 0.2. Each cell represents a particular combination of moneyness and maturity. The top and middle numbers reported for each combination are the mean percentage error (MPE) of the HBS and CAN estimates respectively over 10,000 simulations. The bottom number is a modified canonical estimate (CON) constrained to price an at-the-money option correctly.

with those used in simulations performed by Hutchinson et al. (1994) and Stutzer (1996). Canonical estimates of option price (CAN) are compared to Black- Scholes model prices (HBS), using a historical estimate of volatility from the simulated data, thus ensuring that estimates under each method rely on the same data. The accuracy of CAN and HBS estimates is then eval- uated with reference to the true Black-Scholes price calculated using the known volatility. The simulation procedure is repeated 10,000 times, and the properties of estimates are tabulated. Table I reports results for various combinations of moneyness and maturity.5 The top and middle numbers in each cell represent the mean percentage error (MPE) of HBS and CAN call option estimates relative to the true Black-Scholes price. Without exception, the HBS

5Results are not reported for deep out-of-the-money options with just one week to expiration. The true price for this option is $0.0001, therefore even the slightest pricing error results in an enormous percentage error.

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estimate outperforms the CAN estimate for each combination of moneyness and time to expiration, and this performance edge is most noticeable for deep out-of-the-money options. The accuracy of both HBS and CAN estimates increases monotonically with moneyness. Although HBS estimates show no discernible pattern with maturity, CAN exhibits persistent negative pricing error (i.e., underprices options) for the vast majority of combinations, and this negative bias increases with maturity. Stutzer (1996) reports that the performance of canonical valuation improves when a small amount of option data is used. Specifically, additional constraints can be incorporated into the estimation of risk- neutral probabilities such that one or more options are correctly priced. In Table I, the bottom number in each cell represents this constrained canonical estimate (CON), calculated to correctly price a single at- the money option. This practice of augmenting historical return data with a minimal amount of option data appears highly advantageous. The MPE for CON estimates is a significant improvement over CAN estimates and, in some cases, even outperforms HBS estimates. Curiously, like the CAN estimator, the CON estimator also consistently underprices options, although the magnitude of negative bias is notably smaller. The MPE is a useful statistic in that it shows the average pricing effectiveness of alternative methods (in this case, averaged over 10,000 simulations) and documents the direction of errors. However, the magnitude of pricing errors can be masked to the extent that positive and negative errors cancel each other out. Table II reports an alternative performance measure, mean absolute percentage error (MAPE), adopted by Stutzer (1996). The tenor of the results is similar to MPE in Table I. HBS consistently outperforms CAN, although the performance edge is less dramatic under this alternative metric; in many cases, the MAPE of CAN is only marginally higher than that for HBS. Again, CON produces the lowest MAPE in most cases. MAPE decreases monotonically in moneyness, and it decreases (increases) in maturity for out-of-the-money (in-the-money) options. These findings are largely consistent with a similar analysis performed by Stutzer (1996) over a narrower band of moneyness and maturities.6

6Results for the CON estimator differ from Stutzer (1996) who constrained an out-of-the-money option to be correctly priced (S͞X ϭ 0.95). This article constrains the risk-neutral probabilities to price an at-the-money option correctly.

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TABLE II MAPE of Canonical and Black-Scholes Estimates in a Black-Scholes World

Time to expiration (years) Moneyness (spot/strike) 1͞52 1͞13 1͞41͞23͞41

Deep out-of-the-money n/a 0.2272 0.1088 0.0763 0.0635 0.0559 (0.90) n/a 0.3603 0.1252 0.0847 0.0724 0.0660 n/a 0.3355 0.0878 0.0464 0.0326 0.0254 Out-of-the-money 0.1214 0.0701 0.0498 0.0419 0.0381 0.0354 (0.97) 0.1491 0.0765 0.0543 0.0462 0.0439 0.0427 0.1095 0.0339 0.0142 0.0090 0.0068 0.0055 At-the-money 0.0379 0.0369 0.0341 0.0318 0.0302 0.0289 (1.00) 0.0410 0.0396 0.0378 0.0356 0.0354 0.0353 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 In-the-money 0.0074 0.0174 0.0227 0.0238 0.0238 0.0235 (1.03) 0.0095 0.0199 0.0260 0.0274 0.0284 0.0292 0.0069 0.0086 0.0067 0.0052 0.0044 0.0038 Deep in-the-money 0.0000 0.0007 0.0051 0.0087 0.0107 0.0118 (1.125) 0.0000 0.0015 0.0072 0.0116 0.0141 0.0158 0.0000 0.0014 0.0055 0.0072 0.0074 0.0073

Note. Canonical and HBS estimates are compared to the true Black-Scholes call price, where stock prices are simulated by a geometric Brownian motion with m ϭ 0.1 and s ϭ 0.2. Each cell represents a particular combination of moneyness and maturity. The top and middle numbers reported for each combination are the mean absolute percentage error (MAPE) of the HBS and CAN estimates respectively over 10,000 simulations. The bottom number is a modiﬁed canonical estimate (CON) constrained to price an at-the-money option correctly.

Overall, the simulation results reported in this section support the use of canonical valuation for options. In an environment simulated precisely under the Black-Scholes assumptions, HBS estimates might be expected to outperform nonparametric techniques such as canonical valuation. However, the performance of the CAN estimate is only slightly inferior to HBS, whereas incorporation of a token amount of option data results in CON estimates that clearly outperform the HBS estimate. In practice, of course, it is impossible to verify any speciﬁc functional form for the stock price process, and it is this impossibility that motivates the use of nonparametric techniques such as canonical valuation. One ﬁnal issue explored in this section relates to the sample size required to successfully implement canonical valuation. The simulations conducted to date utilize 200 returns—the HBS option estimate uses these returns to estimate the volatility of the stock, whereas the CAN estimate uses the returns to construct the predictive distribution of stock price. That HBS outperforms CAN suggests that 200 returns

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do a better job at estimating s (and consequently the corresponding lognormal density assumed by the HBS estimate) than they do at approximatingthe full return density required by the nonparametric canonical valuation method. One might question whether the performance edge of HBS over CAN diminishes as the number of returns increases. Table III reports the sensitivity of MAPE under alternative methods to the number of returns used in estimation. All numbers are for a long- dated call option with one year to expiration. The accuracy of all estimation methods improves with sample size. Yet, perhaps surprisingly, HBS maintains its performance edge over CAN even as the number of returns employed approaches 500. In a simulated Black-Scholes world, therefore, it appears that the canonical approach requires the assistance of at least some option data to compete with the standard Black-Scholes model.

TABLE III Sensitivity of Long-Dated Option Estimates to Number of Returns

Number of returns used in estimation Moneyness (spot/strike) 60 100 200 300 400 500

Deep out-of-the-money 0.1025 0.0777 0.0559 0.0449 0.0392 0.0351 (0.90) 0.1145 0.0882 0.0660 0.0557 0.0503 0.0475 0.0482 0.0363 0.0254 0.0208 0.0179 0.0162 Out-of-the-money 0.0650 0.0492 0.0354 0.0284 0.0248 0.0223 (0.97) 0.0748 0.0573 0.0427 0.0359 0.0324 0.0304 0.0105 0.0080 0.0055 0.0046 0.0039 0.0035 At-the-money 0.0530 0.0402 0.0289 0.0232 0.0202 0.0182 (1.00) 0.0622 0.0476 0.0353 0.0298 0.0267 0.0250 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 In-the-money 0.0430 0.0326 0.0235 0.0188 0.0165 0.0148 (1.03) 0.0517 0.0395 0.0292 0.0246 0.0221 0.0206 0.0073 0.0055 0.0038 0.0031 0.0026 0.0024 Deep in-the-money 0.0215 0.0163 0.0118 0.0095 0.0083 0.0074 (1.125) 0.0284 0.0217 0.0158 0.0132 0.0118 0.0110 0.0144 0.0107 0.0073 0.0059 0.0051 0.0046

Note. This table reports the sensitivity of the MAPE of HBS, CAN, and CON estimates to the number of returns used in estimation. Canonical and HBS estimates are compared to the true Black-Scholes call price, where stock prices are simulated by a geometric Brownian motion with m ϭ 0.1 and s ϭ 0.2. The call option has a maturity of one year. The top and middle numbers reported for each cell are the mean absolute percentage error (MAPE) of the HBS and CAN estimates respectively over 10,000 simulations. The bottom number is a modiﬁed canonical estimate (CON) constrained to price an at-the-money option correctly.

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CANONICAL VALUATION IN A STOCHASTIC VOLATILITY WORLD There is a weight of empirical evidence to suggest that the variance of stock returns is stochastic, in direct contrast to the assumptions of Black-Scholes used in the previous analysis. The practical usefulness of a nonparametric technique like canonical valuation, therefore, is best assessed in an environment where the constant-volatility assumption is relaxed. Numerous stochastic volatility models have been suggested in the literature.7 However, with the exception of Heston (1993) and Stein and Stein (1991), these models do not admit closed-form solutions to option price. To document the properties of alternative pricing methodologies, knowledge of the true option price under stochastic volatility is required. For this purpose, Heston’s model is employed in the following simulations because, in addition to giving a closed-form solution, it also accom- modates correlation between the random shocks in the volatility and stock processes. Heston (1993) derives the solution to the price of a European call

option assuming that the stock price, St, follows the process:

ϭ ϩ 1 dSt mSt dt vtSt dz1,t (8)

where the stochastic variance of the stock return vt, is generated by the following Ornstein-Uhlenbeck process:

ϭ Ϫ ϩ 1 dvt k(u vt) dt j vt dz2,t (9)

where k is the speed of mean-reversion, u is the long-run mean variance,

j is the volatility of the volatility generating process, and dz1,t and dz2,t are increments in Wiener processes, which have a correlation coefﬁcient of r. UsingEuler discretizations of (8) and (9), data are simulated under Heston’s stochastic-volatility world out to the option’s expiration time T.8 For consistency with the Black-Scholes world simulations in the previous section, the drift of the stock return, m, and the long-run mean, u, are assumed to be 10% and 4%, respectively. The remainingparameters k, j,

7See, for example, Heston (1993), Hull and White (1987), Melino and Turnbull (1990), Scott (1987), Stein and Stein (1991), and Wiggins (1987). 8One-day time steps are used, and 253 trading days per year are assumed.

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and r are assumed to be 3, 0.40, and Ϫ0.50, respectively. These values are typical of estimates from actual market data (see Lin, Strong, & Xu, 2001; Zhang& Shu, 2003). This simulation is repeated 200 times to generate the distribution of T-year returns. Four estimators are examined under the stochastic volatility world. The HBS estimate is again the Black-Scholes model price using historical volatility estimated from the simulated data. Canonical estimates of option prices (both with and without a constraint) are computed usingthe methodology outlined in the second section. In addition to the HBS estimate, a modiﬁed estimate is obtained by backingout the Black-Scholes implied volatility from the at-the-money option, assumed to be correctly priced. The implied volatility is then used with the Black-Scholes formula for other levels of moneyness to obtain an implied Black-Scholes estimate (IBS). The IBS estimate allows a fairer comparison with the CON estimate because both use the same information (i.e., that the at-the-money option is correctly priced). Thus, the relative comparison between CON and IBS is their respective ability to correct in-the-money and out-of-the- money bias.9 All estimates are compared to the true Heston price for each combination of moneyness and expiration. The entire process is repeated 10,000 times. Tables IV and V report the simulation results. In practice, the Black-Scholes model is known to overprice (underprice) out-of-the- money (in-the-money) options. One explanation is that, in the presence of stochastic volatility, the distribution of stock returns has a fatter left tail than the lognormal assumed by Black-Scholes. Table IV reports that MPE for HBS and IBS is indeed positive (negative) for out-of-the-money (in-the-money) options. It is important to note that MPE for CAN estimates is signiﬁcantly lower than HBS estimates for deep out-of-the- money options. Thus, the canonical technique presents itself as a useful alternative for pricingsuch options. It appears to better capture the fat left tails of stock return distributions under stochastic volatility. Note also that the constrained canonical estimator (CON) clearly outperforms HBS and CAN for all combinations of moneyness and maturity. Although the IBS estimator performs notably better than HBS for out-of-the-money options, the latter hold their own in-the-money. In fact, surprisingly, as the option moves into the money, HBS pricing errors are little different and arguably lower than for CAN estimates. The results for MAPE in Table V are qualitatively similar to those for MPE.

9We are grateful to an anonymous referee for suggesting the IBS estimate.

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TABLE IV MPE of Canonical and Black-Scholes Estimates in a Stochastic-Volatility World

Time to expiration (years) Moneyness (spot/strike) 1͞52 1͞13 1͞41͞23͞41

Deep out-of-the-money n/a 0.5990 0.3804 0.2538 0.1823 0.1429 (0.90) n/a Ϫ0.3138 Ϫ0.0970 Ϫ0.0768 Ϫ0.0738 Ϫ0.0670 n/a Ϫ0.1736 Ϫ0.0059 0.0033 0.0021 0.0017 n/a 0.8420 0.3170 0.1761 0.1185 0.0868 Out-of-the-money Ϫ0.1678 0.0115 0.0607 0.0629 0.0542 0.0484 (0.97) Ϫ0.3298 Ϫ0.0979 Ϫ0.0523 Ϫ0.0488 Ϫ0.0491 Ϫ0.0464 Ϫ0.1267 Ϫ0.0139 Ϫ0.0010 0.0006 0.0006 0.0006 0.1218 0.0652 0.0403 0.0270 0.0200 0.0156 At-the-money Ϫ0.0863 Ϫ0.0268 0.0134 0.0264 0.0266 0.0264 (1.00) Ϫ0.0945 Ϫ0.0490 Ϫ0.0363 Ϫ0.0382 Ϫ0.0401 Ϫ0.0389 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 In-the-money Ϫ0.0224 Ϫ0.0275 Ϫ0.0087 0.0048 0.0088 0.0113 (1.03) Ϫ0.0165 Ϫ0.0225 Ϫ0.0244 Ϫ0.0294 Ϫ0.0324 Ϫ0.0324 Ϫ0.0011 Ϫ0.0003 Ϫ0.0004 Ϫ0.0008 Ϫ0.0007 Ϫ0.0007 Ϫ0.0071 Ϫ0.0154 Ϫ0.0175 Ϫ0.0148 Ϫ0.0120 Ϫ0.0100 Deep in-the-money 0.0000 Ϫ0.0035 Ϫ0.0146 Ϫ0.0150 Ϫ0.0126 Ϫ0.0096 (1.125) Ϫ0.0001 Ϫ0.0013 Ϫ0.0064 Ϫ0.0121 Ϫ0.0161 Ϫ0.0177 Ϫ0.0001 Ϫ0.0003 Ϫ0.0008 Ϫ0.0015 Ϫ0.0018 Ϫ0.0017 Ϫ0.0001 Ϫ0.0032 Ϫ0.0169 Ϫ0.0225 Ϫ0.0222 Ϫ0.0205

Note. Canonical, HBS and IBS estimates are compared to the true Heston call price, where stock prices are simulated in Heston’s (1993) stochastic volatility world (Equations (8) and (9)). Each cell represents a particular combination of moneyness and maturity. The ﬁrst and second numbers reported for each combination are the mean percentage error (MPE) of the HBS and CAN estimates respectively over 10,000 simulations. The third number is a modiﬁed canonical estimate (CON) constrained to price an at-the-money option correctly. The fourth number is an estimate (IBS) from the Black-Scholes formula using the implied volatility of the at-the-money option which is assumed to be correctly priced.

Overall, there is evidence that canonical valuation is increasingly useful when the dynamics of the underlying asset exhibit stochastic volatility. Modifying canonical valuation to incorporate a single market option price is clearly the preferred valuation technique. In both Black- Scholes and Heston worlds, the constrained canonical estimator consistently outperforms CAN, HBS, and IBS estimates.

Sensitivity to Heston Parameters The two key parameters in Heston’s (1993) model that drive the volatility generating process are the speed of mean reversion k and the volatility of the volatility process j. Different values of these parameters can signiﬁ- cantly influence the dynamics of volatility, and are therefore likely to affect the resultingcanonical and HBS estimates. For example, as j

TLFeBOOK 14 Gray and Newman

TABLE V MAPE of Canonical and Black-Scholes Estimates in a Stochastic-Volatility World

Time to expiration (years) Moneyness (spot/strike) 1͞52 1͞13 1͞41͞23͞41

Deep out-of-the-money n/a 0.6317 0.3855 0.2580 0.1872 0.1489 (0.90) n/a 0.5649 0.1859 0.1248 0.1053 0.0922 n/a 0.5807 0.1358 0.0664 0.0444 0.0333 n/a 0.8420 0.3170 0.1761 0.1185 0.0868 Out-of-the-money 0.1893 0.0789 0.0805 0.0768 0.0674 0.0616 (0.97) 0.3344 0.1140 0.0708 0.0628 0.0603 0.0564 0.1671 0.0396 0.0161 0.0098 0.0072 0.0058 0.1218 0.0652 0.0403 0.0270 0.0200 0.0156 At-the-money 0.0874 0.0453 0.0429 0.0465 0.0447 0.0434 (1.00) 0.0954 0.0564 0.0476 0.0473 0.0478 0.0461 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 In-the-money 0.0225 0.0302 0.0283 0.0309 0.0316 0.0319 (1.03) 0.0172 0.0268 0.0319 0.0357 0.0379 0.0376 0.0075 0.0083 0.0065 0.0050 0.0041 0.0035 0.0071 0.0154 0.0175 0.0148 0.0120 0.0100 Deep in-the-money 0.0000 0.0035 0.0149 0.0176 0.0179 0.0178 (1.125) 0.0000 0.0023 0.0092 0.0149 0.0185 0.0201 0.0000 0.0022 0.0058 0.0066 0.0066 0.0064 0.0000 0.0032 0.0169 0.0225 0.0222 0.0205

Note. Canonical, HBS and IBS estimates are compared to the true Heston call price, where stock prices are simulated in Heston’s (1993) stochastic volatility world (Equations (8) and (9)). Each cell represents a particular combination of moneyness and maturity. The ﬁrst and second numbers reported for each combination are the mean absolute percentage error (MAPE) of the HBS and CAN estimates respectively over 10,000 simulations. The third number is a modiﬁed canonical estimate (CON) constrained to price an at-the-money option correctly. The fourth number is an estimate (IBS) from the Black-Scholes formula using the implied volatility of the at-the-money option which is assumed to be correctly priced.

increases, the random component of the volatility generating process has more weight, causing greater movement in volatility. Similarly, as k decreases, volatility mean reverts at a slower rate, also causingvolatility to be driven more by the random component. Thus, when k is low and j is high, volatility is more random and the stock generation process is further removed from the Black-Scholes world. In such a case, it is likely that HBS/IBS estimates are less accurate than when k is high, and/or j is low. A nonparametric technique like canonical valuation may be better equipped to accommodate greater instability in volatility. To examine the sensitivity of pricing errors under alternative methods, the simulation procedure described in the previous section is repeated for assorted combinations of k and j. Values of k range from 2 to 10, and j ranges from 0.10 to 0.50. For each combination, 200 returns are generated under Heston’s model to form the density of future

TLFeBOOK Canonical Valuation of Options 15

FIGURE 1 Sensitivity of pricing errors to parameter values. This ﬁgure graphs the mean pricing errors (MPE) for HBS, CAN, CON, and IBS estimates over 5,000 simulations. Values of k range from 2 to 10, and values of j range from 0.10 to 0.50. The four rows represent MPE for HBS, CAN, CON, and IBS, respectively. The three columns represent out-of-the-money options (0.97), at-the-money options (1.00), and in-the-money options (1.03), respectively. stock prices, from which the price of a call option with three months to expiration is estimated using HBS, IBS, and both canonical approaches. Estimates are compared to the true Heston price to calculate the mean percentage errors across 5,000 simulations.10 Figure 1 shows the sensitivity to values of k and j of the mean percentage error (MPE) under HBS (ﬁrst row), CAN (second row), CON (third row), and IBS (fourth row) estimates. The three columns of

10Fewer simulations are used in this subsection because of the computational demands in running simulations over a wide grid of parameter values. However, even with 5,000 simulations, the graph- ical results display a sufﬁcient degree of smoothness.

TLFeBOOK 16 Gray and Newman

Figure 1 represent out-of-the money options (0.97), at-the-money options (1.00), and in-the-money options (1.03). Examiningrow 1, HBS estimates are generallypositively biased for out-of-the-money and negatively biased for in-the-money options. This conclusion holds across the range of parameter values. A spread of positive and negative pricing errors arises for at-the-money options. Irrespective of moneyness, mispricingis clearly a positive function of j, particularly when k is small. For low k, the volatility of volatility j is the dominant parameter. The higher j, the higher volatility and consequently, the higher the Black-Scholes estimate of option price. Thus, row 1 of Figure 1 shows that, as j increases, HBS overpricingof out-of-the-money options worsens, while HBS underpricingof in-the-money options improves. In row 2 of Figure 1, the (unconstrained) CAN estimates are also highly sensitive to the value of j. As j increases, the magnitude of negative bias increases sharply. The MPEs of CAN estimates are less sensitive to k. Although difficult to read from the graphs, CAN pricing errors improve as k increases. Comparing HBS and CAN estimates, CAN outperforms HBS across the grid of parameter values for out-of-the-money options. This is again encouraging as out-of-the-money options are notoriously difficult to price using the Black-Scholes parametric approach. However, moving into the money, the magnitude of CAN pricing errors is larger than for HBS. This finding is preempted by Table IV, which reports that HBS arguably outperforms CAN as moneyness increase. Figure 1 row 3 clearly illustrates the superior performance of the CON estimator. For out-of-the-money and in-the-money options, the magnitude of CON pricing errors is signiﬁcantly smaller than HBS and CAN estimates.11 Most important, the CON estimator is also highly robust to varying dynamics of the underlying asset. With the possible exception of the low k high j combination, CON pricing errors are extremely small across the grid of parameter values. Turning to the IBS estimator, Figure 1 row 4 is interesting. Table IV shows that MPE for IBS is very respectable, particularly as moneyness increases. However, Figure 1 reveals that IBS performance is not robust across parameter values. MPE is highly sensitive to j, with positive (negative) bias increasing with j for out-of-the-money (in-the-money) options.

11Note that, by construction, the constrained canonical estimate perfectly prices the at-the-money option. The same comment applies to at-the-money IBS estimates.

TLFeBOOK Canonical Valuation of Options 17

Several final general comments can be made regarding Figure 1. First, the magnitude of pricing errors across all estimation methods decreases with moneyness. This is a common ﬁndingin option-pricing literature. Second, of the two parameters, all methods are more sensitive to the volatility of the volatility generating process j, rather than the speed of mean reversion k. Pricingerrors are almost invariant to k values, especially when j is low.

CONCLUSION This study investigates the merit of a nonparametric risk-neutral method for valuing derivative securities known as canonical valuation. Although the Black-Scholes model remains a popular benchmark for the pricing of options because of its ease of implementation, its restrictive parametric assumptions over stock price dynamics give rise to empirical inconsistencies with observed option prices. There are many methods that accommodate these inconsistencies; however, widespread use of such models remains stiﬂed because of their complexity and the requirement of a large quantities of option data as input. Canonical valuation requires no input of option data and is not computationally demanding. This article conducts simulation experiments in Black-Scholes and Heston worlds to explore the potential usefulness of canonical valuation. In a constant-volatility Black-Scholes world, canonical valuation prices call options with less accuracy than an analogous implementation of the Black-Scholes model. The performance edge of HBS estimates is attributable to the fact that the data are simulated precisely in accordance with the Black-Scholes assumptions. However, when estimation of the risk-neutral density under canonical valuation is constrained to correctly price a single at-the-money option, CON clearly outperforms the HBS estimator for most combinations of moneyness and maturity. With the assistance of some token option data, therefore, the canonical approach is able to compete with the Black-Scholes model, even under conditions ideal for the latter. A more useful test of canonical valuation is undertaken in a setting where volatility is stochastic over time. Stock returns are simulated according to Heston’s (1993) stochastic volatility model and canonical and Black-Scholes estimates of option prices are benchmarked against the true Heston price. In this case, the CAN estimator outperforms HBS for deep out-of-the-money options. Perhaps surprisingly, HBS estimates perform well as moneyness increases. Again, the constrained canonical estimator dominates other approaches.

TLFeBOOK 18 Gray and Newman

In summary, the simulation results suggest that, relative to Black- Scholes estimation using historical volatility, the canonical approach has merit. This is particularly the case when a simple constraint is imposed when estimating the risk-neutral density and when the dynamics of the underlying asset depart from Black-Scholes’ constant-volatility assumption. Sensitivity analysis confirms that the accuracy of HBS estimates deteriorates as the asset dynamics become further removed from the Black-Scholes world. Pricing errors are most sensitive to increasing values for the volatility of the volatility process j, yet relatively insensitive to the strength of mean reversion in volatility k. Although the departure of reality from the Black-Scholes world remains an empirical question, this article documents the viability of the constrained canonical estimator even in a constant-volatility setting. An obvious avenue for future research is to investigate the performance of canonical valuation in pricing traded derivatives. In addition, the hedging performance under canonical valuation is another area warranting investigation. Finally, further work is required to explain the consistent negative bias in canonical estimates of option price.

