Quano Opions Uwe Wysup MahFinance AG Waldems, Germany www.mahfinance.com 19 Sepember 2008
Conens 1 Quano Opions 2 1.1 FX Quano Drif Adjusmen.......................... 2 1.1.1 Exensions o oher Models....................... 4 1.2 Quano Vanilla.................................. 5 1.3 Quano Forward................................. 5 1.4 Quano Digial.................................. 6 1.5 Hedging of Quano Opions........................... 6 1.5.1 Vega Posiions of Quano Plain Vanilla Opions............ 7 1.6 Applicaions................................... 8 1.6.1 Performance Linked Deposis...................... 9 1 Quano Opions A quano opion can be any cash-seled opion, whose payoff is convered ino a hird currency a mauriy a a pre-specified rae, called he quano facor. There can be quano plain vanilla, quano barriers, quano forward sars, quano corridors, ec. The Arbirage pricing heory and he Fundamenal heorem of asse pricing, also covered for example in [3] and [2], allow he compuaion of opion values. Oher references: Opions: basic definiions, Opion pricing: general principles, Foreign exchange marke erminology. 1.1 FX Quano Drif Adjusmen We ake he example of a Gold conrac wih underlying XAU/USD in XAU-USD quoaion ha is quanoed ino EUR. Since he payoff is in EUR, we le EUR be he numeraire or domesic or base currency and consider a Black-Scholes model XAU-EUR: ds (3) = (r EUR r XAU )S (3) d + σ 3 S (3) dw (3), (1) USD-EUR: d = (r EUR r USD ) d + σ 2 dw (2), (2) dw (3) dw (2) = ρ 23 d, (3) where we use a minus sign in fron of he correlaion, because boh S (3) and have he same base currency (DOM), which is EUR in his case. The scenario is displayed in Figure 1. The acual underlying is hen Using Iô s formula, we firs obain XAU-USD: S (1) = S(3). (4)
Quano Opions 3 XAU σ σ 3 1 ϕ 23 EUR π ϕ 23 σ 2 ϕ π ϕ12 12 USD Figure 1: XAU-USD-EUR FX Quano Triangle. The arrows poin in he direcion of he respecive base currencies. The lengh of he edges represens he volailiy. The cosine of he angles cos φ ij = ρ ij represens he correlaion of he currency pairs S (i) and S (j), if he base currency (DOM) of S (i) is he underlying currency (FOR) of S (j). If boh S (i) and S (j) have he same base currency (DOM), hen he correlaion is denoed by ρ ij = cos(π φ ij ). d 1 = 1 ( ) 2 ds(2) + 1 2 2 1 = (r USD r EUR + σ 2 2) 1 ( ) 3 (ds(2) ) 2 1 d σ 2 dw (2), (5)
4 Wysup and hence ds (1) = 1 ds (3) + S (3) d 1 = S(3) + S(3) (r EUR r XAU ) d + S(3) + ds (3) d 1 σ 3 dw (3) (r USD r EUR + σ2) 2 d S(3) σ 2 dw (2) + S(3) ρ 23 σ 2 σ 3 d = (r USD r XAU + σ 2 2 + ρ 23 σ 2 σ 3 )S (1) d + S (1) (σ 3 dw (3) σ 2 dw (2) ). Since S (1) is a geomeric Brownian moion wih volailiy σ 1, we inroduce a new Brownian moion W (1) and find ds (1) = (r USD r XAU + σ 2 2 + ρ 23 σ 2 σ 3 )S (1) d + σ 1 S (1) dw (1). (6) Now Figure 1 and he law of cosine imply which yields σ 2 3 = σ 2 1 + σ 2 2 2ρ 12 σ 1 σ 2, (7) σ 2 1 = σ 2 2 + σ 2 3 + 2ρ 23 σ 2 σ 3, (8) σ 2 2 + ρ 23 σ 2 σ 3 = ρ 12 σ 1 σ 2. (9) As explained in he currency riangle in Figure 1, ρ 12 is he correlaion beween XAU-USD and USD-EUR, whence ρ = ρ 12 is he correlaion beween XAU-USD and EUR-USD. Insering his ino Equaion (6), we obain he usual formula for he drif adjusmen ds (1) = (r USD r XAU ρσ 1 σ 2 )S (1) d + σ 1 S (1) dw (1). (10) This is he Risk Neural Pricing process ha can be used for he valuaion of any derivaive depending on S (1) which is quanoed ino EUR. 1.1.1 Exensions o oher Models The previous derivaion can be exended o he case of erm-srucure of volailiy and correlaion. However, inroducion of volailiy smile would disor he relaionships. Neverheless, accouning for smile effecs is imporan in real marke scenarios. See Foreign exchange smile: convenions and empirical facs and Foreign exchange smile modeling for deails. To do his, one could, for example, capure he smile for a muli-currency model wih a weighed Mone Carlo echnique as described by Avellaneda e al. in [1]. This would sill allow o use he previous resul.
