1 IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in hese markes. As is he case wih equiy derivaives, he implied volailiy surface corresponding o vanilla European FX opion prices is neiher fla nor consan. I is herefore widely acceped ha he Black-Scholes GBM model is a poor model for FX markes. As is he case wih equiy derivaives, however, vanilla FX opion prices are quoed and heir Greeks are calculaed using he Black-Scholes framework. I is herefore necessary o undersand how Black-Scholes applies o he FX markes when working wih derivaives in hese markes. We will also consider asses ha are denominaed in a foreign currency bu whose value we wish o deermine in unis of a domesic currency. There are wo differen means of convering he asse s foreign currency value ino domesic currency: i using he prevailing exchange rae and ii using a fixed predeermined exchange rae. The laer mehod leads o he concep of a quano securiy. We will sudy boh mehods in some deail. Because such securiies are exoic, hey are generally no priced in pracice using he Black-Scholes framework. Noneheless, we will consider hem in his framework for wo reasons: i given our knowledge of Black-Scholes, his is he easies way of inroducing he conceps and ii i will afford us addiional opporuniies o work wih maringale pricing using differen numeraires and EMMs. A paricular feaure of FX markes are he riangular relaionships ha exis beween currency riples. For example, he USD/JPY exchange rae can be expressed in erms of he USD/EUR and EURO/JPY exchange raes. These implicaions ough o be borne in mind when working wih FX derivaives. They and oher peculiariies of FX modeling will be sudied furher in he assignmens. 1 Foreign Exchange Modeling Unless oherwise saed we will call one currency he domesic currency and he oher he foreign currency. We will hen le X denoe he exchange rae a ime, represening he ime cos in unis of he domesic currency of 1 uni of he foreign currency. For example, aking he usual marke convenion, if he USD/EUR 1 exchange rae is 1.2, hen USD is he domesic currency, EUR is he foreign currency and 1 EUR coss $1.20. Suppose insead we expressed he exchange rae as he cos in Euro of $1 USD. Then USD would be he foreign currency and EUR would be he domesic currency. Noe ha he designaions domesic and foreign have nohing o do wih where you are living or where he ransacion akes place. They are a funcion only of how he exchange rae is quoed. The erms base and quoe are also ofen used in pracice. The exchange rae hen represens how much of he quoe currency is needed o purchase one uni of he base currency. So 2 quoe = domesic and base = foreign. We will coninue o use domesic and foreign in hese lecure noes. Laer on we will use base currency o refer o he accouning currency. For example, a US based company would ake he USD as is base currency whereas a German company would have he EUR as is base currency. We will le r d and r f denoe he domesic and foreign risk-free raes 3, respecively. Noe ha he foreign currency plays he role of he sock and r f is he dividend yield of he sock. Noe also ha holding he 1 In hese noes when we wrie Curr1/Curr2 we will always ake Curr1 o be he domesic currency and Curr2 o be he foreign currency. Unforunaely, in pracice his is no necessarily he case and he domesic and foreign designaions will depend on he currency pair convenion. 2 Bu beware: I have seen base used in place of he domesic currency as well! 3 We will assume ha r d and r f are consan in hese noes. When pricing long-daed FX derivaives, hese raes are ofen aken o be sochasic. The same applies for long-daed equiy derivaives.