BIBLIOGRAPHY Aït-Sahalia, Y., & Lo, A. (1998). Nonparametric estimation of state-price densi- ties implicit in ﬁnancial asset prices. Journal of Finance, 53, 499–547. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Derman, E., & Kani, I. (1994). Riding on the smile. RISK, 7, 32–39. Foster, F. D., & Whiteman, C. H. (1999). An application of Bayesian option pricing to the soybean market. American Journal of Agricultural Economics, 81, 722–728. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327–343. Hull, J. C., & White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance, 42, 281–300. Hutchinson, J. M., Lo, A. W., & Poggio, T. (1994). A nonparametric approach to pricing and hedging derivative securities via learning networks. Journal of Finance, 49, 771–818. Jackwerth, J. C., & Rubinstein, M. (1996). Recovering probability distributions from option prices. Journal of Finance, 51, 1611–1631. Kon, S. (1984). Models of stock returns—A comparison. Journal of Finance, 39, 147–165. Lin, Y.-N., Strong, N., & Xu, X. (2001). Pricing FTSE 100 index options under stochastic volatility. Journal of Futures Markets, 21, 197–211.

TLFeBOOK Canonical Valuation of Options 19

MacBeth, J., & Merville, L. (1979). An empirical examination of the Black- Scholes call option pricing formula. Journal of Finance, 34, 1173–1186. Melino, A., & Turnbull, S. (1990). The pricing of foreign currency options with stochastic volatility. Journal of Econometrics, 45, 239–265. Merton, R. C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Rubinstein, M. (1985). Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23, 1976, through August 31, 1978. Journal of Finance, 40, 454–480. Rubinstein, M. (1994). Implied binomial trees. Journal of Finance, 49, 771–818. Scott, L. (1987). Option pricing when the variance changes randomly: Theory, estimators, and applications. Journal of Financial and Quantitative Analysis, 22, 419–438. Stein, E., & Stein, J. (1991). Stock price distributions with stochastic volatility. Review of Financial Studies, 4, 727–752. Stutzer, M. (1996). A simple nonparametric approach to derivative security valuation. Journal of Finance, 51, 1633–1652. Stutzer, M., & Chowdhury, M. (1999). A simple nonparametric approach to bond futures option pricing. Journal of Fixed Income, 8, 67–75. Wiggins, J. (1987). Option values under stochastic volatilities. Journal of Financial Economics, 19, 351–372. Zhang, J. E., & Shu, J. (2003). Pricing S&P 500 index options with Heston’s model. Proceedings of IEEE 2003 International Conference on Computational Intelligence for Financial Engineering (CIFE, 2003), March 21–23, 2003, HongKong(pp. 85–92).

TLFeBOOK ON THE ERRORS AND COMPARISON OF VEGA ESTIMATION METHODS

SAN-LIN CHUNG* MARK SHACKLETON

This article discusses convergence problems when calculating Vega (option sensitivity to volatility) that arise from discretization errors embedded in the lattice approach. Four alternative improvements to the traditional binomial method are discussed and investigated for performance. We also propose a new Modified Binomial (MB) Method to calculate Vegas. Numerical results show that although the MB is not the most price accurate of the models, due to its error structure as a function of volatility, it produces the most accurate and fastest Vega estimates. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:21–38, 2005

INTRODUCTION Once the price of an option position has been negotiated and the position established, tracking and maintenance of the hedging properties are essential tasks not only for the buyer but especially for the option seller.

Thanks go to the Editor and Referee for their helpful comments. Financial support from the National Science Council of Taiwan is acknowledged. *Correspondence author, National Taiwan University, 106 Taipei, Taiwan, Republic of China; e-mail: [email protected] Received July 2003; Accepted April 2004

I San-Lin Chung is in the Department of Finance at National Taiwan University in Taipei, Taiwan, Republic of China. I Mark Shackleton is in the Department of Accounting and Finance in the Management School at Lancaster University, United Kingdom.

TLFeBOOK 22 Chung and Shackleton

This is because the option’s risk characteristics change dynamically as the stock price and time to maturity change. Therefore, the so-called “Greeks” (or partial differentials) with respect to model variables must be calculated accurately and repeatedly. For many different option models and numerical estimation methods, there is a body of literature concerning the properties of these Greeks that annotates the problems associated with their numerical pro- cedures. While producing option sensitivities that converge in the number of time steps (grid points or tree nodes), numerical differentiation is hazardous because results converge in an oscillatory fashion (Pelsser & Vorst, 1994; Chung & Shackleton, 2002, for details).1 In addition to the first (second) differentials with respect to price and time variables, Delta, (Gamma), and Theta, there are Greeks that represent differentials with respect to option parameters. The first differential with respect to volatility (and interest rate), Vega (Rho), are also calculated and often quoted even though many option models assume these factors to be constant. This is because these are used in a slightly different, but no less important manner by purchasers and hedgers of options. There are several reasons why an option’s sensitivity to a fixed (or potentially nondynamic) parameter needs to be considered and calculated repeatedly and accurately. It is also clear for option models that explicitly model stochastic volatility and treat it as a variable, why Vega sensitivity is important. First, even if an option model has a fixed volatility, there is the danger of parameter mis-estimation. It may be the case that volatility may be constant but just difficult to estimate accurately with limited data. This means that the sensitivity of a model’s volatility assumption needs to be examined through its Vega to test the sensitivity to any one specific chosen volatility value. Second, even for constant volatility models, potential uncertainty about the exact model specification means that different models may be used not only to estimate the option’s price but also to estimate the impact of the same volatility assumption in different models. Third, for tractability, many practitioners may assume that these last two variables (volatility and interest rates) are known and constant, in the knowledge that this is only an approximation. If the validity of this assumption is questionable, they may use two option positions with

1This is largely due to the fact that changing the number of time steps (or another numerical parameter) changes the number of nodes on either side of the exercise point, therefore the estimated price tends to oscillate around the true price as it converges.

TLFeBOOK Vega Estimation Methods 23

opposite Vega in an attempt to eliminate their volatility risk. Essentially they may estimate Vega from each model with constant volatility, hoping that if volatility actually changes that it will affect both of the mis-specified model prices to the same degree and due to the supposed Vega neutrality—cancel each other out. Finally, there are an increasing number of models that include stochastic rather than fixed volatility. Although in this paper we apply numerical methods to the estimation of Vega sensitivity with fixed volatility, our results relate to other more complex Vega calculation problems. We first review the five methods that are currently used or discussed in the literature. Then we propose a new method, named the Modified Binomial (MB) method and go on to assess the efficiency of each of the five existing methods as well as the new method proposed. We also examine the root mean square error against computational speed (efficiency) of each method. In the final section we present our conclusions.

VEGA ESTIMATION The numerical error of Vega estimates is mainly due to the discretization embedded in the binomial lattice approach, particularly at option expiry where the positioning of nodes with respect to the exercise price is critical. Within the literature, there are at least four proposed solutions that reduce the discretization errors or enhance the rate of convergence compared to the standard tree structure. First, addition of one or more small sections of fine high-resolution lattice within a tree with coarser time and price steps (Figlewski & Gao, 1999). Second, adjustment of the discrete-time solution prior to maturity (Broadie & Detemple, 1996; Heston & Zhou, 2000) or smoothing of the payoffs at maturity (Heston & Zhou, 2000). Finally, allocation and number (the shape and span) of tree nodes (Ritchken, 1995; Tian, 1999; Widdicks, Andricopoulos, Newton, & Duck, 2002). In this section we review the motivation and construction of each method.

Binomial Method In the lattice approach to option pricing and hedging (for example, see Hull, 2000), it is common wisdom in calculating Vega to make a small change, ¢s, to the volatility s and construct a new tree to obtain a new value of the option. This is then used to calculate a numerical derivative. Since the magnitude of the pricing error implicit in the discrete tree

TLFeBOOK 24 Chung and Shackleton

depends on the number of steps in the tree (as well as other things), usually the same number of steps is used so as to minimize potential errors involved in this numerical differentiation. The estimate of Vega is then

P(s ϩ ¢s, n) Ϫ P(s, n) V(s, ¢s, n) ϭ (1) ¢s

where P(s ϩ ¢s, n) and P(s, n) are the estimates of the option price from the original and the new tree, respectively (both with n steps). Note that this is only a numerical differential of the true sensitivity ϭ ¢ V(s, 0, n) lim¢sS0 V(s, s, n) and as such will contain estimation error for the Vega that depends on the size of ¢s and the number of tree steps (n). However, there is a convergence problem similar to that of pricing barrier-type options (Boyle & Lau, 1994; Ritchken, 1995). This method for calculating Vega also faces discretization errors embedded within the lattice approach, particularly with respect to the positioning of final nodes around the exercise threshold. The problem is serious because the two estimates of the option price are obtained using two different trees where the ﬁnal nodes are noncoincident (i.e., same number but differently placed nodes) so that discretization errors in each estimate are imperfectly correlated and do not cancel.

P (s, n) ϭ P (s, n) ϩ e(s, n) est. true (2) ϩ ¢ ϭ ϩ ¢ ϩ Ј ϩ ¢ Pest.(s s, n) Ptrue(s s, n) e (s s, n)

ϩ ¢ Ϫ Pest.(s s, n) Pest.(s, n) V(s, ¢s, n) ϭ ¢s

P (s ϩ ¢s, n) Ϫ P (s, n) eЈ(s ϩ ¢s, n) Ϫ e(s, n) ϭ true true ϩ (3) ¢s ¢s

Thus with imperfectly correlated errors, Vega is always estimated with error. This total error will depend on the way the second error depends on the first. If the best-fit line between them is of unit slope and zero intercept, then their cancellation can be expected and the variance of the residual error will depend on the goodness of ﬁt between the two errors. Hence, investigation of the error dependency is a crucial part of the analysis of Vega estimation. This is conducted in the Numerical Results section.

TLFeBOOK Vega Estimation Methods 25

Adaptive Mesh Model To improve price convergence Figlewski and Gao (1999) propose a method termed the Adaptive Mesh Model (AMM) to reduce the convexity error at the terminal boundary. In their method, one or more small sections of ﬁne high-resolution lattice are added into a tree with coarser time and price steps. This incorporates a ﬁner mesh of values around the critical nodes in the coarse tree and thus achieves greater accuracy by reducing the nonlinearity error. This method has been shown to reduce pricing errors considerably. The cost is that the tree now has different structures in different regions and so is more complex to implement. Furthermore, although the loca- tion of the maximum convexity is know (near the money at maturity), the choice of span around this area is arbitrary. Any number of extra nodes between two and the number of steps already used could be added (in the latter case the tree is again a regular one with twice as many steps). Thus, this method partially smooths the option price with respect to a perturbation of its parameters. With the increased node density around the exercise threshold, price oscillation in n is also reduced. However, in light of Equation (2) the correlation of these errors is shown to be as important as the magnitude, and so the correlation structure of the price errors should also be considered. As will be seen in a latter section a lack of error correlation structure, means that although individually accurate for prices, the AMM method does not actually perform well for Vega estimation.

Binomial Black Scholes The next method is to adjust the discrete-time solution one period prior to maturity. For instance, Broadie and Detemple (1996) proposed the so-called binomial Black Scholes (BBS) method. They augmented the binomial tree method through the addition of Black Scholes prices at the penultimate pricing node, arguing that since one period before maturity the option will revert to either its European value or payoff, these closed-form expressions would be useful in increasing the accuracy of American or other option types. This is akin to using a continuum of new nodes over the last interval to generate a smooth function. Chung and Shackleton (2002) have shown that the numerical differentiation of the BBS prices produces very accurate estimates for

TLFeBOOK 26 Chung and Shackleton

option Deltas and Thetas because the method itself contains a smooth and not discrete function of the stock price and time to maturity. Therefore, it produces option values that are also a smooth function of the initial price. This is especially important when employing numerical differentiation over arbitrarily small changes in a time or price parameter. In this article we further investigate the accuracy of numerical Vegas using the BBS method and show that smoothness in time and stock price space does not necessarily assist estimation of other option sensitivities such as Vega.

Heston Zhou Method The second method, Heston Zhou (HZ), to smooth the option’s payoff at maturity is from Heston and Zhou (2000). They ﬁrst show that the accuracy or rate of convergence of the binomial method depends crucially on the smoothness of the payoff function. Intuitively, if the payoff function at singular (inﬁnite convexity) points can be smoothed, the binomial recursion will be more accurate. They propose an approach that smooths the payoff function of a European option. If g(x) is the payoff function, they suggest setting the smoothed payoff function G(x) as follows

¢ 1 x G(x) ϭ Ύ g(x Ϫ y) dy ¢ 2 x Ϫ¢x where ¢x is the step size of the binomial tree.2 The above transformation is called rectangular smoothing of g(x). The smoothed function G(x) can be easily computed analytically for most payoff functions used in practice. Applying the binomial model to G(x) instead of g(x) yields a rather surprising and interesting result. The associated binomial prices converge now at a rate of 1͞n to the continuous-time limit and this convergence is uniform across the nodes of the binomial tree (in the same paper Heston and Zhou show that without this correction the rate goes with 1͞ 1n ). Against this convergence benefit, this method may suffer greater initial price error, especially for small n.

2 ϭ Ϫ Ϫ 1 2 Heston and Zhou (2000) make a transformation of variables by setting x [ln S (r 2 s )t] and thus model log prices. This smoothing procedure helps numerical differentiation but leads to a price bias. Similar to Jensen’s Inequality, the expectation of a less convex payoff is higher than the more convex payoff.

TLFeBOOK Vega Estimation Methods 27

Ritchken’s Trinomial Method It is possible to reduce the discretization errors embedded in the lattice approach by allocating the nodes of the binomial or trinomial tree so that they match the payoff of the option or satisfy some other specific requirement (e.g., overlapping nodes of two trees as required in calculating Vegas). For example, Ritchken (1995) takes advantage of the flexibility (additional degree of freedom) offered by the trinomial model and proposes an ingenious way to “stretch” the node separation so that one layer of price nodes coincides exactly with the barrier or final exercise price that is problematic. The idea can be applied in constructing trinomial trees for two different volatilities so that their nodes are always coincident. In a standard trinomial tree, the asset price at any given time, can move into three possible states, up, down, or middle, in the next period. ϩ ¢ If St denotes the asset price at time t, then at time t t the prices will be uSt, mSt, or dSt. Parameters are defined as follows

u ϭ els2¢t m ϭ 1 d ϭ eϪls2¢t where l Ն 1 (a dispersion parameter generated by the extra degree of freedom) is chosen freely as long as the resulting probabilities are positive. For fastest convergence, Omberg (1988) suggested setting the ϭ 22p free parameter l 2 (according to a normal density condition). Matching the ﬁrst two moments (i.e., per period mean M and variance © ) of the risk-neutral returns distribution leads to the probabilities associated with these states

u(© ϩ M2 Ϫ M) Ϫ (M Ϫ 1) P ϭ u (u Ϫ 1)(u2 Ϫ 1) ϭ Ϫ Ϫ Pm 1 Pu Pd u2(© ϩ M2 Ϫ M) Ϫ u3(M Ϫ 1) P ϭ d (u Ϫ 1)(u2 Ϫ 1)

2 where3 M ϭ e(rϪq) ¢t and © ϭ M2(es ¢t Ϫ 1) . For the trinomial trees of two different volatilities (s and s ϩ ¢s) to have exactly the same nodes

3q is the dividend yield.

TLFeBOOK 28 Chung and Shackleton

positions, l should be chosen differently for the s and s ϩ ¢s trees so that

els2¢t ϭ el*(sϩ¢s)2¢t (4)

ϭ 22p Therefore, following Omberg we set l 2 but determine l* from s and ¢s accordingly.

ls 22p s l* ϭ ϭ s ϩ ¢s 2 s ϩ ¢s

Now the ﬁnal s and s ϩ ¢s tree nodes are always located at the same points although the probabilities for each branch in the trees are different. This reduces discretization error where nodes and critical boundaries oscillate as a function of step number n. The next section proposes a sixth and new method that also exploits careful node placing to reduce discretization error.

MODIFIED BINOMIAL METHOD We propose a new Modiﬁed Binomial (MB) method to calculate Vegas. Our idea is derived from Amin (1991), who suggested a binomial method to price options where the underlying asset has a time-varying (or functional) volatility. Amin (1991) allowed time step sizes of varying magnitude to cancel out the variable volatility term. Similarly, we increase the number of steps in the s ϩ ¢s tree by two above the s tree so that at maturity all but the two new nodes in the new tree are common and coincident. In other words, if we denote the time step sizes of the s and s ϩ ¢s trees as ¢t, ¢tЈ, we set

T ¢t ϭ n T ¢tЈϭ n ϩ 2 u ϭ es2¢t ϭ e(sϩ¢s)2¢tЈ 1 d ϭ u

¢ ϭ 2n ϩ 2 Ϫ This implies that s ( n 1)s and the volatility increment is ﬁxed in proportion to the volatility by the size of the ﬁrst tree and the number of new nodes at maturity (2), ¢s can still be made arbitrarily

TLFeBOOK Vega Estimation Methods 29

TABLE I Two Binomial Trees (the Larger Reversed in Time to Aid Comparison) With Different Volatilities and Probabilities p, pЈ and All Final Nodes Coincident Except Two. The Four Stock Prices That Generate New Levels Are in Bold.

t ϭ 0 t ϭ ¢t S t ϭ T d t ϭ ¢tЈ t ϭ 0

¢ ϭ T ϩ ¢ ¢ Јϭ T s, t n tree s s, t n ϩ 2 tree S nϩ2 Inadmissable node S0u

a nϩ1 b S0u

n S0u Q a R b

nϪ2 . . . S0u . . . Q o a Ј pS0u R bS0up

Q i nϩ2Ϫi a S0R S0d u bS0 Ϫ Q o a Ϫ Ј 1 pS0d R bS0d 1 p

nϪ2 . . . S0d . . . Q a R b

n S0d

a n+1 b S0d S n+2 Inadmissable node S0d Number of end nodes (n ϩ 1) (n ϩ 3)

(n ϩ 1)(n ϩ 2) (n ϩ 3)(n ϩ 4) Total number of nodes 2 2

small by making n large. Also note that the probability of an up movement as well as the per period discount rate in the new tree are again different.4 The detailed tree structures of the modified binomial method are shown in Table I, note the two extra nodes (inadmissible for the first tree), one at both extremes of the final asset price. Now, as volatility is increased one new node is added above and below the exercise price so that the balance of nodes on either side is less likely to change. Since all but two final nodes are coincident with those in the initial tree, the distance between the exercise threshold and its closest node stays the same when the tree is changed and the potential for oscillation around the exercise threshold is lessened. It is worth discussing that, in the spirit of the extended binomial tree method of Pelsser and Vorst (1994), the modified binomial method

4Probabilities pЈϭ[(er¢tЈ Ϫ d)͞(u Ϫ d)] are smaller than the initial ones p ϭ [(er¢t Ϫ d)͞(u Ϫ d)] because discount factors er¢tЈ are smaller than er¢t[(¢tЈ͞¢t) ϭ (n͞n ϩ 2)] .

TLFeBOOK 30 Chung and Shackleton

and the trinomial method are both essentially one-tree structures because the trees for two different volatilities share almost all of the same nodes. They share the same stock price nodes so that each does not need separate storage during computation, only the option prices need separate storage along with the two probabilities. Therefore, this method and the trinomial method will be more computationally efficiently than the other four methods. All of the six methods discussed in this paper are not mutually exclusive as their features can be combined. For example, the idea of the BBS method can be applied to the trinomial or the modified binomial methods. The AMM method can also be added to the trinomial method to improve accuracy. However, we have considered each method separately to evaluate their individual costs and benefits.

NUMERICAL RESULTS To investigate the convergence properties of Vega by method, numerical values were calculated for the initial and perturbed volatility (s ϭ 40%, ¢ ϭ 2n ϩ 2 Ϫ s s( n 1)) and investigated using differing numbers of total steps in the tree. The same volatility perturbation was used across all methods to ensure a fair comparison. For number of time steps n from 20 to 100 Panels A to F in Figure 1 for the six methods show dollar errors against the known theoretical value (Black Scholes). Panels A to F in Figure 2 show the dollar errors for the two volatilities in scatter graph form for the six methods. Finally, Panels A to F in Figure 3 show the resulting numerical Vega derived from the perturbation around the initial volatility for the six methods (again as a function of number of time steps). Table II shows the root mean square errors for each method for n from 20 to 100 and 500 to 1,000.

Binomial Method It is known from the literature that numerical values oscillate around the true limiting values (the benchmark n S ϱ Black-Scholes value in this case) as n increases. The results here show that this oscillatory convergence is a problem for estimation of Vega as well as other of the so called-Greeks (partial derivative hedge ratios). First, Panel A of Figure 1 reaffirms the result, that neighboring values (in n the number of steps) have pricing errors, which are negatively serially correlated. Panel A of Figure 2 shows that although these

TLFeBOOK Vega Estimation Methods 31

Panel A: Binomial Method Panel B: AMM

0.05 0.05 sigma=0.4 sigma=0.4 0.03 sigma=0.4*sqrt((n+2)/n) 0.03 sigma=0.4*sqrt((n+2)/n) 0.01 0.01

Errors 20

Errors Ϫ Ϫ0.01 20 60 100 0.01 60 100 Ϫ0.03 Ϫ0.03 Ϫ0.05 Ϫ 0.05 number of steps n number of steps n

Panel C: BBS Method Panel D: HZ Method

0.05 0.05 sigma=0.4 sigma=0.4 sigma=0.4*sqrt((n+2)/n) 0.03 sigma=0.4*sqrt((n+2)/n) 0.03 0.01 0.01 Errors Errors Ϫ0.01 20 60 100 Ϫ0.01 20 60 100 Ϫ0.03 Ϫ0.03 Ϫ0.05 Ϫ0.05 number of steps n number of steps n

Panel E: Trinomial Method Panel F: Modiﬁed Binomial Method

0.05 0.05 sigma=0.4 sigma=0.4 0.03 sigma=0.4*sqrt((n+2)/n) 0.03 sigma=0.4*sqrt((n+2)/n) 0.01 0.01 Errors Ϫ0.01 20 60 100 Errors Ϫ0.01 20 60 100 Ϫ0.03 Ϫ0.03 Ϫ 0.05 Ϫ0.05 number of steps n number of steps n

FIGURE 1 Pricing errors as a function of steps.

errors are correlated, they are not perfectly correlated and that there is considerable dispersion around the 45 degree line. Consequently, when (for a fixed level of time steps n) numerical differentiation is employed via a perturbation ¢s ins [Equation (1)], the resulting Vega oscillates around its Black Scholes theoretical value (Panel A of Fig. 3). Convergence is slow in n and the envelope for Vega is roughly a tenth of its level (for small n). Thus as noted by many other authors for option Deltas, the binomial method is also problematic for Vega estimation because of oscillatory convergence.

Adaptive Mesh Model Method The Adaptive Mesh Model (AMM) reduces the pricing error by adding more nodes to the tree in the region near exercise. As can be seen from Panel B of Figure 1, the resulting prices are indeed considerably more

TLFeBOOK 32 Chung and Shackleton

Panel A: Binomial Method Panel B: AMM

0.03 0.03 0.01 0.01 Ϫ0.03 -0.0Ϫ0.01Ϫ 0.01 0.03 Ϫ0.03Ϫ0.01 0.01 0.03 Ϫ0.03 Ϫ0.03 sigma=0.4*sqrt((n+2)/n) sigma=0.4*sqrt((n+2)/n) sigma=0.4 sigma=0.4

Panel C: BBS Method Panel D: HZ Method

0.03 0.03 0.01 0.01 Ϫ Ϫ0.03 Ϫ0.01Ϫ0.01 0.01 0.03 Ϫ0.03 0.01Ϫ0.01 0.01 0.03 Ϫ0.03 Ϫ0.03 sigma=0.4*sqrt((n+2)/n) sigma=0.4*sqrt((n+2)/n) sigma=0.4 sigma=0.4

Panel E: Trinomial Method Panel F: Modiﬁed Binomial Method

0.03 0.03 0.01 0.01 Ϫ0.03 Ϫ0.01Ϫ0.01 0.01 0.03 Ϫ0.04 Ϫ0.03 Ϫ0.02 Ϫ0.01 0.01 0.02 0.03 0.04 Ϫ0.03 Ϫ0.03 sigma=0.4*sqrt((n+2)/n) sigma=0.4*sqrt((n+2)/n) sigma=0.4 sigma=0.4

FIGURE 2 Error correlation across volatilities.

accurate than the Binomial method. For both option estimates (the unperturbed and perturbed volatility) the error envelope is considerably smaller than in the previous case and more importantly, the errors are no longer so negatively serially correlated (across n). Furthermore, the scatter graph of pricing errors under the AMM (Panel B of Fig. 2) is much tighter to the origin, but although it has less dispersion along the 45 degree line, the correlation of these errors is smaller than for the regular Binomial method because the dispersion in the perpendicular to the 45 degree line has not been reduced by the AMM method. Although it does reduce the pricing error, the AMM method inherits and indeed magniﬁes the relative dispersion of the pricing errors. Other methods may produce larger overall errors, but if these errors across two values of s are more dependent they will be easier to eliminate. Thus, the numerical Vega estimates in Panel B of Figure 3 (although not as oscillatory) still only tracks the true Vega within a large (but converging) envelope. This is true for this and the Binomial method

TLFeBOOK Vega Estimation Methods 33

Panel A: Binomial Method Panel B: AMM

10.5 10.5

10 10

9.5 9.5 Vega Vega

9 binomial vega 9 AMM vega true value true value 8.5 8.5 20 40 60 80 100 20 40 60 80 100 number of time steps number of time steps

Panel C: BBS method Panel D: HZ method

9.7 9.7 9.6 9.6 9.5 9.5 9.4 9.4 Vega HZ vega Vega BBS vega 9.3 9.3 true value true value 9.2 9.2 9.1 9.1 20 40 60 80 100 20 40 60 80 100 number of time steps number of time steps

Panel E: Trinomial method Panel F: Modiﬁed Binomial method

9.7 9.7 9.6 9.6 9.5 9.5

9.4 9.4 modified binomial vega Vega Vega 9.3 trinomial vega 9.3 true value true value 9.2 9.2 9.1 9.1 20 40 60 80 100 20 40 60 80 100 number of time steps number of time steps FIGURE 3 Numerical Vegas as a function of steps.

because although the effect of discretization can be reduced, its effect on the error structure and correlation cannot be eliminated. The next pair of methods behave differently and eliminate this last problem but still prove problematic.

