Coupling Smiles Valdo Durrleman Deparmen of Mahemaics Sanford Universiy Sanford, CA 94305, USA Nicole El Karoui Cenre de Mahémaiques Appliquées Ecole Polyechnique 91128 Palaiseau, France November 18, 2006 Absrac The presen paper addresses he problem of compuing implied volailiies of opions wrien on a domesic asse based on implied volailiies of opions on he same asse expressed in a foreign currency and he exchange rae. I proposes an original mehod ogeher wih explici formulas o compue he a-he-money implied volailiy, he smile s skew, convexiy, and erm srucure for shor mauriies. The mehod is compleely free of any model specificaion or Markov assumpion; i only assumes ha jumps are no presen. We also invesigae how he mehod performs on he paricular example of he currency riple dollar, euro, yen. We find a very saisfacory agreemen beween our formulas and he marke a one week and one monh mauriies. Acknowledgemens: The auhors hank Andres Villaquiran for helpful discussions. 1 Inroducion Consider he problem of compuing he implied volailiy of an opion on a domesic asse based on implied volailiies of opions on he same asse expressed in a foreign currency and he exchange rae. I is well known ha he informaion conained in he implied volailiy smile of opions on a given underlying for a given mauriy is exacly he risk-neural disribuion of he random variable modelling he underlying asse s value a mauriy (see [1]). In he presen problem, we have informaion abou he risk-neural disribuions of he foreign asse and he exchange rae a differen fuure daes and we are ineresed in he law of he produc of hese wo random variables. This law is however far from being deermined by he wo marginal disribuions. A his poin, i is emping o cas he problem ino a copula problem, i.e., ha of finding he risk-neural copula of he wo underlying asses. The space of copulas is unforunaely very large and he opion marke gives comparaively lile informaion abou he asses join disribuion. Saisical mehods based on pas join observaions could also be used. These may be manageable wih wo asses bu 1
hey quickly become unracable wih a large number of variables. Once a copula funcion has been chosen, i remains o compue he law of he produc of he wo random variables. As aracive as i migh look, he copula problem focuses only on disribuional properies and herefore compleely forges abou he dynamics of he underlying variables. This is raher unsaisfacory since modern finance is precisely buil on dynamic self-financing sraegies. In he presen paper, we propose a mehod o compue implied volailiies of opions on a domesic asse based on implied volailiies of opions on he same asse expressed in a foreign currency and he exchange rae. The join law is deermined by insananeous correlaions. These can be esimaed from hisorical daa and reinroduce an dynamical picure. The paper s idea is based on resuls of [2] ha relae he spo volailiy process o he shape of he implied volailiy smile around he money and for shor mauriies. These resuls provide a simple way o go back and forh beween he dynamics of he spo (spo volailiy) and is disribuional properies (implied volailiies). The key observaion is ha he spo volailiy of he produc of wo underlying asses is simply he sum heir volailiies. To go from he individual implied volailiies o he implied volailiy of he produc, we use ha simple relaion a he spo volailiy level. Once graned he resuls of [2], he problem is reduced o compuing quadraic variaions. The mehod is herefore very general and flexible. We inend o presen furher applicaions in laer papers. The paper is organized as follows. Firs, we presen he relaions beween spo and implied volailiies. We pu emphasis on heir financial inerpreaions leaving he formal proofs o [2]. The second par provides he formulas o compue he implied volailiy smile of quano opions. Finally, we look a he cross-volailiies problem and see how our soluion performs on he currency riple