Early unwinding of options-futures arbitrage with bid/ask quotations and transaction prices. Joseph K.W. Fung Hong Kong Baptist University

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2 Early unwinding of options-futures arbitrage with bid/ask quotations and transaction prices. Joseph K.W. Fung Hong Kong Baptist University Henry M.K. Mok * The Chinese University of Hong Kong Abstract By early unwinding of initial arbitrage positions simulated from bid/ask quotes and transaction prices, options-futures arbitrageurs can capture extra profits. Profits peak when arbitrageurs apply the dynamic early-unwinding strategy to the bid/ask quotes. Profits are at their lowest when arbitrageurs use the static hold-to-expiration strategy based on transaction prices. However, due to stale quotes, executing trades at prevailing bid/ask quotes can overstate both the size and frequency of arbitrage profits compared to transaction data for either the early-unwinding strategy or the hold-to-expiration strategy. Keywords: early unwinding; bid/ask quotes; hold-to-expiration; options-futures-parity; transaction data. JEL classifications: G13; G19 Acknowledgement: This research has been supported by research grants from The Hong Kong Baptist University and The Chinese University of Hong Kong. We would also like to thank the Hong Kong Futures Exchange for providing the data and Castor Pang for his excellent research assistance. * Corresponding author: Henry M.K. Mok, Department of Decision Sciences & Managerial Economics, The Chinese University of Hong Kong, Shatin, Hong Kong. Phone: (852) , Fax: (852) , henry@baf.msmail.cuhk.edu.hk 2

3 1. Introduction This study extends the early-unwinding strategy to bid/ask quotes supplied by market makers. We compare the size and frequency of actual arbitrage opportunities with those that we derive by simulating hold-to-expiration strategy based on transaction data. Many studies have examined potential arbitrage profitability and its implication on options-futures efficiency (e.g., Lee and ayar, 1993). In efficient markets, investors should quickly take advantage of pure arbitrage opportunity when and if such an opportunity arises. Investors could exploit the price disparity, if any, between the options and futures and secure arbitrage profits. Research also shows that early unwinding the initial arbitrage positions based on transaction prices provides additional profits over and above of the profits generated by the static hold-to-expiration strategy (Fung et al., 1997 and Merrick, 1989). Lee and ayar (1993) and Fung and Chan (1994) use the hold-to-expiration strategy to examine the arbitrage efficiency between the S&P 5 index options and futures contracts. Fung et al. (1997) and Fung and Fung (1997) use transaction data to examine the profitability and efficiency of the Hong Kong Hang Seng Index options and futures contracts. Draper and Fung [22] use the parity condition to examine the arbitrage efficiency between the FTSE-1 index futures and index options. Hemler and Miller (1997) study the efficiency of the options market by using the bid and ask prices of the S&P 5 Index options. Although they find mispricings, they caution that because of stale prices, some quotes might not have been executed. Thus, the apparent arbitrage profits could be illusory. Fung and Mok (21) re-examine the Hang Seng Index options-futures parity condition, using bid/ask quotes and transaction data. They find that under the hold-toexpiration strategy, executing the trades at prevailing bid/ask quotes overstates the 1

4 mispricings that are signaled by transaction data. 1 Moreover, due to stale prices and execution delay, the apparent arbitrage profits could be deceptive. Following another line of research, Cheng et al. (1998) and Draper and Fung [22] simulate the options-futures early-unwinding strategy (Finnerty and Park, 1988; Brennan and Schwartz, 199) using transaction data of the Hang Seng and FSE-1 index derivatives, respectively. They show that by capturing the reversals in pricing errors, a dynamic strategy based on early unwinding provides an incremental profit over and above the static hold-toexpiration strategy. Our study extends these two lines of research by simulating the early-unwinding strategy on 2 months of bid/ask quotes and transaction data (April 1994 to August 1995) obtained from the Hong Kong Futures Exchange. We note that the data are unique, since the exchange stopped providing concurrent bid/ask quote data after the release of this data set. Options and futures are usually traded at ask or bid instead of somewhere in between, as in index portfolio (Harris et al., 1994). When only the transaction data are available, the conventional approach tries to alleviate the problem of misidentification by adding or subtracting a spread from the traded price to impute the bid and ask prices. However, this results in underestimation or overestimation of the prospective arbitrage profits. To alleviate the non-synchronous price problem, we match both the bid/ask prices and the transaction prices within narrow time intervals of one minute to form the optionsfutures trios. Using bid/ask quote data also alleviates the problem of misidentifying arbitrage opportunities in transaction data. Alleviating this misidentification problem is particularly important when a trader adopts the early-unwinding strategy, because it requires identifying the pricing errors at both the initial and the closing of the portfolios. We note that bid and 2

