Price Risk and Bid-Ask Spreads of Currency Options

Transcription

1 Prce Rsk and Bd-Ask Spreads of Currency Optons By Mara E. de Boyre New Mexco State Unversty Department of Fnance, MSC 3FIN P.O. Box Las Cruces, NM Phone: (505) ; Fax: (505) E-mal: Yong O. Km Rsk Management Key Bank 127 Publc Square mal stop:oh Cleveland, Oho Phone: (216) ; Fax: (216) : fax E-mal: Smon J. Pak Penn State Unversty - Great Valley School of Graduate Professonal Studes 30 E. Swedesford Road Malvern, PA Phone: (610) ; Fax: (610) E-mal: sjp14@psu.edu

2 Prce rsk and Bd-Ask Spreads of Currency Optons Abstract Ths paper nvestgates the spread of bd and ask prces of currency optons quoted by Phladelpha Stock Exchange (PHLX) currency opton specalsts. Based on Bessenbder (1994), ths paper suggests that the bd-ask spread for currency optons can be attrbutable to nventory cost. Nonetheless, we fnd that delta and gamma, among other prce rsk measures, are the most sgnfcant explanatory varables n the bd-ask spread of currency optons. Specfcally, we fnd that the bd-ask spread ncreases wth the change n opton prce (delta) and decreases wth the convexty of the opton (gamma). 1

3 Prce rsk and Bd-Ask Spreads of Currency Optons In market mcrostructure lterature, bd-ask spread s made up of three components: order processng costs, nventory holdng costs and asymmetrc nformaton costs. The order processng component, or cost of dealer processng, represents a fee charged by market makers for standng ready to match buy and sell orders. 1 Inventory holdng costs component compensates dealers for the rsk of holdng undversfed portfolos. Wthn the lterature, researchers have created nventory control models that focus on how rsk-averse dealers adjust prces to control ther nventory of an asset. 2 Expectatons are that the bd-ask spread are postvely related (.e., ncreases) to prce and the volatlty of the securty and negatvely related (.e., decreases) to tradng volume. Fnally, the asymmetrc nformaton costs component s to compensate market makers who sustan losses from tradng wth nformed traders. 3 Informatonbased models consder learnng and adverse selecton problems when some market partcpants have prvate nformaton. 4 The bd-ask spread and ts three components have been wdely analyzed usng stock market, foregn exchange market and stock opton market data, but not usng data on currency optons. Emprcal mcrostructure studes usng foregn exchange have been lmted untl recently due to the lack of transacton level data; and to our knowledge no researcher has studed whch prce rsk measures are most sgnfcant when explanng the bd-ask spreads of foregn currency optons. 5 Ths paper contrbutes to the mcrostructure lterature by nvestgatng whch of the prce rsk measures s most sgnfcant when attemptng to explan the spread of bd and ask prces of currency optons quoted by Phladelpha Stock Exchange (PHLX) currency opton specalsts. Unlke stocks traded n the New York Stock Exchange (NYSE), PHLX currency optons are 2

4 nfrequently traded. 6 Ths nfrequency n tradng causes the PHLX currency opton specalsts to take longer to restore hs nventory to the optmal level than for NYSE specalsts, accentuatng the prce rsk of the specalsts unwanted nventores. For ths reason, ths paper postulates that the quoted bd-ask spread of foregn currences optons s manly attrbutable to nventory costs. 7 Gven that nventory costs arse from the prce rsk of the specalst s nventores and that the prce rsk of currency optons s derved from prce rsk of the underlyng currency, one can further stpulate that the prce rsk of the specalst s currency opton nventory (nventory costs) also arses from the exchange rate uncertanty of the underlyng currency. However, t may also depend on the volatlty of the underlyng currency value, nterest rates, and maturty of the opton. When calculatng the prce rsks of currency optons we use the Garman and Kohlhagen (1983) foregn currency European opton-prcng formula adapted for Amercan optons traded n PHLX. 8 Varous senstvty measures of currency opton prces are expressed as Greek symbols: delta and gamma from spot exchange rate changes of the underlyng foregn currency, vega from changes n the volatlty of exchange rate, rho s from changes n domestc and foregn rskless nterest rates, and theta from shortenng maturty of optons (tme value eroson). We fnd that among these prce rsk measures of currency optons, delta and gamma are sgnfcant explanatory varables n the bd-ask spread of currency optons. Specfcally, we fnd that the bd-ask spread ncreases wth the delta and decreases wth the gamma. 3