dollar, euro, yen. 2 Relaing implied and spo volailiies Le us imagine an invesor who can coninuously rade in a sock S and in European opions wrien on i. The implied volailiy a ime of he European opion wih srike K and mauriy T is denoed by Σ (T, K). Because he invesor can diversify he risks in boh he sock and in is possibly random volailiy vecor σ by rading among hese insrumens, he invesor is risk-neural. The dynamics of he sock under he pricing measure is ds S = σ dw where W is a mulidimensional Brownian moion. The spo volailiy is σ = σ. We have assumed, for simpliciy, ha ineres raes are zero. 2
2.1 The four fundamenal conracs and heir relaion o he smile P&L of a dela hedged opion. Le us firs recall a well-known resul abou he profi or loss of a dela hedged opion. Consider he porfolio consising of being long he opion and shor is Black-Scholes dela hedge compued wih he implied volailiy Σ (T, K) as inpu. Over a small ime inerval δ before mauriy, he posiion s profi or loss is 1 ( 2 ΓS2 σ 2 Σ(T, K) 2) δ. Γ is he Black-Scholes gamma compued a Σ(T, K). (For a derivaion of his resul, see [3].) Sraddle Wih ha in mind, le us look a he paricular case where he opion is an a-he-money sraddle (i.e, long an a-he-money call and an a-he-money pu wih same mauriies.) The payoff and he Black-Scholes gamma are ploed agains S in he op lef corner of Figure 1. Figure 1: Payoffs and Black-Scholes gammas. I is easily seen from he Black-Scholes formula ha he price of he a-he-money sraddle is approximaely linear in he implied volailiy Sraddle Σ(T, S). On he oher hand, in view of he large posiive gamma around he money, a dela hedged sraddle will lead o a profi or loss ha is proporional o he realized volailiy 3
σ. ( The invesor is long volailiy. ) We herefore expec Σ(T, S ) and σ o be closely relaed for shor mauriies. The correc relaionship acually is (1a) Σ AT M := lim T Σ (T, S ) = σ. Σ AT M is he limi of a-he-money implied volailiies when he ime o mauriy shrinks o zero. Risk-reversal Nex consider he case where he opion is a risk-reversal (i.e., shor an ou-of-he-money pu wih srike K 1 and long an ou-of-he-money call wih srike K 2 > K 1.) See he op righ corner of Figure 1 for he payoff and gamma of a risk-reversal. When srikes are no oo far from he money, he price of a risk-reversal is approximaely proporional o he difference beween he wo implied volailiies Risk-Reversal Σ(T, K 2 ) Σ(T, K 1 ), or, equivalenly, o he skew. On he oher hand, in view of he posiive gamma a large srikes and negaive gamma a low srikes, is value is in direc relaion wih he correlaion beween moves of S and σ. Indeed, in he region of posiive gamma, he holder of a risk-reversal expecs he volailiy σ o be as high as possible. In oher words, he expecs a posiive correlaion beween moves in S and σ. Likewise, in he region of negaive convexiy, a posiive correlaion will be in he favor of he invesor, since a low sock price will come wih a low volailiy. We herefore expec a relaionship beween skewness and spo/volailiy correlaion. The correc relaionship acually is (1b) S := lim T S Σ K (T, S ) = 1 2σ 2 1 d dσ; ds S S is he skew for shor mauriies, i.e., he shor mauriy limi of he derivaive wih respec o moneyness K/S. To coninue he discussion, we fix some noaion for he sochasic evoluion of σ under he pricing measure: dσ 2 = 2σ δ d + 2σ 2 ν dw. We refer o δ as he drif and o ν = ν as he volailiy of volailiy (V-vol). ν is he coefficien in fron of dw in dσ /σ, i herefore has he dimension of a volailiy. Wih his noaion, (1b) rewries S = ν σ 2σ.. 4