2 Foreign Exchange and Quanos 2 foreign cash accoun is only risk-free from he perspecive of someone who uses he foreign currency as heir uni of accoun, i.e. accouning currency. A domesic invesor who invess in he foreign cash accoun, or any oher asse denominaed in he foreign currency, is exposed o exchange rae risk. Currency Forwards Le F T denoe he ime price of a forward conrac for delivery of he foreign currency a ime T. Then since he iniial value of he forward conrac is 0, maringale pricing implies [ ] 0 = E Q F T X T B T which hen implies F T = Assuming ha ineres raes are consan 4 we obain E Q E Q [ ] XT B T [ ]. 1 B T F T = E Q [X T ] = e r dt E Q [e r dt X T ] a relaionship ha is someimes referred o as covered ineres pariy. Exercise 1 Prove 1 direcly using a no-arbirage argumen. = e r d r f T X 1 Exercise 2 Compue he ime value of a forward conrac ha was iniiaed a ime = 0 wih srike K for delivery a ime T >. The forward currency markes 5 are very liquid and play an imporan role in currency rading. A forward FX rae, F T, is usually quoed as a premium or discoun o he spo rae, X, via he forward poins. FX Swaps An FX swap is a simulaneous purchase and sale of idenical amouns of one currency for anoher wih wo differen value daes. The value daes are he daes upon which delivery of he currencies ake place. In an FX swap, he firs dae is usually 6 he spo dae and he second dae is some forward dae, T. An FX swap is hen a regular spo FX rade combined wih a forward rade, boh of which are execued simulaneously for he same quaniy. FX swaps are ofen called forex swaps. They should no be confused wih currency swaps which are considerably less liquid. A currency swap is ypically a long-daed insrumen where ineres paymens and principal in one currency are exchanged for ineres paymens and principal in anoher currency. FX swaps are regularly used by insiuions o fund heir FX balances. Forwards and FX swaps are ypically quoed in erms of forward poins which are he difference beween he forward price and he spo price. Using 1 we see F T X = e r d r f T 1 X 4 We could also allow ineres raes ha are i deerminisic or ii sochasic bu Q-independen of X T in our derivaion of 1. Bu a no-arbirage argumen using zero-coupon bonds could sill be used o derive 1 in general. See Exercise 1. 5 Exchange-raded currency fuures also exis. 6 Bu oher value daes are also possible. I is possible, for example, o rade forward-forward swaps where boh value daes are beyond he spo dae, or om-nex swaps.
3 Foreign Exchange and Quanos 3 When ineres raes are idenical he forward poins are zero. As he ineres rae differenial ges larger, he absolue value of he forward poins increases. Currency Opions A European call opion on he exchange rae, X, wih mauriy T and srike K pays max0, X T K a ime T. If we assume ha X has GBM dynamics hen he same Black-Scholes analysis ha we applied o opions on socks will also apply here. In paricular he Q-dynamics of he exchange rae are given by and he price of he currency opion saisfies dx = r d r f X d + σ x X dw x 2 CX, = e r f T X Φd 1 e r dt KΦd 2 3 where d 1 = log X K + rd r f + σx/2t 2 σ x T and d 2 = d 1 σ T. where σ x is he exchange rae volailiy. All he usual Black-Scholes Greeks apply. I is worh bearing in mind, however, ha foreign exchange markes ypically assume a sicky-by-dela implied volailiy surface. This means ha as he exchange rae moves, he volailiy of an opion wih a given srike is also assumed o move in such a way ha he volailiy skew, as a funcion of dela, does no move. The noional of he opion is he number of foreign currency unis ha he opion holder has he righ o buy or sell a mauriy. So for example, if he noional of a call opion is N, hen he ime value of he opion is N CX,. Dela and he Opion Premium When rading a currency opion, he price of he opion may be paid in unis of he domesic currency or i may be paid in unis of he foreign currency. This siuaion never arises when rading sock opions. For example, if you purchase an opion on IBM hen you will pay for ha opion in dollars, i.e. he domesic currency, and no in IBM sock which plays he role of he foreign currency. Noe ha his is a maer of pracice as here is nohing in heory o sop me paying for he IBM opion in IBM sock. In currency markes, however, and depending on he currency pair, i migh be quie naural o pay for he currency opion in he foreign currency. When compuing your dela i is imporan o know wha currency was used o pay for he currency opion. Reurning o he