Binomial Black Scholes Method The so-called Binomial Black Scholes (BBS) method smooths out the option value function through the addition of the continuous Black Scholes pricing function one period before maturity. Thus, as the number of time steps is varied there are no ﬁnal nodes that can oscillate around the payoff condition. In terms of price accuracy, the BBS performs well (Panel C of Fig. 1). Furthermore, the errors while small in magnitude are also highly correlated. However, the errors decrease at different rates as n increases,

TLFeBOOK 34 Chung and Shackleton

TABLE II RMS (Root-Mean-Squared) Errors of Six Vega Estimates

Number of steps 20 40 60 80 100 500 1000

Binomial 0.4817 0.2727 0.3313 0.2211 0.2254 0.0743 0.0766 (0:00.054) (0:00.086) (0:00.132) (0:00.192) (0:00.266) (0:05.083) (0:19.974) Adaptive 0.1818 0.1311 0.1075 0.0927 0.0890 0.0406 0.0307 Mesh Model (0:00.062) (0:00.117) (0:00.204) (0:00.320) (0:00.469) (0:09.696) (0:38.093) Binomial 0.2971 0.1511 0.1010 0.0759 0.0609 0.0124 0.0060 Black Sholes (0:00.132) (0:00.584) (0:02.383) (0:04.326) (0:06.243) (0:48.092) (1:49.478) Heston 0.1452 0.0736 0.0507 0.0382 0.0303 0.0061 0.0030 Zhou (0:00.052) (0:00.082) (0:00.125) (0:00.195) (0:00.265) (0:05.085) (0:19.730) Trinomial 0.1502 0.0767 0.0498 0.0378 0.0309 0.0061 0.0031 (0:00.057) (0:00.107) (0:00.200) (0:00.317) (0:00.458) (0:10.127) (0:39.822) Modiﬁed 0.1118 0.0546 0.0414 0.0282 0.0231 0.0044 0.0025 Binomial (0:00.050) (0:00.078) (0:00.130) (0:00.187) (0:00.265) (0:04.984) (0:19.460)

1 243 ϭ 2 g 2 ϭ Ϫ Note. RMS 243 iϭ1ei defines the root-mean-squared errors for European puts where the error ei (V *i Vi) ϩ ͞ 0.5 depends on Vi is the accurate (closed form) European Vega, V*i the estimated Vega using s and s((n 2) n) . There are 243 parameter sets used: S ϭ 40, and combinations of K ϭ 35, 40, 45, s ϭ 0.2, 0.3, 0.4, T ϭ 1, 4, 7 months, r ϭ 3, 5, 7%, and q ϭ 2, 5, 8%. The CPU time (in minutes, seconds, hundred of seconds) required to value all 243 options is also given.

therefore the difference in errors is not expected to be zero. This leads to the situation where Equation (1) is inconsistent due to the non-zero mean error difference. The best situation for the two errors is that their best ﬁt line is of unit slope and passes through the origin with a tight ﬁt. Then both errors are highly dependent and their difference is much smaller. The scatter graphs in Panel C of Figure 2 contain the 45 degree line for reference. As can be seen for Panel C of Figure 2, this is not the case for the BBS method so the numerical Vega contains error bias. Thus, the numerical Vega (in Panel C of Fig. 3) although stable across choice of n is inconsistent for all but the highest number of step. This is to say that the option price difference for the higher volatility less the lower is too low compared to the volatility difference itself.

Heston Zhou Method Like the BBS, this method applies smoothing. The D Panels tell a similar story. Price errors themselves are large and declining almost monotonically with n (Panel D of Fig. 1). Although highly correlated across the two volatilities, these errors are not consistent. Their regres- sion line is not of unit slope; it is greater than one so that the error difference again is not zero.

TLFeBOOK Vega Estimation Methods 35

Unlike the previous case, the higher volatility price contains a higher mean error than the lower case as can be seen in Panel D of Figure 2. As a consequence, the Vega estimate is overestimated and converges from above but only slowly (Panel D of Fig. 3). Both of these last two methods that smooth the value function (BBS and HZ) thus lead to less precise Vega estimates even though they may produce prices that are in themselves more accurate. This is because of the structure present in the errors. The next two methods seek to exploit some of the similarity properties of the tree node structure in each volatility case and therefore produce a smaller overall error.

Trinomial Method The trinomial tree method performs quite well in Vega estimation. Price errors are highly correlated and have a best fit very close to the unit slope, zero intercept line. Thus, the errors cancel out well in the numerical Vega estimation and although upward biased (the higher volatility option has the higher error) they converge quite well as n increases. These results are somewhat akin to those of the Binomial method since two binomial steps (with recombination) produce a very similar tree to that of a trinomial. Looking at Panel A of Figure 1, if alternate points as a function of n (say for even n) were considered the results would be similar to the trinomial tree. Indeed, the (upper) envelopes of the price and Vega curves for the Binomial Method are similar to the curves for the Trinomial method in Panel E of Figures 1 and 3. Therefore, it may seem as if the results for the Binomial method could have been substantially improved through the use of even n ordinates only, which would be equivalent to comparing two trinomial trees similar to Ritchken’s method. However, when we tested this method it only seemed to eliminate the oscillatory convergence in the special case where the initial stock price S equaled the exercise price K. For other S͞K ratios, the Vega still converged in an oscillatory fashion. Overall, as can be seen from Panel E of Figures 1 to 3, the Trinomial method while containing some bias, works quite well in terms of Vega estimation.

Modiﬁed Binomial Method Finally, the Modified Binomial (MB) Method is presented. Because it does not add more nodes near expiry or attempt to smooth out the value function by an insertion of a functional form near expiry, the pricing errors are large and suffer the regular oscillatory convergence problem.

TLFeBOOK 36 Chung and Shackleton

Computational time in seconds (log scale) 0.01 0.10 1.00 10.00 100.00 1000.00 1

0.1

0.01 B RMS error (log scale) AMM BBS HZ T MB 0.001

FIGURE 4 RMS error against computational time for six Vega estimation methods.

For this method to work the volatility perturbation has to be chosen to ϭ 2n ϩ 2 5 depend on n in a particular fashion (with s 40% and s n ). Panel F of Figure 1 shows the price of two options. However, because of the positioning of the ﬁnal nodes, these errors are both highly correlated and nearly on the unit slope zero intercept line as can be seen from the scatter plot in Panel F of Figure 2.

COMPARISON OF RMS ERRORS AND TIMES Table II and Figure 4 show root mean squared errors for 243 option prices for each of the six methods. For all 35 ϭ 243 combinations of parameters drawn from combinations of K ʦ 535, 40, 456, T ʦ 51, 4, 76 months, s ʦ 520, 30, 406%, r ʦ 53, 5, 76%, q ʦ 52, 5, 86% (dividend ¢ ϭ 2n ϩ 2 Ϫ yield) numerical Vegas were calculated using s s( n 1) for the perturbation. The errors against the Black Scholes continuous value were squared, averaged, and square rooted to produce an aggregate RMS statistic. This was done repeatedly for differing numbers of time steps n ʦ 520, 40, . . . , 100, 500, 10006 to examine the convergence properties. The time taken for computation is also shown in minutes, seconds, and hundredths of seconds.

5This choice of volatility perturbation is required for the MB method to work but has also been adopted for all other methods to ensure comparability of results. It also has the useful property of decreasing as n increases in a manner that does not distort the numerical differentiation.

TLFeBOOK Vega Estimation Methods 37

Each method almost always yields a lower RMS for increased n (the binomial method being the most striking exception for lower values of n). Computational time always increases with n and all methods require a similar magnitude of time to compute apart from the BBS (which requires normal integrals). However, the AMM and the trinomial method take about twice the time of the binomial, HZ and MB methods because of their extra tree complexity. For many values of n and especially for large n, the binomial method and AMM perform the worst in terms of RMS errors. This is because of the poor correlation between the errors for differing volatilities. Surprisingly, the BBS method also performs poorly for low n as well, however, its performance improves as n increases although this comes with increased computational burden. Although the BBS price errors are well correlated across the volatility and its perturbed value, their slope is not unity so that the Vegas calculated using this method contain bias. The final three methods (Trinomial, HZ, and MB) are quite similar in terms of RMS performance and speed, however (like the AMM method) the Trinomial calculation that contains more nodes and paths takes longer to evaluate than the HZ and MB methods. The new Modified Binomial method performs quite similarly to the HZ method but its errors are always lower for comparable computational times. However, because of the exceptionally good correspondence of pricing errors (they fall on the unit slope line in Fig. 2F), the MB Vega estimates in Panel F of Figure 3, although oscillatory, have low error, are within a tight envelope and converge quickly. This is in contrast to all previous methods. In many applications, this MB method would be highly suitable for robust, quick, and accurate Vega calculation. This is because it uses only two extra (and uniquely different) nodes so the maximum amount of nodes are common to both trees, i.e., its simple tree design lends it more intuitive appeal. This is remarkable given the lack of price accuracy inherent in the Modified Binomial method.

CONCLUSION In conclusion, the model that produces the most accurate option prices may not be the one that is best for calculating partial derivatives via numerical differentiation. This is because the error in the Vega depends not only on the magnitude of the errors present in the two price estimates, but critically on their correlation structure. Models that produce individually accurate but jointly inaccurate prices such as the AMM or BBS will not be good for estimating option

TLFeBOOK 38 Chung and Shackleton

Vegas (and other Greeks). Models that are tailored speciﬁcally to reduce the discretization error near a termination boundary however, do much better in terms of Vega estimation. These models, such as the Modiﬁed Binomial (MB) presented here, are the ones that are also most likely to produce the best Vega results for other option types such as American and Barrier, where no closed form benchmark formulae are available.

BIBLIOGRAPHY Amin, K. I. (1991). On the computation of continuous time option prices using discrete approximations. Journal of Financial and Quantitative Analysis, 26, 477–495. Boyle, P., & Lau, S. H. (1994). Bumping up against the barrier with the binomial model. Journal of Derivatives, 1, 6–14. Broadie, M., & Detemple, J. (1996). American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies, 9, 1211–1250. Chung, S. L., & Shackleton, M. (2002). The Binomial Black-Scholes Model and the Greeks. Journal of Futures Markets, 22, 143–153. Figlewski, S., & Gao, B. (1999). The Adaptive Mesh Model: A new approach to efﬁcient option pricing. Journal of Financial Economics, 53, 313–351. Heston, S., & Zhou, G. (2000). On the rate of convergence of discrete-time contingent claims. Mathematical Finance, 10(1), 53–75. Hull, J. (2000). Options, futures and other derivatives (4th ed.). Englewood Cliffs, NJ: Prentice Hall. Omberg, E. (1988). Efficient discrete time jump process models in option pricing. Journal of Financial Quantitative Analysis, 23, 161–174. Pelsser, A., & Vorst, T. (1994). The Binomial Model and the Greeks. Journal of Derivatives, 1, 45–49. Ritchken, P. (1995). On pricing barrier options. Journal of Derivatives, 3, 19–27. Tian, Y. (1999). A ﬂexible binomial option pricing model. The Journal of Futures Markets, 19, 817–843. Widdicks, M., Andricopoulos, A. D., Newton, D. P., & Duck, P. W. (2002). On the enhanced convergence of standard lattice methods for option pricing. The Journal of Futures Markets, 22, 315–338.

TLFeBOOK THE GLOBAL MARKET FOR OTC DERIVATIVES: AN ANALYSIS OF DEALER HOLDINGS

EKATERINA E. EMM GERALD D. GAY*

We provide a descriptive examination of the trading activities of one of the most important intermediaries in global financial markets—the OTC derivatives dealer. These dealers play a central role in the provision of derivative products and in the intermediation of market risks faced by financial and nonfinancial firms alike. Utilizing a unique database, we analyze the derivatives holdings of 264 dealers spanning 34 countries over the period 1995–2001. We document the geographic composition of dealers on both country and regional levels as well as analyze trends in dealer holdings on an aggregate and individual product level. We further analyze the extent of global merger activity among dealers and resulting consolidation effects. Finally, we investigate at the individual dealer level

The authors gratefully acknowledge the helpful comments and suggestions of Milind Shrikhande, Ufuk Ince, and Anna Agapova along with those of an anonymous reviewer. We have beneﬁted from discussions with Andrei Osonenko of Swaps Monitor Publications regarding the data used in this analysis. *Correspondence author, Department of Finance, J. Mack Robinson College of Business, Georgia State University, MSC 4A1264, Gilmer Street SE, Unit 4, Atlanta, GA 30303; e-mail: [email protected] Received September 2003; Accepted February 2004

I Ekaterina E. Emm is Genevieve Albers Visiting Fellow at Seattle University in Seattle, Washington. I Gerald D. Gay is a Professor and Chairman in the Department of Finance at J. Mack Robinson College of Business at Georgia State University in Atlanta, Georgia.

TLFeBOOK 40 Emm and Gay

INTRODUCTION In this article we provide a descriptive look at the trading activities of one of the most important intermediaries in global financial markets—the over-the-counter (OTC) derivatives dealer. The proliferation and use of derivatives during the past two decades, especially those traded OTC, has been among the most spectacular developments in financial markets. According to recent estimates by the Bank of International Settlements (2003), as of year-end 2002 the global outstanding notional amount of OTC derivatives (e.g., swaps, forwards, and options) had grown to over $141.7 trillion.1 Clearly, the growth in this market has been driven by the needs of firms seeking risk management solutions and the ability of financial engineers and dealers to respond. This study furthers our understanding of the structure of this important market and complements an existing literature that has largely focused on the demand for hedging services. In particular, an extensive literature has emerged that analyzes rationales for hedging and the demand for derivatives, typically presented from the perspective of the corporate end user.2 In contrast, the supply side has received little atten- tion, especially in regards to the activities of OTC derivatives dealers who play a central role in the provision of derivative products and in the intermediation of market risks faced by financial and nonfinancial firms alike.3 The study is also important in that it addresses the extent to which this market has evolved along international dimensions and in specific product offerings.

1By comparison, the BIS also reports the global exchange-traded derivatives market at $23.9 trillion notional outstanding as of year-end 2002. A comparison of the market structures of OTC and futures exchange trading is discussed in Kamara (1988) and Stulz (2003). 2For reviews of this literature see, for example, Smithson (1998), Allayannis and Ofek (2001) and the seminal papers of Smith and Stulz (1985), Froot, Scharfstein, and Stein (1993) and DeMarzo and Duffie (1995). Also, Bartram, Brown, and Fehle (2003) provide a recent examination of the hedging practices and determinants of derivatives usage of a comprehensive and global sample of nonnonfinancial firms. 3Studies by Sun, Sundaresan, and Wang (1993), Kambhu, Keane, and Benadon (1996), and Malhotra (1997) discuss the role of derivatives dealers in facilitating the passage of price risk from end users to other market participants. Dodd (2002) describes various organizational forms along which derivatives dealers provide intermediation services. These include traditional dealer markets wherein bids and offers are quoted (often orally over the telephone or on electronic bulletin boards) with transactions negotiated on a bilateral basis, as well as electronic trading platforms in which bids and offers are posted and trades are executed against these quotes.

TLFeBOOK Global Market for OTC Derivatives 41

To these ends we analyze the derivatives holdings of 264 dealers spanning 34 countries over the period 1995–2001. Our analysis uses a unique database obtained from Swaps Monitor Publications, Inc. This Database of Dealer Outstandings provides what we understand to be the most comprehensive collection of disaggregated holdings information of OTC derivatives dealers and thus permits identification of specific dealers and their derivatives holdings on a longitudinal basis.4 Position information includes not only total derivative holdings (measured in notional dollars outstanding), but in most instances “asset group” breakdowns for interest rate, currency, equity, commodity, and credit derivatives. For many dealers, further breakdowns within these asset groups are reported for specific “product lines” including swaps, forwards, and option contracts. Our investigation begins with a global perspective on the size of this market and trends that have occurred in both aggregate holdings and subgroupings over the 1995–2001 period. We find that the overall OTC derivatives market has grown substantially, having more than doubled in size from our initial 1995 reported numbers. The growth is primarily driven by trade in interest rate derivatives, which make up roughly three- fourths of the market. However, despite observing growth in most of the other derivative asset groups, trade in currency derivatives has been relatively flat with total positions actually somewhat below peak levels of 1997, a pattern we attribute to the consolidation of several currencies into the euro. With respect to specific product lines, we find that swaps are overall the most prevalent interest rate derivative. However, forward contracts are the leading instrument among the currency derivatives, whereas options lead in the case of both equity and commodity derivatives. Next, we document the distribution of dealers by country and by geographic region (e.g., North America, Europe, and Asia/Pacific) and make comparisons on the basis of the number of active dealers and global market share. We find that although the United States is the leading country in terms of both the number of dealers and global market share, there is substantial activity conducted by dealers based in Germany, Japan, Britain, and France. In fact, on a geographic basis, the

4The only other data source known to us that is similar in this regard is The Ofﬁce of Comptroller of Currency’s (OCC) Bank Derivatives Report, which is based on information collected from Reports of Condition and Income (call reports). The main drawback of OCC data set, however, is that it is limited to banking organizations in the U.S.

TLFeBOOK 42 Emm and Gay

European region has the largest contingent of dealers and market share. Using individual dealer market shares, we compute levels of and show trends in dealer concentration on both a global level and within the United States. On both levels, dealer concentration has grown significantly over our sample period. For example, the 4- and 20-firm concentration ratios for the global population of dealers have risen from 14% to 28% and 48% to 67%, respectively. In the United States, the statistics are even higher, with ratios having risen to 69% and 98%, respectively. We briefly discuss potential concerns that these statistics may suggest regarding systemic risk in the financial system. We also analyze the extent of global merger activity that has occurred among derivatives dealers and discuss various effects on industry structure. Our final set of analyses focuses on the extent of asset group and product line offerings of dealers. Attesting to their market breadth, the number of dealers making markets in the various asset groups is large and is greatest in interest rate and currency derivatives (an average of 197 and 201 dealers, respectively). This is followed by equity derivatives (116 dealers), commodity derivatives (66 dealers), and credit derivatives (27 dealers). Across each of the North American, European, and Asian/Pacific regions, comparable percentages of dealers make markets in interest rate and currency derivatives (more than 90% of dealers in each region) and equity derivatives (more than 50% participation). However, significant regional differences exist in commodity and credit derivatives, with North American and Asian/Pacific dealers being the most active, and European dealers the least active. Consistent with these findings, dealers in the North America and the Asia/Pacific regions offer the largest array of individual product lines. The rest of this article is organized as follows: the next section, Database Description, describes the data set on which we base our analysis. Aggregate Dealer Holdings provides statistics on the size and trends of aggregate dealer holdings along with breakdowns for holdings in various derivative asset groups and product lines. Geographic Composition and Concentration Levels documents the geographic composition of dealer activity along with concentration levels, while Merger Activity presents our merger analysis. In Extent of Asset Group and Product Line Participation, we analyze, at the individual dealer level, the extent of participation in each of the various asset groupings and product lines. The last section provides concluding remarks.

TLFeBOOK Global Market for OTC Derivatives 43

DATABASE DESCRIPTION Our study uses the June 2003 edition of the Database of Dealer Outstandings, published by Swaps Monitor, Inc.5 Our analysis focuses on all dealers reported in the database disclosing derivatives holdings information over the 1995–2001 period. This entails 264 dealers spanning 34 countries. Swaps Monitor states that their database consists of “all reli- ably accurate data that have been publicly disclosed by all dealers since 1994.” Sources used by Swaps Monitor to derive derivatives holdings include “audited financial statements, regulatory filings, reports to share- holders, or other documents subject to similar standards of accuracy.” To be considered a dealer for inclusion in the database, at least one of three criteria must be met: The firm is a primary member of the International Swaps and Derivatives Association (ISDA) the firm has total derivatives for trading purposes of at least $10 billion; or the firm has commodity or equity derivatives for trading purposes of at least $1 billion. All positions are reported in notional U.S. dollar amounts with non-U.S. dollar holdings converted to dollars at the prevailing exchange rate as of the balance sheet date.6 For reference to the interested reader, we present in Appendix A an alphabetic listing of all dealers that appear in the database for the 1995–2001 period. For each dealer, we identify their country of origin and all years for which position information is reported. In addition, we indicate those years when a dealer ceased reporting because of being acquired by or merging with another dealer (denoted by M), and in the last column we identify the acquiring firm. We also note those dealers and years in which Swaps Monitor has indicated that either disclosure

5Swaps Monitor Publications, Inc. is a private company located at 401 Broadway, Suite 610, New York, NY 10013; telephone 212-625-9380. Until 1997 the ﬁrm published its annual Database of Users of Derivatives, which focused on the derivatives holdings of end-users. The ﬁrm has since continued to serve as a leading industry vendor of derivatives data focusing primarily on the provision of quantitative information regarding the activities of derivatives dealers. 6A number of entities are now conducting regular surveys of various segments of the OTC derivatives markets. Unfortunately, information on individual dealer holdings is not made available as surveyed information is kept on a confidential basis with only aggregated information reported. Among the more comprehensive surveys are those conducted by the International Swaps and Derivatives Association (ISDA), the Bank for International Settlements (BIS), and the British Bankers’ Association (BBA). Initiated in 1989, the ISDA Market Survey is conducted semiannually and covers the holdings of their primary membership. The BIS publishes 2 surveys: The Regular OTC Derivatives Market Statistics, which has been conducted semiannually since June 1998, and The Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity which has been published triennially since 1995. Both surveys are based on information collected by the central banks of the G10 countries on major banks and dealers. Finally, the BBA has conducted their Credit Derivatives Survey every two years since 1996.

TLFeBOOK 44 Emm and Gay

has ceased or is not available (denoted by DC). These omissions occur mainly in 2001 and were caused, in large part, by changes in reporting requirements. The implementation of Financial Accounting Standard (FAS) 133 and revisions to International Accounting Standard (IAS) 39 led many firms to report market or replacement values rather than notional holdings.7 Finally, we indicate any change in name that a dealer may have experienced. Swaps Monitor reports dealers’ OTC holdings on three levels. First, for each dealer, information is provided on their total notional outstandings for all derivatives positions combined. Second, for most dealers, Swaps Monitor provides breakdowns of total dealer holdings in each of ﬁve “asset” groups: interest rate, currency, equity, commodity, and credit derivatives. Third, for about three-fourths of all dealers, further breakdowns of these asset group holdings are provided along three product lines: swaps, forwards, and options (with the exception of credit derivatives). While acknowledging that our data set does not contain an exhaustive listing and breakdown of all derivatives holdings of all dealers, we still report and discuss our ﬁndings as though it captures a reasonable representation of the global OTC dealer market.

AGGREGATE DEALER HOLDINGS To provide context to our subsequent analysis, we begin by presenting the reader with an annual summary of global dealer holdings with breakdowns for each derivative asset group and, within each group, each product line.8 These annual totals for the period 1995–2001, reported in terms of notional dollar holdings, are provided in Table I. Before com- menting on the ﬁndings, we note that total dealer holdings will exceed total actual outstanding positions resulting from interdealer transactions. That is, when a dealer’s counterparty is another dealer, as opposed to an end user, the derivative will show up on the books of both dealers. Although our database does not provide information about counterparties, statistics reported in various editions of the ISDA Operations Benchmarking Survey indicate that dealers consider 35% to 40% of their customers to be professional counterparties (e.g., other dealers). Still,

7To ascertain the potential magnitude of these 2001 omissions on our 2001 analysis, we computed the year 2000 aggregate market share of these same dealers. This computed to be 8.9% of the global total of which 6.1% was attributed to two dealers, Goldman Sachs and Morgan Stanley. 8As discussed above, because of incomplete disclosure by some dealers, totals for product lines within an asset group will not necessarily equal the total reported for that asset group, nor will totals across asset groups equal the total reported outstanding.