Buerfly spread Consider now a buerfly spread, where he invesor is long an ou-of-he-money call, an ou-of-he-money pu, and shor an a-he-money sraddle. See he boom lef corner of Figure 1 for he payoff and gamma of a buerfly spread. When srikes are no oo far from he money, he price of a risk-reversal is approximaely proporional o he convexiy of he smile Buerfly Spread Σ(T, K 1 ) 2Σ(T, S) + Σ(T, K 2 ) As before, we relae he price o he profi or loss of he rade. Firs, if S moves up (say, o 110, in case of he figure) a long posiion in he buerfly spread will mosly look like a long posiion in a risk-reversal (similar gamma profiles). Also, if S moves down, a long posiion in he buerfly spread will mosly look like a shor posiion in a risk-reversal. The price of he buerfly spread should herefore be proporional o he correlaion beween changes in S and changes in he price of a risk-reversal, or, as we jus saw, changes in skewness S. Second, assume ha he spo/volailiy correlaion is zero. Suppose ha he V- vol ν is large. This implies ha σ varies a lo, eiher up or down. If σ urns ou o be large as well, hen he spo experiences a large volailiy and moves up and down, bu in each case he spo will be in a region of posiive gamma. Since σ is large, he rade will resul in a profi. Similarly, in case σ urns ou o be small, S will no be volaile and will say around is curren value, which is in he region of negaive gamma. The rade again resuls in a profi because volailiy is small. The price of he buerfly spread should herefore be proporional o ν. Third, le us look a wha migh happen if he spo/volailiy correlaion is no zero. Whaever he sign, his will lower he profi of he invesor in a buerfly spread. Indeed, i may lead o low volailiy in regions of posiive gamma and high volailiy in he region of negaive gamma. The price of he buerfly spread should herefore be proporional o S or S 2. The correc relaionship acually is (1c) C := lim S 2 2 Σ T K (T, S ) = 2 2 3σ 2 1 d ds; ds S + ν2 3σ 2S2 σ S. C is he smile s convexiy for shor mauriies, i.e., he limi of he second derivaive wih respec o moneyness as he ime o mauriy shrinks o 0. The reason for he las erm on he righ-hand side ( S ) comes from he fac ha we differeniae wih respec o moneyness and no log-moneyness. Calendar spread Le us finally sudy he erm srucure of he implied volailiy surface. To his end, we consider a calendar spread, where he invesor is shor an a-he-money sraddle wih mauriy T 1 and long anoher a-he-money sraddle wih longer mauriy T 2 > T 1. The price of he calendar spread is approximaely proporional o he difference beween he wo implied volailiies Calendar Spread Σ(T 2, S) Σ(T 1, S) Because of he posiive gamma beween T 1 and T 2, he profi or loss of he calendar spread is proporional o a relaive increase of volailiy σ. This longer erm effec is 5
relaed o he drif and we expec he price of he calendar spread o be proporional o δ. On he oher hand, his long erm effec will be less pronounced in case he V-vol is high because he random flucuaions will be more imporan. We herefore expec he price of he calendar spread o be proporional o ν. Also, a non zero spo/volailiy correlaion will dampen he long erm effec by driving he spo far away from is curren value o regions of posiive bu small gamma as volailiy changes. The correc relaionship acually is (1d) Σ M := lim T T (T, S ) = 1 ( ) δ σ 2 C 3σ S 2. 2 M is he smile s erm srucure for shor mauriies, i.e., he limi of he derivaive wih respec o mauriy T as he ime o mauriy shrinks o 0. Equaions (1a) (1d) are consequence of no arbirage resricions among he spo and European call or pu opions. They are rigorously derived in [2] under a se of assumpions, which essenially insure ha he implied volailiy surface is smooh and behaves nicely for shor mauriies. In paricular, hey insure ha he limis of derivaives showing in he above equaions do exis in some precise sense. These condiions do no depend on any paricular model bu hold, for insance, in he case of sochasic volailiy models wih analyic coefficiens. Precise saemens and assumpions can be found in [2]. Le us jus sress ha he marke informaion is modelled by he filraion generaed by a finie dimensional Wiener process. 