sock analogy, suppose you paid for an IBM call opion in IBM sock ha you borrowed in he sock-borrow marke. Then I would inheri a long dela posiion from he opion and a shor dela posiion posiion from he opion paymen. My overall ne dela posiion will sill be long why?, bu less long ha i would have been if I had paid for i in dollars. The same is rue if you pay for a currency opion in unis of foreign currency. When an opion premium is paid in unis of he foreign currency and he dela is adjused o reflec his, we someimes refer o i as he premium-adjused dela. As is probably clear by now, currencies can be quie confusing! And i akes ime working in he FX markes before mos people can ge compleely comforable wih he various marke convenions. The one advanage ha currencies have over socks from a modeling perspecive is ha currencies do no pay discree dividends and so he ofen ad-hoc mehods ha have been developed for handling discree dividends are no needed when modeling currencies. Oher Delas There are oher definiions of dela ha are commonly used in FX-space. These alernaive definiions arise because i he value of currency derivaives can be expressed naurally in domesic or foreign currency unis and ii forwards play such an imporan role in FX markes. In addiion o he regular Black-Scholes dela which is he usual dela sensiiviy and he premium-adjused dela we also have:
4 Foreign Exchange and Quanos 4 The forward dela. This is he sensiiviy of he opion price wih respec o changes in he value of he underlying forward conac wih he same srike and mauriy as he opion. The premium-adjused forward dela. This is he same as he forward dela bu you need o adjus for he fac ha he opion premium was paid in unis of he foreign currency. Noe ha all of he dela definiions we have discussed hus far have implicily assumed ha we wan o hedge he domesic value of he opion. However, we migh also wan o hedge he foreign value of he opion. Example 1 Hedging Domesic Value v Foreign Value Suppose an FX rader in Ausralia buys a JPY/AUD call opion from an FX rader in Japan. If he Ausralian rader hedges he AUD value of he opion and he JPY rader hedges he JPY value of he opion, hen hey will no pu on equal and opposie hedges. For example, suppose he value of he opion is 100 AUD and he curren AUD/JPY exchange rae is 65 JPY per AUD. If he Ausralian rader has hedged he AUD value of he opion and he exchange rae moves from 65 o 67, say, his posiion will sill be worh 100 AUD. However, he JPY value of his posiion will have changed from = 6, 500 JPY o = 6, 700 JPY. So hedging he AUD value of his posiion clearly does no hedge he JPY value of his posiion. As a resul, we have anoher se of delas ha correspond o hedging he foreign value of he opion. There are also sicky-by-dela versions of all hese delas. Clearly hen, one mus be very careful in specifying and inerpreing a dela. The defaul dela ha is quoed in he marke place will, as always, depend on he currency pair and marke convenions. Trading sysems will generally be capable of reporing hese differen delas. Wha Does A-he-Money ATM Mean? The usual definiion of a-he-money in equiy-space is ha he srike of he opion under consideraion is equal o he curren spo price. Bu here are oher alernaive definiions ha commonly occur in FX-space. They are: 1. A-he-money spo which is our usual definiion. 2. A-he-money forward. 3. A-he-money value neural. This is he srike, K, such ha he call value wih srike K equals he pu value wih he same srike, K. 4. A-he-money dela-neural. This is he srike, K, where he dela of he call opion wih srike K is equal o minus he dela of he pu opion wih he same srike K. Bu as here many differen definiions of dela, we have many possibiliies here. Building an FX Volailiy Surface The consrucion of a volailiy surface in FX space is unforunaely much more difficul han in equiy-space. The simple reason for his is ha he liquid opions in he FX markes are quoed for a given dela and no for a given srike. In paricular, FX markes ypically quoe volailiy prices for 1. ATM opions. However, he paricular definiion of ATM will depend on he currency-pair and marke convenions dela and 10-dela srangles. A srangle is a long call opion ogeher wih a long pu opion. 3. And 25-dela and 10-dela risk-reversals. A risk reversal is a long call opion ogeher wih a shor pu opion. In each of he srangles and risk reversals, he srike of he pu opion will be lower han he srike of he call opion. These srikes hen need o be deermined numerically using combinaions of he quoed volailiies and he designaed delas. Once he various srikes and heir implied volailiies have been calculaed, curve fiing
