Chapter 7 - Arbitrage in FX Markets

Transcription

1 Rauli Susmel Dep. o Finance Univ. o Houson FINA 4360 Inernaional Financial Managemen Chaper 7 - Arbirage in FX Markes Las Lecure We wen over eec o governmen on S FX rae regimes: Fixed, ree loa & mixed. CB serilized (no eec on domesic Money Markes) and non-serilized inervenions. This Lecure Eec o arbirage on S Arbirage Deiniion: I involves no risk and no capial o your own. I is an aciviy ha akes advanages o pricing misakes in inancial insrumens in one or more markes. Tha is, arbirage involves (1) Pricing misake (2) No own capial (3) No Risk Noe: The deiniion we used presens he ideal view o (riskless) arbirage. Arbirage, in he real world, involves some risk (he lower, he closer o he pure deiniion o arbirage). We will call his arbirage pseudo arbirage. There are 3 ypes o arbirage: (1) Local (ses uniorm raes across banks) (2) Triangular (ses cross raes) (3) Covered (ses orward raes) 1. Local Arbirage (One good, one marke) I ses he price o one good in one marke. Law o one price: he same good should rade or he same price in he same marke. Example: Suppose wo banks have he ollowing bid-ask FX quoes: Bank A Bank B USD/GBP Taking boh quoes ogeher, Bank A sells he GBP oo low relaive o Bank B s prices. (Or, conversely, Bank B buys he GBP oo high relaive o Bank A s prices). This is he pricing misake! Skech o Local Arbirage sraegy: (1) Borrow USD 1.51 (<= No own capial!) (2) Buy a GBP rom Bank A (a ask price S A,ask = USD 1.51) CH-7.1

2 (3) Sell GBP o Bank B (a bid price S B,bid = USD 1.53) (4) Reurn USD 1.51 and make a π = USD.02 proi (1.31% per USD 1.51 borrowed) Local Arbirage Noes: All seps should be done simulaneously. Oherwise, here is risk! (Prices migh change). Bank A and Bank B will noice a book imbalance: - Bank A: all aciviy a he ask side (buy GBP orders i.e., GBP undervalued a S A,ask ) - Bank B: all aciviy a he bid side (sell GBP orders i.e., GBP overvalued a S B,bid ). Boh banks will noice he imbalance and hey will adjus he quoes. For example, Bank A will increase S A,ask and Bank B will reduce S B,bid, say o USD/GBP and USD/GBP, respecively. 2. Triangular Arbirage (Two relaed goods, one marke) Triangular arbirage is a process where wo relaed goods se a hird price. In he FX Marke, riangular arbirage ses FX cross raes. Cross raes are exchange raes ha do no involve he USD. Mos currencies are quoed agains he USD. Thus, cross-raes are calculaed rom USD quoaions i.e., he mos liquid quoes. The cross-raes are calculaed in such a way ha arbirageurs canno ake advanage o he quoed prices. Oherwise, riangular arbirage sraegies would be possible. Example: Suppose Bank One gives he ollowing quoes: S JPY/USD, = 100 JPY/USD S USD/GBP, = 1.60 USD/GBP S JPY/GBP, = 140 JPY/GBP Take he irs wo quoes. Then, he implied (no-arbirage) JPY/GBP quoe should be: S I JPY/GBP, = S JPY/USD, x S USD/GBP, = 160 JPY/GBP > S JPY/GBP, A S JPY/GBP, = 140 JPY/GBP, Bank One undervalues he GBP agains he JPY (wih respec o he irs wo quoes). This is he pricing misake! Skech o Triangular Arbirage (Key: Buy undervalued GPB wih he overvalued JPY): (1) Borrow USD 1 (2) Sell USD/Buy JPY a S JPY/USD, = 100 JPY/USD i,e, sell he USD or JPY 100. (3) Sell JPY/Buy GBP a S JPY/GBP, = 140 JPY/GBP i.e., sell JPY 100 or GBP (4) Sell GBP/Buy USD a S USD/GBP, = 1.60 USD/GBP i.e., sell he GPB or USD (5) Reurn loan, keep prois: π: USD (14.29% per USD borrowed). The riangle: JPY Sell USD a Sell JPY a S = 100 JPY/USD S = 140 JPY/GBP USD GBP Sell GBP a S = 1.60 USD/GBP CH-7.2