5 ask prices are firm quotations. These prices are actually executable unless the market maker revises the quotes after a time lapse, or there is a substantial market movement, in which case the market maker calls out not held and revises his quotes. Thus, the profit opportunities based on the bid/ask quotes could be short-lived and difficult to exploit. If the bid/ask quotes are good until update (Helmer and Miller, 1997), and if traders assume that the price disparity can be exploited at the prevailing quotes, then arbitrage profits can be enhanced vis-à-vis transaction prices. Moreover, as a corollary to the incremental early-unwinding profits generated from the hold-to-expiration strategy using transaction data, we postulate that unwinding the put-call-futures arbitrage portfolio secured at bid/ask quotes early will generate an incremental profit over and above that of the static hold-to-expiration strategy based on bid/ask quote data. We compare the arbitrage profitability of the early-unwinding and hold-to-expiration strategies for two classes of arbitrageurs, namely that of the members and non-members, to that of the benchmark group of zero-transaction-cost arbitrageurs. We find that the earlyunwinding strategy adds extra profits to all arbitrage groups, over and above that of the holdto-expiration strategy. The arbitrage profit is highest when arbitrageurs apply the dynamic early-unwinding strategy to the bid/ask quotes and is lowest for the static hold-to-expiration strategy on transaction prices that are adjusted for spread costs. The paper proceeds as follows. Section 2 outlines the options-futures arbitrage profit triggers for the early-unwinding strategy and the hold-to-expiration strategy using bid/ask data. Section 3 describes the data and the transaction costs. Section 4 reports the empirical results. Section 5 concludes the paper. 2. The put-call futures parity condition 3

6 A. The put-call futures parity condition with bid/ask quotes The put-call-futures parity relation is not unique to trading and market impact costs. Instead, effective arbitrage should retain the bid and ask prices of the contracts, for instance the futures contract, within a no-arbitrage band determined by the cost of arbitrage. The following equations express the upper ( F U futures prices. ) and lower ( F L ) bounds for the bid and ask U a b r t t F X C P = + ( )( + ) (1) L b a r t t F X C P = + ( )( + ) (2) where X is the common exercise price for the options; t 2 -t is the time-to-maturity or the holding period for the three options and futures contracts until maturity; t and t 2 are in fractions of a year; C a and C b are, respectively, the ask price and the bid price for the call option contract at current time t ; P a and Pb are, respectively, the ask price and the bid price for the put option contract at current time t ; and r is the risk-free rate. If the bid futures price is above the upper bound ( F U ), the arbitrageur should sell short a futures contract at the bid price F b, hedge the position by creating a synthetic long futures position by buying a call at C a the ask futures price is below the lower bound ( F L futures contract at F a, and going short a put at Pb. On the other hand, if ), the arbitrageur should go long on a, and hedge the exposure with a synthetic short futures position created by going short on a call at C b and buying a put at P a. B. The pricing errors with the hold-to-expiration strategy for bid/ask quotes 4

7 Profitable arbitrage requires that the pricing error be larger than the cost for executing the portfolio. Factoring in the transaction cost (Γ ) and the opportunity cost for margin deposits (M), we can write the upper ( e + ) and the lower ( e ) pricing errors for the hold-to-expiration strategy based on bid/ask quote data as follows: + b U e = ( F F ) - (Γ+M) > (3) L a e = ( F F ) - (Γ+M) > (4) where M r t t 2 = k + ( 1 ) and k is the total margin deposits (Fung and Fung, 1997). C. The early-unwinding condition for bid/ask quotes Early unwinding an initial arbitrage position can be profitable if the sign of the initial error reverses itself and the magnitude of the error is no less than the marginal cost for early unwinding. The marginal cost for early unwinding comprises the incremental trading cost (τ) and the interest savings (m) due to early unwinding by releasing the margin deposits before the natural expiration of the initial arbitrage portfolio. We extend the early-unwinding triggers in Cheng, Fung, and Pang (1998) to the context of bid/ask prices. We set a condition for early unwinding an initial short-futures arbitrage portfolio that is met when the ask price of the futures ( F a 1 ) falls below the lower price bound for the futures ( F L 1 ) at an intermediate time t 1, where t < t 1 < t 2, by a magnitude no less than the marginal cost of early unwinding. That is, L a F F τ m (5) 1 1 L b a r t t where F1 X C1 P = [ + ( )( + ) ] 5