TLFeBOOK TABLE I Global Dealer Holdings of OTC Derivatives: 1995–2001 (Notional Amounts in Millions of U.S. Dollars)

1995 1996 1997 1998 1999 2000 2001

Panel (a): Aggregate holdings by asset group and corresponding product lines Total Derivatives 77,506,608 92,519,317 110,058,173 138,212,369 146,274,676 161,470,977 163,177,229 Interest rate derivatives 40,684,054 53,696,915 67,736,797 95,239,212 107,981,851 121,292,657 120,677,847 Interest rate swaps 23,759,432 32,774,316 42,023,401 62,806,422 71,974,329 85,513,984 89,540,172 Interest rate forwards 5,691,174 6,237,371 7,914,908 9,072,842 11,152,715 9,989,006 11,336,604 Interest rate options 5,943,583 8,411,520 9,881,377 14,138,893 15,016,579 14,501,067 15,954,142 Currency derivatives 26,127,605 30,964,206 35,140,466 34,232,223 26,288,986 27,252,407 28,213,563 Currency swaps 2,016,376 2,522,180 2,788,281 3,406,985 3,842,552 4,714,237 6,391,315 Currency forwards 18,074,372 20,480,435 20,356,345 20,684,292 15,840,683 15,468,354 15,227,129 Currency options 2,510,345 3,805,983 5,706,037 4,965,583 3,167,199 3,320,318 3,533,331 Equity derivatives 799,759 848,868 1,366,470 2,427,094 2,655,399 3,029,986 2,367,393 Equity swaps 39,682 53,979 88,325 119,095 169,660 190,862 274,759 Equity forwards 15,423 17,733 35,992 51,374 120,047 51,242 56,894 Equity options 640,726 656,335 1,030,517 1,982,536 2,049,303 2,516,334 1,893,702 Commodity derivatives 457,596 556,619 542,569 728,062 1,082,630 2,933,788 663,963 Commodity swaps 27,606 41,617 44,132 82,569 95,057 306,887 112,852 Commodity forwards 121,865 140,339 141,496 150,637 210,273 111,785 71,150 Commodity options 91,237 94,314 107,910 158,988 204,549 361,551 188,664 Credit derivatives 0 17,538 54,447 159,486 314,749 573,622 785,191 Panel (b): Aggregate product line holdings All swaps 25,843,096 35,392,092 44,944,139 66,415,071 76,081,598 90,725,970 96,319,098 All forwards 23,902,834 26,875,878 28,448,741 29,959,145 27,323,718 25,620,387 26,691,777 All options 9,185,891 12,968,152 16,725,841 21,246,000 20,437,630 20,699,270 21,569,839

Note. The table presents aggregate position holdings of OTC derivatives dealers. Total holdings as well as breakdowns for various asset groups and product lines are provided. The sample includes all dealers reported in the Database of Dealer Outstandings published by Swaps Monitor Publications, Inc. TLFeBOOK 46 Emm and Gay

for purposes of our analysis and comments on industry structure, it is important to account for all dealer holdings. As reported in Panel A of Table I, we see that over reported global dealer holdings grew from $77.5 trillion to over $163 trillion over the 1995 through 2001 period. The largest asset group was interest rate derivatives, which, for example, in 2001 exceeded $120 trillion and composed about 74% of the global total. Currency derivatives composed the second largest group with $28 trillion in dealer holdings as of year-end 2001, or 17%. However, the growth in currency derivatives (aside from currency swaps) has not kept pace with that of other groups. To illustrate, in 1997, dealer holdings of currency derivatives peaked in excess of $35 trillion and were about 32% of the global total at that time. By 1999 the total had fallen to $26.2 trillion (18%). One reason for this decline is the introduction of the euro, which has reduced trading in a number of derivatives based on the former individual currencies of the various European Union countries.9 The third largest asset group was equity derivatives, followed by commodity and credit derivatives. In Panel B we present the cumulative yearly totals for each of the various product lines (e.g., swaps, forwards, and options) after summing their respective totals in each asset group. Swaps have the largest annual totals and have been the most rapidly growing product line. In 2001, dealers reported swaps totals of $96.3 trillion, or approximately 60%, of all dealer holdings. This was followed by forward contracts, with reported holdings of $26.7 trillion, and options, with $21.6 trillion. Although swaps, in general, were the most popular overall product line, there were notable exceptions within the various asset groups. As shown in Panel A, among currency derivatives, forward contracts were consistently the most favored product line, and in some years, even currency options had larger totals than currency swaps. Further, in the equity and commodity groups, option totals typically exceeded both those of their swap and forward contract counterparts.

GEOGRAPHIC COMPOSITION AND CONCENTRATION LEVELS We next inspect the global layout of dealer operations by first looking at the number of dealers headquartered in each country as well as aggregate country-level holdings relative to global totals. We then report global

9A similar decline was observed during this time frame in the volume of exchange-traded currency futures at the Chicago Mercantile Exchange and which also has been attributed to the advent of the euro. For a discussion, see “Back to the Futures in Chicago,” Business Week Online, July 14, 2003.

TLFeBOOK Global Market for OTC Derivatives 47

and U.S. concentration levels based on holdings of individual dealers. We also note that dealer activity levels are attributed to the home country of the dealer, although many dealers will conduct trades through ofﬁces in other countries.10

Geographic Distribution and Market Share Table II provides a look at dealer activity by country (and geographic region) over the period 1995–2001. The table is presented in three panels: In Panel A we list the number of dealers in each country by year;11 in Panel B we report the percentage of global derivatives holdings of all dealers in each country by year; and in Panel C we present a rank order comparison of country dealer activity using each of the two measures. In reviewing Panel A, we find that the United States leads all countries in terms of having the largest number of dealers with annual totals typically ranging from the mid- to upper 40s to a peak of 49 dealers in 1997 and 1998. Following the United States with the most dealers are Germany and Japan, each having comparable totals (mid-20s). These countries are then followed by Britain and France, which are also both comparable in dealer totals (mid- to lower teens). As shown in the bottom row of Panel A, the total number of dealers in each year has generally been in excess of 200, with a peak number of dealers occurring in 1997 when 237 dealers reported. By region, Europe has the greatest number of dealers with roughly 50% of the global total and approximately twice the total number of dealers found in North America (i.e., United States and Canada). Although led by Germany, France, and Great Britain, the dominance of the European region in terms of numbers of dealers is also a result of several other countries— such as Italy, Switzerland, Belgium, Denmark, and Austria—having a significant number of dealers. Within the Asia/Pacific region, Japan has the greatest number of dealers, followed distantly by Australia and Singapore. Initially surprising to us was the absence of dealers based in Hong Kong. Though we understand that a significant amount of derivatives activity takes place in Hong Kong, it appears to be originated through the local-based

10To illustrate, our referee cites the case of Canadian Imperial Bank of Commerce. While the dealer is based in Toronto, Canada, it conducts many of its trades in London. 11As discussed earlier, a number of U.S. and European dealers did not report notional holdings in 2001, thus causing 2001 to be an exception in parts of our analysis. We have repeated much of our analysis both with and without the 2001 data and obtain, except where otherwise noted, qualitatively similar ﬁndings.

TLFeBOOK 48 Emm and Gay

TABLE II Geographic Composition of OTC Derivatives Dealers: 1995–2001

Country 1995 1996 1997 1998 1999 2000 2001

Panel A: Number of dealers by country and geographic region North America Canada 8 8 8 8 8 7 7 US 47 47 49 49 46 41 29 Total 55 55 57 57 54 48 36 Europe Austria 6 7 6 5 5 3 1 Belgium 9 9 9 4 4 5 3 Britain 15 16 15 15 14 12 9 Czech 0 1 1 1 1 1 0 Denmark 4 4 4 7 7 5 4 Finland 3 3 3 3 3 3 3 France 16 15 15 15 13 12 10 Germany 26 28 28 26 25 25 21 Greece 1 1 1 2 2 2 0 Ireland 2 2 2 2 2 2 1 Israel 1 1 1 1 1 1 0 Italy 11 11 13 12 8 7 2 Netherlands 6 5 5 5 5 4 3 Norway 3 3 3 3 3 2 1 Poland 0 1 2 2 2 2 1 Portugal 4 4 4 5 4 4 1 Russia 0 1 0 0 0 0 0 Spain 5 5 5 5 4 3 3 Sweden 4 4 4 3 3 3 2 Switzerland 5 6 7 6 6 6 3 Total 121 127 128 122 112 102 68 Asia/Paciﬁc Australia 6 7 7 7 7 6 6 China 1 1 1 0 0 0 0 Hong Kong 0 1 0 0 0 0 0 Japan 22 24 27 24 24 24 23 Korea 1 1 1 1 1 1 0 Malaysia 0 0 1 1 1 1 1 Singapore 3 4 4 4 4 4 1 Thailand 1 1 1 1 1 1 0 Total 34 39 42 38 38 37 31 Other countries Bahrain 2 3 3 3 3 3 3 Brazil 1 1 1 1 1 1 0 Saudi Arabia 1 2 2 2 2 2 0 South Africa 3 4 4 4 6 6 2 Total 710101012125

Grand total 217 231 237 227 216 199 140

TLFeBOOK Global Market for OTC Derivatives 49

TABLE II (Continued)

Country 1995 1996 1997 1998 1999 2000 2001

Panel B: Global market share by country and geographic region (in percentage) North America Canada 3.79 % 3.56 % 3.34 % 2.99 % 2.81 % 2.56 % 3.09 % US 27.80 28.93 32.55 34.30 35.30 37.67 33.13 Total 31.60 32.50 35.89 37.29 38.11 40.23 36.22 Europe Austria 0.47 0.41 0.53 0.32 0.46 0.16 0.04 Belgium 1.51 1.79 1.88 1.46 1.55 1.37 1.62 Britain 9.43 9.69 9.06 8.57 4.76 7.98 9.50 Czech – 0.00 0.00 0.01 0.01 0.00 – Denmark 1.08 1.27 1.27 1.18 1.13 0.81 0.79 Finland 0.51 0.68 0.61 0.39 0.28 0.56 0.48 France 13.18 12.07 11.24 10.85 12.15 11.12 11.81 Germany 6.52 8.16 9.36 10.21 13.03 12.74 15.66 Greece 0.01 0.00 0.00 0.02 0.02 0.02 – Ireland 0.19 0.19 0.18 0.16 0.13 0.11 0.03 Israel 0.01 0.01 0.02 0.01 0.02 0.02 – Italy 1.09 1.27 1.36 1.00 1.32 1.20 0.89 Netherlands 2.06 2.09 2.22 2.69 3.23 2.96 3.27 Norway 0.35 0.44 0.42 0.49 0.34 0.20 0.15 Poland – 0.00 0.00 0.00 0.01 0.02 0.02 Portugal 0.12 0.12 0.19 0.15 0.10 0.09 0.03 Russia – 0.00 ––––– Spain 0.82 0.85 0.80 0.66 0.79 0.77 1.12 Sweden 1.63 1.80 1.54 1.09 0.93 0.72 0.59 Switzerland 6.89 7.27 7.07 10.39 6.26 5.56 7.42 Total 45.85 48.11 47.75 49.64 46.50 46.41 53.43 Asia/Paciﬁc Australia 1.28 1.25 1.22 0.87 0.94 0.85 0.95 China 0.03 0.03 0.03 –––– Hong Kong – 0.03 ––––– Japan 20.79 17.56 14.64 11.80 14.02 12.13 9.10 Korea 0.04 0.05 0.08 0.04 0.04 0.02 – Malaysia – – 0.00 0.00 0.00 0.00 0.01 Singapore 0.04 0.06 0.05 0.04 0.04 0.08 0.14 Thailand 0.02 0.02 0.01 0.01 0.01 0.01 – Total 22.21 19.01 16.04 12.76 15.04 13.10 10.19 Other countries Bahrain 0.05 0.07 0.04 0.03 0.13 0.03 0.05 Brazil 0.02 0.09 0.05 0.03 0.03 0.05 – Saudi Arabia 0.02 0.03 0.02 0.02 0.02 0.01 – South Africa 0.26 0.19 0.22 0.23 0.17 0.16 0.11 Total 0.34 0.38 0.33 0.31 0.34 0.26 0.16

Grand total 100% 100% 100% 100% 100% 100% 100%

Note. The global market share of derivatives holdings for each dealer is calculated based on the dealer’s notional amount outstanding of its reported total derivatives. A dash, “–”, indicates no reporting dealers, and 0.00 indicates a number less than 0.005%. (Continued)

TLFeBOOK 50 Emm and Gay

TABLE II Geographic Composition of OTC Derivatives Dealers (Averaged over 1995–2001) (Continued)

Ranking

By number of dealers By market share (%) Difference Rank Average Rank Average in ranking Country (1) (2) (3) (4) (1–3)

Panel C: Comparison of country rankings US 1 44.0 1 32.81 0 Germany 2 25.6 4 10.81 ؊2 Japan 3 24.0 2 14.29 1 Britain 4 13.7 5 8.43 ؊1 France 4 13.7 3 11.77 1 Italy 6 9.1 11 1.16 ؊5 Canada 7 7.7 7 3.16 0 Australia 8 6.6 13 1.05 ؊5 Belgium 9 6.1 9 1.60 0 Switzerland 10 5.6 6 7.26 4 Denmark 11 5.0 12 1.08 ؊1 Austria 12 4.7 16 0.34 ؊4 Netherlands 12 4.7 8 2.65 4 Spain 14 4.3 14 0.83 0 South Africa 15 4.1 18 0.19 ؊3 Portugal 16 3.7 20 0.11 ؊4 Singapore 17 3.4 21 0.06 ؊4 Sweden 18 3.3 10 1.19 8 Finland 19 3.0 15 0.50 4 Bahrain 20 2.9 22 0.06 ؊2 Norway 21 2.6 17 0.34 4 Ireland 22 1.9 19 0.14 3 Saudi Arabia 23 1.6 27 0.02 ؊4 Poland 24 1.4 31 0.01 ؊7 Greece 25 1.3 30 0.01 ؊5 Israel 26 0.9 28 0.01 ؊2 Korea 26 0.9 24 0.04 2 Thailand 26 0.9 29 0.01 ؊3 Brazil 26 0.9 23 0.04 3 Czech 30 0.7 32 0.00 ؊2 Malaysia 30 0.7 33 0.00 ؊3 China 32 0.4 25 0.03 7 Russia 33 0.1 34 0.00 ؊1 Hong Kong 33 0.1 26 0.03 7

Note. This panel presents a rank order comparison of two measures: the average annual number of dealers and their average annual global market share based on statistics reported in Panels A and B, respectively. Both measures are calculated for each country over the 1995–2001 period. A positive (negative) difference in rankings indicates the degree of improvement (lowering) of a country’s ranking on the basis of market share relative to its ranking by number of dealers.

TLFeBOOK Global Market for OTC Derivatives 51 operations and subsidiaries of a number of U.S. and European-based dealers, such as HSBC (originally the Hong Kong and Shanghai Banking Corporation), Standard Chartered, JPMorganChase and Deutsche Bank (see, e.g., Farooqi, 2002). “Other” countries, which include Bahrain, Brazil, Saudi Arabia, and South Africa, have a small contingent of dealers. In Panel B of Table II we report for each country and year the cumulative notional holdings of all dealer positions expressed as a percentage of the global total. Because position information is reported in the database on a consolidated basis, we attribute the entirety of a dealer’s positions to its home country, though we recognize that dealers may engage in significant cross-border activity. Loosely speaking, we see that dealer activity measured on the basis of market share appears somewhat consistent with that using simple number of dealer totals. One difference is that U.S. dealer activity becomes relatively larger. That is, the United States has only about one-fifth of the global number of dealers, but those dealers conduct more than one-third of the business. Using the year 2000 as a basis for reference, the United States is shown to have approximately 38% market share of the global total. Other leading countries include Germany (13%) followed by Japan (12%), France (11%), Britain (8%) and Switzerland (6%). Together, these six countries account for roughly 88% of the global total. By geographic region, in year 2000 Europe accounted for the largest market share with 46%, followed by North America with 40%, and the Asia/Pacific region with 13%. Looking at trends over our sample period, the market shares of North America and Europe have been somewhat stable with slight increases. The North American increase is driven by the United States and, within Europe, largely by Germany. Further, these increases coincide with a decline in the market share held by Japanese dealers, which has shown a continual drop from a peak of 21% in 1995 to a 9% share in 2001. Japan’s 21% share in 1995 placed it second in the world, behind only the 28% share of U.S. dealers. In Panel C we provide additional perspective as to the relative number of dealers and global market share of each country by comparing each of the two ranking measures. Using data from Panels A and B, we first calculate the average annual number of dealers and the average annual market share for the period 1995–2001 and then report the rank ordering based on each of these two measures. In the last column we compute the difference in rankings under the two measures. A positive (negative) difference in rankings indicates the degree of improvement

TLFeBOOK 52 Emm and Gay

(lowering) of a country’s ranking on the basis of market share relative to its ranking by number of dealers. The same five countries rank in the top 5 under both measures, though Germany notably fell two spots under the market share measure relative to the number of dealers measure. For other countries ranked in the top 20, countries whose rankings by market share fell signiﬁcantly below their rankings by numbers of dealers include Italy and Australia (Ϫ5 each), Austria, Portugal and Singapore (Ϫ4 each) and South Africa (Ϫ3). Countries showing sizeable positive increases in relative rankings using market share include Sweden (ϩ8), Switzerland, Netherlands, Finland, and Norway (ϩ4 each).

Global and U.S. Dealer Concentration Levels Using our global market share percentages computed at the individual dealer level, we next estimate for each year the N-dealer concentration ratios. We repeat these calculations after restricting our sample to U.S. dealers and U.S. market totals. These statistics for years 1995 and 2000 are presented graphically in Figure 1.12 As we observe in Figure 1, for both samples the levels of concen- trations have grown significantly over the years as indicated by the upward shift in the two sets of curves. To illustrate, for the global sample of dealers (see the GL’95 and GL’00 curves) in 1995 the four-firm ratio was 14%, the eight-firm ratio was 23%, and the 20-firm ratio was 48%. In 2000 these percentages had grown to 28%, 42%, and 67%, respectively. For U.S. dealers, concentration levels (see US’95 and US’00) are significantly higher than those measured on a global basis and have also increased over time. In 1995 the four-firm U.S. concentration ratio was 45%, the eight-firm ratio was 70%, and the 20-firm ratio was 98%. By 2000 these had grown to 69%, 89%, and 98%, respectively. In Table III we provide information regarding the leading dealers underlying each pair of concentration curves illustrated in Figure 1. In Panel A we list the 10 leading dealers on a global basis along with their relative market shares for years 1995 and 2000.13 Similarly, Panel B lists the 10 leading U.S. dealers in 1995 and 2000 and their U.S. market

12The results for all other years are available upon request. Also, we present results for the year 2000 instead of 2001 because of the absence of holdings data in 2001 of two large dealers (Goldman Sachs and Morgan Stanley) as discussed earlier in footnote 7. 13Smithson (1995, 1996) reports ranking information of leading OTC dealers for the earlier period of 1992–1994.

TLFeBOOK Global Market for OTC Derivatives 53

100% 90% 80% 70% 60% 50% 40% 30% Cumulative market share 20% 10% 0% 01234567891011121314151617181920 N-dealer concentration ratio

GL '95 GL '00 US '95 US '00

FIGURE 1 Global and U.S. dealer concentration ratios: 1995 versus 2000. Note. The ﬁgure shows N-dealer concentration ratios for the global and U.S. samples of OTC derivatives dealers, using corresponding market share percentages computed at the individual dealer level. The concentration ratios are presented for years 1995 and 2000. The global-market sample contains 217 dealers in 1995 and 199 in 2000. The U.S.-market sample contains 47 dealers in 1995 and 41 in 2000. All statistics are shown for the first 20-firm concentration ratios. GL ’95 and GL ’00 refer to the 1995 and 2000 global concentration ratios, respectively; while US ’95 and US ’00 refer to the 1995 and 2000 U.S. concentration ratios, respectively.

shares. Comparing the 2000 rankings with those in 1995, we see large changes reﬂecting, in part, several incidences of industry consolidation. The most signiﬁcant of these involved Chase Manhattan Bank. In 1995 Chase Manhattan had a global ranking of 21st and market share of 1.68%. Following its subsequent mergers and takeovers involving Chemical Bank, Robert Fleming (U.K.) and J.P. Morgan, JPMorganChase had become by the year 2000 the global leader with a 13.9% global market share and 36.9% U.S. market share. We explore dealer merger activity in greater detail in the next section. Although the concentration numbers just presented appear high, we present them solely as another indicator of industry structure. We acknowledge that a number of concerns have been expressed by policy makers and market observers with respect to the growth and size of the derivatives market and level of concentration therein. Many of these have centered on systemic risk concerns, that is, the risk that a default by a major dealer could cause a domino effect, affecting not only the well-being of immediate counterparties, but spreading and ultimately

TLFeBOOK 54 Emm and Gay

TABLE III Leading Global and U.S. Dealers: 1995 versus 2000

1995 2000

Market Market RankingDealer share (%) RankingDealer share (%)

Panel A: Global dealer rankings based on global market share 1 Chemical (US) 3.97 1 JP Morgan Chase (US) 13.90 2 JP Morgan (US) 3.81 2 Deutsche Bank (Germany) 5.61 3 Societe Generale (France) 3.02 3 Citigroup (US) 4.65 4 Citicorp (US) 2.76 4 Bank of America (US) 3.87 5 Fuji Bank (Japan) 2.47 5 BNP Paribas (France) 3.75 6 Credit Suisse (Switzerland) 2.44 6 Goldman Sachs (US) 3.70 7 NatWest Bank (Britain) 2.42 7 Royal Bank of Scotland 3.47 8 Credit Lyonnais (France) 2.38 (Britain) 9 Swiss Bank Corporation 2.36 8 Fuji Bank (Japan) 3.19 (Switzerland) 9 UBS (Switzerland) 2.96 10 Industrial Bank of Japan 2.36 10 Societe Generale (France) 2.88 (Japan) Panel B: U.S. dealer rankings based on U.S. market share 1 Chemical 14.29 1 JP Morgan Chase 36.90 2 JP Morgan 13.71 2 Citigroup 12.35 3 Citicorp 9.93 3 Bank of America 10.27 4 Bankers Trust 7.03 4 Goldman Sachs 9.82 5 Merrill Lynch 6.45 5 Morgan Stanley 6.40 6 BankAmerica 6.43 6 Merrill Lynch 6.19 7 Goldman Sachs 6.19 7 Lehman Brothers 5.62 8 Chase Manhattan 6.04 8 American International 1.81 9 Lehman Brothers 5.61 Group 10 Salomon 4.75 9 Berkshire Hathaway 1.46 (General Re) 10 Bank One Corporation 1.29

threatening the entire ﬁnancial system.14 However, we note the number of safeguards in place to help prevent such occurrences including regulatory initiatives, such as bank examinations and capital adequacy standards. More important, the dealer community has been proactive in making important advances with the development and use of master agreements, bilateral netting agreements, collateral arrangements, and other risk-mitigation arrangements. For further discussion of these market-based mechanisms for addressing counterparty risk, see, for example, the Group of Thirty (1993), Gay and Medero (1996), and

14See Hentschel and Smith (1995) for an analysis of why systemic risk concerns attributable to derivatives have been overstated.

TLFeBOOK Global Market for OTC Derivatives 55

Weinstein (2003). Also, Bomfim (2002) finds empirical evidence that netting agreements and other credit enhancement mechanisms used in swaps markets have been successful in mitigating counterparty credit risk during periods of market turmoil.

MERGER ACTIVITY We next investigate the extent and implications of merger activity among dealers (see Appendix A for information on dealers that experienced mergers). During the 1995–2001 time period, we identify a total of 54 mergers among dealers. A breakdown of the level of yearly merger activity is provided in the first row of Table IV. The years 1997 and 1999 had the highest occurrence of mergers with 14 and 15 mergers, respectively. We alternatively measure the magnitude of merger activity by computing separately the cumulative premerger market shares of both target and acquiring firms. That is, for each year, we sum the market shares of target firms and also that of the acquirers. These statistics are reported in the second and third rows, respectively, of Table IV. For target firms, with the exception of 1996 and 2001, merger activity in each year was comparable in magnitude with the cumulative premerger market share in the 5%–6% range. For acquirers, 2000 and 1998 were particularly active years, with acquirers having premerger market shares of 16.9% and 9.4%, respectively. The year 2000 results are driven largely by the completion on December 31, 2000. of the merger between JPMorgan and Chase

TABLE IV Dealer Merger Activity: 1995–2001

1995 1996 1997 1998 1999 2000 2001 Number of mergers (54 in total) 5 2 14 8 15 9 1

Target ﬁrms 6.66% 0.08% 4.75% 5.68% 6.52% 5.81% 0.02%

Acquiring ﬁrms 4.64% 3.11% 6.71% 9.42% 7.93% 16.86% 0.49% market share Total 11.30% 3.19% 11.46% 15.10% 14.45% 22.67% 0.52% Cumulative premerger

Note. The table presents the annual number of mergers and the corresponding cumulative pre-merger market shares of the acquiring and target ﬁrms. The sample is comprised of 54 dealer mergers during the period 1995–2001 as reported in the Database of Dealer Outstandings published by Swaps Monitor Publications, Inc.