3 Noe: Bank One will noice a book imbalance (all he aciviy involves selling USD or JPY, selling JPY or GBP, selling GBP or USD.) and will adjus quoes. Say: S JPY/USD, (say, S JPY/USD, = 93 JPY/USD). S USD/GBP, (say, S USD/GBP, = 1.56 USD/GBP). S JPY/GBP, (say, S JPY/GBP, = 145 JPY/GBP). There is convergence beween S I JPY,GBP, & S JPY,GBP, : S I JPY,GBP, (= S JPY,USD, x S USD/GBP, ) S JPY,GBP, Again, all he seps in he riangular arbirage sraegy should be done a he same ime. Oherwise, we ll be acing risk and wha we are doing should be considered pseudo-arbirage. I does no maer which currency you borrow (USD, GBP, JPY) in sep (1). As long as he sraegy involves he sep Sell JPY/Buy GBP (ollowing he direcion o he arrows in he riangle above!), you should ge he same proi as a %. 3. Covered Ineres Arbirage (Four insrumens -wo goods per marke-, wo markes) Open he hird secion o he WSJ: Brazilian bonds yield 10% and Japanese bonds 1%. Q: Why wouldn' capial low o Brazil rom Japan? A: FX risk: Once JPY are exchanged or BRL (Brazilian reals), here is no guaranee ha he BRL will no depreciae agains he JPY. The only way o avoid his FX risk is o be covered wih a orward FX conrac. Inuiion: Suppose we have he ollowing daa: i JPY = 1% or 1 year (T=1 year) i BRL = 10% or 1 year (T=1 year) S =.025 BRL/JPY We consruc he ollowing sraegy, called carry rade, o proi rom he ineres rae dierenial: Today, a ime =0, we do he ollowing (1)-(3) ransacions: (1) Borrow JPY 1,000 a 1% or 1 year. (A T=1 year, we will need o repay JPY 1,010.) (2) Conver o BRL a S =.025 BRL/JPY. Ge BRL 25. (3) Deposi BRL 25 a 10% or 1 year. (A T=1 year, we will receive BRL ) Now, we wai 1 year. A ime T=1 year, we do he inal sep: (4) Exchange BRL or JPY a S T. Problem wih his carry rade: Today, we do no know S T=1-year. Noe: - I S T =.022 JPY/BRL, we will receive JPY 1250, or a proi o JPY I S T =.025 JPY/BRL, we will receive JPY 1100, or a proi o JPY I S T =.027 JPY/BRL, we will receive JPY 1019, or a proi o JPY 9. - I S T =.030 JPY/BRL, we will receive JPY 916, or a proi o JPY -74. We are acing FX risk. Tha is, (1)-(4) is no an arbirage sraegy. CH-7.3