8 and m = r t k 1 + ) 365 r (1 + ) t t2 t1 ( 1 For arbitrageurs who face significant costs of margin deposits, the interest saving (m) from early unwinding may offset the incremental trading cost (τ). Thus it is not necessary to require F L < F. However, the class of arbitrageurs that uses interest-bearing securities to a 1 1 satisfy the margin requirement would not bear any interest cost. Thus, m equals zero and it requires the condition F L < F. a 1 1 Early unwinding of an initial short-futures arbitrage position generates an incremental profit. Thus, the total (equals initial plus incremental) arbitrage profit from early unwinding is the sum of the initial profit based on the static hold-to-expiration strategy (equation 3) and the incremental profit from early unwinding: e + + L [( F F1 ) τ + ] (6) a 1 m Similarly, the total arbitrage profit on the early unwinding of an initial long-futures arbitrage position is equal to the profit of the initial position based on the static hold-toexpiration strategy (equation 4) plus the incremental profit from early unwinding: e b U + [( F1 F1 ) τ + m] (7) To make our comparisons, we also investigate the profitability of the early unwinding strategy with transaction prices. We do this by replacing the bid/ask data by transaction data and factoring in estimates of the spread cost component (η) into equations (3) to (7). After these modifications, the total profit from initial short and long futures arbitrage position subsequent to early-unwinding of transaction prices can be written, respectively, as follows: + * * η + [( F1 F1 ) τ φ1 + m] ; [( F1 F1 ) τ φ1 + m] > (8) 6

9 * * η + [( F1 F1 ) τ φ1 + m] ; [( F1 F1 ) τ φ1 + m] > (9) where φ 1 is the total spread cost of the futures and options contracts at the time of earlyunwinding. η andη stand for the magnitude of the initial upper and lower + bound + * violations, respectively. η = F F + Γ + φ + M ) and ( > η * = [( F Γ φ M ) F ] >. φ is the spread cost at t. F 1 is the transaction price of the futures contract at t 1. * F 1 is the fair value of the futures price relative to the traded r t t * prices of the options, where F1 X C1 P = [ + ( )( + ) ]. 3. Data and transaction costs We obtain 2 months (April 1, 1994 to August 31, 1995) of time-stamped bid/ask quotes and transaction data of the Hang Seng Index (HSI) options and futures contracts from the Hong Kong Futures Exchange. Both the index futures and the options are traded through an open-outcry system, with the trading of the index options facilitated by a market-making system. The futures market has been highly liquid and in 1997 was the eighth most traded contract in the world. The options are European style. They share identical expiration cycles with the futures contracts, thus allowing for a direct application of the options-futures parity relation. To alleviate the nonsynchronous price problem, we match both the bid/ask prices and the transaction prices within narrow time intervals of one minute to form the options-futures trios. We obtain one-week and one-month Hong Kong Inter-Bank Offer Rates (HIBOR) from Data Stream. We interpolate the appropriate interest rate for a particular holding period from the two rates and use it as the risk-free rate of interest (r) for that holding period. 7

10 Members of the Hong Kong Exchange bear the lowest trading costs since the members could use interest-bearing securities as margin deposits. For non-members, we estimate the total margin deposit per arbitrage portfolio at HK$8, (1,6 index points). During our study period, we estimate the round-trip transaction costs per arbitrage portfolio of the hold-to-expiration strategy at 1.11 and index points, respectively, for members and non-members, and the marginal trading costs for early unwinding to members and non-members at.31 and 2.3 index points, respectively. We aggregate the bid/ask spread of the futures contract at five index points and one-half of the spread per contract per side to estimate the total spread cost per arbitrage trade. We average the spread costs for the options according to their degree of moneyness. We estimate that the costs are between 12 and 53 index points and that deep-in-the-money options have the highest bid/ask spread (see Fung and Mok (21) for details). 4. Results and discussions A. Early-unwinding dtrategy with bid/ask quotes By assuming that trades can be executed at the prevailing quotes, the earlyunwinding strategy adds extra profits to all arbitrage groups over and above that of the holdto-expiration strategy (Tables 1 and 2). The total (equals initial plus incremental) earlyunwinding profits ( , and index points, respectively) for the three classes of investors (Table 1) are more than 5 percent larger than that derived from transaction prices (Table 2). [ISERT TABLE 1 HERE] [ISERT TABLE 2 HERE] 8