5 I. Prce Rsk Measures of Currency Optons and Bd-Ask Spread. a. The Garman-Kohlhagen Formula and Prce Rsk Measures The prce rsk measures are calculated usng the Garman-Kohlhagen formula for European optons, whch s modfed for Amercan optons. 9 The Garman-Kohlhagen formula treats the foregn nterest rate as a contnuous dvdend yeld. For European calls and puts, the formulae are, respectvely, c p = = r f τ rτ Se N( d1) Xe N( d 2 ) r Se f τ rτ N d ) + Xe N( ( 1 d 2 ) (1) where S s the spot exchange rate for the underlyng currency (the dollar value of one unt of the underlyng currency), r s the rskless nterest rate for the dollar, r f s the rskless nterest rate for the underlyng currency, J s the tme to maturty, X s the exercse prce, N(.) s the cumulatve normal densty functon, and d 1 = ln 2 ( S ) + ( r r f + ) X σ τ σ 2 τ d 2 = d1 σ τ (2) where F s the volatlty of the spot exchange rate for the underlyng currency. Sgns of prce rsk measures 10 of foregn currency optons (Greek symbols) are: c p Delta call = > 0, Delta put = < 0 S S 2 2 c p Gamma call = > 0, Gamma = > 0 2 put = Gamma 2 call S S 4

6 Vega call = c p > 0, Vega put = > 0 σ σ Rho call = c p > 0, Rho put = < 0 r r c p RhoF call = < 0, RhoF put = > 0 r r f f c, Theta call = =? τ p Theta put = < 0 τ 11,12 B. Relaton between Bd-Ask Spread of Currency Optons and Prce Rsk Measures Bd-ask spread of currency optons can be attrbuted mostly to nventory costs. As we argued prevously, currency opton specalsts who often have to accommodate transtory order mbalance must carry unwanted nventores. Dealers can control nventory levels to a certan degree by lowerng the bd and ask quotes when nventory level exceeds the optmal level and rasng the prce quotes when nventory level s below the optmal level. However, currency optons n PHLX are nfrequently traded; and the PHLX specalsts are lkely to hold the undesrable level of nventores longer than NYSE specalsts. Ths mples that the same level of unwanted nventores s lkely to pose more problems for PHLX specalsts than for NYSE specalsts even f prce rsk for one unt of nventores s same for both currency opton and stock n queston. Consequently, bd-ask spread quoted by PHLX currency opton specalsts would be more senstve wth respect to prce rsk of currency optons than the spread quoted by NYSE specalsts. We consder n ths paper optons on four currences, Brtsh Pound (BP), Deutsche Mark (DM), Japanese Yen (JY), and Swss Franc (SF), snce those four currency optons are most 5

7 frequently traded n PHLX durng We examne senstvtes of bd-ask spread wth respect to measures of prce rsk of call and put optons, respectvely, on each currency n the followng regresson equatons: bd ask spread call, = a0 + a1deltacall, + a2gammacall, + a3vegacall, a 4Rhocall, + a5rhofcall, + a6thetacall, + ε call, + (3) bd ask spread, + (4) put, = b0 + b1 Delta put, + b2gamma put, + b3vega put b 4Rho put, + b5rhofput, + b6theta put, + ε put, where = BP, DM, JY, SF. Both the delta and the gamma measure the mpact of spot exchange rate changes on the opton prce. We predct that bd-ask spread for call (put) s postvely (negatvely) related to delta, and that bd-ask spread for both call and put optons s negatvely related to gamma. Delta measures the slope of the optons. For a call opton, the larger the delta (whch s postve), the larger the change n the opton prce for a small change n the underlyng spot exchange rate, thereby ndcatng hgher prce rsk of the call opton. Hence, the specalst sets wder bd-ask spread for a call opton wth the larger delta,.e., a 1 > 0. For a put opton, the larger delta (whch s negatve) ndcates the flatter slope of the put opton prce. Hence, the larger the delta of the put, the smaller the mpact of the spot exchange rate change on the put prce, and consequently the specalsts sets the narrower spread,.e., b 1 < 0. Delta predcts accurately how much opton prce would change for a small change n the underlyng exchange rate. However, t becomes a less accurate predctor of opton prce change for a large exchange rate change because of the convexty of opton prces. For a large exchange rate ncrease (decrease), the delta under predcts (over predcts) the ncrease (decrease) n the call opton prce whle t over predcts (under predcts) the decrease (ncrease) n the put opton prce. 6