4 Now, a ime =0, we can use he FX orward marke o insure a cerain exchange rae or he JPY/BRL. Suppose we ge a quoe o F,1-yr =.026 JPY.BRL. A ime =0, we re-do sep (4): (4 ) Sell BRL orward a.026 JPY/BRL. (We will receive JPY 1058, or a sure proi o JPY 48.) => We are acing no FX risk. Tha is, (1)-(4 ) is an arbirage sraegy (covered arbirage). Now, insead o borrowing JPY 1,000, we will ry o borrow JPY 1 billion (and make a JPY 48M proi) or more. Obviously, no bank will oer a.026 JPY/BRL orward conrac! Ineres Rae Pariy Theorem Q: How do banks price FX orward conracs? A: In such a way ha arbirageurs canno ake advanage o heir quoes. To price a orward conrac, banks consider covered arbirage sraegies. Review o Noaion: i d = domesic nominal T days ineres rae. i = oreign nominal T days ineres rae. S = ime spo rae (direc quoe, or example USD/GBP). F,T = orward rae or delivery a dae T, a ime. Noe: In developed markes (like he USA), all ineres raes are quoed on annualized basis. We will use annualized ineres raes (The exbook is compleely misaken when i quoes periodic raes!!) Now, consider he ollowing (covered) sraegy: (1) A ime 0, we borrow rom a oreign bank 1 uni o a oreign currency (FC) or T days. A ime=t, We pay he oreign bank (1 + i x T/360) unis o he FC. (2) A ime 0, we exchange FC 1 a S or 1 uni o FC we ge S. (3) We deposi S in a domesic bank or T days. A ime T, we receive S (1 + i d x T/360) (in DC). (4) A ime 0, we buy a T days orward conrac o exchange domesic currency (DC) or FC a a F,T. A ime T, we exchange he DC S (1 + i d x T/360) or FC, using F,T. We ge S (1 + i d x T/360)/F,T unis o FC. This sraegy will no be proiable i, a ime T, wha we receive in FC is less or equal o wha we have o pay in FC. Tha is, arbirage will ensure ha S (1 + i d x T/360)/F,T = (1 + i x T/360). Solving or F,T, we obain he ollowing expression: F, T S (1 i xt (1 i d xt / 360) / 360) (Ineres Rae Pariy Theorem or IRPT) CH-7.4

5 The IRP heory, also called covered IRPT, as presened above was irs clearly exposed by John Maynard Keynes (1923). I is common o use he ollowing linear IRPT approximaion: F,T S [1 + (i d - i ) x T/360]. This linear approximaion is quie accurae or small i d & i (say, less han 10%). Noes: Seps (1) and (4) simulaneously done produce a FX swap ransacion! In his case, we buy he FC orward a F,T and go sell he FC a S. We can hink o (F,T - S ) as a proi rom he FX swap. We ge he same IRPT equaion i we sar he covered sraegy by (1) borrowing DC a i d ; (2) exchanging DC or FC a S ; (3) deposiing he FC a i ; and (4) selling he FC orward a F,T. Example: IRPT a work. Daa: S = 106 JPY/USD. i d=jpy =.034. i =USD =.050. F,1-year =? Using he IRPT ormula: F,1-year-IRPT = 106 JPY/USD x (1+.034)/(1+.050) = JPY/USD. Using he linear approximaion: F,1-year-IRPT = 106 JPY/USD x ( ) = JPY/USD. The approximaion error is less han 0.08%. Noe: I a bank ses F A,1-year = JPY/USD arbirageurs canno proi rom he bank s quoes. Arbirageurs can proi rom any violaion o IRPT. Example 1: Violaion o IRPT 1 - Undervaluaion o orward FC (=USD, in his example). Suppose IRPT is violaed. Bank A oers: F A,1-year=100 JPY/USD. F A,1-year = 100 JPY/USD < F,1-year-IRPT = JPY/USD (a pricing misake!): Bank A undervalues he orward USD agains he JPY. Take advanage o Bank A s undervaluaion: Buy USD orward a F A,1-yr. Skech o a covered arbirage sraegy: (1) Borrow USD 1 rom a U.S. bank or one year a 5%. (2) Conver USD o JPY a S = 106 JPY/USD (3) Deposi he JPY in a Japanese bank a 3.4%. (4) Cover. Buy USD orward/sell orward JPY a F A,1-yr =100 JPY/USD Cash lows a ime T=1 year, (i) We ge: JPY 106 x (1+.034)/(100 JPY/USD) = USD CH-7.5