11 The average profit for same-day unwinding is higher than unwinding the positions in other days (Tables 1 and 2). Our results indicate that the holding period for the arbitrage trades is short. The size of the arbitrage profit from the early unwinding of the long futures strategy is larger than that of the short futures strategy. Also, because of higher transaction costs for non-members, the number of profitable positions (Table 1) for non-members (n=69) is smaller than that of the member arbitrageurs (n=1,681). B. Early-unwinding and the hold-to-expiration strategies Table 3 also shows that when arbitrageurs execute trades at prevailing bid/ask quotes, both the size and frequency of arbitrage profits are larger than signaled by transaction data under either the early-unwinding strategy or the hold-to-expiration strategy. For example, the numbers of early-unwinding trades based on bid/ask quotes for members and nonmembers (n=1681 and n=69, respectively) are much larger than those of early-unwinding trades based on transaction prices (n=7 and n=1, respectively). (We note that this remarkably small number of observations is due partly to the large bid/ask spread in transaction prices.) The total early unwinding profits derived from bid/ask quotes are higher than those derived from transaction data. 2 At the same time, the hold-to-expiration arbitrage profits simulated from prevailing bid/ask quotes (27.41, and index points, respectively) are higher than those signaled by transaction prices (18.91, and index points, respectively) for the zero-transaction-cost, and the member and non-member arbitrageurs. [ISERT TABLE 3 HERE] In general, the total arbitrage profit derived from early unwinding of the initial arbitrage portfolio based on bid/ask quotes is the highest. Second highest profits are 9

12 generated by the early-unwinding profit derived from transaction prices, followed by the hold-to-expiration strategy with bid/ask quotes, and lastly the static hold-to-expiration strategy with transaction prices (with adjustment for spread cost). 5. Conclusion Our study extends the early-unwinding strategy to firm bid/ask quotes. We compare our results from using this strategy to the size and frequency of arbitrage opportunities that we derive from the hold-to-expiration strategy based on transaction data. When we assume that trades are executed at the prevailing quotes, we find that the early-unwinding strategy adds extra profits to all arbitrageur groups, over and above that of the hold-to-expiration strategy. The profitability of the options-futures arbitrage peaks when arbitrageurs apply the dynamic early-unwinding strategy to the bid/ask quotes, and at its lowest for the static holdto-expiration strategy that uses transaction prices that are adjusted for spread costs. Our results also indicate that the size of the arbitrage profit from early unwinding of the long-futures strategy is larger than that of the short-futures strategy. A caveat is that the apparent arbitrage opportunities could be the result of stale quotes. Bid/ask quotes overstate both the size and frequency of arbitrage profits signaled by transaction data under both the early-unwinding strategy and the hold-to-expiration strategy. 1

13 References Bae, K.H., Chan, K. & Cheung, Y.L. (1998). The profitability of index futures arbitrage: Evidence from bid-ask quotes. Journal of Futures Markets 18(7), Brennan, M. J. & Schwartz, E. S. (199). Arbitrage in stock index futures. Journal of Business 63, Cheng, L. T.W., Fung, J. K.W. & Pang, C. W.S. (1998 ). Early unwinding strategy in index options-futures arbitrage. The Journal of Financial Research (XXI,), Draper, Paul and Fung, J. K.W. (22), A study of arbitrage efficiency between the FTSE- 1 index futures and options contracts, Journal of Futures Markets 22(1), Finnerty J. E. & Park, H. Y. (1988). How to profit from program trading: Empirical evidence on the potential return from aggressive program trading, Journal of Portfolio Management Fung, J. K.W. & Chan, K.C. (1994). On the arbitrage-free pricing relationship between index futures & index options: A note. The Journal of Futures Markets 14, Fung, J. K.W., Cheng, L. T.W. & Chan, K.C. (1997). The intraday pricing efficiency of Hong Kong Hang Seng Index options and futures markets. The Journal of Futures Markets 17, Fung, J. K.W.& Fung, A. K.W. (1997). Mispricing of futures contracts: A study of index futures versus index options contracts. Journal of Derivatives (Winter), Fung, J. K.W. & Mok, H. M.K. (21). Index options-futures arbitrage: A comparative study with bid/ask and transaction data. Financial Review 36,