2.2 Implied volailiies approximaion We can use formulas (1a) (1d) o compue an approximaion of he implied volailiy smile in a given model. By model, we mean a paricular sochasic evoluion for σ. Indeed, firs parameerize he sochasic differenial equaion by δ and ν, dσ 2 = 2σ δ d + 2σ 2 ν dw. Then, compue Σ AT M, S, C, and M using (1a) (1d). We ge he following approximaion for he implied volailiy smile, valid for < T : (2) Σ (T, K) = Σ AT M + (T )M + K S S S + ( K S S ) 2 C 2 + error erms. This expansion is no exacly Taylor s expansion because wo derivaives are missing and i is herefore no easy o quanify he error erms. However, his approximaion 2 will be bes if boh T and K S S are small compared o he ypical scales of 2 he problem. Tha means 1. The ime scale is given by he inverse of he K S S volailiy so T small means σ 2 T 1. An example: Heson model Heson s model reads dσ 2 = κ ( µ σ 2 ) d + εσ d W 6
where W is a Brownian moion wih correlaion ρ wih ha driving he sock. This model is easily rewrien using a wo dimensional Wiener process W and (1a) (1d) lead o he following: S = ερ 4σ C = ε2 ( ) 2 5ρ 2 ερ 24σ 3 4σ M = κ ( ) µ σ 2 ε 2 4σ 48σ 3 ( 2 ρ 2 /2 ) + ερσ 8. These formulas give a lo of insigh abou he roles of he model parameers. 2.3 From implied o spo volailiies The nice feaure of (1a) (1d) is ha hey can be invered o give informaion abou he underlying process when we observe he smile dynamics. We saw ha he dynamics of he smile is well summarized by he dynamics of he shor-daed and a-he-money implied volailiy Σ AT M, he skewness S, he convexiy C, and he erm srucure M. Given ha we have an idea abou heir dynamics, here is wha we can say abou he underlying process S and is volailiy σ. The insananeous volailiy is precisely he implied volailiy of he a-he-money and shor-daed opion (3a) σ = Σ AT M. The sochasic differenial equaion driving σ can be expressed using hese derivaives: (3b) dσ 2 = 2σ ( 2M + σ 2 C + 3σ S 2 ) d + 2σ 2 ν dw, where ν is he V-vol and, as usual, ν = ν. The inner produc beween σ and ν is given, up o a proporionaliy facor, by S : (3c) σ ν = 2σ S. Finally, ν is relaed o he join quadraic variaion beween he smile s slope and he reurn on S: (3d) ν 2 = 3σ (C + S ) + 6S 2 2 1 ds; ds σ d S (3a), (3c), and (3d) are enough o characerize ν and σ up o roaions. Since Wiener measure is invarian under hem, he process (S, σ) is compleely specified.. 7
Example To illusrae how his mehodology could be pu o use, we look a daa on opions on he euro-yen exchange rae. The daa we used is explained in secion 4.1. Here, we simply compue S EURJP Y and C EURJP Y AT M,EURJP Y and plo hem agains Σ in Figure 2. I shows ha, while S EURJP Y should be modelled as a sochasic process on is own, C EURJP Y can reasonably be modelled as a simple power funcion of he exchange rae s volailiy. Figure 2: S EURJP Y and C EURJP Y vs. Σ AT M,EURJP Y from 06-Oc-2003 o 30-Sep-2005 2.4 Maringale represenaion and hedging The previous asympoic resul on implied volailiies is ighly linked o an asympoic resul on he maringale represenaion of opion prices. We recall i here. I can be skipped in a firs reading since we will only use i in secion 3.4. Le us consider a call or a pu opion wih srike K and mauriy T. Is ime- 8
price, O (T, K), is O (T, K) = E { (S T K) ± F } (Recall ha ineres raes are zero.) Under he pricing measure O (T, K) is a maringale and we would like o ge an expression for is maringale represenaion. In [2], i is shown ha i can be wrien as follows: do (T, K) = ( (T, K) + Vega (T, K)α (T, K)) ds + Vega (T, K)β (T, K)dW where and Vega are respecively he opion s Black dela and vega evaluaed a implied volailiy Σ (T, K), α (T, K) = 1 1 dσ(t, K); ds Σ (T, K). σ 2 S d S K α (T, K) is acually chosen such ha he remaining volailiy risk is uncorrelaed wih S, i.e, β (T, K) is a vecor process saisfying Moreover, β (T, K)σ = 0. β (T, K) 2 = d d Σ(T, K) 1 σ 2 ( 1 dσ(t, K); ds d S ) 2. We would like o use his maringale represenaion o hedge he opion wih a posiion in S and oher opions. The adjused dela hedge is well idenified in he firs erm in he maringale represenaion. The