5 Foreign Exchange and Quanos 5 echniques are hen used o build he enire volailiy surface. So building an FX implied volailiy surface is considerably more complicaed han building an equiy implied volailiy surface. This appears o be a legacy of how FX opions markes developed and in paricular, of how risk reversals and srangles became he liquid securiies in he marke. Unil recenly here have been very few if any deailed descripions of he various convenions in FX markes and how FX volailiy surfaces are buil. More recenly, however, some papers 7 have been published on his opic. They can be consuled for furher edious bu necessary deails. 2 The Triangle Relaionships The fac ha all currency pair combinaions can be raded imposes resricions on he dynamics of he various exchange raes. To see his we will inroduce some new noaion. In paricular, le Then i is easy o see ha X a/b := # of unis of currency A required o purchase 1 uni of currency B. X a/b = X a/c 1 X a/b = X b/a X c/b 4 The advanage of his noaion is ha i follows he usual rules of division, i.e. a/c c/b = a/b. The ideniy in 4 clearly defines he price process for any currency pair once he price processes for he oher wo pairs are specified. Example 2 Geomeric Brownian Moions Suppose X a/c where W a/c implies and X c/b and W c/b boh follow GBM s under some probabiliy measure, P, so ha dx a/c = µ a/c X a/c d + σ a/c X a/c dw a/c dx c/b = µ c/b X c/b d + σ c/b X c/b dw c/b are P-Brownian moions wih dw a/c dx a/b = where σ a/b := 5 dw c/b = ρ d. Then Iô s Lemma applied o 4 µ a/c + µ c/b + σ a/c σ c/b ρ X a/b d + σ a/b X a/b dw a/b σ a/c 2 + σ c/b 2 + 2ρσ a/c σ c/b 6 and where W a/b is also 8 a P-Brownian moion. The volailiy, σ a/b, is herefore compleely deermined by he volailiies and correlaion of he oher wo exchange raes. In pracice, we of course know ha FX raes do no follow GBM s. In paricular, he implied volailiies for each of he hree currency pairs will be a funcion of srike and ime-o-mauriy and so 6 is of lile pracical value. However, we emphasize again ha if we have a good model for he dynamics of X a/c and X c/b hen 4 uniquely deermines he dynamics of X a/b. Convering o Base Currency Delas Suppose now ha you have bough a call opion on X a/b T and ha he opion payoff is denominaed in he base domesic currency, A. Le CX a/b denoe he ime value in currency A of his opion as a funcion of he 7 See, for example, FX Volailiy Smile Consrucion by D. Reiswich and U. Wysup See also he forhcoming book Foreign Exchange Opion Pricing: A Praciioner s Guide by I.J. Clark and published by Wiley. 8 See 13 and he paragraph following i for a jusificaion of his saemen.
6 Foreign Exchange and Quanos 6 curren exchange rae, X a/b. Then his opion could be dela-hedged in he usual Black-Scholes manner. Noe, however, ha if we dela-hedge he opion we are hedging he currency A value of he opion. In cerain circumsances, however, and as menioned earlier, we may wish o hedge or monior he currency C value of his opion. For example, a rading desk of a US bank may be responsible for all EUR/JPY rading. The rading book and is P&L will hen be denominaed in EUR, say 9. The desk, however, may also wan o monior he USD value of is posiions and P&L if USD is he base currency of he bank. Afer all, here is no much poin in being he greaes EUR/JPY rader in he world and earning billions of JPY if he value of JPY agains he USD has collapsed and is now worhless. So le V c X c/a, X c/b denoe he ime currency C value of he opion. Then, using 4 and 5 we obain V c X c/a, X c/b = X c/a = X c/a C = = X c/a C, X a/b C, X a/c, X a/c X a/c C X c/b, Xc/b X c/a X c/b 7. 8 According o 8, or 7 if we prefer, he currency C value of he opion now has wo dela exposures: i a dela o he X c/a exchange rae and ii a dela o he X c/b exchange rae. We obain V c X c/a V c X c/b = C Xc/b X c/a = C X C X 9 10 where C X is he usual Black-Scholes dela. These quaniies can hen be used o hedge he currency C value of he opion. Or we migh jus hedge he X c/a exposure and leave he X c/b exposure un-hedged. This way of viewing he value of he opion is paricularly useful if he rading book conains posiions in many currency pairs and we ulimaely only care abou he value of he porfolio or he P&L in one paricular currency. I is also useful when some of he currency pairs, e.g. X a/b, are no liquid and we