6 (ii) We pay: USD 1 x (1+.05) = USD 1.05 Proi = П = USD USD 1.05 = USD.046 (or 4.6% per USD borrowed) Aer one year, he U.S. invesor realizes a risk-ree proi o USD. 046 per USD borrowed. =oday T = 1 year Borrow 1 USD 5% USD 1.05 Deposi JPY % JPY Noe: Arbirage will ensure ha Bank A s quoe quickly converges o F,1-yr = JPY/USD. Example 2: Violaion o IRPT 2 - Overvaluaion o orward FC (=USD). Now, suppose Bank X oers: F X,1-year=110 JPY/USD. Then, F X,1-year > F,1-year-IRPT (pricing misake!) The orward USD is overvalued agains he JPY. Take advanage o Bank X s overvaluaion: Sell USD orward. Skech o a covered arbirage sraegy: (1) Borrow JPY 1 rom or one year a 3.4%. (2) Conver JPY o USD a S = 106 JPY/USD (3) Deposi he USD a 5% or one year (4) Cover. Sell USD orward/buy orward JPY a F X,1-yr=110 JPY/USD. Cash lows a T=1 year: (i) We ge: USD 1/106 x (1+.05) x (110 JPY/USD) = JPY (ii) We pay: JPY 1 x (1+.034) = JPY π = JPY JPY = JPY.0556 (or 5.56% per JPY borrowed) Noe: Arbirage will ensure ha Bank A observes a lo o buying USD orward a F A,1-yr =100 JPY/USD. Bank A will quickly increase he quoe unil i converges o F,1-yr = JPY/USD. IRPT: Assumpions Behind he covered arbirage sraegy -seps (1) o (4)-, we have implicily assumed: (1) Funding is available. Sep (1) can be execued. (2) Free capial mobiliy. No barriers o inernaional capial low i.e., sep (2) and, laer, sep (4) can be implemened. (3) No deaul/counry risk. Seps (3) and (4) are sae. (4) Absence o signiican ricions. Typical examples: ransacion coss & axes. Small ransacions coss are OK, as long as hey do no impede arbirage. We are also implicily assuming ha he orward conrac or he desired mauriy T is available. This may no be rue. In general, he orward marke is liquid or shor mauriies (up o 1 year). For many currencies, say rom emerging marke, he orward marke may be liquid or much shorer mauriies CH-7.6

7 (up o 30 days). IRPT and he Forward Premium Consider linearized IRPT. Aer some algebra, (F,T -S )/S (i d - i )x T/360. Le T=360. Then, p = [(F,T -S )/S ] x 360/T i d - i. p measures he annualized reurn rom a long (shor) posiion in he FX spo marke and a shor (long) posiion in he FX orward marke. Tha is, i measures he reurn rom an FX Swap ransacion. We say i: p>0 premium currency ( he FC rades a a premium agains he DC or delivery in T days ) p<0 discoun currency ( he FC rades a a discoun ) Equilibrium: p exacly compensaes (i d - i ) No arbirage opporuniies No capial lows (because o pricing misakes). Example: Violaions o IRPT and Capial Flows B - Go back o Example 1 p = [(F,T -S )/S ] x 360/T = [( )/106] x 360/360 = => USD rades a a discoun. p = < (i d - i ) = Arbirage possible (pricing misake!) capial lows! Check Seps (1)-(3) in Example 1: Foreign (U.S.) capial lows o Japan (capial inlows o Japan). A - Go back o Example 2 p = [(F,T -S )/S ] x 360/T = [( )/106] x 360/360 = => USD rades a a premium. p = > (i d - i ) = Arbirage possible (pricing misake!) capial lows! Check Seps (1)-(3) in Example 2: Domesic (Japanese) capial lows o USA (capial oulows). i d -i B (Capial inlows) - Example 1 Graph 7.1: IRPT Line IRPT Line A (Capial oulows) - Example 2 p (orward premium) Consider a poin under he IRPT line, say A (like in Example 2): p > i d -i (or p + i > i d ) a long spo/shor orward posiion has a higher yield han borrowing abroad a i and invesing a home a i d CH-7.7