14 Harris, L., Sofianos, G. & Shapiro, J. E. (1994). Program trading and intraday volatility. The Review of Financial Studies 7, Hemler, L. M. & Miller, T. W. Jr. (1997). Box spread arbitrage profits following the 1987 market crash: Real or illusory? Journal of Financial and Quantitative Analysis 32, Lee, J. H. & ayar,. (1993). A transactions data analysis of arbitrage between index options and index futures. Journal of Futures Markets 13, Merrick J. J. Jr. (1989). Early unwindings and rollovers of stock index futures arbitrage programs: Analysis and implications for predicting expiration day effects, Journal of Futures Markets 9,

15 Table 1: Arbitrage profit of the early unwinding strategy with bid/ask quotes Total Same Day Unwinding on-same Day Unwinding Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit Zero Cost a Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit Total ** {12.42} ** {5.} ** {14.86} ** {5.8} ** {1.7} ** Longfutures Strategy c Shortfutures Strategy d ** ** {15.} {1.38} ** ** {4.99} {5.} ** ** {15.31} {12.73} ** ** {5.3} {5.8} ** ** {14.56} {1.6} [.3] ** ** Member b Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit Total ** {15.77} [.3] ** {8.9} [.5] ** {18.86} [.3] ** {8.91} [.5] ** {13.68} [.9] ** 26.6 Longfutures Strategy c Shortfutures Strategy d ** ** {23.88} [.5] {13.83} [.3] ** ** {6.46} [.6] 28.9 {8.91} [.5] ** ** {23.87} [.5] {18.77} [.3] ** ** {12.13} [.6] 28.9 {8.91} [.5] ** ** {25.21} [.9] {8.98} [.65] ** ** {4.92} [.2] 78.1 {4.1} [.3] 135. {4.92} [.2] {4.4} [.43] 77.1 {4.4} [.64] {8.9} [.43] 14

16 onmember Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit b Total ** {22.7} [.5] ** {21.9} [3.51] ** {22.57} [.5] ** {23.88} [3.51] ** {24.21} [.5] ** Longfutures Strategy c Shortfutures Strategy d ** ** {22.57} [.5] {22.7} [.5] ** ** {19.} [3.71] 27.8 {22.89} [3.51] otes: The table is calculated based on equations (6) and (7). ** Indicates statistical significance at the.5 level ** ** {22.57} [.1] {23.} [.5] a Zero-cost refers to the class of arbitrageurs who do not incur any transaction costs ** ** { } [3.71] 27.8 {23.88} [3.51] ** ** {4.97} [.5] 5.44 {5.34} [.11] ** ** {11.91} [3.75] 67.1 {9.47} [3.82] 64.1 {13.91} [3.75] b Members and on-members refer to members and non-members of the Hong Kong Futures Exchange, respectively. c d Long-futures signal refers to the profitability of a long-futures strategy following a long-futures arbitrage signal. Short-futures signal refers to the profitability of a short-futures strategy following a short-futures arbitrage signal. 15