Quano Opions 5 1.2 Quano Vanilla Common among Foreign exchange opions is a quano plain vanilla paying Q[φ(S T K)] +, (11) where K denoes he srike, T he expiraion ime, φ he usual pu-call indicaor aking he value +1 for a call and 1 for a pu, S he underlying in FOR-DOM quoaion and Q he quano facor from he domesic currency ino he quano currency. We le µ = r d r f ρσ σ, (12) be he adjused drif, where r d and r f denoe he risk free raes of he domesic and foreign underlying currency pair respecively, σ = σ 1 he volailiy of his currency pair, σ = σ 2 he volailiy of he currency pair DOM-QUANTO and ρ = σ2 3 σ 2 σ 2 (13) 2σ σ he correlaion beween he currency pairs FOR-DOM and DOM-QUANTO in his quoaion. Furhermore we le r Q be he risk free rae of he quano currency. Wih he same principles as in Pricing formulae for foreign exchange opions we can derive he formula for he value as v = Qe r QT φ[s 0 e µt N (φd + ) KN (φd )], (14) d ± = ln S 0 + ( µ ± 1σ2) T K 2 σ, (15) T where N denoes he cumulaive sandard normal disribuion funcion and n is densiy. 1.3 Quano Forward Similarly, we can easily deermine he value of a quano forward paying Q[φ(S T K)], (16) where K denoes he srike, T he expiraion ime, φ he usual long-shor indicaor, S he underlying in FOR-DOM quoaion and Q he quano facor from he domesic currency ino he quano currency. Then he formula for he value can be wrien as v = Qe r QT φ[s 0 e µt K]. (17) This follows from he vanilla quano value formula by aking boh he normal probabiliies o be one. These normal probabiliies are exercise probabiliies under some measure. Since a forward conrac is always exercised, boh hese probabiliies mus be equal o one.
6 Wysup 1.4 Quano Digial A European syle quano digial pays QII {φst φk}, (18) where K denoes he srike, S T he spo of he currency pair FOR-DOM a mauriy T, φ akes he values +1 for a digial call and 1 for a digial pu, and Q is he pre-specified conversion rae from he domesic o he quano currency. The valuaion of European syle quano digials follows he same principle as in he quano vanilla opion case. The value is v = Qe r QT N (φd ). (19) We provide an example of European syle digial pu in USD/JPY quano ino EUR in Table 1. Noional Mauriy European syle Barrier Theoreical value Fixing source 100,000 EUR 3 monhs (92days) 108.65 USD-JPY 71,555 EUR ECB Table 1: Example of a quano digial pu. The buyer receives 100,000 EUR if a mauriy, he ECB fixing for USD-JPY (compued via EUR-JPY and EUR-USD) is below 108.65. Terms were creaed on Jan 12 2004 wih he following marke daa: USD-JPY spo ref 106.60, USD-JPY ATM vol 8.55%, EUR-JPY ATM vol 6.69%, EUR-USD ATM vol 10.99% (corresponding o a correlaion of -27.89% for USD-JPY agains JPY-EUR), USD rae 2.5%, JPY rae 0.1%, EUR rae 4%. 1.5 Hedging of Quano Opions Hedging of quano opions can be done by running a muli-currency opions book. All he usual Greeks can be hedged. Dela hedging is done by rading in he underlying spo marke. An excepion is he correlaion risk, which can only be hedged wih oher derivaives depending on he same correlaion. This is normally no possible. In FX he correlaion risk can be ranslaed ino a vega posiion as shown in [4] or in he secion on Foreign exchange baske opions. We illusrae his approach for quano plain vanilla opions now.