TLFeBOOK 56 Emm and Gay

Manhattan. This was by far the largest merger among derivatives dealers to date and served to solidify Chase’s position as the largest derivatives dealer in the world. To illustrate, in 1999 Chase Manhattan held the top position with 8.1% of the global dealer market share, while JP Morgan ranked third with 5.4%. In addition to having the highest global market share based on all holdings, Chase also ranked first in interest rate derivatives, third in currency derivatives, twelfth in equity derivatives, fourth in commodity derivatives, and third in credit derivatives. Following the merger, JPMorganChase’s global share rose to 13.9% and it ranked first in interest rate, currency, and credit derivatives; second in equity derivatives; and seventh in commodity derivatives. In Table V we report on the nature of the merger activity on a geographic basis (i.e., U.S., European, and other). We first note that most merger activity has occurred among European dealers. This entailed 35 of the 54 total mergers followed by nine mergers among U.S. dealers. Second, we observe only a few mergers that did not involve either U.S. or European firms (see “other”), because U.S. and European dealers were involved in all but 3 of the 54 mergers. Third, the number of inter- regional mergers of dealers was relatively few. There were four instances of European acquirers taking over U.S. firms, but only two instances of a U.S. dealer acquiring a European target. The most significant case of the former was the Deutsche Bank acquisition of Bankers Trust in 1999. These two dealers had premarket shares of 3.4% and 1.7%, which corresponded to global rankings of fifth and eigh- teenth, respectively.

TABLE V Geographic Breakdown of Dealer Mergers: 1995–2001

Acquirer

US European Other Total

US 9 4 0 13 European 2 35a 1b 38 Other 0 0 3c 3 Target Total 11 39 4 54

Note. The table provides a locational breakdown of acquirers and targets involved in 54 dealer mergers over the period 1995–2001 as reported in the Database of Dealer Outstandings published by Swaps Monitor Publications, Inc. aEuropean mergers were intercountry and 26 were intracountry. Out of nine intercountry mergers, three mergers involved a non-European Union acquirer that was the same Finnish dealer. bThis was a merger between Bahraini and British derivatives dealers. cTwo mergers were between Japanese derivatives dealers and one was between Australian dealers.

TLFeBOOK Global Market for OTC Derivatives 57

We next investigate the extent to which merger activity had an effect on the subsequent product line offerings of the merged entity. For each merger we computed four statistics regarding the subsequent number of product line offerings of the merged entity measured one year (or immediately thereafter) following the merger. These include (1) the number of product lines that had been offered by the target that were subsequently dropped by the acquirer (lines dropped), (2) the number of product lines that had been offered by the acquirer and were exited (lines exited), (3) the number of new products lines offered by the merged entity that had not been formerly offered by the acquirer but were offered by the target (lines added), and (4) the number of new product lines that had not been formerly offered by either the target or acquirer (lines blossomed). We ﬁnd that for product lines dropped, of the 54 mergers there was one line dropped in ﬁve cases, and two lines dropped in two cases. In the remaining 47 mergers, there were no lines dropped. For product lines exited, three acquirers exited one of their existing product lines, while one acquirer exited two product lines. For product lines added, there was one addition in four mergers, and two additions in two mergers. Finally, for product lines blossomed, in nine instances there was an increase of one new product line, and in one instance the merger was followed by the addition of two new product lines. Thus, in sum, it appears that neither the offering of new product lines nor savings from reducing product lines appear to be a primary motive for or a consequence of the majority of the mergers.

EXTENT OF ASSET GROUP AND PRODUCT LINE PARTICIPATION In this section we further analyze the extent of dealer participation in each of the various (1) asset groupings, and (2) product lines. As mentioned earlier, Swaps Monitor reports breakdowns of dealer holdings in ﬁve asset groups: interest rate, currency, equity, commodity, and credit derivatives. Also when available, further breakdowns within each asset group (with the exception of credit derivatives) are also provided along three product lines: swaps, forwards, and options.

Asset Group Analysis We compute and report in Table VI the average annual number of dealers reporting positions for each asset group over the 1995–2001 period. This is done for both the global universe of dealers as well as for each

TLFeBOOK 58 Emm and Gay

TABLE VI Numbers of OTC Derivatives Dealers by Asset Group and Geographic Region (Averaged over 1995–2001)

North Global America Europe Asia/Paciﬁc Other

Interest rate derivatives 197.3 48.9 107.4 32.7 8.3 Currency derivatives 201.7 47.3 110.6 34.7 9.1 Equity derivatives 116.6 29.6 64.4 19.3 3.3 Commodity derivatives 66.7 35.9 19.7 10.3 0.9 Credit derivatives 27.9 14.1 8.4 5.3 0.0 Total number of dealers 209.6 51.7 111.4 37.0 9.4

Note. The table presents the average annual number of dealers reporting positions in each asset group over the period 1995–2001. The numbers are reported for the global population of dealers and those in various geographic regions.

geographic region, that is, North America, Europe, Asia/Pacific, and other. On the global level, the two asset groups having the greatest degree of dealer participation are interest rate and currency derivatives. These groups have comparable averages of 197.3 and 201.7 participating dealers. Equity derivatives run a distant third (116.6 dealers), followed by commodity derivatives (66.7 dealers), and then credit derivatives (27.9 dealers). These orderings are also generally observed within each of the geographic regions. One exception is that within North America, the number of dealers offering commodity derivatives exceeds that for equity. Although credit derivatives have the smallest number of dealers, they have shown rapid growth in recent years. Our yearly analysis shows that dealers offering credit derivatives had grown from zero in 1995 to 59 in 2001, attesting to the rise in popularity of these products. To explore further the nature of dealer participation in various asset group combinations, we address two related questions. First, what proportion of dealers makes markets in various combinations of derivative asset groups? For example, what fraction of all dealers makes markets in both interest rate and currency derivatives? Second, if we observe a dealer who makes a market in one asset group, what is the likelihood that the same dealer makes a market in another specified group? For example, given that a dealer offers commodity derivatives, what is the likelihood that the same dealer also offers equity derivatives? We perform these calculations on a yearly basis for all pairwise group combinations and report averages across years. The calculations are conducted

TLFeBOOK Global Market for OTC Derivatives 59

TABLE VII Derivatives Dealers Offerings: Pair-wise Product Proportions (Averaged over 1995–2001)

Asset group Interest rate Currency Equity Commodity Credit

Panel A: Global Interest rate 94.14 Currency 92.64 96.25 Equity 55.42 55.01 55.62 Commodity 31.42 31.02 25.02 31.83 Credit 13.22 13.09 12.61 9.34 13.29

Panel B: North America Interest rate 94.48 Currency 89.23 91.44 Equity 57.18 55.52 57.18 Commodity 68.51 66.85 49.17 69.34 Credit 27.07 26.52 26.24 26.24 27.35

Panel C: Europe Interest rate 96.41 Currency 96.41 99.23 Equity 57.82 57.82 57.82 Commodity 17.69 17.69 14.74 17.69 Credit 7.56 7.56 7.31 1.67 7.56

Panel D: Asia/Paciﬁc Interest rate 88.42 Currency 87.26 93.82 Equity 50.97 50.97 52.12 Commodity 26.64 26.64 26.25 27.80 Credit 14.29 14.29 12.74 11.20 14.29

Panel E: Other Interest rate 87.88 Currency 87.88 96.97 Equity 34.85 34.85 34.85 Commodity 9.09 9.09 9.09 9.09 Credit 0.00 0.00 0.00 0.00 0.00

Note. The table presents the average fraction of dealers who make markets in various pair-wise derivative asset group combinations over the 1995–2001 period. The panels are constructed using the global sample of dealers and those in each of the geographic regions.

using the global population of dealers and separately for each geographic region. The results for both questions are presented in Tables VII and VIII, respectively. In Panel A of Table VII, for the global set of dealers, we see that 92.6% of all dealers offered both interest rate and currency derivatives,

TLFeBOOK 60 Emm and Gay

TABLE VIII Derivatives Dealers Offerings: Conditional Product Proportions (Averaged over 1995–2001)

Asset group North Conditioned Paired Global America Europe Asia/Paciﬁc Other

Interest rate Currency 98.41 94.35 100.00 98.45 100.00 Equity 59.34 60.41 60.59 53.03 41.96 Commodity 34.08 72.35 18.51 29.42 12.63 Credit 15.72 30.42 9.66 16.39 0.00 Currency Interest rate 96.29 97.33 97.37 92.20 90.48 Equity 57.79 60.58 59.07 52.43 37.30 Commodity 32.96 72.73 18.04 28.83 10.71 Credit 15.41 30.80 9.56 16.08 0.00 Equity Interest rate 99.62 100.00 100.00 84.10 85.71 Currency 98.90 97.25 100.00 84.10 85.71 Commodity 45.72 86.93 25.54 36.49 24.29 Credit 22.94 49.27 14.21 18.01 0.00 Commodity Interest rate 98.79 98.53 100.00 97.94 57.14 Currency 97.57 96.01 100.00 97.94 57.14 Equity 78.64 71.50 83.17 70.16 57.14 Credit 28.65 39.60 11.39 19.15 0.00 Credit Interest rate 85.47 84.92 57.14 71.43 – Currency 84.77 83.33 57.14 71.43 – Equity 82.00 82.53 55.09 39.25 – Commodity 66.91 82.53 10.04 35.15 –

Note. The table reports the average fraction of dealers who, given that they make markets in one asset group, also make a market in a second asset group. These proportions are averaged over the 1995–2001 period for the global sample of dealers and those in each geographic region.

which is by far the highest reported pairwise combination. (Note that numbers along the diagonal simply represent the fraction of dealers offering derivatives in one speciﬁc asset group.) More than one-half of all dealers (55%) made markets in both interest rate and equity derivatives as well as in both currency and equity derivatives. Approximately 31% offered both commodity and interest rate derivatives as well as commodity and currency derivatives, whereas only 25% of dealers offered both equity and commodity products. Across each of the geographic regions (Panels B–E), we see a fairly high percentage of dealers offering both interest rate and currency derivatives. These products appear to be staple offerings of most dealers anywhere. Comparable fractions of dealers (more than 50%) in

TLFeBOOK Global Market for OTC Derivatives 61

North America, Europe, and Asia/Pacific are also offering combinations of equity and interest rate as well as equity and currency derivatives. Notable differences between regions emerge in the areas of commodity and credit derivatives. For example, about two-thirds of North American dealers offered both commodity and interest rate derivatives and commodity and currency derivatives. However, only one-fourth (27%) of Asia/Pacific dealers offered these combinations as did only one-sixth (18%) of European dealers. About one-fourth of North American dealers offered credit derivatives in combination with other derivative groups. These fractions fall to about 14% for Asia/Pacific dealers and to only 7% for European dealers. In the region labeled other, although a reasonable amount of equity derivative activity was observed, dealers otherwise primarily offered only interest rate and currency derivatives. In Table VIII we report on the fraction of dealers who, given that they make markets in one asset group, also make a market in a second group. These percentages are again reported for dealers composing our global population as well as those in each geographic region. Our overall findings are consistent with those in our discussion of Table VII, but we would add a few comments. First, it appears that if a dealer is observed to offer equity derivatives, there is a very high probability that the dealer also offers interest rate and currency derivatives. Second, if a dealer offers commodity derivatives, there is a high probability that the same dealer also offers interest rate, currency, and equity derivatives. Third, a somewhat similar result is also found with respect to credit derivatives in North America. We further explore the various arrays of products offered by dealers in the following section.

Product Lines Analysis For each year we identify those dealers who provided a complete disclosure of their level of participation in swaps, forwards, and options in all asset groups. For each dealer disclosing such information, we then tally the number of individual product lines that the dealer reported positions in. For example, a dealer making markets in interest rate swaps, forwards, and options (three product lines), currency swaps and forwards (two product lines), and commodity options (one product line) would be said to make markets in a total of six product lines. The maximum number of product lines for years 1996–2001 would be 13: three each for interest rate, currency, equity, and commodity derivatives, and one for credit derivatives for which no further breakdowns are provided.

TLFeBOOK 62 Emm and Gay

In 1995, prior to the advent of credit derivatives, the maximum number of product lines would be 12. We are able to make these determinations for approximately 75% of all dealers in each year and thus can draw insights into the array of dealer offerings at the individual product level and structural differences in such offerings over time and across regions. In Panel A of Table IX we report the annual means and standard deviations of the number of derivative product lines for our global sample and each regional set of dealers. As shown in the ﬁrst row, dealer product lines for the global sample have steadily increased from an average of 6.7 in 1995 to 8.6 in 2001, an increase of 1.9 product lines. Within

TABLE IX Trends in Product Line Offerings of Dealers by Geographic Region: 1995–2001

1995 1996 1997 1998 1999 2000 2001

Panel A: Average number of product lines offered by year (standard deviation in parentheses) Global 6.7 6.7 7.3 7.4 7.8 7.9 8.6 (3.2) (3.1) (3.1) (3.1) (3.3) (3.4) (3.6) Number of observations 157 174 186 180 168 151 108 North America 9.3 9.3 9.9 9.7 9.3 9.0 10.1 (3.7) (3.7) (4.2) (4.2) (4.3) (4.5) (4.0) Europe 6.5 6.6 6.7 6.6 7.0 7.1 7.2 (2.2) (2.2) (2.1) (2.2) (2.3) (2.3) (2.2) Asia/Paciﬁc 3.8 4.1 7.0 7.8 8.9 9.8 10.8 (2.0) (2.1) (2.6) (2.2) (3.4) (3.6) (3.7) Other 3.0 4.8 3.2 4.4 3.9 4.0 2.7 (1.0) (3.1) (3.0) (3.6) (1.8) (2.3) (2.1) Panel B: Number of dealers offering the full array of all product lines Global 25 2 22 21 23 25 31 North America 242 2221181615 Europe 10 0 0 1 0 0 Asia/Paciﬁc 00004916 Other 0000000

Note. The table reports in Panel A the average number of product lines offered by dealers globally and in each geographic region and, in Panel B, the number of dealers offering the full array of all 13 product lines (12 in 1995 prior to the advent of credit derivatives). For computing the number of product line offerings, the total sample of dealers available from the Swaps Monitor Database of Dealer Outstandings was restricted to dealers who provided complete disclosure of their level of participation in swaps, forwards and options in each asset group.

TLFeBOOK Global Market for OTC Derivatives 63 regions, dealers in North America and, in later periods, the Asia/Pacific region typically offer the most products. In North America product offerings have expanded slightly from an average of 9.3 per dealer in 1995 to 10.1 in 2001. The offerings of Asian/Pacific dealers have grown remark- ably, rising from 3.8 product lines in 1995 to 10.8 in 2001. European dealers have shown only a slight growth, from 6.5 product lines in 1995 to 7.2 in 2001. In Panel B of Table IX we tabulate the number of dealers in each year who offered the full array of all 13 product lines. Until recently, these dealers have been primarily North American (specifically, U.S. dealers). However, Japanese dealers are now making significant advances in this respect. To illustrate, in 2001 we find 31 dealers who offered the full array of all 13 product lines, including 16 Japanese dealers and 15 from the United States. No other country had a dealer offering the full array of product lines. By comparison, in 1995 there were a global total of 25 dealers who offered all 12 products then available (prior to the introduction of credit derivatives). Of these 25 dealers, 24 were from the United States, and the only other was from Austria. Japan had no dealers offering all 12 products, although there were five Japanese dealers offering 11 products at the time. To further identify the source of the changes in the number of product line offerings between 1995 and 2001, we next compute and compare the fraction of dealers that offered each individual product line in each of these two years. These are reported in Table X both for the global sample and each of the three primary geographic regions. To help illustrate how to interpret this table, consider first the results for the global sample of dealers. In 1995 for the asset group “Interest Rate Derivatives,” 0.89, or 89%, of all dealers offered interest rate swaps (S), 75% offered interest rate forwards, and 82% offered interest rate options (O). Summing these three fractions gives 2.46, which can be interpreted as the average number of interest rate product lines offered by dealers in 1995. Comparing these numbers to those in 2001, we see that the percentage of dealers offering interest rate swaps increased to 95%, an increase of 6% over 1995. This increase of 6% can also be interpreted as contributing 0.06 to the cumulative overall increase in average product lines offered of 1.94 (see last column), a relatively small proportion. For all interest rate derivatives (swaps, forwards, and options combined), the change totaled 0.29, again a small component. For the entire global sample, the asset group providing the largest contribution to the 1.94 overall product line increase was equity derivatives (0.54) followed by

TLFeBOOK TABLE X Changes in Dealer Product Line Offerings: 1995 versus 2001

Interest rate derivatives Currency derivatives Equity derivatives Commodity derivatives Credit Cumulative Year S F O Total S F O Total S F O Total S F O Total derivatives total

Panel A: Breakdown of the number of offerings by product line and asset group Global 1995 0.89 0.75 0.82 2.46 0.80 0.89 0.82 2.51 0.30 0.34 0.43 1.07 0.18 0.22 0.24 0.64 0.00 6.68 2001 0.95 0.87 0.93 2.75 0.89 0.89 0.89 2.67 0.45 0.49 0.67 1.61 0.33 0.37 0.41 1.11 0.48 8.62 Change 0.06 0.12 0.11 0.29 0.09 0.00 0.07 0.16 0.15 0.15 0.24 0.54 0.15 0.15 0.17 0.47 0.48 1.94 North 1995 0.92 0.87 0.92 2.72 0.85 0.92 0.90 2.67 0.69 0.64 0.72 2.05 0.62 0.64 0.62 1.87 0.00 9.31 America 2001 0.96 0.92 0.92 2.80 0.96 0.92 0.92 2.80 0.64 0.64 0.64 1.92 0.64 0.64 0.68 1.96 0.64 10.12 Change 0.04 0.05 0.00 0.08 0.11 0.00 0.02 0.13 ؊0.05 0.00 ؊0.08 ؊0.13 0.02 0.00 0.06 0.09 0.64 0.81

Europe 1995 0.95 0.86 0.84 2.65 0.83 0.86 0.85 2.54 0.24 0.31 0.43 0.98 0.06 0.10 0.16 0.32 0.00 6.49 2001 0.96 0.85 0.96 2.78 0.87 0.87 0.87 2.61 0.24 0.33 0.65 1.22 0.02 0.09 0.15 0.26 0.33 7.20 Change 0.01 ؊0.01 0.12 0.13 0.04 0.01 0.02 0.07 0.00 0.02 0.22 0.24 ؊0.04 ؊0.01 ؊0.01 ؊0.06 0.33 0.71 Asia/1995 0.69 0.23 0.61 1.53 0.65 0.96 0.61 2.23 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.76 Paciﬁc 2001 0.96 0.92 0.92 2.80 0.96 0.96 0.92 2.85 0.77 0.73 0.77 2.27 0.73 0.73 0.73 2.19 0.69 10.80 Change 0.27 0.69 0.31 1.27 0.31 0.00 0.31 0.62 0.77 0.73 0.77 2.27 0.73 0.73 0.73 2.19 0.69 7.04

Note. For each derivatives asset group, we report the proportion of all dealers reporting positions in each speciﬁed product line. In addition, within each of the ﬁve asset groups, we present the average total number of product lines offered by dealers. The last column reports the average cumulative total of product lines offered per dealer. In the table, “S” stands for swaps, “F” stands for forwards, and “O” stands for options. Our sample consists of 157 dealers in 1995 and 108 in 2001. TLFeBOOK Global Market for OTC Derivatives 65

credit derivatives (0.48) and commodity derivatives (0.47). Currency derivatives provided the smallest incremental change, as dealers showed an average increase of only 0.16 product lines. For North American dealers, apart from the offering of credit derivatives, there was little expansion in the number of other offerings. Of the 0.81 overall increase in average product lines offered, 0.64 can be attributed to credit derivatives. For European dealers, the overall increase was 0.71 primarily a consequence of credit derivatives (0.33) and equity options (0.22). Asia/Pacific dealers showed by far the greatest overall increase. Signiﬁcant increases occurred in all asset groups and individual product lines (with the exception of currency forwards). A most significant development among Asian/Pacific dealers related to complete absence in 1995 of the offering of either equity or commodity derivatives. However, by 2001, approximately 75% of dealers in this region were offering a full array of equity and commodity derivatives.

CONCLUSION An extensive literature has emerged that offers theoretical rationales for corporate hedging as well as attempts to provide empirical validation through inspection of firms’ derivatives usage. We complement this literature by focusing on an important supplier of such hedging services, the OTC derivatives dealer. Our analysis is facilitated by a unique longitudinal database, which allows us to inspect and track individual dealer positions. Specifically, we analyze the structure of the OTC derivatives market during the 1995–2001 period with respect to dealer holdings and market share, geographic differentials, merger activity, and the mix of product offerings. The market for OTC derivatives, which had its fledgling beginnings in the early 1980s, has grown into a truly global market. This growth was evidenced by global dealer holdings having more than doubled during the 1995–2001 period of our investigation, a development, we believe, that can be attributed to a regulatory framework that has focused largely on self-regulation and market discipline and that in turn has played a vital part in promoting financial innovation. We observe that the growth in dealer holdings has primarily occurred in interest rate derivatives that now consist of approximately three-fourths of the OTC market. With respect to specific product lines within the various derivative asset groups, we find that swaps are the most prevalent interest derivative, while forward contracts lead among currency derivatives, and options lead in the case of both equity and

TLFeBOOK 66 Emm and Gay

commodity derivatives. On a geographic basis, the United States is the leading country in terms of both the number of dealers and global market share. Among regions, however, Europe has the largest contingent of dealers and market share, led by dealers in Germany, Britain, and France. As a result of merger activity and other factors, dealer concentration levels rose significantly over our sample period. To illustrate, the four- and 20-firm concentration ratios for the global population of dealers rose from 14% to 28% and 48– to 67%, respectively, and were even higher in the %. We find that the breadth of asset group and product line offerings of dealers is large and growing. Globally, the number of dealers making markets in interest rate and currency derivatives averaged 197 and 201 dealers, respectively. This was followed by equity derivatives (116 dealers), commodity derivatives (66 dealers), and credit derivatives (27 dealers). Across each of the North American, European, and Asian/Pacific regions, comparable percentages of dealers make markets in interest rate and currency derivatives (more than 90% of dealers) and equity derivatives (more than 50% participation). However, significant regional differences exist in commodity and credit derivatives, with North American and Asian/Pacific dealers being the most active, and European dealers the least active. Consistent with these findings, dealers in the North America and the Asia/Pacific regions offer the largest array of individual product lines. We conclude by noting that studies such as this should provide additional guidance, and to some extent restraint, to policy makers and other market observers who have expressed concerns over the growing size of this market. Certainly, additional analysis regarding levels of credit exposure and replacement values of positions held by dealers are logical extensions of the work presented here. Further, analysis of the degree to which increasing product line diversification may ameliorate systemic risk concerns would be beneficial. Still, we share the belief that the growth of the global economy has been and will continue to be greatly assisted by the wide availability of OTC derivative products that enable participants to better manage risks.

TLFeBOOK APPENDIX A This appendix (Table A.I) provides a complete alphabetic listing of all derivatives dealers that appear in the Swaps Monitor Database of Dealer Outstandings for the 1995–2001 period. To be considered a dealer in the Swaps Monitor Database, a dealer must meet at least of one of three criteria: The firm is a primary member of ISDA; the firm has total derivatives for trading purposes of at least $10 billion; or the firm has commodity or equity derivatives for trading purposes of at least $1 billion. The sample includes 264 derivatives dealers that represent 34 countries. In the appendix, DC indicates that disclosure has since ceased, and M indicates that the dealer merged or was acquired.

