This paper examines the impact of transaction costs on the distribution-free properties of European foreign exchange options traded on the Philadelphia Exchange (PHLX) from Aug. 2005-July 2006. Empirical tests were conducted using intra-daily option quotes observed at ve minutes intervals. Deviations from put-call parity (PCP) were examined to determine the role of alternative transaction cost measures. A more formal statistical procedure was also employed to test the factors determining deviations from PCP. The results extended previous studies that examined the impact of transaction costs on PCP deviations for stock index options, e.g., Vipul (2008), Garray, Ordonez and Gonzalez (2003) and Wagner, Ellis and Dubofsky (1996), and Hoque, Chan and Manzur (2008) for foreign currency options. The paper is organized as follows. In Section 2, transaction cost measures are speci ed and discussed. Section 3 gives an overview of the distribution free properties of option prices being considered: a lower boundary condition; and, put-call parity. Section 4 describes the data used in this study. Section 5 provides empirical

Produced by The Berkeley Electronic Press, 2010 1 results for deviations from distribution free conditions. Section 6 contains a review of a more formal econometric analysis. The nal section presents conclusions of the paper.

At least since Gould and Galai (1974), the role of transaction costs in determining deviations from the PCP relationship have been of interest. In foreign exchange options markets, these costs are of various types: bid-ask spreads commissions; and, other types of costs such as clearing fees, exchange fees, and exercise fees. The actual impact of different types of transaction costs are market and trader speci c and costs associated with the marginal trader are dif cult to determine due to the lack of precise data about speci c trades. Studies going back to Demsetz (1968), Phillips and Smith (1980) and Stoll (1989), identi ed the bid- ask spread as the most important implicit transaction cost. However, in assessing deviations from put-call parity, it is not apparent when a trade is initiated to exploit an observed deviation whether the trade will be closed out prior to the expiration date. Holding the trade to expiration and delivering on the underlying positions involve different transaction costs than a trade closed out prior to that date which requires additional transactions costs, especially bid-ask spreads, to be assessed. To study the impact of different transaction cost measures associated with put-call parity deviations, the following different categories of transaction costs are considered: (A) minimum transactions costs, where only the bid-ask spreads on the initial trade are considered; (B) costs in (A) plus additional estimated bid-ask spreads associated with closing out the call, put and exchange rate positions prior to expiration; and, (C) maximum transactions costs, including bid-ask spreads from (A) and (B) plus ‘other transaction costs’ that includes commissions and other xed costs. Recognizing that ‘other transaction costs’ are more dif cult to estimate, the identi able costs for a one-lot, PHLX options contract are taken as:

In turn, because ‘maximum transaction costs’ re ect costs associated one- lot trading, this measure can be considered as the upper bound for transaction costs associated with put-call parity deviations.

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In uence of Transaction Costs on Foreign Exchange Option Contracts: Intra-Daily Tests: 1-18 3

In this section, two distribution-free properties of currency option prices are examined: a lower boundary condition (LBC); and, put-call parity (PCP). Assessing the role of transactions costs requires the trading strategies associated with these relationships to be speci ed using appropriate bid and ask rates. Active execution requires that purchases (borrowing) be made at the ask and sales (lending) be made at the bid.

Lower Boundary Condition

The lower boundary condition used in this study is obtained from the standard lower boundary condition for a European currency option. In the case of a call, this condition can be stated, e.g., Poitras (2002, p.378):

C t >S , X , T @ t max > 0, S t B> RT* @ X B> RT @@ (1)

where: C t >S , X , T @ { C t call option price in domestic (US) currency at time t; T = time to expiration time of the option measured in fractions of a year; RT and RT* are the time t domestic and foreign currency interest rates with maturity date T; B>RT @ exp> RT @ and B> RT* @ exp> RT* @ , are the continuously compounded present value of US$1 or FX$1 to be paid at T; St = foreign exchange rate ($US/FX$) at time t; and, X = option exercise price in domestic currency. Violation of this condition allows for: S t B> RT* @ units of FX to be borrowed at RT* ; the funds converted to domestic currency at St to purchase the call at Ct , and invest X B>RT @ . By assumption that the LBC is violated, there is an arbitrage pro t remaining because the investment will mature to X which will be used to exercise the call for ST which is used to settle the loan. The LBC is a relatively weak arbitrage condition compared to PCP. If actual time dated prices used by traders could be continuously observed, it is almost certain that few if any LBC deviations would be found. As such, tests of the LBC condition serve to benchmark sample noise that can arise from a variety of factors such as data error, lack of synchronous observations, variations in accuracy across data providers and so on. The European lower boundary restriction on call and put option prices can empirically be re-expressed as inequalities:

C t >S , X , T @ B>RT @ 1 t C t t S t X t max>0, S t X @ (2)

Pt >S , X , T @ B>RT @ 1 t Pt t X S t t max>0, X S t @ (3)

where: Pt >S , X , T @ { Pt put option price in domestic currency; and, B>RT @ 1 is the future value of $1 borrowed at t. To achieve equivalence with the distribution free lower boundary condition for a European currency option, the

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Due to B>RT @ 1 these conditions are slightly weaker upper bound on option prices than the standard European LBC.