17 Table 2: Arbitrage profit of the early unwinding strategy with transaction prices Total Same Day Unwinding on-same Day Unwinding Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit o Spread Cost Total ** Longfutures Strategy c Shortfutures Strategy d ** ** 1.81 Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit 212. {165.44} [285.65] 212. {9.86} 13.3 {9.39} ** ** ** {7.73} {9.1} {6.91} ** ** ** {9.99} 212. {9.97} 61.9 {1.5 } ** ** ** {9.66} 8.32 {8.49} {9.73} ** ** ** {9.65} (9.83) (9.13) ** ** [12.44] ** [11.72] {7.29} {9.6} {5.71} Zero Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit Cost a Total ** Longfutures Strategy c Shortfutures Strategy d ** ** {6.15} [.4] 93.7 {8.82} [.24] {4.39} [.4] ** ** ** {5.36} [.21] {4.4} [.36] 58.8 {7.22} [.21] ** ** {5.55} [.24] {5.55} [.24] ** ** {5.11} [1.45] {5.11} [1.45] (6.35) (5.39) [27.62] n.a. n.a. n.a. n.a. n.a. n.a (4.3) [21.15] {6.48} [.4] 93.7 {3.36 } [.6] {4.39} [.4] (6.35) (3.1) [14.44] (5.6) [15.6] {5.36} [.21] {4.4} [.36] 58.8 {7.22} [.21] 16

18 Member b Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit Total ** {27.94} [5.79] ** {9.66} [.57] {23.91 } {41.42 } (2.73) {27.94} [5.79] (2.29) {6.73} [.57] Longfutures Strategy c Shortfutures Strategy d ** ** {36.3} [11.79] {17.46} [5.79] ** ** {9.66} [.57] 24.8 {9.35} [3.83] [11.79] 36.3 {23.91 } [11.79] [9.66] {41.42 } [9.66] () n.a. n.a. n.a. n.a. n.a. n.a (2.75) {56.25} [56.25] {17.46} [5.79] 1.57 () (2.61) {.57} [.57] 24.8 {9.35} [3.83] onmember Initial Profit Unwinding Profit Initial Profit Unwinding Profit Initial Profit Unwinding Profit b Total {14.46} [14.46] {1.96} [1.96] n.a. n.a. n.a. n.a. n.a. n.a {14.46} [14.46] Longfutures Strategy c Shortfutures Strategy d {14.46} [14.46] {1.96} [1.96] n.a. n.a. n.a. n.a. n.a. n.a {14.46} [14.46] 1.96 {1.96} [1.96] 1 n.a {1.96} [1.96] n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. otes: The table is calculated based on equations (8) and (9). ** Indicates statistical significance at the.5 level. a Zero-cost refers to the class of arbitrageurs who do not incur any transaction costs. b Members and on-members refer to members and non-members of the Hong Kong Futures Exchange, respectively. c d Long-futures signal refers to the profitability of a long-futures strategy following a long-futures arbitrage signal. Short-futures signal refers to the profitability of a short-futures strategy following a short-futures arbitrage signal. 17

19 Table 3: Average profit of the early unwinding and hold-to-expiration strategies Bid/ask Quotes Transaction Prices Zero-Cost Arbitrageurs Initial Incremental Initial Incremental Early- Unwinding Strategy (29.51) ** n= (26.53) ** n= (6.78) ** n= (6.39) ** n=89 Hold-to- Expiration Strategy a (62.83) ** n= (11.38) ** n=257 Member Arbitrageurs Initial Incremental Initial Incremental Early- Unwinding Strategy (28.14) ** n= (26.83) ** n= (3.6) ** n= (1.95) n=7 Hold-to- Expiration Strategy a (62.35) ** n= (11.8) ** n=23 on-member Arbitrageurs Initial Incremental Initial Incremental Early- Unwinding Strategy (19.86) ** n= (24.96) ** n= (n.a.) n= (n.a.) n=1 Hold-to- Expiration Strategy a (5.57) ** n= (8.86) ** n=97 otes: Figures for the early-unwinding strategies are reproduced from Tables 1 and 2 for easy references. + Total arbitrage profit from early unwinding is the sum of the initial profit based on the static holdto-expiration strategy and the incremental profit from early unwinding. ** Indicates statistical significance at the.5 level. s in parentheses. a Figures on the hold-to-expiration strategy are obtained from Fung and Mok (21) which used the same data set. 18

20 otes 1 This result is different from Bae, Chan and Cheung (1998) and the difference is explained in Fung and Mok (2). Suffice it to say here is that Bae, Chan and Cheung (1998) adopted a ten-minute time interval as their matching criterion for the options-futures trios that could introduce significant legging risk and biases. 2 Ex-ante analysis also shows that the potential arbitrage opportunities are short-lived and disappear within five minutes (Fung and Mok (21)). Thus, some of the apparent arbitrage opportunities are deceptive, which suggests that the bid/ask quote data should be interpreted with care. 11