8 Gamma measures convexty of optons. When the specalst flls a sell order resultng n a long poston n the optons, convexty of (call or put) opton prce s a desrable feature. Thus, the specalst quotes the hgher bd prce for a hgh gamma opton than for a low gamma opton. However, when the specalst flls a buy order resultng n the short poston, t s an undesrable feature. Thus, the specalst quotes the lower ask prce for a hgh gamma opton than for a low gamma opton. Hence, we expect that the larger gamma results n a smaller bd-ask spread,.e., a 2 < 0 and b 2 < 0. Vega measures how much opton prce would change when the volatlty of exchange rate changes. However, Black-Scholes (1973) or Garman-Kohlhagen formula assumes the constant volatlty. In order to examne the effects of exchange rate volatlty changes on opton prces wthn the framework of Garman and Kohlhagen model, we assume that the volatlty s expected to reman constant over the lfe of optons. Thus, the larger the vega, the larger the mpact of changes n exchange rate volatlty on the opton prce. Hence, we expect that the larger vega results n a larger bd-ask spread,.e., a 3 > 0 and b 3 > 0. Currency opton value also depends on both domestc and foregn nterest rates. Changes n domestc or foregn nterest rate affect almost mmedately the exchange rate. The ndrect effect of nterest rate changes on the opton value through exchange rate changes s already captured n delta and gamma. The rho correspondng to the domestc nterest rate (Rho) only measures the drect effect of nterest rate changes on currency opton value; that s, t measures the senstvty of opton value to the domestc nterest rate. The prce of currency call (put) optons ncreases (decreases) wth domestc nterest rate. When domestc nterest rate ncreases, the present value of the exercse prce that call (put) opton holders pay (receve) upon exercsng the opton decreases, thereby ncreasng (decreasng) the value of currency calls (puts). 7

9 Therefore, the rho correspondng to the domestc nterest rate s postve for calls and negatve for puts. The larger the rho correspondng to the domestc nterest rate, the larger (smaller) the mpact of domestc nterest rate changes on the prce of calls (puts). Hence, we expect that the larger rho correspondng to the domestc nterest rate results n a larger (smaller) bd-ask spread for calls (puts),.e., a 4 > 0 and b 4 < 0. The prce of currency call (put) optons decreases (ncreases) wth foregn nterest rate. When foregn nterest rate ncreases holdng exchange rate constant, the value of foregn currency, not ncludng nterest payment, decreases, thereby decreasng (ncreasng) the value of currency calls (puts). The rho correspondng to the foregn nterest rate (RhoF) s negatve for calls and postve for puts. It measures the senstvty of opton value to changes n the foregn nterest rate. The larger the rho correspondng to the foregn nterest rate, the smaller (larger) the mpact of foregn nterest rate changes on the prce of calls (puts). Thus, we expect that the larger rho correspondng to the domestc nterest rate results n a smaller (larger) bd-ask spread for calls (puts),.e., a 5 < 0 and b 5 > 0. Fnally, the prce of currency optons depends on the maturty of optons. Theta measures the tme value eroson. As the tme to maturty decreases, the opton becomes less valuable. The more negatve the theta, the larger the value loss of optons s wth the passage of tme. The specalst wll prefer optons wth a lower rate of tme value eroson,.e., a larger (and hence smaller absolute value) theta. Therefore, an opton wth a larger theta s lkely quoted at a hgher bd and a hgher asked compared to an opton wth a lower theta. However, t s not clear the effect of theta on bd-ask spread,.e., sgns of a 6 and b 6 are ambguous. 8