Put-Call Parity

The PCP arbitrage condition is a distribution free property for put and call options with the same exercise price and expiration date, e.g., Poitras, Veld and Zabolotnyuk (2009). More precisely:

C t >S , X , T @ X B>RT @ S t B> RT* @ Pt >S , X , T @ (5)

The two arbitrage trading strategies associated with PCP depend on whether the call or put is overpriced. When C > SB* + P - XB then a conversion strategy is executed where the call is sold and XB is borrowed, with the proceeds used to buy the put and the balance converted at St and used to buy the foreign B*. This generates an arbitrage pro t by assumption at T. This trade is sometimes referred to as the long PCP strategy. Similarly, the arbitrage when the put option is overpriced, P > XB + C - SB* is the reversal strategy also known as the short PCP strategy.

Deviation Test Framework

Bid and ask prices play a critical role in determining the impact of transactions costs on arbitrage trading. In particular, a conversion strategy would involve: borrowing the present value of the exercise price at the domestic offer rate with maturity date T; writing a call at the bid price; buying a put with the same X and T at the ask price; exchanging the remaining balance into foreign currency at ask rate (US$/C$); and, investing the proceeds into an appropriate xed income security with maturity date T at the foreign bid rate.1 Given this, using minimum transactions costs (A) absence of arbitrage requires:

. Bid and ask rates for foreign exchange depend on the FX quoting method. When the US$ is the domestic currency, then the exchange rate into FX$ is quoted as $US/$FX. For option prices and prices in general, the bid must always be not greater than the ask. However, for foreign exchange transactions, whether the bid rate for selling US and buying FX is greater than the ask rate for buying US and selling FX depends on the quotation method. For example, when $US/C$ = .9575 is the bid, then $US/C$ = .9600 could be the ask; in this case bid > ask. However, selling $US and buying Canadian takes place at the ask rate because the bid will receive more C$ per US$. In effect, the bid (ask) quote based on C$/US$ is translated into ask (bid) quotes based on US$/C$.

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In uence of Transaction Costs on Foreign Exchange Option Contracts: Intra-Daily Tests: 1-18 5

ST conv C tb X B>RT @s Pt a S ta B RT* > @ d0l (6)

Conversely, a reversal strategy involves: borrowing the present value of one unit foreign currency at ask rate; exchanging these funds from foreign into domestic currency at the bid rate (US$/C$); writing a put at the bid price; buying a call with the same X and T at ask price; and, investing the balance into an appropriate xed income security with maturity date T. The absence of arbitrage condition for the reversal strategy is:

ST rev > @ C Ptb S tb B RT* s t a X B>RT @l d 0 (7)

In both cases, the assumption is that the arbitrage will be held until the expiration and maturity date (T). In calculating the empirical deviations when other transactions costs are also included, the following arbitrage pro t function is used for the conversion strategy:

Similarly, for the reversal strategy:

where TC is xed transactions costs per trade and i refers to one of two cases (B and C) for calculating such costs. In contrast to minimum transaction costs (A), case B includes bid-ask costs for trades closed out prior to maturity. This is a more complicated task because transactions costs for actual close-out trades cannot be accurately determined, e.g., Bodurtha and Courtadon (1986, 1995). In practice, there is a timing option that permits traders to sell at the ask and buy at the bid for some, but not necessarily all, of the positions involved in the trade. With this in mind, additional bid-ask transaction costs (B) for the conversion and reversal trades are determined as:

TC B , t > C ta C tb @ > Pt a Ptb @ > S ta S tb @ (10)

For the maximum transaction costs case (C), $26.24 per contract is added to TCB, to obtain TCC where the per contract cost is appropriately adjusted to re ect the notional principal of the contract. Tests of arbitrage relationships such as PCP depend fundamentally on the quality of the data employed. In particular, any study involving bid and ask option prices has to confront the dif culty of determining, say, the ‘ask’ price from the available spectrum of market and limit quotes available at a given point in time. Being a less restrictive condition than PCP, tests of the LBC serve to benchmark the data quality used in PCP tests. With this in mind, the following absence of arbitrage LBC conditions can be developed for both puts and calls:

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S LBC , call C ta B>RT @l S tb X d 0 S LBC , put Pta B>RT @l X S ta d 0 (11)

For deviations associated with transaction cost measures B and C, these LBC conditions are augmented to include TCLB and TCLC (e.g., S LBC , call , B S LBC , call TCL B ,c,t ) where:

TCL B,c,t > Cta Ctb @ > S ta S tb @ TCL B, p,t > Pta Ptb @ > S ta S tb @ (12)

In other words, a distinction is made for transactions costs associated with the LBC condition for calls, which uses the bid-ask spread for calls, and conversely for puts. In both cases, the two arbitrage pro t functions using TCLC involve addition of $26.25 per contract to TCLB ,c,t and TCLB , p ,t .

This study uses intra-day price quotes for the British pound, Swiss franc and Euro options traded on the Philadelphia Stock Exchange (PHLX). These data are obtained from Reuter’s database through SIRCA for the trading period from 1 August 2005 to 31 July 2006. Each intra-day observation contains the options price (bid and ask), strike price, maturity date and trading time indicated with a time-stamp. Put and call data are matched at ve minute intervals. For example, put-call pairs traded at 09:05, 09:10, 09:15 and so on are included in the sample. The Structured Query Language (SQL) was used to obtain the synchronised put- call pairs as given in Tables 1 and 2. Due to signi cant differences in trading volume, the number of pair-wise put-call observations varies for each currency. For the British pound, a ‘small sample’ of 618 synchronized put-call pairs were obtained from 13,414 and 285,211 observations for calls and puts, respectively. For the Swiss franc, there are 41,212 put-call pairs and a ‘large sample’ of 197,815 put-call pairs for the Euro. Available option maturities are for 3, 6, 9, and 12 months. The expiration months are March, June, September and December. If the expiration month has 5 Fridays, the options expire on the third Friday, otherwise it is the second Friday of the expiration month. The option contract size was £10000, SF10000 and £10000 for British pound, Swiss franc and Euro, respectively. Signi cantly, the daily closing spot FX rate (bid and ask) and daily closing Eurodollar and Eurocurrency interest rates (bid and ask), obtained from DATASTREAM, were employed instead of the corresponding time dated spot FX rate and interest rates. While the prices for put and call pairs were time synchronized, because the FX and interest rates used to determine the PCP http://epublications.bond.edu.au/ijbf/vol7/iss2/1 6

In uence of Transaction Costs on Foreign Exchange Option Contracts: Intra-Daily Tests: 1-18 7 deviations were not synchronized, the deviations did not necessarily represent actual arbitrage opportunities. However, due to PHLX market organization and trading conventions used to price and trade PHLX foreign exchange options,2 the use of synchronized data for rates would produce deviations that are primarily due to noise arising from, say, dif culties in determining actual bid and ask prices at which trades could actually be executed. Descriptive statistics for spot FX rates and interest rates based on daily data are presented in Table 3. For the British pound, the mid-spot FX rate mean, i.e., [(spot bid mean + spot ask mean)/2] was $1.7857, slightly higher than the strike price mean reported in Table 2 by $0.0857 (= $1.7857 – $1.7000). In contrast, the difference between the mid-spot price mean and strike price mean is $0.0069 and $0.0183 for Swiss franc and Euro, respectively. This indicates that, on average, these put-call option pairs are typically trading at or near the money.

This section presents empirical results for LBC and PCP deviations for different categories of transaction costs. The number of LBC deviations is reported in Table 4 and summarized in Table 5. The mean pro t ($) per contract for individual currencies is calculated as the average of the arbitrage pro t ($) for each LBC deviation, multiplied by the contract size. Similarly, mean pro t ($) for all currencies is calculated as summation of total arbitrage pro t ($) for each currency (British pound, Swiss franc and Euro) divided by total number of PCP deviations for all currencies. As can be seen in the minimum transactions cost case (Panel A), the LBC deviations for British pound, Swiss franc and Euro, respectively, are: for call options are 0, 0.45 percent, and 0.53 percent and, for all currencies jointly, 0.52%; for put options, LBC deviations represent 0, 2.56, and 2.69 per cent and, for all currencies taken together, 2.64%. Consistent with Shastri and Tandon (1985), the results for all currencies jointly indicate that LBC deviations in put prices are higher than those for call prices. For all currencies taken together, the average pro t for both call and put options is about $US25 (or about 1.5 percent) per contract due to the deviation from the LBC for calls and puts. Comparing these results with round-trip bid-ask spreads (B) and total transactions costs (C), the number and percentage of LBC deviations is progressively reduced with the ratio of % put deviations to % call deviations staying relative constant at approximately ve times more put deviations than for calls.