TABLE A.I

Dealer Name Country Years Listed Merged With

1 Abbey National Britain 1995 1996 1997 1998 1999 2000 2001 2 ABN-Amro Netherlands 1995 1996 1997 1998 1999 2000 2001 3 Allied Irish Banks Ireland 1995 1996 1997 1998 1999 2000 DC 4 Amalgamated Banks of South Africa South Africa 1995 1996 1997 1998 1999 2000 2001 5 AMB Holdings South Africa 1999 2000 DC 6 Ambac US 1995 1996 1997 1998 1999 2000 2001 7 Amerada Hess US 1996 1997 1998 1999 2000 2001 8 American Electric Power US 1998 1999 2000 DC 9 American Express US 1995 1996 1997 1998 1999 2000 DC 10 American International Group US 1995 1996 1997 1998 1999 2000 2001 11 Aon Corporation US 1995 1996 1997 1998 1999 DC DC 12 Aozora Bank1 Japan 1995 1996 2000 2001 13 Aquila, Inc.2 US 1995 1996 1997 1998 1999 2000 2001 14 Arab Banking Corporation Bahrain 1995 1996 1997 1998 1999 2000 2001 15 Argentaria Spain 1995 1996 1997 1998 1999 MMBanco Bilbao Vizcaya 16 Artesia3 Belgium 1995 1996 1997 1998 1999 2000 M Dexia 17 Asahi Bank Japan 1995 1996 1997 1998 1999 2000 2001 18 ASLK-CGER Bank Belgium 1995 1996 1997 MMMMFortis 19 Australia and New Zealand Bank Australia 1995 1996 1997 1998 1999 2000 2001 20 Avista Energy US 1997 1998 1999 2000 2001 21 Bacob Bank Belgium 1995 1996 1997 MMMMArtesia 22 Baden-Wurttembergische Bank Germany 1995 1996 1997 1998 1999 2000 2001 23 Banca Commerciale Italiana Italy 1995 1996 1997 1998 1999 MMBanca Intesa

TLFeBOOK (Continued) TABLE A.I (Continued)

Dealer Name Country Years Listed Merged With

24 Banca di Roma Italy 1995 1996 1997 1998 1999 2000 2001 25 Banca Intesa Italy 1997 1998 1999 2000 2001 26 Banca Monte dei Paschi di Siena Italy 1995 1996 1997 1998 1999 2000 DC 27 Banca Nazionale del Lavoro Italy 1995 1996 1997 1998 1999 2000 DC 28 Banca Nazionale dell’Agricoltura Italy 1995 1996 1997 1998 DC DC DC 29 Banco Bilbao Vizcaya Argentaria4 Spain 1995 1996 1997 1998 1999 2000 2001 30 Banco Central Hispanoamericano Spain 1995 1996 1997 1998 MMMSantander 31 Banco Comercial Portugues Portugal 1995 1996 1997 1998 1999 2000 DC

32 Banco di Napoli Italy 1995 1996 1997 1998 1999 MMIstituto Bancario San Paolo di Torino 33 Banco Espirito Santo Portugal 1995 1996 1997 1998 1999 2000 2001 34 Banco Portugues do Atlantico Portugal 1995 1996 1997 1998 MMMBanco Comercial Portugues 35 Banco Santander Central Hispano5 Spain 1995 1996 1997 1998 1999 2000 2001 36 Banco Totta & Acores Portugal 1995 1996 1997 1998 1999 2000 DC 37 Bangkok Bank Thailand 1995 1996 1997 1998 1999 2000 DC 38 Bank Austria Austria 1995 1996 1997 1998 1999 2000 M HypoVereinsbank 39 Bank Brussels Lambert Belgium 1995 1996 1997 MMMMING Group 40 Bank Degroof Belgium 2000 41 Bank fur Arbeit und Wirtschaft Austria 1995 1996 1997 1998 1999 2001 42 Bank Handlowy Poland 1996 1997 1998 1999 2000 M Citigroup 43 Bank Hapoalim Israel 1995 1996 1997 1998 1999 2000 DC 44 Bank of America6 US 1995 1996 1997 1998 1999 2000 2001 45 Bank of China China 1995 1996 1997 DC DC DC DC 46 Bank of Ireland Ireland 1995 1996 1997 1998 1999 2000 2001 47 Bank of Montreal Canada 1995 1996 1997 1998 1999 2000 2001 48 Bank of New York US 1995 1996 1997 1998 1999 2000 2001 49 Bank of Nova Scotia Canada 1995 1996 1997 1998 1999 2000 2001 50 Bank of Scotland Britain 1995 1996 1997 1998 1999 2000 2001 51 Bank of Tokyo Japan 1995 MMMMMMMitsubishi Bank 52 Bank of Yokohama Japan 1996 1997 1998 1999 2000 2001 53 Bank One Corporation7 US 1995 1996 1997 1998 1999 2000 2001 TLFeBOOK 54 Bank Rozwoju Eksportu Poland 1997 1998 1999 2000 2001 55 Bankers Trust US 1995 1996 1997 1998 1999 MMDeutsche Bank 56 Bankgesellschaft Berlin Germany 1995 1996 1997 1998 1999 2000 2001 57 Banque Cantonale Voudoise Switzerland 1995 1996 1997 1998 1999 2000 2001 58 Banque Indosuez France 1995 MMMMMMCredit Agricole 59 Banque Populaire France 1995 1996 1997 1998 1999 2000 2001 60 Banque Worms France 1995 1996 1997 1998 1999 2000 DC 61 Barclays Britain 1995 1996 1997 1998 1999 2000 2001 62 Bayerische Hypotheken-und Germany 1995 1996 1997 MMMMHypoVereinsbank Wechsel Bank 63 Bayerische Landesbank Germany 1995 1996 1997 1998 1999 2000 2001 64 Bear Stearns US 1995 1996 1997 1998 1999 2000 2001 65 Berkshire Hathaway8 US 1995 1996 1997 1998 1999 2000 DC 66 BfG Bank Germany 1995 1996 1997 1998 MMMSkandinaviska Enskilda Banken 67 BHF-Bank Germany 1995 1996 1997 1998 1999 2000 2001 68 BNP Paribas9 France 1995 1996 1997 1998 1999 2000 2001 69 BP Amoco Britain 1997 1998 1999 2000 DC 70 Caisse Centrale des Banques France 1995 1996 1997 1998 MMMNatexis Banques Populaires Populaires 71 Caisse d’ Epargne10 France 1995 1996 1997 1998 1999 2000 2001 72 Caisse des Depots France 1995 1996 1997 1998 1999 2000 2001 73 Caixa Geral de Depositos Portugal 1998 1999 2000 DC 74 Caja de Madrid Spain 1995 1996 1997 1998 1999 2000 2001 75 Canadian Imperial Bank of Canada 1995 1996 1997 1998 1999 2000 2001 Commerce 76 Cassa di Risparmio delle Italy 1995 1996 1997 MMMMBanca Intesa Provincie Lombarde 77 Cera Bank Belgium 1995 1996 1997 MMMMKredietbank 78 Ceskoslovenka Czech 1996 1997 1998 1999 2000 DC 79 Charterhouse Britain 1996 MMMMMCredit Commercial de France 80 Chemical US 1995 MMMMMMJP Morgan Chase 81 Christiania Bank Norway 1995 1996 1997 1998 1999 MMNordea 82 Chuo Mitsui Trust & Banking11 Japan 1997 1998 1999 2000 2001 83 Chuo Trust & Banking Japan 1997 1998 1999 MMMitsui Trust & Banking 84 Ciroﬁ SIM S.p.A. Italy 1998 DC DC DC 85 Citigroup12 US 1995 1996 1997 1998 1999 2000 2001

(Continued) TLFeBOOK TABLE A.I (Continued)

Dealer Name Country Years Listed Merged With

86 Colonial13 Australia 1995 1996 1997 1998 1999 MMCommonwealth Bank of Australia 87 Commerce Asset-Holding Berhad Malaysia 1997 1998 1999 2000 2001 88 Commerzbank Germany 1995 1996 1997 1998 1999 2000 2001 89 Commonwealth Bank of Australia Australia 1995 1996 1997 1998 1999 2000 2001 90 Corpcapital Bank South Africa 1999 2000 91 CPR France 1995 1996 1997 1998 1999 2000 DC 92 Credit Agricole France 1995 1996 1997 1998 1999 2000 2001 93 Credit Commercial de France France 1995 1996 1997 1998 1999 MMHSBC Holdings 94 Credit Lyonnais France 1995 1996 1997 1998 1999 2000 2001 95 Credit Suisse Switzerland 1995 1996 1997 1998 1999 2000 2001 96 Creditanstalt Austria 1995 1996 1997 MMMMBank Austria 97 Credito Italiano Italy 1995 1996 1997 1998 1999 2000 DC 98 Dai-Ichi Kangyo Bank Japan 1995 1996 1997 1998 1999 2000 2001 99 Daiwa Bank Japan 1997 1998 1999 2000 2001 100 Daiwa Securities Japan 1995 1996 1997 DC DC DC DC 101 Den Danske Bank Denmark 1995 1996 1997 1998 1999 2000 2001 102 Den Norske Bank Norway 1995 1996 1997 1998 1999 2000 2001 103 DePfa Germany 1995 1996 1997 1998 1999 2000 2001 104 Deutsche Bank Germany 1995 1996 1997 1998 1999 2000 2001 105 Deutsche Girozentrale Germany 1995 1996 1997 1998 1999 2000 DC 106 Development Bank of Singapore Singapore 1995 1996 1997 1998 1999 2000 2001 107 Dexia14 Belgium 1995 1996 1997 1998 1999 2000 2001 108 Donaldson Lufkin & Jenrette US 1995 1996 1997 1998 1999 2000 M Credit Suisse 109 Dresdner Bank Germany 1995 1996 1997 1998 1999 2000 2001 110 Duke Energy US 1995 1996 1997 1998 1999 2000 DC 111 Dynegy US 1995 1996 1997 1998 1999 2000 2001 112 DZ Bank15 Germany 1995 1996 1997 1998 1999 2000 2001 113 EFG Eurobank Greece 1998 1999 2000 DC 114 El Paso Energy16 US 1995 1996 1997 1998 1999 2000 2001 115 Enron Corp US 1995 1996 1997 1998 1999 2000 DC 116 Erste Bank17 Austria 1995 1996 1997 1998 1999 2000 DC

TLFeBOOK 117 First National Bank of South Africa South Africa 1995 1996 1997 DC DC DC DC 118 First Security US 1995 1996 1997 1998 1999 2000 M Wells Fargo 119 First Union US 1995 1996 1997 1998 1999 2000 2001 Wachovia 120 Fleet Financial US 1995 1996 1997 1998 1999 MMFleetBoston 121 FleetBoston18 US 1995 1996 1997 1998 1999 2000 2001 122 Fortis Belgium 1995 1996 1997 1998 1999 2000 2001 123 Fuji Bank Japan 1995 1996 1997 1998 1999 2000 2001 124 Generale Bank Belgium 1995 1996 1997 MMMMFortis 125 Giro Credit Austria 1995 1996 MMMMMErste Bank 126 Goldman Sachs US 1995 1996 1997 1998 1999 2000 DC 127 Gruppo CRT Italy 1995 1996 1997 1998 MMMCredito Italiano 128 Gruppo Generali Italy 2000 129 Gulf International Bank Bahrain 1995 1996 1997 1998 1999 2000 2001 130 Gulf Investment Bank Bahrain 1996 1997 1998 1999 2000 2001 131 Hambros Bank Britain 1995 1996 1997 MMMMSociete Generale 132 Hamburgische Landesbank Germany 1995 1996 1997 1998 1999 2000 2001 133 HBOS19 Britain 1995 1996 1997 1998 1999 2000 2001 134 Hokkaido Takushoku Japan 1995 1996 1997 DC DC DC DC 135 HSBC Holdings Britain 1995 1996 1997 1998 1999 2000 2001 136 HypoVereinsbank20 Germany 1995 1996 1997 1998 1999 2000 2001 137 IKB Deutsche Industriebank Germany 1995 1996 1997 1998 1999 2000 2001 138 INA Italy 1997 1998 DC DC DC 139 Industrial Bank of Japan Japan 1995 1996 1997 1998 1999 2000 2001 140 ING Group Netherlands 1995 1996 1997 1998 1999 2000 2001 141 Investec Group South Africa 1996 1997 1998 1999 2000 2001 142 Istituto Bancario San Paolo di Torino Italy 1995 1996 1997 1998 1999 2000 DC 143 Istituto Mobiliare Italiano Italy 1995 1996 1997 MMMMIstituto Bancario San Paolo di Torino 144 Joyo Bank Japan 1996 1997 1998 1999 2000 2001 145 JP Morgan US 1995 1996 1997 1998 1999 2000 M JP Morgan Chase 146 JP Morgan Chase21 US 1995 1996 1997 1998 1999 2000 2001 147 Julius Baer Group Switzerland 1997 1998 1999 2000 DC 148 Jyske Bank Denmark 1995 1996 1997 1998 1999 2000 2001 149 Kapital Holding22 Denmark 1995 1996 1997 1998 1999 MMDen Danske Bank 150 KBC Bank23 Belgium 1995 1996 1997 1998 1999 2000 2001 151 KeyCorp US 1995 1996 1997 1998 1999 2000 2001 152 Korean Development Bank24 Korea 1995 1996 1997 1998 1999 2000 DC 153 Kreditanstalt fur Wiederaufbau Germany 1995 1996 1997 1998 1999 2000 2001 154 Labouchere Netherlands 1995 1996 1997 1998 1999 MMDexia

TLFeBOOK (Continued) TABLE A.I (Continued)

Dealer Name Country Years Listed Merged With

155 Landesbank Baden-Wurttemberg25 Germany 1995 1996 1997 1998 1999 2000 2001 156 Landesbank Hessen-Thuringen Germany 1995 1996 1997 1998 1999 2000 2001 157 Landesbank Rheinland-Pfalz Germany 1995 1996 1997 1998 1999 2000 2001 158 Landesbank Sachsen Girozentrale Germany 1996 1997 1998 1999 2000 DC 159 Landesgirokasse Germany 1995 1996 1997 MMMMSudwestdeutsche Landesbank 160 Lehman Brothers US 1995 1996 1997 1998 1999 2000 2001 161 LG & E Energy US 1995 1996 1997 1998 1999 DC DC 162 Lloyds TSB26 Britain 1995 1996 1997 1998 1999 2000 2001 163 Macquarie Bank Australia 1995 1996 1997 1998 1999 2000 2001 164 Maple Partners27 Canada 1995 1996 1997 1998 1999 DC DC 165 MCN Energy US 1995 1996 1997 1998 1999 2000 DC 166 Mees Pierson Netherlands 1995 MMMMMMFortis 167 Mellon Bank US 1995 1996 1997 1998 1999 2000 2001 168 Merrill Lynch US 1995 1996 1997 1998 1999 2000 2001 169 Mitsubishi Tokyo Financial Group28 Japan 1995 1996 1997 1998 1999 2000 2001 170 Mitsubishi Trust & Banking Japan 1995 1996 1997 1998 1999 2000 2001 171 Morgan Stanley US 1995 1996 1997 1998 1999 2000 DC 172 Moscow Narodny Bank Britain 1995 1996 1997 1998 1999 2000 DC 173 Natexis Banques Populaires29 France 1995 1996 1997 1998 1999 2000 2001 174 National Australia Bank Australia 1995 1996 1997 1998 1999 2000 2001 175 National Bank of Canada Canada 1995 1996 1997 1998 1999 2000 2001 176 National Bank of Greece Greece 1995 1996 1997 1998 1999 2000 DC 177 National City Corporation US 1995 1996 1997 1998 1999 2000 2001 178 NationsBank US 1995 1996 1997 1998 MMMBank of America 179 NatWest Bank Britain 1995 1996 1997 1998 1999 MMRoyal Bank of Scotland 180 Nedcor South Africa 1998 1999 2000 DC 181 Niagara Mohawk Holdings US 2000 DC 182 NIB Capital Bank30 Netherlands 1995 1996 1997 1998 1999 2000 DC 183 Nikko Securities Japan 1995 1996 1997 1998 1999 2000 2001 184 NM Rothschild & Sons Britain 1995 1996 1997 1998 1999 2000 2001 185 Nomura Holdings31 Japan 1995 1996 1997 1998 1999 2000 2001 TLFeBOOK 186 Nord LB Germany 1995 1996 1997 1998 1999 2000 2001 187 Nordbanken Sweden 1995 1996 1997 MMMMNordea 188 Nordea32 Finland 1995 1996 1997 1998 1999 2000 2001 189 Norinchukin Bank Japan 1995 1996 1997 1998 1999 2000 2001 190 Northern Trust US 1995 1996 1997 1998 1999 2000 2001 191 Nykredit Denmark 1998 1999 2000 2001 192 Okobank Finland 1995 1996 1997 1998 1999 2000 2001 193 Osterreichische Postsparkasse Austria 1996 1997 1998 1999 2000 DC 194 Overseas Chinese Banking Singapore 1995 1996 1997 1998 1999 2000 DC Corporation 195 Overseas Union Bank Singapore 1996 1997 1998 1999 2000 DC 196 Paine Webber US 1995 1996 1997 1998 1999 2000 M Union Bank of Switzerland 197 PanEnergy Corp US 1995 MMMMMMDuke Energy 198 Paribas33 France 1995 1996 1997 1998 MMMBNP Paribas 199 Peregrine Investments Hong Kong 1996 DC DC DC DC DC 200 PG&E Corporation US 1998 1999 2000 DC 201 PNC US 1995 1996 1997 1998 1999 2000 2001 202 Postbank34 Germany 1995 1996 1997 1998 1999 2000 DC 203 Prudential US 1995 1996 1997 1998 1999 2000 2001 204 Rabobank Netherlands 1995 1996 1997 1998 1999 2000 2001 205 Raiffeisen Zentralbank Osterreich Austria 1995 1996 1997 1998 1999 2000 DC 206 Reliant Energy35 US 1996 1997 1998 1999 2000 DC 207 Republic New York US 1995 1996 1997 1998 1999 MMHSBC Holdings 208 Riyad Bank Saudi Arabia 1996 1997 1998 1999 2000 DC 209 Robert Fleming Britain 1995 1996 1997 1998 1999 2000 M JP Morgan Chase 210 Rossiyskiy Kredit Bank Russia 1996 DC DC DC DC DC 211 Royal Bank of Canada Canada 1995 1996 1997 1998 1999 2000 2001 212 Royal Bank of Scotland Britain 1995 1996 1997 1998 1999 2000 2001 213 Sakura Bank Japan 1995 1996 1997 1998 1999 2000 2001 214 Sal. Oppenheim jr Germany 1995 1996 1997 1998 1999 2000 2001 215 Salomon Smith Barney36 US 1995 1996 1997 1998 MMMCitigroup 216 Sampo37 Finland 1995 1996 1997 1998 1999 2000 2001 217 Saudi American Bank Saudi Arabia 1995 1996 1997 1998 1999 2000 DC 218 Saudi International Bank Britain 1995 1996 1997 1998 1999 MMGulf International Bank 219 Schroders Britain 1995 1996 1997 1998 1999 2000 DC 220 Sempra Energy US 1997 1998 1999 2000 2001

(Continued) TLFeBOOK TABLE A.I (Continued)

Dealer Name Country Years Listed Merged With

221 SGZ Bank Germany 1996 1997 1998 1999 2000 M DZ Bank 222 Shinkin Central Bank38 Japan 1995 1996 1997 1998 1999 2000 2001 223 Shinsei Bank39 Japan 1995 1996 1997 1998 1999 2000 2001 224 Shoko Chukin Bank Japan 1997 1998 1999 2000 DC 225 Skandinaviska Enskilda Banken Sweden 1995 1996 1997 1998 1999 2000 2001 226 Smith Barney US 1995 1996 1997 MMMMSalomon Smith Barney 227 Societe Generale France 1995 1996 1997 1998 1999 2000 2001 228 Spar Nord Bank Denmark 1998 1999 2000 DC 229 St. George Bank Australia 1996 1997 1998 1999 2000 2001 230 Standard Bank of South Africa South Africa 1995 1996 1997 1998 1999 2000 DC 231 Standard Chartered Britain 1995 1996 1997 1998 1999 2000 2001 232 State Street US 1995 1996 1997 1998 1999 2000 2001 233 Sumitomo Bank Japan 1995 1996 1997 1998 1999 2000 2001 234 Sumitomo Trust & Banking Japan 1995 1996 1997 1998 1999 2000 2001 235 Sun Life Financial40 Canada 1995 1996 1997 1998 1999 2000 2001 236 SunTrust Banks US 1995 1996 1997 1998 1999 2000 2001 237 Svenska Handelsbanken Sweden 1995 1996 1997 1998 1999 2000 2001 238 Swedbank Sweden 1995 1996 1997 1998 1999 2000 DC 239 Swiss Bank Corporation Switzerland 1995 1996 1997 MMMMUnion Bank of Switzerland 240 Sydbank Group Denmark 1998 1999 2000 2001 241 Texas Utilities US 1997 1998 1999 2000 2001 242 Tokai Bank Japan 1996 1997 1998 1999 2000 2001 DC 243 Toronto-Dominion Bank Canada 1995 1996 1997 1998 1999 2000 2001 244 Total Fina Elf41 France 1995 1996 1997 1998 1999 2000 2001 245 Toyo Trust & Banking Japan 1995 1996 1997 1998 1999 2000 2001 246 Trinkaus & Burkhardt Germany 1995 1996 1997 1998 1999 2000 2001 247 UBS42 Switzerland 1995 1996 1997 1998 1999 2000 2001 248 UFJ Holdings43 Japan 1995 1996 1997 1998 1999 2000 2001 249 Unibanco Brazil 1995 1996 1997 1998 1999 2000 DC 250 Unibank Denmark 1995 1996 1997 1998 1999 MMNordea

TLFeBOOK 251 Union Bank of Norway Norway 1995 1996 1997 1998 1999 2000 DC 252 Union Europeenne de CIC France 1995 1996 1997 1998 1999 2000 2001 253 United Overseas Bank Group Singapore 1995 1996 1997 1998 1999 2000 DC 254 Wachovia US 1995 1996 1997 1998 1999 2000 2001 255 Wells Fargo US 1995 1996 1997 1998 1999 2000 2001 256 Westdeutsche Genossenschafts Germany 1995 1996 1997 1998 1999 2000 2001 Zentralbank 257 Westdeutsche Landesbank Germany 1995 1996 1997 1998 1999 2000 2001 258 Westpac Australia 1995 1996 1997 1998 1999 2000 2001 259 Williams Companies US 1995 1996 1997 1998 1999 2000 2001 260 Yamaichi Securities Japan 1995 1996 1997 DC DC DC DC 261 Yasuda Trust & Banking Japan 1995 1996 1997 1998 1999 2000 2001 262 Zurcher Kantonalbank Switzerland 1995 1996 1997 1998 1999 2000 DC 263 Zurich Financial Services Switzerland 1998 1999 2000 DC 264 Zurich Group Switzerland 1996 1997 DC DC DC DC

Note. 13formerly, State Bank of New South Wales 25formerly, Sudwestdeutsche Landesbank 34formerly, Deutsche Siedlungs-und 1formerly, Nippon Credit Bank 14formerly, Credit Communal 26formerly, Lloyds Landesrentenbank 2formerly, UtiliCorp United 15formerly, DG Bank 27formerly, First Marathon 35formerly, Houston Industries 3formerly, Banque Paribas Belgique 16formerly, El Paso Natural Gas 28formerly, Bank of Tokyo-Mitsubishi and 36formerly, Salomon 4formerly, Banco Bilbao Vizcaya 17formerly, Die Erste Osterreichische Mitsubishi Bank 37formerly, Leonia and Postipankki 5formerly, Santander Spar-Casse 29formerly, Natexis, Credit National BFCE, 38formerly, Zenshinren Bank 6formerly, BankAmerica 18formerly, BankBoston and Credit National 39formerly, Long-Term Credit Bank 7formerly, First Chicago NBD 19formerly, Halifax 30formerly, De Nationale Investeringsbank 40formerly, Clarica and Mutual Group 8formerly, General Re Corporation 20formerly, Bayerische Vereinsbank 31formerly, Nomura Securities 41formerly, Elf Aquitaine 9formerly, Banque Nationale de Paris 21formerly, Chase Manhattan 32formerly, Merita Nordbanken and Merita 42formerly, Union Bank of Switzerland 10formerly, CCCEP 22formerly, BG Bank and Bikuben Bank Group 43formerly, Sanwa Bank 11formerly, Mitsui Trust & Banking 23formerly, Kredietbank 33formerly, Compagnie Financiere de 12formerly, Citicorp 24formerly, Korea Development Bank Paribas TLFeBOOK 76 Emm and Gay

BIBLIOGRAPHY Abken, P. A. (1993, March/April). Over-the-counter financial derivatives: Risky business? Federal Reserve Bank of Atlanta Economic Review, 1–22. Allayannis, G., & Ofek, E. (2001, April). Exchange rate exposure, hedging, and the use of foreign currency derivatives. Journal of International Money and Finance, 20, 273–296. Bank of International Settlements. (2003, May 8). OTC derivatives market activity in the second half of 2002. Basel, Switzerland: Monetary and Economic Department. Bartram, S. M., Brown, G. W., & Fehle, F. R. (2003). International evidence on financial derivatives usage. Working paper, University of North Carolina, Chapel Hill, North Carolina. Bomfim, A. N. (2002). Counterparty credit risk in interest rate swaps during times of market stress. Working paper, Board of Governors of the Federal Reserve System, Washington, D.C. DeMarzo, P. M., & Duffie, D. (1995). Corporate incentives for hedging and hedge accounting. Review of Financial Studies, 8, 743–771. Dodd, R. (2002). The structure of OTC derivatives markets. The Financier, 9, 1–5. Farooqi, S. (2002, December). Interbank derivatives survey 2002: On top of the market. Asia Risk, 17–26. Froot, K. A., Scharfstein, D. S., & Stein, J. C. (1993). Risk management: Coordinating corporate investment and financing policies. Journal of Finance, 48, 1629–1658. Gay, G. D., & Medero, J. (1996). The economics of derivatives documentation: Private contracting as a substitute for government regulation. Journal of Derivatives, 3, 78–89. Group of Thirty. (1993, July). Derivatives: Practices and principles. Washington, DC: Global Derivatives Study Group. Hentschel, L., & Smith, C. W., Jr. (1995). Controlling risks in derivatives markets. Journal of Financial Engineering, 4, 101–125. Kamara, A. (1988, Winter). Market trading structures and asset pricing: Evidence from the treasury-bill markets. The Review of Financial Studies, 357–375. Kambhu, J., Keane, F., & Benadon, C. (1996, April). Price risk intermediation in the over-the-counter derivatives markets: Interpretation of a global survey. FRBNY Economic Policy Review, 1–15. Malhotra, D. K. (1997, Spring). An empirical examination of the interest rate swap market. Quarterly Journal of Business and Economics, 19–29. Smith, C. W., Jr., & Stulz, R. M. (1985, December). The determinants of firms’ hedging policies. Journal of Financial and Quantitative Analysis, 391–405. Smithson, C. W. (1995). Managing financial risk 1995 yearbook. CIBC Wood Gundy, 55–59. Smithson, C. W. (1996). Managing financial risk 1996 yearbook. CIBC Wood Gundy, 91–94.