Quano Opions 7 1.5.1 Vega Posiions of Quano Plain Vanilla Opions Saring from Equaion (14), we obain he sensiiviies v [ σ = QS 0e ( µ r Q)T n(d + ) ] T φn (φd + )ρ σt, v σ = QS 0e ( µ r Q)T φn (φd + )ρσt, v ρ = QS 0e ( µ r Q)T φn (φd + )σ σt, v = v ρ σ 3 ρ σ 3 = v σ 3 ρ σ σ = QS 0 e ( µ r Q)T φn (φd + )σ σt σ 3 σ σ = QS 0 e ( µ r Q)T φn (φd + )σ 3 T = QS 0 e ( µ r Q)T φn (φd + ) σ 2 + σ 2 + 2ρσ σt. Noe ha he compuaion is sandard calculus and repeaedly using he ideniy S 0 e µt n(φd + ) = Kn(φd ). (20) The undersanding of hese Greeks is ha σ and σ are boh risky parameers, independen of each oher. The hird independen risk is eiher σ 3 or ρ, depending on wha is more likely o be known. This shows exacly how he hree vega posiions can be hedged wih plain vanilla opions in all hree legs, provided here is a liquid vanilla opions marke in all hree legs. In he example wih XAU-USD-EUR he currency pairs XAU-USD and EUR-USD are raded, however, here is no liquid vanilla marke in XAU-EUR. Therefore, he correlaion risk remains unhedgeable. Similar saemens would apply for quanoed socks or sock indices. However, in FX, here are siuaions wih all legs being hedgeable, for insance EUR-USD-JPY. The signs of he vega posiions are no uniquely deermined in all legs. The FOR-DOM vega is smaller han he corresponding vanilla vega in case of a call and posiive correlaion or pu and negaive correlaion, larger in case of a pu and posiive correlaion or call and negaive correlaion. The DOM-Q vega akes he sign of he correlaion in case of a call and is opposie sign in case of a pu. The FOR-Q vega akes he opposie sign of he pu-call indicaor φ. We provide an example of pricing and vega hedging scenario in Table 2, where we noice, ha dominaing vega risk comes from he FOR-DOM pair, whence mos of he risk can be hedged.
8 Wysup daa se 1 daa se 2 daa se 3 FX pair FOR-DOM XAU-USD XAU-USD XAU-USD spo FOR-DOM 800.00 800.00 800.00 srike FOR-DOM 810.00 810.00 810.00 quano DOM-Q 1.0000 1.0000 1.0000 volailiy FOR-DOM 10.00% 10.00% 10.00% quano volailiy DOM-Q 12.00% 12.00% 12.00% correlaion FOR-DOM DOM-Q 25.00% 25.00% -75.00% domesic ineres rae DOM 2.0000% 2.0000% 2.0000% foreign ineres rae FOR 0.5000% 0.5000% 0.5000% quano currency rae Q 4.0000% 4.0000% 4.0000% ime in years T 1 1 1 1=call -1=pu FOR 1-1 1 quano vanilla opion value 30.81329 31.28625 35.90062 quano vanilla opion vega FOR-DOM 298.14188 321.49308 350.14600 quano vanilla opion vega DOM-Q -10.07056 9.38877 33.38797 quano vanilla opion vega FOR-Q -70.23447 65.47953-35.61383 quano vanilla opion correlaion risk -4.83387 4.50661-5.34207 quano vanilla opion vol FOR-Q 17.4356% 17.4356% 8.0000% vanilla opion value 32.6657 30.7635 32.6657 vanilla opion vega 316.6994 316.6994 316.6994 Table 2: Example of a quano plain vanilla. 1.6 Applicaions The sandard applicaion are performance linked deposi or performance noes as in [5]. Any ime he performance of an underlying asse needs o be convered ino he noional currency invesed, and he exchange rae risk is wih he seller, we need a quano produc. Naurally,
Quano Opions 9 an underlying like gold, which is quoed in USD, would be a defaul candidae for a quano produc, when he invesmen is in a currency oher han USD. 1.6.1 Performance Linked Deposis A performance linked deposi is a deposi wih a paricipaion in an underlying marke. The sandard is ha a GBP invesor waives her coupon ha he money marke would pay and insead buys a EUR-GBP call wih he same mauriy dae as he coupon, srike K and noional N in EUR. These parameers have o be chosen in such a way ha he offer price of he EUR call equals he money marke ineres rae plus he sales margin. The srike is ofen chosen o be he curren spo. The noional is ofen a percenage p of he deposi amoun A, such as 50% or 25%. The annual coupon paid o he invesor is hen a pre-defined minimum coupon plus he paricipaion p max[s T S 0, 0] S 0, (21) which is he reurn of he exchange rae viewed as an asse, where he invesor is proeced agains negaive reurns. So, obviously, he invesor buys a EUR call GBP pu wih srike K = S 0 and noional N = pa GBP or N = pa/s 0 EUR. Thus, if he EUR goes up by 10% agains he GBP, he invesor ges a coupon of p 10% p.a. in addiion o he minimum coupon. Example. We consider he example shown in Table 3. In his case, if he EUR-GBP spo fixing is 0.7200, he addiional coupon would be 0.8571% p.a. The break-even poin is a 0.7467, so his produc is advisable for a very srong EUR bullish view. For a weakly bullish view an alernaive would be o buy an up-and-ou call wih barrier a 0.7400 and 75% paricipaion, where we would find he bes case o be 0.7399 wih an addiional coupon of 4.275% p.a., which would lead o a oal coupon of 6.275% p.a. Composiion ˆ From he money marke we ge 49,863.01 GBP a he mauriy dae. ˆ The invesor buys a EUR call GBP pu wih srike 0.7000 and wih noional 1.5 Million GBP. ˆ The offer price of he call is 26,220.73 GBP, assuming a volailiy of 8.0% and a EUR rae of 2.50%. ˆ The deferred premium is 24,677.11 GBP. ˆ The invesor receives a minimum paymen of 24,931.51 GBP.