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Following Hoque et al. (2008), a more formal statistical analysis was employed to examine the impact of transaction costs on the validity of PCP. Under the null hypothesis that PCP is valid, regression equations were estimated for the conversion and reversal strategy, respectively. Each equation is estimated using all the sample data, e. g. 197,815 put-call pairs are used to estimate each of two Euro regressions. After conducting stationarity tests using the standard augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests to check whether a unit root is present in the data series involved, regressions for the conversion strategy were speci ed as:

C tb Pt a © > @ l O 0 O1 §¨ S ta B RT* X B>RT @s ·¸ K t ¹ (13)

Regressions for the reversal strategy were also estimated using:

Ptb C ta © > @¹ s T 0 T 1 §¨ X B>RT @l S tb B RT* ·¸ K t (14)

where the equation errors are speci ed to accommodate serial correlation and GARCH effects. A

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Lagrange multiplier (LM) test was conducted for the presence of GARCH(r,s) errors in _t . Once the presence of GARCH errors was con rmed by the LM test, an appropriate lag order was determined by further diagnostic tests. The regressions for the conversion and reversal strategies were initially estimated using ordinary least squares regression. The regression equations were then re-estimated, incorporating adjustments for serial correlation and GARCH effects. With appropriate error term adjustments, the estimate of O0 is not statistically different than 0 for any currency, while O1 is statistically less than 1 being: .980 [.00051] for the Swiss franc; and, .974 [.00029] for the Euro with [coef cient standard errors] adjusted for serial correlation and GARCH effects. As with the conversion strategy results, T0 = 0 cannot be rejected at any reasonable signi cance level. The estimates of T1 are again statistically less than 1 being: .991 [.00041] for the Swiss franc; and,. 967 [.00034] for the Euro. To interpret these results observe that the independent variable is, more-or- less, the moneyness of the options. The zero intercepts indicate that bid prices are systematically higher than warranted by PCP, likely to account for market marker pro ts. To see this, consider ATM options in the conversion trade. Because the bid price is below the ask for a given option, the use of C b P a as the dependent variable subtracts the higher ask price from the lower bid price. Because ATM means the independent variable will be more-or-less zero, it is expected that the intercept would be negative in this case; just as a positive intercept would be expected if C a P b were used as the independent variable. Interpretation of the regression slopes being slightly less than 1 is more complicated. Given that most options are short dated, and foreign and domestic interest rate levels are not substantially different, symmetric pricing with respect to moneyness for puts and calls implies a coef cient equal to 1 when the intercept is zero. Because all the sample data were used in each regression, the estimated value cannot be attributed to the convex impact of moneyness on option pricing arising from a bias in the selection of speci c put-call pairs for the sample. All the sample data is used for each regression. Recalling that the results in Tables 8 and 9 indicated overpricing of OTM calls and ITM puts, the regression results extend these observations to the full sample where the number of deviations for the conversion strategy outnumber the reversal deviations (e.g. 67,205 to 46, 987 for the Euro). This provides a larger (smaller) number of observations where it is most likely that OTM calls (ITM puts) will be over-priced.

This paper empirically examined two distribution free properties for currency options traded on Philadelphia Stock Exchange (PHLX) from 1 August 2005 to 31 July 2006. The two fundamental no-arbitrage conditions are a lower boundary condition (LBC) and put-call parity (PCP). Deviations from arbitrage conditions were investigated using alternative measures of transaction costs.

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Three measures of transaction costs were considered: (i) minimum transaction costs that involve only initial bid-ask spreads, (ii) transaction costs associated with trades closed out prior to expiration; and (iii) a total transaction costs measure. Even using minimum transaction costs calculated with closing spot FX and interest rate quotes, less than 1% (3%) of call (put) option prices involve deviations from the LBC. The LBC deviations are measured in fractions of a percent as alternative, more expensive, measures of transaction costs are employed. In contrast, PCP deviations for the conversion and reversal trades are 33.86% and 23.43%, respectively, for all currencies jointly. Using round trip transactions costs, the PCP deviations are reduced on average, to 13.32% and 12.60% percent, respectively; reducing to 4.99 and 3.09 percent of all put-call pairs when total transaction costs are taken into account.

Author information: The submitting author is Professor Geoffrey Poitras of Simon Fraser University, Canada. He may be contacted at poitras@sfu.ca. The co-authors of the paper are respectively teaching staff Ariful Haque of University of Southern Queensland and Assoc Professor Meher Manzur of Curtin University, meher.manzur@cbs.curtin.edu.au.