10 II. Data Descrpton Transactons data for currency optons are obtaned from the currency prcng hstory tape of the Phladelpha Stock Exchange. The tape has most of the transactons data for all the currency optons traded at the PHLX from January 3, 1984 through May 30, Each lne representng a transacton reports: a tcker symbol, the trade date, transacton tme, expraton month, strke prce, put or call, opton premum, bd, ask, and concurrent Telerate quotes for spot, bd and ask exchange rate. There are about 1.6 mllon trades recorded n the tape for the entre perod. To reduce the computatonal requrement, only one year data s selected for emprcal estmaton. The transacton data for the calendar year 1996 s selected because t s the latest perod for whch trade data s avalable. There are a total of 78 tcker symbols n the1996 data set wth 2,232,802 contracts traded n 57,857 transactons wthn 254 tradng days. 15 The data s further restrcted to the four most frequently traded Amercan currency optons and the correspondng European optons. The selected tcker symbols are: Brtsh pounds (XBP/ CBP), Deutsche Mark (XDM/CDM), Japanese Yen (XJY/CJY) and Swss Franc (XSF/CSF). These are the regular optons exprng on Frday before the thrd Wednesday of the expraton month. The data for the selected 8 tcker symbols n 1996 show a total of 1,120,043 contracts traded n 35,158 transactons wthn 254 tradng days (See Table I). The selected data set s cleaned up by elmnatng the followng ncomplete records: () records wth mssng spot exchange rates; () records wth mssng bd or ask prce for optons; () records wth mssng exercse prces; (v) records wth opton premum outsde the bd-ask quotes,.e., opton premum hgher than asked prce, or lower than bd prce; (v) records wth 9

11 quoted bd-ask spread greater than maxmum permtted 16 ; and (v) duplcate records. There are a total of 27,574 usable currency optons records. Insert Table I here The nterest rates used are from Datastream. They are the daly quotes of the Brtsh Bankers Assocaton London nterbank settlement fxng rates (BBAISR) for maturtes rangng from 1 month to 12 months for all four currences and the U.S. dollar. The values for delta, gamma, vega, Rho, Rhof and theta are calculated usng Garman- Kohlhagan formula for European optons and the recursve ntegraton method [Hwang, Subrhamanum, and Yu (1996)] by modfyng the Garman-Kohlhagen formulae for Amercan optons. 17 The exchange rate volatlty of a currency on a gven date s assumed to be the average mpled volatlty computed from the currency optons traded on the mmedately precedng date. The currency optons used to calculate the mpled volatlty were restrcted to the optons near-at-the-money wth strke prces wthn 10% of the spot prces and maturty longer than one week. Two records were deleted from the sample due to computatonal non-convergence, leavng a total of 27,572 records for regresson analyss. III. Regresson Results The regresson results for bd-ask spread of call optons, equaton (3), are n Table II. The regresson results for bd-ask spread of put optons, equaton (4), are n Table III. Table IV lsts the regresson statstcs for all eght equatons. 10

12 Insert Table II here Insert Table III here Insert Table IV here The constant term s hghly sgnfcant n all the equatons. The results for delta, gamma and vega are hghly sgnfcant at less than 0.1% sgnfcance level. Delta s hghly sgnfcant n all the equatons wth postve coeffcents for call optons and negatve for put optons, as expected. Gamma s negatve n all the equatons, as expected, and sgnfcant n all but one equaton, Yen (YN) call optons (sgnfcant at 7.2%). Vega s sgnfcant at or less than 0.1% for all the equatons. The sgns are postve as expected for all, except for Brtsh Pound (BP) call and BP put. The negatve sgn (and sgnfcant at less than 0.1%) for vega n BP equatons for call and put can not be explaned. 18 Theta, Rho, and Rhof have mxed results n sgnfcance and n sgns. These results support those found wthn the stock opton lterature. Leland (1985), Merton (1989), and Boyle and Vorst (1992), theoretcally examne the mpact of the underlyng stock spreads on the hedgng costs mposed on opton dealers. The researchers conclude that the opton bd-ask spread s postvely related to the spread of the underlyng asset and to the senstvty of the opton to changes n volatlty (vega). Jameson and Wlhelm (1992) examne the effects of opton's gamma, the error n delta hedgng, and vega, the uncertanty n volatlty that can not be hedged, on stock opton spreads. The authors fnd that both varables explan the statstcal sgnfcance of the opton spread. 11