8 Arbirage is possible! (There is a pricing misake). Covered Arbirage Sraegy: (1) Borrow DC a i d or T days (2) Conver DC o FC (he long posiion in FC) (3) Deposi FC a i or T days (4) Sell orward FC a F,T (he shor orward posiion in FC) Tha is, oday, a a poin like A, domesic capial ly o he oreign counry: Wha an invesor pays in domesic ineres rae, i d, is more han compensaed by he high orward premium, p, and he oreign ineres rae, i. IRPT: Evidence Saring rom Frenkel and Levich (1975), here is a lo o evidence ha suppors IRPT. For example, Graph 7.2 plos he daily ineres rae dierenial agains he annualized orward premium. They plo very much along he 45 line. Moreover, he correlaion coeicien beween hese wo series is 0.995, highly correlaed series! Graph 7.2: IRPT Line USD/GBP (daily, ) Using inra-daily daa (10 inervals), Taylor (1989) also ind srong suppor or IRPT. A he ickby-ick daa, Akram, Rice and Sarno (2008, 2009) show ha here are shor-lived (rom 30 seconds up o 4 minues) deparures rom IRP, wih a poenial proi range o per uni. The shor-lived naure and small proi range poin ou o a airly eicien marke, wih he daa close o he IRPT line. There are siuaions, however, where we observe signiican and more persisen deviaions rom he CH-7.8

9 IRPT line. These siuaions are usually aribued o moneary policy, credi risk, unding condiions, risk aversion o invesors, lack o capial mobiliy, deaul risk, counry risk, and marke microsrucure eecs. For example, during he inancial crisis here were several violaions o IRPT (in Graph 7.2, he poin well over he line (-.0154,-.0005) is rom May 2009). These violaions are aribued o unding consrains i.e., diiculies o do sep (1): borrow. See Baba and Parker (2009) and Grioli and Ranaldo (2011). Chaper 7 - Appendix Taylor Series Deiniion: Taylor Series Suppose is an ininiely oen diereniable uncion on a se D and c D. Then, he series T (x, c) = Σ n [ (n) (c)/n!] (x - c) n is called he (ormal) Taylor series o cenered a, or around, c. Noe: I c=0, he series is also called MacLaurin Series. Taylor Series Theorem Suppose C n+1 ([a, b]) -i.e., is (n+1)-imes coninuously diereniable on [a, b]. Then, or c [a,b] we have: ( x) T( x, c) R where R n 1 In paricular, he T (x, c) or an ininiely oen diereniable uncion converges o i he remainder R (n+1) (x) converges o 0 as n. Example: 1 s -order Taylor series expansion, around c=0, o (x) = log(1+xd), where d is a consan (x) = log(1+x d) (x 0 =0) = 0 (x) = d/(1+x d) (x 0 =1) = d => 1 s -order Taylor s series ormula (n=1): log(1+x d) T(x; c) = 0 + d (x-0) = x d => i d=1, hen log(1+x) x Applicaion: IRP Approximaion We sar wih IRP: Now, ake logs: 1 ( x) n! / // (n) c 0 c 1 c x x c x c x c 2 0 x x 0! F (n 1 ), T p n x p dp (1 id xt S (1 i xt 1! / 360) / 360) 2! log( F, ) log( S ) log(1 i xt / 360) log(1 i xt T d / 360) n! 0 n R Aer simple algebra: log( F, ) log( S ) log(1 i xt / 360) log(1 i xt T d log( F / 360) S, T ) log( ) F, T S S CH-7.9

10 Recall ha log changes can approximae percenage changes: Then, using he approximaion or log(1+xd) = x d, we ge Solving or F,T, ges he linearized approximaion o IRP. F, T S S i d xt / 360 i xt / 360 CHAPTER 7 - BONUS COVERAGE: IRPT wih Bid-Ask Spreads Exchange raes and ineres raes are quoed wih bid-ask spreads. Consider a rader in he inerbank marke: She will have o buy or borrow a he oher pary's ask price. She will sell or lend a he bid price. There are wo roads o ake or arbirageurs: borrow domesic currency or borrow oreign currency. Bid s Bound: Borrow Domesic Currency (1) A rader borrows DC 1 a ime =0, and repays 1+i ask,d a ime=t. (2) Using he borrowed DC 1, she can buy spo FC a (1/S ask, ). (3) She deposis he FC a he oreign ineres rae, i bid,. (4) She sells he FC orward or T days a F bid,,t This sraegy would yield, in erms o DC: (1/S ask, ) (1+i bid, ) F bid,,t. In equilibrium, his sraegy should yield no proi. Tha is, (1/S ask, ) (1+i bid, ) F bid,,t (1+i ask,d ). Solving or F bid,,t, F bid,,t S ask, [(1+i ask,d )/(1+i bid, )] = U bid. Ask s Bound: Borrow Foreign Currency (1) The rader borrows FC 1 a ime =0, and repay 1+i ask,. (2) Using he borrowed FC 1, she can buy spo DC a S ask,. (3) She deposis he DC a he oreign ineres rae, i bid,d. (4) She buys he FC orward or T days a F ask,,t Following a similar procedure as he one deailed above, we ge: F ask,,t S bid, [(1+i bid,d )/(1+i ask, )] = L ask. CH-7.10