TLFeBOOK Global Market for OTC Derivatives 77

Smithson, C. W. (1998). Questions regarding the use of price risk management by industrial corporations. Working paper, CIBC World Markets, New York. Stulz, R. M. (2003). Risk management & derivatives. Mason, OH: Thompson- Southwestern. Sun, T., Sundaresan, S., & Wang, C. (1993, August). Interest rate swaps: An empirical investigation. Journal of Financial Economics, 77–99. Weinstein, J. D. (2003, May). Master netting agreement developments in energy industry. Futures and Derivatives Law Report, 1–6.

TLFeBOOK IS IT TIME TO REDUCE THE MINIMUM TICK SIZES OF THE E-MINI FUTURES?

ALEXANDER KUROV* TATYANA ZABOTINA

On the Chicago Mercantile Exchange (CME), so-called “E-mini” index futures contracts trade on the electronic GLOBEX trading system along- side the corresponding full-size contracts that trade on the open outcry ﬂoor. This paper ﬁnds that the current minimum tick sizes of the E-mini S&P 500 and E-mini Nasdaq-100 futures contracts act as binding constraints on the bid-ask spreads by not allowing the spreads to decline to competitive levels. We also find that, while exchange locals trade very actively on GLOBEX, they do not tend to act as liquidity suppliers. Taken together, our empirical results suggest that it is time for the CME to consider decreasing the minimum tick sizes of the S&P 500 and Nasdaq-100 E-mini futures contracts. A tick size reduction is likely to result in lower trading costs in the E-mini futures markets. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:79–104, 2005

We thank Jeffrey Bacidore, Grigori Erenburg, Dennis Lasser, Kristian Rydqvist, Nancy Scannell, and two anonymous referees for helpful comments and suggestions. We are also grateful to the Commodity Futures Trading Commission (CFTC) staff for their help in obtaining the data used in this paper. Errors or omissions are our responsibility. *Correspondence author, West Virginia University, B&E Building, 1600 University Ave., Room 219, Morgantown, West Virginia, 26506-6025; e-mail: [email protected] Received June 2003; Accepted February 2004

I Alexander Kurov is an Assistant Professor at the College of Business and Economics at West Virginia University in Morgantown, West Virginia. I Tatyana Zabotina is a Visiting Assistant Professor at the College of Business and Management at the University of Illinois at Springﬁeld in Springﬁeld, Illinois.

TLFeBOOK 80 Kurov and Zabotina

INTRODUCTION Domowitz and Steil (1999), Domowitz (2002), and Conrad, Johnson, and Wahal (2003) have shown that automating trade execution leads to significant reductions in trading costs. Technological innovation and demand in the marketplace for efficient trade execution are fueling the momentum in financial markets away from open outcry towards electronic trading. The shift to automated execution has been particularly pronounced in futures markets.1 Although open outcry trading still accounts for a large proportion of the futures trading volume in the United States, both the Chicago Mercantile Exchange (CME) and the Chicago Board of Trade (CBOT) have gradually moved towards offering products traded on electronic platforms. In particular, the CME offers so-called “E-mini” futures contracts that trade on the electronic GLOBEX system.2 The E-mini S&P 500 index futures contracts were introduced in September 1997 and similar E-mini futures on the Nasdaq-100 index were introduced in June 1999. The E-mini contracts are sized at one-fifth of their full-size counterparts traded on the CME floor. Since E-mini trading is particularly affordable to traders with small margin accounts, the E-mini contracts quickly became popular among retail traders.3 The primary contributions of this study are twofold. First, we show that the bid-ask spreads of the E-mini S&P 500 and E-mini Nasdaq-100 futures are constrained by the current minimum tick sizes. We maintain that the binding minimum tick sizes are likely to impede price competition on GLOBEXand may contribute to higher transaction costs in the E-mini futures compared to their floor-traded counterparts. Second, we use trade data with attached trader type identification codes to examine the extent to which exchange locals, clearing members and off-exchange traders demand and supply liquidity on GLOBEX. The main focus of this study is to examine the impact of the current minimum price increments, called tick sizes, on trading costs in the E-mini markets. The minimum tick size of the E-mini S&P 500 index futures is 0.25 index points ($12.50), while for the regular S&P 500

1The London International Financial Futures Exchange (LIFFE), Marché a Terme International de France (MATIF), and the Sydney Futures Exchange (SFE) moved to electronic trading in the last ﬁve years. 2GLOBEX was introduced by the CME in 1992. Until 1997, it was used primarily as an off-hours trading system. 3According to former CME Chairman, Scott Gordon, “The largest growing, the fastest growing seg- ment of our business comes from individual investors that are trading our E-mini products online. It’s by far and away the largest growth area.” Reported in Carlson (2000).

TLFeBOOK Tick Sizes of the E-Mini Futures 81 futures it is only 0.1 index points ($25). The minimum tick sizes of the regular and E-mini Nasdaq-100 futures contracts are equal at 0.5 index points ($50 and $10, respectively). We ﬁnd that the bid-ask spreads of the E-mini contracts rarely exceed the minimum tick sizes, suggesting that the current tick sizes act as binding constraints on the spreads. The bid-ask spread represents the market maker’s compensation for providing liquidity. The costs of liquidity provision, and therefore the spreads, generally vary with market conditions. Assuming that traders rarely use below-cost pricing, our finding that the bid-ask spreads in the E-mini markets are constant at the level of one tick implies that the spreads are above the competitive levels. If the tick sizes are reduced, competition of limit order traders is likely to lead to narrower bid-ask spreads in the E-mini markets. The current binding minimum tick sizes are likely to impede price competition on GLOBEX and lead to suboptimal executions for many limit order traders. When the bid-ask spread is equal to one tick, a limit order that improves the price effectively becomes a market order that hits the best bid or lifts the best offer and is executed immediately. To earn the spread, a trader may submit a limit order that matches the best price. That order is placed at the end of the queue in the limit order book. The trader is unable to increase the probability of execution by tightening the spread. Thus, instead of being presented with a continuum of possible choices, the E-mini traders are faced with a limited number of alternatives. Despite these negative aspects, the E-mini futures are successful products for a number of reasons. In particular, these contracts are affordable to retail traders and offer immediate and anonymous execution. In further analysis, we examine supply and demand of liquidity in the E-mini markets. It is feasible that at the inception of the E-mini contracts the minimum tick sizes were established as sufficiently large in order to attract liquidity by creating arbitrage opportunities between the E-minis and the full-size contracts4 and to create an incentive for the CME locals to act as market makers on GLOBEX.5 Exchange locals play the role of informal market makers in open outcry futures markets by frequently buying at the bid and selling at the offer. When the E-mini contracts were introduced, GLOBEX terminals were installed on the

4For the E-mini S&P futures the arbitrage between the pit and GLOBEX is likely to play a larger role because of the larger tick size of the E-mini contract. 5The bid-ask spread of the regular S&P 500 contract was one tick when the contract started to trade. This binding minimum tick size was appropriate, since it probably helped to attract market makers to that market (e.g., Grossman & Miller, 1988).

TLFeBOOK 82 Kurov and Zabotina

CME floor on the perimeter of the trading pits, so that the exchange locals could observe the open outcry trading, while making the market in the electronic system. But do the exchange locals tend to act as market makers on GLOBEX? We ﬁnd that locals trading E-mini contracts, who enjoy substantial privileges compared to off-exchange traders, are at least as likely to demand as supply liquidity. The locals appear to trade aggressively, relying on the access to the pit dynamics, while a substantial proportion of liquidity is supplied by limit orders of off-exchange traders. Exchange locals still act as liquidity suppliers in a substantial proportion of E-mini trades. Assuming that the locals submit limit orders primarily to earn the bid-ask spread, a reduction of the E-mini tick sizes and a subsequent decline in spreads may have a negative effect on the supply of liquidity on GLOBEX. The dynamic equilibrium nature of this limit order market, however, is likely to ensure that any possible reduction in liquidity is temporary. For example, Biais, Hillion, and Spatt (1995) show that limit order traders actively compete and quickly provide liquidity when liquidity dries up.6 Therefore, if the tick sizes of the E-mini contracts are reduced and, consequently, the bid-ask spreads decline, the supply of liquidity on GLOBEX is unlikely to suffer. A tick size that is too small is likely to increase negotiation costs. Furthermore, in open outcry markets a small tick size often leads to errors in trade processing. GLOBEXis an electronic limit order book market with five best bids and five best offers visible to all traders. Therefore, negotiation costs are not likely to be significant and trade- processing errors are not an issue of concern. A coarse price resolution, however, may preclude trading at mutually agreeable prices in some cases, and the potential gains from trade may be lost (see Brown, Laux, & Schachter, 1991). Furthermore, a large minimum price increment may impede price discovery given that movements of the unobservable equilibrium price are often smaller than the tick size.7 Altering the characteristics of an already successful contract is not without risk, and exchanges are reluctant to make such changes. Nevertheless, our results suggest that the beneﬁts of decreasing the minimum tick sizes of the E-mini contracts are likely to outweigh the possible costs discussed below. This paper is intended to initiate a discussion on the issue. The remainder of the paper is organized as follows. The second section provides a literature review and some institutional detail. The third

6Theoretical papers discussing trading costs and supply of liquidity in limit order markets include Foucault (1999) and Foucault, Kadan, and Kandel (2003), among others. 7Beaulieu, Ebrahim, and Morgan (2003) report empirical evidence supporting this notion.

TLFeBOOK Tick Sizes of the E-Mini Futures 83

section describes the data used in the paper and the descriptive statistics of our sample. The fourth section discusses the methodology and the empirical results. The last section provides a summary and conclusion.

LITERATURE REVIEW AND BACKGROUND

Literature on Tick Size Reductions A number of theoretical papers including Seppi (1997), Anshuman and Kalay (1998), and Cordella and Foucault (1999) examine the issue of optimal tick size. Several papers also model the effects of changes in the minimum tick size on market liquidity. Ronen and Weaver (2001) show that if the tick size is reduced the bid-ask spread will decline even if the tick is not binding. A version of Glosten (1994) model with discrete prices, as in Sandas (2001), predicts that a reduction of minimum tick size in a limit order market leads to lower bid-ask spreads and lower depth at the best quotes, although cumulative depth is unaffected. Alternatively, Foucault et al. (2003) show that reducing the tick size may or may not lead to a reduction in the bid-ask spread in limit order markets, depending on the proportion of patient traders in the market. Many markets reduced minimum tick sizes in recent years and there is a large empirical literature on the effects of tick size reductions on market liquidity. Nearly all of these studies report a decline in the bid-ask spreads.8 The bid-ask spread is a convenient measure of execution costs for relatively small orders. It is important, however, to also consider a potential impact of tick size changes on other liquidity characteristics including market depth. Focusing mainly on the bid-ask spreads and quoted depths at the inside quotes, a number of studies deliver a mixed verdict on the impact of tick size reductions on market liquidity. For example, Bacidore (1997), Porter and Weaver (1997), and Chakravarty, Harris, and Wood (2001), among others, show that both the bid-ask spread and quoted depth decline after a tick size reduction. However, Ahn, Cao, and Choe (1996) and Ronen and Weaver (2001) find that after a tick reduction from eighths to sixteenths on the AMEX, the bid- ask spreads declined, but the market depth did not decrease. Studies that look beyond the depth at best bid and offer ﬁnd that tick size reductions often result in lower trading costs even for trades that exceed the quoted depth at the inside quotes. MacKinnon and Nemiroff (1999) observe that the move by the Toronto Stock Exchange

8Harris (1997a) provides a review of arguments and empirical evidence concerning decreasing the minimum tick size.

TLFeBOOK 84 Kurov and Zabotina

(TSE) to decimal pricing did not lead to an increase of the price impact of trading. Coupled with a decline in the bid-ask spreads, this finding implies that liquidity improved after decimalization at the TSE. In another recent paper, Bessembinder (2003) finds no evidence of systematic intraday price reversals on either NYSE or Nasdaq after decimalization. Such reversals of quote changes would be expected if decimalization had hurt liquidity supply. Goldstein and Kavajecz (2000) show that depths in the NYSE limit order book declined throughout the book after the tick reduction from eighths to sixteenths. Bacidore, Battalio, and Jennings (2003), however, find that while displayed liquidity in the NYSE limit order book declined after decimalization, the execution quality for all order sizes did not deteri- orate. They suggest that non-displayed liquidity available on the exchange floor accounts for this result. Bacidore et al. (2003) and Chan and Hwang (2001) point out that the results of Goldstein and Kavajecz (2000) are likely to be explained by specialists “stepping ahead of the book” more frequently after the tick size reduction. Chan and Hwang (2001) show that the market liquidity improved after a tick size reduction on the Hong Kong Stock Exchange, a pure limit order market. A number of studies examine the impact of tick size reductions on execution costs of institutional traders. Jones and Lipson (2001) show that realized execution costs of institutional traders increased after the NYSE tick size reduction from eighths to sixteenths. Alternatively, the results of Chakravarty, Panchapagesan, and Wood (2002) suggest that decimalization did not increase execution costs for institutions trading NYSE stocks.

Competitive Considerations The CME currently has an exclusive license to trade S&P 500 index futures contracts. This exclusive license, along with a ﬁrst-mover advantage, allows the CME to enjoy a competitive advantage in its high volume index futures markets by precluding competition for order flow from other market centers.9,10 One example of how the CME has used its competitive advantage is its decision to halve the denomination and double the tick size of the regular S&P 500 futures contracts in November 1997,

9Holder, Tomas, and Webb (1999) show that being ﬁrst to list a futures contract is the most important determinant of the competitive outcome when several exchanges offer equivalent futures contracts. 10Although no other exchanges list equivalent index futures contracts, the CBOT’s DJIA index futures contract represents a source of indirect competition for the CME.

TLFeBOOK Tick Sizes of the E-Mini Futures 85

soon after the E-mini S&P 500 futures were introduced. Bollen, Smith, and Whaley (2003) find that this measure increased the welfare of the exchange members, while increasing the trading costs of the CME’s customers. Eurex, the Swiss-German electronic derivatives market, has announced plans to launch a U.S. exchange offering a full range of financial derivatives based on U.S. underlying assets. This new exchange may become a source of competition for the CME in the future if it lists similar futures contracts. Existing research shows that competition between market centers leads to lower trading costs. For example, Mayhew (2002), Anand and Weaver (2002), and De Fontnouvelle, Fishe, and Harris (2003) find that competition among options exchanges reduces bid-ask spreads. Similarly, Boehmer and Boehmer (2003) and Tse and Erenburg (2003) show that the bid-ask spreads of exchange- traded funds (ETFs) declined significantly after entrance of the NYSE into the ETF market. Even without competing markets, the side-by-side approach used by the CME gives its customers a choice between the floor and GLOBEX.11 But do the current characteristics of the CME’s futures contracts allow for unimpeded competition between the floor and GLOBEX? Given that five E-minis have to be traded to replicate one regular contract, many institutional traders currently prefer to trade the regular futures to save on brokerage commissions and fees, which are charged on a per contract basis. Lowering the tick sizes of the E-mini contracts is likely to lead to lower bid-ask spreads, rendering these contracts more attractive to institutional traders.

DATA AND DESCRIPTIVE STATISTICS

Data This study employs computerized trade reconstruction (CTR) data and time and sales data for the regular S&P 500 and Nasdaq-100 futures. We also use trade data for the E-mini S&P 500 and the E-mini Nasdaq- 100 futures. These data are obtained from the Commodity Futures Trading Commission (CFTC). The CTR data and E-mini trade data

11In its 1999 Annual Report, the CME stated that the exchange provides the customers with the best of both open outcry and electronic platforms and leaves it to the customers to choose between the two systems. According to Savage (2002), CME’s Chairman Emeritus Leo Melamed also emphasizes that the choice between the open outcry and electronic systems must be made by the marketplace itself.

TLFeBOOK 86 Kurov and Zabotina

contain the contract ticker symbol, trade date, trade time to the nearest second, the contract month, buy/sell code, number of contracts traded, trade price, customer type indicator (CTI), CTI of the opposite side of the trade, session indicator (pit or GLOBEX) and in timestamp (when the order is received on the trading floor or entered into GLOBEX). CTI is designated from 1 to 4 as follows: CTI1 are trades executed for a floor trader’s personal account (local trade), CTI2 are trades executed for a clearing firm’s account, CTI3 are trades executed for a personal account of another floor trader, and CTI4 are trades executed for an account of an outside customer.12 Our sample period extends over the 248 trading days from January 2, 2002 to December 31, 2002. Days with more than one hour of data miss- ing, such as shortened pre-holiday days, are removed from the sample. For every trading day, only the most actively traded contract in each market is considered.13

Summary Statistics Table I reports summary statistics for regular and E-mini futures. Trades in the E-mini contracts occur much more frequently than trades in the regular futures. For example, trading frequency of the E-mini Nasdaq-100

TABLE I Summary Statistics of Regular and E-mini S&P 500 and Nasdaq-100 Futures for January 2, 2002 to December 31, 2002

S&P 500 Nasdaq-100

Regular E-mini Regular E-mini

Mean number of trades per day 15,454 113,930 3,874 64,579 Mean number of trades per minute 38.2 281.3 9.6 159.5 Mean trading volume (contracts) 74,488 449,071 15,748 212,266 Mean trading volume ($ billion)a 18.40 21.47 1.86 4.87 Mean trade size (contracts) 4.82 3.94 4.07 3.29 Mean trade size ($ ‘000)a 1,190.3 188.5 480.8 75.4 Mean open interest (contracts) 572,728 252,577 64,240 128,777 Mean open interest ($ billion)a 141.59 11.99 7.32 2.95 Mean $ volume market share 45.3% 54.7% 27.1% 72.9%

Note. All statistics are for regular trading hours. aBased on closing prices.

12Daigler and Wiley (1998) provide a detailed discussion of the four CTI categories. 13Trading activity shifts from the futures contract approaching expiration to the next available contract during the second week of the expiring contract’s month when the exchange redesignates the lead contract.

TLFeBOOK Tick Sizes of the E-Mini Futures 87

futures exceeds that of the regular Nasdaq-100 futures by a factor of about sixteen. The open interest in the E-mini futures is substantially smaller than the daily trading volume. At the same time, for the regular futures the open interest far exceeds the daily trading volume. This suggests that many hedgers still prefer to trade the full-size contracts to save on brokerage commissions and exchange fees, which are charged on a per contract basis. Alternatively, the hedge horizon may tend to be longer for traders using full-size contracts. Both E-mini S&P 500 and E-mini Nasdaq-100 futures contracts appear to have been successful products. Table I shows that the E-minis have a large dollar volume market share ranging from 54.7% for E-mini S&P 500 to 72.9% for E-mini Nasdaq-100 futures. Trading volumes of both E-mini contracts have been growing steadily since the introduction of these contracts.

Distribution of Trading Volume by CTI Type Our paper considers liquidity supply and demand on GLOBEX. It is therefore important for our analysis to examine the distribution of trading volume among different trader types. Table II reports the proportions of daily volume for various counterparty combinations in regular and E- mini futures. Consistent with prior literature (e.g., Manaster & Mann, 1996; Ferguson & Mann, 2001), trades between exchange locals and off- exchange customers account for the largest proportion of trades and volume among all four considered contracts.

TABLE II Distribution of the Total Trading Volume by CTI Counterparty Combination for Regular and E-mini S&P 500 and Nasdaq-100 Futures for January 2, 2002 to December 31, 2002

Local (CTI1) with Customer (CTI4) with

CTI1 CTI2 CTI3 CTI4 CTI2 CTI3 CTI4 other

S&P 500 futures 14.6% 3.2% 8.2% 53.1% 2.5% 3.2% 13.8% 1.4% E-mini S&P 500 25.3% 18.3% 0.1% 32.1% 11.3% 0.04% 9.7% 3.2% Nasdaq-100 16.2% 6.4% 3.7% 50.5% 5.3% 5.0% 11.2% 1.7% E-mini Nasdaq-100 17.8% 18.4% 0.1% 31.3% 14.4% 0.1% 13.9% 4.0%

Note. The CTI categories include local traders (CTI1), clearing members (CTI2), other floor traders (CTI3), and off- exchange customers (CTI4). "Other" refers to trades between clearing members, trades between other ﬂoor traders and trades between clearing members and other ﬂoor traders. The percentages trading volume are calculated for each day and then averaged across days. All statistics are for regular trading hours.

TLFeBOOK 88 Kurov and Zabotina

Massimb and Phelps (1994) and Locke and Sarkar (2001) suggest that exchange locals are likely to abandon automated futures markets. In contrast, Table II shows that the locals participate in a large proportion of trades in the E-mini markets. The locals trade very actively by using GLOBEX terminals located on the perimeter of the trading pits on the CME floor. Kurov and Lasser (2004) suggest that the access of locals trading E-mini contracts to the open outcry ﬂoor ensures that they have an informational advantage over the off-exchange traders and are able to trade proﬁtably on GLOBEX. Interestingly, clearing members account for a larger proportion of the total trading in the E-minis than in regular futures. For example, clearing members participate in about 30% of trading volume in the E-mini S&P 500 futures compared to only about 6% of trading volume in the regular S&P 500 futures. The access of clearing members to the trading floor may create an incentive for them to trade on GLOBEX, leading to their high trading activity in the E-mini markets.

EMPIRICAL TESTS AND RESULTS

Estimated Bid-Ask Spreads of Regular and E-mini Futures To analyze the trading costs in the regular and E-mini markets, we begin by calculating customer execution spreads. This direct measure of transaction costs in futures markets is suggested by Locke and Venkatesh (1997) and used by Ferguson and Mann (2001), among others. The execution spread is calculated as mean customer buy price minus mean customer sell price for a ﬁve-minute interval, with prices weighted by trade size. The mean execution spread reported in Table III is close to zero for both E-mini futures. Furthermore, the execution spreads in trades with locals are even smaller, suggesting that locals trading E-mini contracts tend to use market orders to quickly take desirable positions and pay the bid-ask spread at least as often as they earn it. In contrast, for the regular futures the customer execution spread in trades with locals is greater than the all-trade spread, suggesting that the role of locals on the trading ﬂoor is closer to traditional market making than on GLOBEX. The execution spreads are calculated by aggregating across all customer orders, including limit and market orders. Traders using market orders demand liquidity and pay the bid-ask spread, while traders using limit orders supply liquidity and tend to earn the spread. Therefore, the execution spread results discussed above suggest that off-exchange traders in the E-mini markets are about equally likely to use market and

TLFeBOOK Tick Sizes of the E-Mini Futures 89

TABLE III Execution Costs in Regular and E-mini S&P 500 and Nasdaq-100 Futures for January 2, 2002 to December 31, 2002

S&P 500 Nasdaq-100

Regular E-mini Regular E-mini

Panel A: Regular trading hours Mean all-trade execution spreada 0.0107% 0.0014% 0.0259% 0.0035% Mean against-local execution spreada 0.0136% Ϫ0.0001% 0.0323% Ϫ0.0005% Mean estimated bid-ask spread (index points) 0.226 0.256 1.038 0.515 Mean estimated bid-ask spread (percent)a 0.023% 0.027% 0.090% 0.045% Number of observationsb 314,338 4,414,178 151,074 2,248,732 Panel B: 7:15–8:15 a.m. (CST) Mean estimated bid-ask spread (index points) 0.222 0.289 0.846 0.653 Mean estimated bid-ask spread (percent)a 0.023% 0.031% 0.074% 0.058% Number of observationsb 33,016 66,389 9,200 33,684

Note. Execution spread is calculated as mean customer buy price minus mean customer sell price for a 5-minute interval, with prices weighted by trade size. Bold text indicates that the execution spread is statistically signiﬁcant at the 1% level. The bid-ask spread is estimated using the CFTC estimator, which is calculated as the average absolute value of price reversals. To minimize inﬂuence of large price changes, which are unlikely to be related to bid-ask bounce, price changes are eliminated from calculation of estimated bid-ask spread if they are equal to or exceed in absolute value: 5 index points for regular and E-mini S&P 500 futures; 20 index points for regular Nasdaq-100 futures during the regular trading hours; 10 index points for regular Nasdaq-100 futures during the preopening period; 10 index points for E-mini Nasdaq-100 futures. Removing large price changes does not affect the results qualitatively. The results showing the impact of omitting large price changes on the estimated spreads are available upon request. aPercent of contract value. bNumber of observations used to calculate the bid-ask spreads.

limit orders. Given that the bid-ask spread is the cost of immediacy, the effective spread for liquidity-demanding orders is a more informative measure of trading costs than the aggregate execution spread.14 Our data do not contain bid-ask quotes.15 Therefore, we estimate effective spreads for liquidity-demanding orders using the estimator suggested by Wang et al. (1994) and Wang, Yau, and Baptiste (1997).16 This estimator, which is used by the CFTC, is calculated as the average opposite

14For example, Demsetz (1968) notes that the bid-ask spread represents the cost of trading without delay. Similarly, Grossman and Miller (1988) argue that the cost of trading immediately rather than waiting to trade is the essence of market liquidity. Most studies of bid-ask spreads in equity markets estimate effective spreads for market orders. 15Time and sales data for regular futures include bid-ask quotes when the quote price differs from the price of the previous trade. However, these bid-ask quotes are not recorded systematically and we remove them from the analysis. 16We repeated calculation of the bid-ask spreads using the spread estimator suggested by Bhattacharya (1983). The results (not reported but available upon request) were similar. According to ap Gwilym and Thomas (2002), correlations of the Bhattacharya (1983) estimator with effective and realized spreads are in excess of 0.75.