13 In summary, the emprcal results support the hypothess that the bd-ask spread of currency optons ncreases wth the absolute value of delta and decreases wth gamma for all currency optons. The bd-ask spread s found to be postvely related to vega for call and put optons on Deutsche Mark (DM), Japanese Yen (JY) and Swss Franc (SF) and negatvely for call and put on Brtsh Pound (BP). The effects of all other Greek letters, theta, Rho, and RhoF, are mxed n sgnfcance and sgn. IV. Concluson The determnants of the spread of bd and ask prces of currency optons are examned n ths paper usng the Phladelpha Stock Exchange (PHLX) currency opton data. Foregn currency markets are drven prmarly by macroeconomc nformaton, bd-ask spread of currences can be attrbuted manly to nventory costs, and thus, bd-ask spread of optons of foregn currences as well. Pror studes have demonstrated that the prce rsk of the specalst s currency opton nventory (nventory costs) can be best explaned by the prce rsk of the currency optons. Ths study shows that the bd-ask spreads can be mostly explaned by two measures of prce rsk, delta and gamma whch are due to the uncertanty of the underlyng exchange rates. The most sgnfcant emprcal fndng s that the bd-ask spread ncreases wth the absolute value of delta and decreases wth the gamma, ndcatng that the uncertanty n the underlyng currency value s the most sgnfcant determnant of the bd-ask spread of currency optons. The study also fnds that other prce rsk measures such as vega, whch s due to changes n the volatlty of the exchange rate, Rho, whch s due to changes n the domestc nterest rate, 12

14 and theta, whch s due to dfferent terms of the optons, are also sgnfcantly related to the bdask spreads for many currency optons. 13

15 Table I 1996 Transactons for the Eght Tcker Symbols The selected tcker symbols are: Brtsh pounds (XBP/ CBP), Deutsche Mark (XDM/CDM), Japanese Yen (XJY/CJY) and Swss Franc (XSF/CSF). These are the regular optons exprng on Frday before the thrd Wednesday of the expraton month. The data for the selected 8 tcker symbols n 1996 show a total of 1,120,043 contracts traded n 35,158 transactons wthn 254 tradng days. Symbol Contract Volume Number of Records Daly Average Transacton Frequency XJY (Yen, Amercan) 209,556 7, XDM (DM, Amercan) 351,541 7, XSF (SF, Amercan) 102,462 6, CSF (SF, European) 111,443 5, XBP (BP, Amercan) 248,130 4, CJY (Yen, European) 41,920 1, CDM (DM, European) 37,843 1, CBP (BP, European) 17,148 1, Total for 8 Tcker symbols n ,120,043 35, Total for all 78 tcker symbols n ,232,802 57,

16 Table 2 Regresson Equatons for Bd-Ask Spreads n 1996 Call Trades Regresson results for bd-ask spread of call optons usng the followng equaton: bd ask spread = a + a Delta + a Gamma + a Vega + a Rho + a RhoF + call, 0 1 call, 2 call, 3 call, 4 call, 5 call, a6thetacall, + εcall, The varable delta predcts how much opton prce would change for a small change n the underlyng exchange rate whle Gamma measures convexty of optons. Rho and RhoF measures the senstvty of opton value to the domestc nterest rate and the senstvty of opton value to changes n the foregn nterest rate, respectvely. Vega measures how much opton prce would change when the volatlty of exchange rate changes and Theta measures the tme value eroson. Equaton for BP Call Equaton for DM Call Equaton for JY Call Equaton for SF Call Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Const 1.55E E E E Delta 6.06E E E E Gamma -2.36E E E E Rho -2.76E E E E RhoF -2.74E E E E Vega -1.41E E E E Theta -1.05E E E E