can only hedge hem by rading in X c/a and X c/b. Remark 1 If we are assuming sicky-by-dela implied volailiy surfaces hen we migh wan o include addiional erms in 9 and 10 corresponding o C. For example, if X c/a σ a/b changes and we hold consan X c/b hen X a/b will change accordingly wih a corresponding change in σ a/b due o he sicky-by-dela assumpion. This would require an addiional erm on he righ-hand-side of 9. Similarly, a change in X c/b whils holding X c/a consan would require an addiional erm on he righ-hand-side of 10. Wheher or no such erms would be included in pracice probably depends on he currencies in quesion, he preferences of he rader and he flexibiliy of he sysems. Exercise 3 Creae an Excel spreadshee o price and hedge currency opions. You can assume he opions are on he JPY/EUR exchange rae. Simulae values of he exchange rae hrough ime and execue he usual dela-hedging sraegy. Compare he final USD value of he hedge porfolio which is denominaed in JPY wih he final USD value of he porfolio where he USD-value of he opion was dynamically hedged. You can assume he exchange raes all follow GBM s wih consan ineres raes, volailiies and correlaion. These parameers should be inpus in he spreadshee. 9 Alhough i could jus as easily be JPY.
7 Foreign Exchange and Quanos 7 3 Opions on Foreign Asses Sruck in Domesic Currency Le S denoe he ime price 10 of he foreign asse in unis of he foreign currency. Then he payoff of he opion is given by Payoff = max0, X T S T K. 11 Le us wrie he domesic risk-neural dynamics for S, wih he usual domesic cash accoun as numeraire, as ds = µ s S d + σs dw 12 wih dw dw x = ρ d. We assume he exchange rae dynamics coninue o follow 2. In general we could le he correlaion coefficien, ρ, be sochasic bu we will assume here ha i is a consan, ρ, for all. Noe ha he underlying price, X S is he ime value of he foreign sock measured in unis of he domesic currency. We will also assume ha S pays a coninuous dividend yield of q. Remark 2 From he perspecive of a foreign invesor wih foreign cash accoun as numeraire, he foreign asse s risk-neural drif in 12 would be µ s = r f q. This is no rue for he domesic invesor, however, since S is no he domesic price of a raded asse nor 11 is he foreign cash accoun a domesic numeraire. In fac i is S X ha is he domesic price of a raded asse. This raded asse pays a dividend yield of q and so i is his securiy ha has a drif of r d q under he domesic risk-neural dynamics, i.e. he Q-dynamics aking he domesic cash accoun as numeraire. A simple applicaion of Iô s Lemma implies ha Z := X S has dynamics given by dz = r d r f + µ s + ρσ x σ d + σ x dw x + σ dw Z σx = r d r f + µ s + ρσ x σ d + σ z dw x + σ dw σ z σ z 13 where σ z := σ 2 x + σ 2 + 2ρσ x σ. The quadraic variaion of he second erm in parenheses on he righ-hand-side of 13 equals d. We can herefore apply Levy s Theorem and wrie Z as a Q-GBM wih drif r d q and volailiy, σ z. Noe ha we didn care abou he drif in 13 since he Q-drif of Z mus be r d q. The opion price is hen given by he Black-Scholes formula wih risk-free rae r d, dividend yield q and volailiy σ z. Exercise 4 Argue ha he value of µ s saisfies µ s = r f q ρσ x σ. 14 Remark 3 In pracice, a more sophisicaed model would generally be used o price his opion, depending as i does on he join dynamics of X and S. This is no rue of our firs example where we priced a vanilla currency opion. In his case he correc volailiy o be used in he Black-Scholes formula could be read off he volailiy surface. I is also no rue of he opion in he exercise below where he volailiy surface can again be used o obain he correc opion price. Exercise 5 How would you price an opion on a foreign asse ha has been sruck in foreign currency. Hin: This is easy! 10 Noe ha in he U.S a securiy whose ime value is given by X S is called an American Deposiory Receip or ADR. If, for example, S represens he ime value of a share of Volkswagen denominaed in Euro, hen a Volkswagen ADR allows US invesors o easily inves in Volkswagen wihou having o rade in he foreign exchange markes. Similarly a European Deposiory Receip or EDR allows European invesors o inves direcly in non-euro-denominaed securiies. Noe, however, ha he holder of an ADR or EDR is sill exposed o foreign exchange risk. This is in conras o he holder of a quano securiy which we will discuss laer. 11 In paricular, an EMM-numeraire pair for a foreign invesor is no an EMM-numeraire for a domesic invesor. Bu see Exercise 6.