TLFeBOOK 90 Kurov and Zabotina

direction absolute price change. Price changes in the same direction as the preceding price changes are discarded to reduce the impact of changes in the underlying futures price unrelated to the bid-ask bounce. The estimated bid-ask spreads are reported in Table III. The average daily estimated bid-ask spread of the E-mini S&P 500 futures is 0.256 index points, suggesting that the spread rarely exceeds one tick. Somewhat surprisingly, the bid-ask spread of this E-mini contract is higher than that of the regular S&P 500 futures. As mentioned above, previous research that compares open outcry and electronic markets shows that the trading costs in electronic markets tend to be lower. On the other hand, the average bid-ask spread of the regular Nasdaq-100 futures exceeds that of the corresponding E-mini contracts by a factor of two. The bid-ask spread expressed as a percentage of contract value is about 0.027% for the E-mini S&P 500 contract and about 0.045% for the E-mini Nasdaq-100 contract. These percentage spreads are similar in magnitude to the bid-ask spreads of the most liquid ETFs reported by Boehmer and Boehmer (2003). In order to examine the effects of the binding minimum tick sizes of the E-mini futures, we estimate bid-ask spreads of both regular and both E-mini futures contacts for the periods since the introduction of the E-mini contracts. The estimated bid-ask spreads are presented in Figure 1. It is instructive to compare the average daily spread graphs for the regular and E-mini S&P 500 futures. The graph for the regular S&P 500 futures shows a large variation in the bid-ask spread depending on changing market conditions between October 1997 and December 2002. In contrast, the similar graph for the E-mini S&P 500 futures exhibits much less temporal variation. This ﬁnding suggests that the larger minimum tick size of the E-mini S&P 500 contract became a binding constraint on the spread soon after the contract started trading. As this E- mini market matured, periods with bid-ask spread exceeding the minimum tick size of 0.25 index points became increasingly rare. The graphs of intraday variation in the estimated bid-ask spread for the year 2002 presented in panels A and B of Figure 1 tell a similar story. Consistent with prior research (e.g., Wang et al., 1994), the graph for the regular S&P 500 futures shows substantial intraday variation in spreads. At the same time, the similar graph for the E-mini S&P 500 futures is nearly ﬂat. Figure 1 also reports the average estimated bid-ask spreads for the regular and E-mini Nasdaq-100 futures. The temporal variation in the estimated spreads for both contracts was substantial until about May 2001. After May 2001 the bid-ask spread of the regular Nasdaq-100 futures has been close to one index point, which corresponds to two ticks. The bid-ask

TLFeBOOK Panel A. Regular S&P 500 futures Panel B. E-mini S&P 500 futures Estimated average bid-ask spread for October 1, 1997 to December 31, 2002 Estimated average bid-ask spread for October 1, 1997 to December 31, 2002 1.0 1.00 0.9 0.8 0.75 0.7 0.6 0.5 0.50 0.4 Index Points Index Index Points Index 0.3 0.25 0.2 0.1 0.0 0.00 Jul-98 Jul-99 Jul-00 Jul-01 Jul-02 Jul-98 Jul-99 Jul-00 Jul-01 Jul-02 Oct-97 Apr-98 Oct-98 Apr-99 Oct-99 Apr-00 Oct-00 Apr-01 Oct-01 Apr-02 Oct-02 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Oct-97 Apr-98 Oct-98 Apr-99 Oct-99 Apr-00 Oct-00 Apr-01 Oct-01 Apr-02 Oct-02 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02

Intraday variation of the estimated bid-ask spread Intraday variation of the estimated bid-ask spread for January 2, 2002 to December 31, 2002 for January 2, 2002 to December 31, 2002

0.5 0.50

0.4

0.3

0.25 0.2 Index Points Index Index Points Index 0.1

0.0 0.00 8:30-8:45 8:45-9:00 9:00-9:15 9:15-9:30 9:30-9:45 9:45-10:00 8:30-8:45 8:45-9:00 9:00-9:15 9:15-9:30 9:30-9:45 10:00-10:15 10:15-10:30 10:30-10:45 10:45-11:00 11:00-11:15 11:15-11:30 11:30-11:45 11:45-12:00 12:00-12:15 12:15-12:30 12:30-12:45 12:45-13:00 13:00-13:15 13:15-13:30 13:30-13:45 13:45-14:00 14:00-14:15 14:15-14:30 14:30-14:45 14:45-15:00 15:00-15:15 9:45-10:00 10:00-10:15 10:15-10:30 10:30-10:45 10:45-11:00 11:00-11:15 11:15-11:30 11:30-11:45 11:45-12:00 12:00-12:15 12:15-12:30 12:30-12:45 12:45-13:00 13:00-13:15 13:15-13:30 13:30-13:45 13:45-14:00 14:00-14:15 14:15-14:30 14:30-14:45 14:45-15:00 15:00-15:15

FIGURE 1 Estimated CFTC bid-ask spreads. Note: The bid-ask spread is estimated using the CFTC estimator, which is calculated as the average absolute value of price reversals. Chicago time shown. TLFeBOOK Panel C. Regular Nasdaq-100 futures Panel D. E-mini Nasdaq-100 futures Estimated average bid-ask spread for July 1, 1999 to December 31, 2002 Estimated average bid-ask spread for July 1, 1999 to December 31, 2002 5.5 5.5 5.0 5.0 4.5 4.5 4.0 4.0 3.5 3.5 3.0 3.0 2.5 2.5 2.0

2.0 Points Index Index Points Index 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 Jul-02 Jul-01 Jul-00 Jul-99 Jul-02 Jul-01 Jul-00 Jul-99 Jan-02 Jan-01 Jan-00 Mar-02 Mar-01 Mar-00 Jan-02 Jan-01 Jan-00 Nov-02 Nov-01 Nov-00 Nov-99 Sep-02 Sep-01 Sep-00 Sep-99 May-02 May-01 May-00 Mar-02 Mar-01 Mar-00 Nov-02 Nov-01 Nov-00 Nov-99 Sep-02 Sep-01 Sep-00 Sep-99 May-02 May-01 May-00

Intraday variation of the estimated bid-ask spread Intraday variation of the estimated bid-ask spread for January 2, 2002 to December 3, 2002 for January 2, 2002 to December 31, 2002 1.5 1.5

1.0 1.0

Index Points Index 0.5

0.0 0.0 8:30-8:45 8:45-9:00 9:00-9:15 9:15-9:30 9:30-9:45 9:45-10:00 8:30-8:45 8:45-9:00 9:00-9:15 9:15-9:30 9:30-9:45 10:00-10:15 10:15-10:30 10:30-10:45 10:45-11:00 11:00-11:15 11:15-11:30 11:30-11:45 11:45-12:00 12:00-12:15 12:15-12:30 12:30-13:00 13:00-13:15 13:15-13:30 13:30-13:45 13:45-14:00 14:00-14:15 14:15-14:30 14:30-14:45 14:45-15:00 15:00-15:15 15:15-15:30 9:45-10:00 10:00-10:15 10:15-10:30 10:30-10:45 10:45-11:00 11:00-11:15 11:15-11:30 11:30-11:45 11:45-12:00 12:00-12:15 12:15-12:30 12:30-12:45 12:45-13:00 13:00-13:15 13:15-13:30 13:30-13:45 13:45-14:00 14:00-14:15 14:15-14:30 14:30-14:45 14:45-15:00 15:00-15:15 TLFeBOOK FIGURE 1 (Continued) Tick Sizes of the E-Mini Futures 93

spread of the E-mini contract, on the other hand, has settled at the level of 0.5 index points, i.e., one tick. The market makers in the floor-traded contract appear to have been reluctant to reduce spreads below two ticks, while the spread in the E-mini contract is apparently bounded by the minimum tick size. The graph of intraday variation in bid-ask spreads shows that for the floor-traded contract the average spreads are highest in the first 15 minutes of trading. At the same time, the bid-ask spread of the E-mini Nasdaq-100 futures is essentially flat intraday. We also examined intraday variations in volume and volatility for each of the four futures contracts under consideration. Consistent with prior research, our results show that volume and volatility in both regular and both E-mini futures contracts follow a U-shaped pattern.17 There are sound theoretical and empirical reasons to believe that spreads in limit order markets are affected by volatility. Models of Handa and Schwartz (1996) and Handa, Schwartz, and Tiwari (2003) suggest that the bid-ask spreads in limit order markets decrease with transitory volatility and dispersion of traders’ valuations and increase with funda- mental volatility. Ahn, Bae, and Chan (2001) find that transitory volatility is associated with lower spreads in the Hong Kong Stock Exchange, since transitory volatility encourages limit order traders to submit orders more actively. The finding that the E-mini spreads do not seem to be affected by changes in price volatility suggests that the current levels of bid-ask spreads, which are very close to one tick for both E-mini contracts, represent sufficient compensation to liquidity providers even under the most adverse market conditions. If the minimum tick sizes in these two contacts are reduced, the bid-ask spreads are likely to decline. In their analysis of the regular S&P 500 futures, Huang and Stoll (1998) show that as trading volume increases the price increment chosen by the market tends to decline. The binding minimum tick sizes of the E-mini contracts, however, do not allow the traders to use smaller price increments despite a large increase in the trading volume over the last few years.

Estimated Bid-Ask Spreads of the Regular and E-mini Futures During the Preopening Period The minimum tick size of the E-mini S&P 500 futures exceeds the tick size of the full-size S&P 500 contracts by a factor of 2.5. In order to further examine the impact of the large minimum tick size of the E-mini

17These results are not reported to conserve space but are available upon request.

TLFeBOOK 94 Kurov and Zabotina

contract we look at the bid-ask spreads during the one-hour period of 7:15–8:15 am (CST) before the start of the regular pit trading hours. During the regular trading hours, the full-size contracts are traded in the open outcry pits. On the other hand, before 8:15 am (CST) both S&P 500 and Nasdaq-100 regular contracts trade on GLOBEX. This allows us to examine how the larger tick size of the E-mini S&P 500 futures affects the bid-ask spreads, while controlling for the trading mechanism. Trading on GLOBEX is already fairly active during the one- hour preopening period. For example, the average number of trades per minute during this period is about 11 for the regular S&P 500 futures and about 27 for their E-mini counterparts. Panel B in Table III reports the results. The average estimated bid- ask spread of E-mini S&P 500 futures is about 0.29 index points, exceeding the spread of the corresponding regular contract by about 30%. At the same time, the bid-ask spread of the E-mini Nasdaq-100 futures is substantially lower than that of the regular contract. This ﬁnding appears to support the conclusion that the bid-ask spread of the E-mini S&P 500 contract is constrained by its relatively large minimum tick size.

Price Clustering and Potential Impact of a Tick Size Reduction on Front-Running and Quote Matching In addition to estimating the bid-ask spread, we examine clustering of trade prices in both regular and both E-mini contract markets. The results reported in Figure 2 show that trades in the E-mini futures occur with roughly equal frequency on all possible ticks (four ticks for the E-mini S&P 500 and two ticks for the E-mini Nasdaq-100 futures).18 This evidence is consistent with the earlier ﬁnding that the bid and ask prices are separated by one tick. At the same time, trades in both regular contracts appear to be clustered on certain ticks. This clustering is especially prominent in the regular Nasdaq-100 futures with about 98% of trades occurring at full index points. Market microstructure theory warns that a small minimum tick size may hurt liquidity because it makes front-running of limit orders and quote-matching more proﬁtable and easily implemented. Seppi (1997) suggests that if traders with superior information or market access are

18Price clustering on GLOBEX is currently not pronounced, but it is possible that if the E-mini tick sizes are reduced the price clustering will increase. Such price clustering may limit the effect of tick size reduction on the bid-ask spreads on GLOBEX.

TLFeBOOK Panel A. S&P 500 futures

Regular E-mini

0.35 33.1% 0.30 27.0% 29.3% 25.1% 0.30 0.25 23.7% 24.2%

0.25 0.20 0.20 0.15 0.15 10.9% 9.4% 0.10 0.10 7.4% 7.0% 0.05 0.05 0.8% 0.7% 1.0% 0.6% 0.00 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.00 0.25 0.50 0.75

Panel B. Nasdaq-100 futures

Regular E-mini

1.00 98.0% 1.00

0.80 0.80

0.60 0.60 51.9% 48.1%

0.40 0.40

0.20 0.20

2.0% 0.00 0.00 0.0 0.5 0.0 0.5

FIGURE 2 Percent of trade prices at each tick increment for regular and E-mini S&P 500 and Nasdaq-100 futures

TLFeBOOK for January 2, 2002 to December 31, 2002. 96 Kurov and Zabotina

able to “step ahead” of the limit orders of other traders, those limit orders are more likely to receive execution when the price moves away from them. Quote-matchers try to identify informed orders in the limit order book and take liquidity from such orders by slightly improving the limit price. If the price moves against the quote matchers, they are often able to liquidate the position by trading with the limit order they attempt to front-run. Harris (1997b) argues that quote-matchers are parasitic traders because they take liquidity from large patient traders, forcing the large traders to hide their orders. Quote-matching behavior is likely to occur in markets that strictly enforce time precedence. This strategy may be difficult to implement, however, because it is hard to identify informed orders, especially if trading is anonymous. Harris (1996) shows that informed traders tend to split their orders into multiple parts and quickly get in and out of the market in an effort to conceal their information and make quote-matching unprofitable. The price clustering results reported in Figure 2 suggest that quote- matching does not seem to be a frequently used strategy in the regular S&P 500 and Nasdaq-100 futures markets despite the fact that open outcry markets preserve time precedence by allowing traders to quote only when they improve the standing quote. In the regular S&P 500 futures, most of the trades occur at full index points and 0.5 index points. At the same time, trades at 0.9, 0.1, 0.4, and 0.6 index points are much less common than those farther away from the 0.0 and 0.5. This evidence suggests that quote-matchers do not step in front of other traders who prefer to price their trades at 0.0 and 0.5 index points. As mentioned above, trades in the regular Nasdaq-100 futures at 0.5 index points occur very infrequently. We also examined the distribution of trades for the regular Nasdaq-100 futures during the first half of the year 2000, before their tick size was increased from 0.05 to 0.5 index points. With about 99.4% of trades occurring at full index point and about 0.5% of trades at 0.5 index points, trades at other tick increments were extremely infrequent. Furthermore, trades at 0.05 and 0.95 index points were not more frequent than trades at some other increments.19 These price clustering results support the conclusion that quote-matching is not frequently employed by floor traders. The results of Kurov and Lasser (2004) suggest that exchange locals trading E-mini contracts may be able to use order flow information from the trading floor to time their trades on GLOBEX, potentially undercutting limit orders of other traders to speed up execution.

19These results are not reported for brevity but are available from the authors upon request.

TLFeBOOK Tick Sizes of the E-Mini Futures 97

Quote-matching may also occur on GLOBEXbecause it is an electronic market with an open limit order book. Therefore, it is possible that a reduction of tick sizes of the E-mini contracts will lead to increased use of front- running and quote-matching strategies. However, the results discussed in the next section show that the locals tend to trade with market orders. Given that their informational advantage comes from observing the pit dynamics, and is therefore short lived, the locals are likely to continue using market orders regardless whether the minimum tick sizes are reduced.

Supply and Demand of Liquidity on GLOBEX and Potential Impact of a Tick Size Reduction on Liquidity Supply One possible beneﬁt of high bid-ask spreads dictated by a binding minimum tick size is that they represent compensation to liquidity providers. According to the traditional view of liquidity provision in open outcry futures markets supported by Silber (1984) and Kuserk and Locke (1993), exchange locals play the role of informal market makers. Chang and Locke (1996) show that clearing members also play an important role in liquidity provision by absorbing a substantial proportion of the customer order imbalances. Grossman and Miller (1988) observe that, in futures markets, the proﬁt a market maker earns from a quick turn- around often equals the minimum tick size. As a result, if the minimum tick size is too low, floor traders may be unable to recoup fixed costs. Grossman and Miller conclude that the minimum tick rule is important because it supports a minimum level of profit to market makers and guarantees provision of liquidity. Tick size reductions often result in decline of the bid-ask spreads, thus reducing the incentive for market makers to provide liquidity. However, Wood (2000) argues that smaller tick sizes encourage pseudo- dealers, or “naturals,” to provide liquidity by allowing them to compete for execution. As opposed to dealers, pseudo-dealers trade, for example, to rebalance their portfolios. Pseudo-dealers have lower liquidity provision costs than dealers because any given pseudo-dealer does not have to be present in the market during the whole trading day. Furthermore, in a market dominated by pseudo-dealers the same traders at various times supply and provide liquidity. Wood (2000) suggests that the ability of pseudo-dealers to trade directly without intermediation leads to lower trading costs. In support of this view, CFTC (2002) notes that once sus- tainable market liquidity is established, it often becomes unnecessary to maintain a market maker structure.

TLFeBOOK 98 Kurov and Zabotina

Liquidity on GLOBEXis supplied by limit order providers and con- sumed by impatient traders using market orders. If the E-mini tick sizes are reduced and the bid-ask spreads decline, the limit order traders may not be able to receive sufficient compensation for liquidity provision. Liquidity in the E-mini markets may decline as a result. If most of the limit orders are submitted by CME locals, the locals may need to maintain relatively high bid-ask spreads to recover their fixed investments. In this section, we consider whether the exchange locals tend to act as market makers on GLOBEX. To find out whether liquidity on GLOBEX is supplied primarily by exchange locals, we need to classify trades by type of initiator. Since the E-mini trade data do not include bid-ask quotes, we use the tick rule to identify trades initiated by different types of traders.20 For example, a trade is classified as buyer-initiated if it occurs on an up-tick. If in a buyer-initiated trade the buyer is a local, then the trade is classified as a local-initiated buy trade. If a trade occurs on a zero-tick, i.e., if its price is equal to the price of its preceding trade, we classify it using the GLOBEX order submission times. The submission times are reported for both sides of each trade. Therefore, the trader whose order was submitted last initiated the trade.21 The initiating trader in each trade is classified as a liquidity demander and the other counterparty as a liquidity supplier. Similar to Frino and Jarnecic (2000), once the trades are classified we calculate proportions of trades and volume in which the different trader types act as liquidity demanders and as liquidity suppliers. The results are presented in Table IV. In the E-mini S&P 500 futures locals act as liquidity suppliers in 44.8% of all trades and 48.5% of all trading volume. They act as liquidity demanders in 42.7% of trades and 51.6% of volume, suggesting that local-initiated trades tend to be relatively large. Thus, in the E-mini S&P 500 futures the exchange locals are about equally likely to supply as to demand liquidity. In the E-mini Nasdaq-100 futures locals demand liquidity in 42.9% of trades and 47.6% of volume, while they supply liquidity in only 36.6% of trades and 37.2% of volume. Therefore, on a net basis the locals are liquidity demanders in the E-mini Nasdaq-100 market. The small customer execution spreads reported in Table IV lend further support to the conclusion that the exchange locals tend to trade aggressively to exploit their

20Aitken and Frino (1996) and Finucane (2000) show that the tick rule performs well when zero-tick trades are removed. 21The proportion of non-zero tick trades classiﬁed identically by the tick rule and order submission times exceeded 98%. This suggests that the submission times can be used to classify trades accurately for non-zero tick trades.

TLFeBOOK Tick Sizes of the E-Mini Futures 99

TABLE IV Liquidity Supply and Demand in S&P 500 and Nasdaq-100 E-mini Futures for January 2, 2002 to December 31, 2002

Liquidity demand Liquidity supply

CTI1 CTI2 CTI3 CTI4 CTI1 CTI2 CTI3 CTI4

Panel A. E-mini S&P 500 futures Trades 42.7% 15.0% 0.11% 42.2% 44.8% 19.2% 0.13% 35.9% Volume 51.6% 16.1% 0.07% 32.2% 48.5% 19.8% 0.07% 31.6% Panel B. E-mini Nasdaq-100 futures Trades 42.9% 16.2% 0.25% 40.6% 36.6% 20.9% 0.18% 42.2% Volume 47.6% 17.1% 0.15% 35.2% 37.2% 23.8% 0.11% 39.0%

Note. The CTI categories include local traders (CTI1), clearing members (CTI2), other floor traders (CTI3), and off-exchange customers (CTI4). A trader demands liquidity in a particular trade if the trader initiates the trade. Non-zero tick trades are classiﬁed by type of initiator using the tick rule. Zero-tick trades are classified by using the reported GLOBEX order submission times. Nonclassiﬁable trades (i.e., trades that occurred on a zero tick and have equal reported submission times) are excluded from calculation. These trades account for about 8.8% and 10.5% of the total number of trades for the E-mini S&P 500 and E-mini Nasdaq-100 futures, respectively. All statistics are for regular trading hours.

informational advantage derived from their access to the open outcry order ﬂow.22 Consistent with Chang and Locke (1996), clearing members appear more likely to supply than demand liquidity. For example, in the E-mini S&P 500 futures they demand liquidity in 16.1% of the total volume and supply liquidity in 19.8% of the total volume. Clearing members participate in about 30% of trading volume, but (based on CFTC’s Commitments of Traders Data)23 they account for more than half of the total open interest in the E-mini contracts. If clearing members were acting primarily as market makers on GLOBEX by using scalping tactics to earn the spread, one would expect an opposite result, i.e., a relatively high trading volume along with a relatively low open interest. Therefore, it is likely that a substantial proportion of clearing member trades is undertaken for hedging purposes, with limit rather than market orders used to reduce transaction costs. This conclusion is consistent with Daigler and Wiley (1998), who suggest that the clearing members are primarily hedgers.

22The E-mini futures were originally created for retail customers. While our results show that exchange locals do not tend to act as market makers in the E-mini markets, it is possible that locals make markets to retail customers. We thank a referee for pointing this out. Unfortunately, our dataset does not allow us to test this hypothesis. 23The Commitments of Traders data are available at http://www.cftc.gov/.

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Our findings imply that exchange locals do not tend to act as market makers in the E-mini futures markets, while they continue to enjoy substantial privileges including lower GLOBEX and clearing fees and exclusive access to important trading information on the open outcry floor. CFTC (2002) suggests that granting market access privileges to market makers in perpetuity may often lead to a situation where the public costs of such privileges eventually outweigh the public benefits afforded by the market makers. The E-mini futures markets seem to be an example of such a disconnect between public costs and public benefits of privileges granted to market makers.

SUMMARY AND CONCLUSION The CME employs a “side-by-side” trading model in which so-called “E-mini” versions of several floor-traded index futures contracts trade on the electronic GLOBEX trading system. This paper examines the impact of the minimum tick sizes on the bid-ask spreads in the E-mini S&P 500 and E-mini Nasdaq-100 futures markets. We find that the tick sizes of the E-mini contracts act as binding constraints on the bid-ask spreads by not allowing the spreads to decline to competitive levels. Given the resulting relatively high trading costs on GLOBEX, many institutional traders are likely to prefer to trade the full-size index futures contracts traded on the CME floor. We also find that the exchange locals trading E-mini contracts, who continue to enjoy substantial privileges compared to off-exchange customers, are at least as likely to demand as supply liquidity. This finding contradicts the prevailing notion that locals play the role of market makers in futures markets. Furthermore, this result supports the notion that electronic limit order markets are likely to develop adequate liquidity without the presence of designated market makers. In its 1997 letter to the CME regarding several changes to contract specifications proposed by the CME, which included increasing the minimum tick size of the forthcoming E-mini S&P 500 futures contract from 0.1 to 0.25 index points, the CFTC suggested that the CME “should monitor the trading in the E-mini contracts to determine if the minimum tick is too large, and adjust the minimum tick if warranted.”24 Our empirical results suggest that the time has come for the CME to consider reducing the minimum tick sizes of the E-mini S&P 500 and E-mini Nasdaq-100 contracts. Such a measure is likely to result in a reduction of trading costs in the two major U.S. index futures markets.

24U.S. Commodity Futures Trading Commission, September 8, 1997.

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