17 Table III Regresson Equatons for Bd-Ask spreads n 1996 Put Trades Regresson results for bd-ask spread of call optons usng the followng equaton: bd ask spread put, = b0 + b1 Deltaput, + b2gammaput, + b3vegaput, + b4rhoput, + b5 RhoFput, + b6thetaput, + ε put, The varable delta predcts how much opton prce would change for a small change n the underlyng exchange rate whle Gamma measures convexty of optons. Rho and RhoF measures the senstvty of opton value to the domestc nterest rate and the senstvty of opton value to changes n the foregn nterest rate, respectvely. Vega measures how much opton prce would change when the volatlty of exchange rate changes and Theta measures the tme value eroson. Equaton for BP Put Equaton for DM Put Equaton for JY Put Equaton for SF Put Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Coeffcent t-stat Const 1.43E E E E Delta -2.54E E E E Gamma -1.26E E E E Rho -3.74E E E E RhoF -3.82E E E E Vega -1.59E E E E Theta 2.42E E E E

18 Table IV Regresson Statstcs BP Call DM Call JY Call SF Call BP Put DM Put JY Put SF Put NOB Y-mean 1.48E E E E E E E E-04 Y-sdt 6.54E E E E E E E E-04 SSR 5.93E E E E E E E E-04 Var of Res 3.05E E E E E E E E-08 SER 5.53E E E E E E E E-04 R R2-adj F-stat 1.31E E E E E E E E+02 LLF 1.19E E E E E E E E+04 17

19 References Admat, Anat R., and Paul Pflederer, 1988, A Theory of Intraday Patterns: Volume and Prce Varablty, Revew of Fnancal Studes, 1, Amhud, Y., and H. Mendelson, 1980, Dealershp Markets: Market Makng wth Inventory, Journal of Fnancal Economcs, 8, Back, K., 1993, Asymmetrc Informaton and Optons, Revew of Fnancal Studes, 6, Bessembnder, Hendrk, 1994, Bd-Ask Spreads n the Interbank Foregn Exchange Markets, Journal of Fnancal Economcs, 35, Boyle, P.P. and Vorst, T., 1992, Opton Replcaton n Dscrete Tme wth Transacton Cost, Journal of Fnance, 47(1), Copeland, T., and D. Gala, 1983, Informaton Effects on the Bd-Ask Spread, Journal of Fnance, 38, Demsetz, Harold, 1968, The Cost of Transactng, Quarterly Journal of Economcs, 82 (1), Easley, Davd and Maureen O'Hara, 1987, Prce, Trade Sze, and Informaton n Securtes Markets, Journal of Fnancal Economcs, 19, and P. S. Srnvas, 1998, Opton Volume and Stock Prces: Evdence on Where Informed Traders Trade, Journal of Fnance, 53(2), Foster, F.D., and S. Vswanathan, 1994, Strategc Tradng Wth Asymmetrcally Informed Traders and Long-Lved Informaton, Journal of Fnance and Quanttatve Analyss, 29, Garman, M., and S. Kohlhagen, 1983, Foregn currency opton values, Journal of Internatonal Money and Fnance, 2(3), Glassman, Debra, 1987, Exchange Rate Rsk and Transacton Costs: Evdence from Bd-Ask Spreads, Journal of Internatonal Money and Fnance, 6(4), Glosten, L., and P. Mlgrom, 1985, Bd, Ask and Transacton Prces n a Specalst Market wth Heterogeneously Informed Traders, Journal of Fnancal Economcs, 14, Ho, T., and H. Stoll, 1981, Optmal Dealer Prcng Under Transactons and Return Uncertanty, Journal of Fnancal Economcs, 9,

20 , 1983, The Dynamcs of Dealer Markets Under Competton, Journal of Fnance, 8, Jameson, Melvn and Wllam Wlhelm, 1992, Market Makng n the Optons Markets and the Costs of Dscrete Hedge Rebalancng, Journal of Fnance, 43, Kyle, Albert S., Contnuous Auctons and Insder Tradng, Econometrca, 53(6), Lee, Tae Hwy, 1994, Spread and Volatlty n Spot and Forward Exchange Rates, Journal of Internatonal Money and Fnance, 13 (3), Leland, Hayne, 1985, Opton Prcng and Replcaton wth Transactons Costs, Journal of Fnance, 40(5), Madhavan, A., 1992, Tradng Mechansms n Securtes Markets, Journal of Fnance, 47, Merton, R.C., 1989, On the Applcaton of the Contnous-Tme Theory of Fnance to Fnancal Intermedaton and Insurance, The Geneva Papers on Rsk and Insurance, 14, Stoll, H., 1978, The Supply Dealer Servces n Securty Markets, Journal of Fnance, 33, Subrahmanyam, A., 1991, Rsk Averson, Market Lqudty, and Prce Effcency, Revew of Fnancal Studes, 4, Tnc, S., 1972, The Economcs of Lqudty Servces, Quarterly Journal of Economcs, 86,