10 Wysup Noional 5,000,000 GBP Sar dae 3 June 2005 Mauriy Number of days 91 Money marke reference rae EUR-GBP spo reference 0.7000 Minimum rae Addiional coupon S T Fixing source 2 Sepember 2005 (91 days) 4.00% ac/365 2.00% ac/365 30% 100 max[s T 0.7000,0] 0.7000 ac/365 EUR-GBP fixing on 31 Augus 2005 (88 days) ECB Table 3: Example of a performance linked deposi, where he invesor is paid 30% of he EUR-GBP reurn. Noe ha in GBP he daycoun convenion in he money marke is ac/365 raher han ac/360. ˆ Subracing he deferred premium and he minimum paymen from he money marke leaves a sales margin of 254.40 GBP (awfully poor I admi). ˆ Noe ha he opion he invesor is buying mus be cash-seled. Variaions. There are many variaions of he performance linked noes. Of course, one can hink of European syle knock-ou calls or window-barrier calls. For a paricipaion in a downward rend, he invesor can buy pus. One of he frequen issues in Foreign Exchange, however, is he deposi currency being differen from he domesic currency of he exchange rae, which is quoed in FOR-DOM (foreign-domesic), meaning how many unis of domesic currency are required o buy one uni of foreign currency. So if we have a EUR invesor who wishes o paricipae in a EUR-USD movemen, we have a problem, he usual quano confusion ha can drive anybody up he wall in FX a various occasions. Wha is he problem? The payoff of he EUR call USD pu [S T K] + (22) is in domesic currency (USD). Of course, his payoff can be convered ino he foreign currency (EUR) a mauriy, bu a wha rae? If we conver a rae S T, which is wha we could do in he spo marke a no cos, hen he invesor buys a vanilla EUR call. Bu here, he invesor receives a coupon given by
Quano Opions 11 p max[s T S 0, 0] S T. (23) If he invesor wishes o have performance of Equaion (21) raher han Equaion (23), hen he payoff a mauriy is convered a a rae of 1.0000 ino EUR, and his rae is se a he beginning of he rade. This is he quano facor, and he vanilla is acually a self-quano vanilla, i.e., a EUR call USD pu, cash-seled in EUR, where he payoff in USD is convered ino EUR a a rae of 1.0000. This self quano vanilla can be valued by invering he exchange rae, i.e., looking a USD-EUR. This way he valuaion can incorporae he smile of EUR-USD. Similar consideraions need o be aken ino accoun if he currency pair o paricipae in does no conain he deposi currency a all. A ypical siuaion is a EUR invesor, who wishes o paricipae in he gold price, which is measured in USD, so he invesor needs o buy a XAU call USD pu quanoed ino EUR. So he invesor is promised a coupon as in Equaion (21) for a XAU-USD underlying, where he coupon is paid in EUR, his implicily means ha we mus use a quano plain vanilla wih a quano facor of 1.0000. References [1] Avellaneda, M., Buff, R., Friedman, C., Grandechamp, N., Kruk, L. and Newman, j. (2001). Weighed Mone Carlo: A new Technique for Calibraing Asse- Pricing Models. Inernaional Journal of Theoreical and Applied Finance, vol 4, No. 1, pp. 91-119. [2] Hakala, J. and Wysup, U. (2002). Foreign Exchange Risk. Risk Publicaions, London. [3] Shreve, S.E. (2004). Sochasic Calculus for Finance I+II. Springer. [4] Wysup, U. (2001). How he Greeks would have hedged Correlaion Risk of Foreign Exchange Opions, Wilmo Research Repor, Augus 2001. Also in Foreign Exchange Risk, Risk Publicaions, London 2002. [5] Wysup, U. (2006). FX Opions and Srucured Producs. Wiley Finance Series.
Index correlaion, FX, 5 currency riangle, 4 law of cosine, 4 performance linked deposi, 8 quano digial, 6 quano drif adjusmen, 2 quano facor, 2 quano forward, 5 quano opions, 2 quano plain vanilla, 8 quano vanilla, 4 self-quano, 10 vega, quano plain vanilla, 6 12