8 Foreign Exchange and Quanos 8 Exercise 6 Le Q f be an EMM for a foreign invesor corresponding o some foreign numeraire, X f say. Noe ha X f is hen denominaed in unis of he foreign currency. Show ha Q f can also be used as an EMM for a domesic invesor. Wha is he corresponding domesic numeraire? Is he converse also rue, i.e., can any domesic EMM also be used as an EMM for a foreign invesor? 4 Quano-Securiies We now consider securiies whose payoff is based on S T convered ino unis of domesic currency a a fixed exchange rae, X. Such a securiy is called a quano. We will consider quano forwards and quano opions and price hem wihin he Black-Scholes framework. Quanos are raded very frequenly in pracice, paricularly in he srucured producs marke. Consider, for example, an opion on a baske of hree socks: IBM, Toyoa and Siemens. The hree sock prices are denominaed in USD, JPY and EUR, respecively. Bu wha currency should he opion payoff be denominaed in? Suppose we choose US dollars bu we don wan any exchange rae exposure. Then a call opion wih srike K migh have a payoff given by max 0, 1 IBM T + 1 SIE T + 1 T OY T 3 IBM 0 3 SIE 0 3 T OY 0 K and we say ha Siemens and Toyoa have been quano ed ino US dollars a fixed exchange raes of 1/3 SIE 0 and 1/3 T OY 0, respecively. Anoher marke where quanos are frequenly raded is he commodiy marke. Mos commodiies are priced in US dollars bu non-us invesors ofen wish o rade commodiies denominaed in unis of heir domesic currency. For example, a European invesor who wishes o buy an opion on crude oil bu wihou he USD/EUR exchange risk could buy a quano opion where he oil is quano ed ino EUR. For example, a call opion on oil wih srike K and mauriy T would normally have a ime T payoff of max0, O T K which is denominaed in USD since he oil price, O T, is denominaed in USD. If he opion is quano ed ino Euro, however, hen he payoff is again max0, O T K, bu now i is denominaed in Euro despie he fac ha O T is denominaed in USD. Quano Forwards Consider a quano-forward conrac where XS T is exchanged a mauriy, T, for F q unis of he domesic currency. If we ener ino his conrac a ime, hen we know ha F q := F q is chosen so ha he iniial value of he conrac is 0. Using he usual domesic cash accoun as numeraire, we immediaely obain ha [ 0 = E Q e r dt XS ] T F q so ha F q = XE Q [S T ]. We herefore need he Q-dynamics of S in order o compue F q. Bu from 12 and 14, we know he Q-dynamics of S are ds = r f q ρσ x σs d + σs dw 15 and so we obain why? F q = XS e r f q ρσ x σt. 16 Exercise 7 Show ha he fair ime- value of receiving XS T a ime T is given by Hin: This requires a one-line argumen given 15 or 16. V = XS e r f r d q ρσ x σt. 17
9 Foreign Exchange and Quanos 9 Noe ha V is he value in unis of he domesic currency of a non-dividend 12 paying raded asse. Noe also ha F q = e r dt V 18 which is he usual form of a forward price. Tha is, F q is he ime- forward price for ime-t delivery of a domesic non-dividend paying securiy ha is worh V T a ime T. The sandard replicaing porfolio argumen using V and he domesic cash accoun can be used as an alernaive proof of 18. The quesion also arises as o how o dynamically replicae he ime-t payoff V T = XS T using he domesic cash accoun, he foreign cash accoun and he foreign asse. In fac a each ime,, he replicaing sraegy consiss of 1. Invesing V unis of he domesic currency in he