21 Footnotes 1 Accordng to Demsetz (1968), these costs may nclude, but are not lmted to, subscrptons to specalzed electronc nformaton and tradng systems. Please see Tnc (1972) for more on nventory holdng cost component. 2 Inventory holdng costs models have been fashoned by Demsetz (1968), Stoll (1978), Amhud and Mendelson (1980), and Ho and Stoll (1981, 1983). 3 Copeland and Gala (1983), Kyle (1985), Glosten and Mlgrom (1985), Easley and O Hara (1987), Admat and Pflederer (1988), Madhavan (1992) and Foster and Vswanathan (1994) have provded poneerng models of adverse selecton n securtes tradng. 4 When a dealer receves a trade, he wll revse hs expectatons and set spreads to protect hmself aganst nformed traders. 5 Glassman (1987), Admat and Pflederer (1988), and Subrahmanyam (1991) make a sgnfcant contrbuton to the lterature by provdng models that not only show the proportonal relatonshp between exchange rate volatlty and bd-ask spreads but also predcts the change n volume and volatlty at the openng and closng of the tradng day and determne how lqudty traders affects prce effcency. 6 For example, t s not uncommon to observe that for more than two hours there are no trades of optons on Brtsh Pound whch s one of most frequently traded currency optons n PHLX. 7 We follow Bessenbder s (1994) concluson that bd-ask spreads of currences can be attrbuted manly to nventory costs. 8 Please note that even though other studes have focused on the spread of the underlyng assets and opton prce and volume, the ntent of ths paper s to calculate the prce rsk of currency optons and determne whch of the prce rsk measure(s) s (are) most sgnfcant. 20

22 9 The majorty of the PHLX currency optons are Amercan optons. For numercal computatons of Amercan optons, we use the recursve ntegraton method [Hwang, Subrhamanum, and Yu (1996)] by modfyng the Garman-Kohlhagen formulae. 10 Theta s not a rsk measure because there s no uncertanty n the passage of tme, but t measures the tme value eroson of an nventory of currency optons. 11 Detaled expressons of Greek symbols can be found n most optons text books. 12 Gammas for both European call and put are postve and have the same expresson. The sgn of theta for European call currency opton s usually negatve except n-the-money European call on a currency wth a very hgh nterest rate. 13 The dataset obtaned from PHLX covers the perod from January 1984 through May The year 1996 was selected for emprcal test because t was the last year wth a full year s data. 14 There are two mssng perods n 1985: January 25 - February 21 and October 25 - November The 78 symbols are: ADW AZW BPU BPW CAD CAZ CBP CBX CBY CBZ CCD CCV CDM CDW CDZ CFF CFV CJV CJY CJZ CSF CSY CSZ DMW DMZ EAW ECD ECU ECW EDA EDM EDZ EFF EJY EJZ EPO ESU ESW ESZ FFW JYW JYZ MXZ MYW MYX MYZ PMW PMX SFW SFZ XAD XAZ XBP XBX XBY XBZ XCD XCV XDA XDB XDC XDD XDE XDM. 16 Refer to the "Study Gude for Phladelpha Stock Exchange Foregn Currency Opton Partcpants for the PHLX FCO Qualfcaton Examnaton for Specalsts, Regstered Opton Traders and Floor Brokers." 17 The numercal calculatons are based on 4 pont Rchardson extrapolaton wth geometrc sequence, power of tme-nterval squared, exercsable once, twce, quadruple, and octuple. Ths 21

23 method mproves computatonal accuracy several folds over arthmetc sequence method. The computatonal detals are not ncluded here, but are avalable from the authors upon request. 18 One possble explanaton s that vega may be related to volume for the Brtsh Pound (BP). If the volatlty of BP were to jump up there may be more demand for optons for the purpose of hedgng actvtes. In ths case, specalsts who antcpate that nventory holdng perod wll be short may not be as concerned about an ncrease n prce rsk of optons as before. 22