orhogonal par is a pure volailiy risk ha can be hedged wih a posiion in oher opions. In he case of a-he-money and shor-daed opions we furher know ha (4) lim T β (T, S ) = σ ν 2S σ. 3 Quano opions This secion is devoed o quano opions, i.e., equiy linked forex opions. These are discussed in deails in [4] in he consan volailiy case. We are going o apply he resuls of he previous secion o he coupling of he smile of a sock in he foreign marke and he smile of he exchange rae o ge he smile of opions on he sock in he domesic marke. We now assume deerminisic (bu non necessarily zero) ineres raes. This is a minor assumpion; in he shor mauriy regime, he effec of ineres raes can be alogeher negleced. To use he resuls presened in secion 2, we simply have o replace S by he forward price. 9
3.1 Noaions Consider a foreign economy where he forward conrac for delivery a ime T of some underlying asse rades a F f (T ) a ime. Is volailiy is denoed by σ f : df f (T ) F f (T ) = σf dw f where W f is Brownian moion under he foreign pricing measure. The ype of quano opions we consider here are opions on he foreign forward conrac wih srike in he domesic currency. As usual, we inroduce he forward exchange rae from he foreign o he domesic currency for delivery a ime T. We denoed i by X (T ) and is volailiy by σ X : dx (T ) X (T ) = σx dw d where W d is Brownian moion under he domesic pricing measure. Sandard arbirage argumens show ha X (T )F f (T ) is nohing else han he arbirage price of he forward conrac on he foreign underlying asse for delivery a ime T and denominaed in domesic currency, F d (T ), i.e., Is volailiy is denoed by σ d and F d (T ) = X (T )F f (T ). df d (T ) F d (T ) = σd dw d. Moreover, he change of measure from he foreign o he domesic measures is given by (5) W f = W d Then, 0 σ d = σ f + σ X, σ X s ds. and if le γ = σ X σ f be he insananeous covariance beween reurns on X and F f, we ge he following expression for (σ d ) 2 : (6) (σ d ) 2 = (σ f ) 2 + 2γ + (σ X ) 2. The quano opion can be hough of as an opion in he domesic marke on he domesic forward conrac F d. If we assume ha σ d is a deerminisic funcion of ime, hen, a call or a pu quano opion is priced and hedged using Black s formula. 10
3.2 Bringing smiles ino he picure We now wish o incorporae he smile effecs on boh he foreign marke and he foreign exchange marke ino he pricing and hedging of quano opions. In general, σ d is no deerminisic and we would like o compue he volailiy parameer o be plugged ino Black s formula. As in secion 2.3, we inroduce he sochasic differenial equaions driving σ f and σ X : ) d(σ f ) 2 = 2σ f (2M f + (σ f ) 2 C f + 3σ f (S f ) 2 d + 2(σ f ) 2 ν f dw f and d(σ X ) 2 = 2σ X ( 2M X + (σ X ) 2 C X + 3σ X (S X ) 2) d + 2(σ X ) 2 ν X dw d. We someimes wrie δ f and δ X for he erms in parenhesis in he drifs. We shall also need o specify he dynamics of he insananeous correlaion ρ = σ f σ X /(σ f σ X ): dρ = µ ρ d + ω ρ dw d. I will be useful o compue he dynamics of he insananeous covariance γ = σ f σ X : dγ = [γ ( δ f σ f + δx σ X 1 2 ν f ν X 2 ν f σ X ) + + σ f σ X The drif of γ will be denoed by µ and is diffusion erm by ω. ( µ ρ + ω ρ (ν f + ν X )) ] d [ ] γ (ν f + ν X ) + σ f σ X ω ρ dw d. 3.3 Formulas for S d, C d, and M d We now compue S d, C d, and M d in erms of he corresponding quaniies for F f and X and various insananeous covariances. Using Iô s calculus, we ge a sochasic differenial equaion for (σ d ) 2 from (6) (7) ( ( ) ) ) d(σ d ) 2 = 2 µ + σ f δ f σ f ν f σ X + σ X δ X d+2 ((σ f ) 2 ν f + (σ X ) 2 ν X + ω dw d Therefore, F d s V-vol is (8) ν d = (σf ) 2 + γ (σ d ) 2 ν f + (σx ) 2 + γ (σ d ) 2 ν X + σf σ X (σ d ) 2 ωρ. From his, we can compue he hree derivaives of he implied volailiy surface. Proposiion 3.1. S d, C d, and M d S d = (σf ) 2 + γ ( 2σ f 2(σ d ) 3 S f + ν f σ X ) are given by he following + (σx ) 2 + γ ( 2σ X 2(σ d ) 3 S X + ν X σ f ) + σf σ X 2(σ d ) 3 ωρ σ d 11