11 Graph 7.2: Trading bounds or he Forward bid and he Forward ask. F ask,,t F bid,,t L ask U bid F,T Example: IRPT bounds a work. Daa: S = USD/GBP i USD = 7¼-½, i GBP = 8 1/8 3/8, F,one-year = USD/GBP. Check i here is an arbirage opporuniy (we need o check he bid s bound and ask s bound). i) Bid s bound covered arbirage sraegy: 1) Borrow USD 1 a 7.50% or 1 year => we will repay USD a T=1 year 2) Conver o GBP => we ge GBP 1/ = GBP ) Deposi GBP a 8.25% 4) Sell GBP orward a 1.64 USD/GBP => we ge (1/1.6620) x ( )x1.64 = USD => No arbirage opporuniy. For each USD we borrow, we lose USD ii) Ask s bound covered arbirage sraegy: 1) Borrow GBP 1 a 8.375% or 1 year => we will repay GBP a T=1 year 2) Conver o USD => we ge USD ) Deposi USD a 7.250% 4) Buy GBP orward a USD/GBP => we ge x( )x(1/1.6450) = GBP => No arbirage opporuniy. For each GBP we borrow, we lose GBP Noe: The bid-ask orward quoe is consisen wih no arbirage. Tha is, he orward quoe is wihin he IRPT bounds. Check: U bid = S ask, [(1+i ask,d )/(1+i bid, )] = x[1.0750/ ] = USD/GBP F bid,,t = USD/GBP. L ask = S bid, [(1+i bid,d )/(1+i ask, )] = x[1.0725/ ] = USD/GBP F ask,,t = USD/GBP. CH-7.11

12 CHAPTER 7 BRIEF ASSESMENT 1. Assume he ollowing inormaion: S.USD/AUD =.8 USD/AUD S,USD/GBP = 1.40 USD/GBP S,AUD/GBP = 1.80 AUD/GBP Is riangular arbirage possible? I so, explain he seps ha would relec riangular arbirage, and compue he proi rom his sraegy (expressed as a % per uni borrowed). Explain how marke orces move o eliminae riangular arbirage s prois. 2. Diicul. Le s complicae riangular arbirage, by inroducing bid-ask spreads. Assume he ollowing inormaion: S,USD/AUD = USD/AUD S,USD/GBP = USD/GBP Calculae an arbirage-ree cross rae (AUD/GBP) quoe (wih bid-ask spread). 3. Assume he ollowing inormaion: S = 1.10 USD/EUR i EUR = 1.50% i USD = 2.75% T = 180 days (A) Deermine he arbirage-ree 180-day orward rae (use IRP). (B) Suppose Bank Q oers F Q,180 = 1.12 USD/EUR. Given his inormaion, is covered ineres arbirage possible? Design a covered arbirage sraegy and calculae is prois. (C) Suppose Bank P oers F P,180 = 1.08 USD/EUR. Given his inormaion, is covered ineres arbirage possible? Design a covered arbirage sraegy and calculae is prois. 4. Assume he ollowing inormaion: S = 1.40 USD/GBP F Q,270 = 1.42 USD/GBP i GBP = 2.50% i USD = 2.75% T = 180 days Calculae p (he orward premium) and he ineres rae dierenial. Wha kind o capial lows he U.K. economy will experience? CH-7.12