foreign asse. 2. Invesing V unis of he domesic currency in he foreign-cash accoun, i.e. borrowing in he foreign currency o fund he purchase of he foreign asse in sep Invesing V unis of he domesic currency in he domesic cash accoun. Noe ha he firs wo componens of he hedging sraegy ensure ha he sraegy has no FX exposure. This mus be he case as he asse we seek o replicae, V T = XS T, has no FX exposure. Noe also ha he ime -value of he replicaing porfolio is V, as expeced. Exercise 8 Use Iô s Lemma and 17 o show ha he replicaing sraegy is indeed as we have described. Hin: You migh sar by assuming ha a unis of he domesic currency are invesed in he foreign asse, b unis of domesic currency are invesed in he foreign cash accoun and c unis of domesic currency are invesed in he domesic cash accoun. The ime- domesic currency value of his porfolio is W = a + b + c. The replicaing sraegy mus be self-financing 13 and so he dynamics of he sraegy mus saisfy dw = a dreurn on foreign asse +b dreurn on foreign cash accoun +c dreurn on domesic cash accoun. Noe, for example, ha dreurn on foreign asse = dx e q S /X e q S. Why? Use his and oher corresponding erms o compue dw and hen compare wih he dynamics of V which are obained using 17. See Back s A Course in Derivaive Securiies Secion 6.6 if you are unsure. Quano Opions We are now in a posiion o price quano opions. In fac, given our earlier analysis pricing a quano opion is sraighforward. Le us assume in paricular ha we seek o price a call opion wih payoff in unis of he domesic currency given by Quano Opion Payoff = max 0, XST K. Maringale pricing saes ha he ime- price, P say, of his securiy is given by P = E Q [e r dt max 0, XST K ] where he Q-dynamics of S saisfy 15. Noe ha we can rewrie 15 as ds = r d q f S d + σs dw 19 where q f := q + r d r f + ρσ x σ. Bu hen we see ha he quano opion price is jus X imes he Black-Scholes price of a call opion on a sock wih iniial value S, ime-o-mauriy T, volailiy σ, srike K/ X and dividend yield q f. Remark 4 An alernaive mehod of calculaing he quano opion price is o view i as an opion on he non-dividend securiy wih ime -value V. We can again use he Black-Scholes formula bu his ime using a zero-dividend yield and an iniial value of V 0 = XS 0 e r f r d q ρσ xσt. 12 Noe ha even hough V is non-dividend paying, he underlying securiy S, may well pay dividends. 13 All replicaing sraegies are, by definiion, assumed o be self-financing.
10 Foreign Exchange and Quanos 10 I is also sraighforward o replicae he payoff of he quano opion using he foreign asse, he foreign cash accoun and he domesic accoun. This is done by adoping he usual Black-Scholes dela-hedging sraegy using he domesic cash accoun and he porfolio wih value V as he underlying risky asse. A each ime, we hold dela unis of he porfolio and borrow he difference beween he cos of he dela unis and he value of he opion. The porfolio is consruced using seps 1 o 3 as described earlier. Remark 5 Noe ha he key o valuing quano ed securiies is obaining he domesic risk-neural dynamics of hese securiies. Everyhing else follows from he usual argumens. Remark 6 Banks and oher marke makers are very relucan o quano asses from high-risk economies. Afer all, who would wan o sell an opion ha pays in US dollars max0, Z T K where Z T is he reurn on he Zimbabwe sock marke!?
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