C d = 6(Sd ) 2 S d σ d + (νd ) 2 + (σf ) 2 + γ [ ( ) (σ f 3σ d 3(σ d ) 5 ) 2 3σ f (C f + S f ) + 18(S f ) 2 (ν f ) 2 +2σ f 1 ds f ; dx + σ f 1 d νf σ X d X d σ ; df d ( ) ] + 2ν f X F d σ X 5σ f S f + σ X S X + ν f σ X [ + (σx ) 2 + γ (σ X 3(σ d ) 5 ) ( 2 3σ X (C X + S X ) + 18(S X ) 2 (ν X ) 2) + 2σ X 1 ds X ; df f d F f +σ X 1 d νx σ f ; df d ( ) ] + 2ν X d σ f F d σ f 5σ X S X + σ f S f + ν X σ f + (νx σ f )(ν f σ X ) 3(σ d ) 3 γ ( ) 2 2σ f 3(σ d ) 5 S f + ν f σ X 2σ X S X ν X σ f σ f + σ X 1 d(ω ρ σ d ); df d 3(σ d ) 5 d F d + 2γ ( ) 3(σ d ) 5 ωρ σ d 2σ f S f + ν f σ X + 2σ X S X + ν X σ f M d = (σd ) 2 C d 2 3σd (S d ) 2 2 + 1 2σ d [ 2σ f M f + (σ f ) 3 C f + 3(σ f S f ) 2 + 2σ X M X ] +(σ X ) 3 C X + 3(σ X S X ) 2 + µ (σ f ) 2 ν f σ X These formulas are remarkable in he sense ha hey do no depend on a sochasic volailiy model. The equaion giving S d has he following inerpreaion. The produc of he volailiy and he smile s slope in he domesic marke is an average of he same quaniies in he foreign and foreign exchange markes plus correcion erms involving insananeous covariances. These correcion erms have an ineresing role. Assume ha he X-smile and F f -smile have a U-shape wih minimum a he money. This means ha S X = S f = 0. This also means, in view of (3c), ha each forward is uncorrelaed wih is volailiy. On he oher hand, he same migh no be rue in he domesic marke if he forwards are correlaed wih each oher s volailiy, or if he insananeous covariance beween he forwards is correlaed wih he domesic forward. Proof. To compue S d, use (1b) o ge S d = νd σ d 2σ d, Then replace σ d = σ f + σ X and ν d from (8). Finally, use (1b) for S f and S X. To compue C d, sar by using (1b) and (1c) o ge C = 6S2 S + ν2 + 2 1 d(σ 3 S); df σ 3σ 3σ 5 d F and hen compue he quadraic covariaion from he formula jus found for S d. Finally, o compue M d, find he drif of (σ d ) 2 from (7) and use (1d). 12
3.4 Hedging In his secion, we sudy how an opion on F d can be hedged. As explained in secion 2.4 we compue he adjused dela, which gives he number of domesic conracs o be held. We denoe by Π d (T, K) he porfolio consising in being shor one opion and long he adjused dela number of domesic conracs. The risk of his porfolio is a pure volailiy risk: d Πd (T, K) B d (T ) = Vega d β d (T, K)dW d wih β d (T, K)σ d = 0. B d (T ) denoes he price of he domesic zero coupon bond wih mauriy T. Using (4), his remaining risk is approximaed (for a-he-money and shor-daed opion) by Thanks o (8), i rewries ( d Πd (T, K) Vega d B d (T ) d Πd (T, K) B d (T ) (σ f ) 2 + γ σ d Vega d ( σ d ν d 2S d σ d ) dw d. ν f + (σx ) 2 + γ ν X σ d + σf σ X σ d ω ρ 2S d σ d ) dw d. Par of his risk can be approximaely replicaed wih posiions in dela hedged opions in he foreign marke Π f (T f, K f ), dela hedged opions in he foreign exchange marke Π X (T X, K X ), and in F f, X and F d d Πd (T, K) σf + ρ σ X B d (T ) σ d ( + 2Vega d σ f + ρ σ X σ d S f Vega d Vega f df f F f : d Πf (T f, K f ) B f (T f ) + σx + ρ σ f σ d + σx + ρ σ f σ d S X dx X S d Vega d Vega X df d F d d ΠX (T X, K X ) B d (T X ) ) + σf σ X 2σ d ω ρ dw d From he las formula, i becomes apparen ha his scheme is acually a rade on he volailiy of he correlaion beween X and F f. Indeed, if his correlaion is deerminisic, ω ρ = 0. If correlaion is sochasic and ω ρ canno be wrien as a linear combinaion of σ f, σ X, ν f, and ν X, par of he correlaion s volailiy risk canno be hedged. 4 Applicaion o cross-volailiies As an illusraion of Proposiion 3.1, we look a he problem of cross-volailiies. Take he hree major currencies EUR, USD, and JPY, and heir corresponding exchange raes EURJPY, EURUSD, and USDJPY. They saisfy he no arbirage relaion EURJPY = EURUSD USDJPY. We are going o reconsruc he implied volailiy smile on exchange rae EURJPY from he smiles on EURUSD and USDJPY. In his case, he smile on EURJPY is also observable so ha we can see how Proposiion 3.1 performs in pracice. 13.
4.1 The daa The OTC marke is paricularly well suied for our purposes. Conracs have mauriies 1 week, 1, 2, 3, and 6 monhs, and 1 and 2 years. We will mosly focus on conracs having one week or one monh mauriy since our resuls are valid for shor mauriies. The available conracs are sraddles, risk reversals, and buerfly spreads. As i cusomary in hese markes, he srikes are quoed in erms of he corresponding dela. Sraddles are a-he-money forward. Risk reversals and buerfly spreads have srikes a ±25 or ±10 dela. (For conracs wih a mauriy of one week, only ±25 dela risk reversals and buerfly spreads are available.) The rule o compue he implied volailiies given mid-marke quoes for he a-he-money sraddle (a), 25-dela risk reversal (r), and 25-dela buerfly spread (b) is repored in Figure 3. Figure 3: Rule o compue implied volailiies given mid-marke quoes for he ahe-money sraddle (a), 25-dela risk reversal (r), and 25-dela buerfly spread (b). There are 520 daa poins from 06-Oc-2003 o 30-Sep-2005. Figure 4 displays he observed volailiy smiles agains moneyness for he hree exchange raes a a paricular dae. 4.2 Resuls We firs compue he observed S and C by fiing a parabola hrough he mid-marke quoes, and M by linear inerpolaion beween a-he-money implied volailiies. We hen use he formulas of Proposiion 3.1 o reconsruc hese quaniies for he EURJPY based on EURUSD and USDJPY. The various insananeous correlaions are compued wih a 30 rading day window. We did no esimae he correlaion ρ beween reurns on EURUSD and USDJPY, bu insead used he implied correlaion based on (6) wih he a-he-money implied volailiies. The drif of ρ, which appears 14
Figure 4: Implied volailiies for each exchange rae on Sepember 23, 2005. A cross + denoes a mid-marke quoe. in µ and which only plays a role in M, is se o 0 because we do no have a reasonable way of esimaing i wihou knowing he risk premia. We hen use he same mehod o reconsruc EURUSD implied volailiies from he EURJPY and USDJPY smiles, and USDJPY implied volailiies from he EURJPY and EURUSD smiles. To apply Proposiion 3.1 in hese cases, we need o compue he smile of 1/F based on ha of F. We make use of he following formulas ha are consequence of he domesic call/foreign pu symmery: S 1/F = S F C 1/F + S 1/F = C F + S F M 1/F = M F. They can also be recovered from (1a) (1d) using he fac ha σ 1/F = σ F. Figures 5, 6, 7, 8, 9, and 10 repor resuls for he period from 01-Mar-2004 o 06-Oc-2005. In each graph, we plo wo esimaed quaniies agains he acually observed values. We firs compue S, C, and M based on he formulas of Proposiion 3.1, i is called he full esimaor. The parial esimaor is obained by seing o 0 all insananeous covariances. The overall agreemen very good. I is beer wih one week mauriy, as i was expeced, since we are using asympoic formulas valid in he shor mauriy regime. Using esimaes of he insananeous covariances does no seem o really improve he qualiy of he esimaors. Surprisingly, our formulas predic a somewha more convex EURUSD smile. A firs explanaion could be ha opion raders in he EURUSD 15
marke may no rade opions involving he Japanese Yen. In oher words, markes may be segmened. Anoher explanaion could be ha we are seeing he effec of jumps. Finally, o complemen he resuls above, we focus on he period from 31-Dec- 2003 o 24-Mar-2004. The reconsrucion formula does no work very well during his period as seen in Figure 11 for he EURJPY a one week mauriy. For insance, i predics negaive values for he smile s convexiy. The problem probably comes from he markes exreme volailiy during he period as seen in Figure 12: we ploed he daily reurns and he one week a-he-money implied volailiies. Due o he suddenly low a-he-money volailiies, he hisorical insananeous covariance beween σ USDJP Y and ρ USDJP Y/EURUSD exhibis a very usual behavior. The boom par of Figure 12 displays his insananeous covariance agains anoher one ha is more sable. 5 Conclusion The presen paper addresses he problem of compuing implied volailiies of opions wrien on a domesic asse based on implied volailiies of opions on he same asse expressed in a foreign currency and he exchange rae. I proposes an original mehod ogeher wih explici formulas o compue he ATM implied volailiy, he smile s skew, convexiy, erm srucure for shor mauriies. The mehod is compleely free of any model specificaion or Markov assumpion. I only assumes ha jumps are no presen. We also invesigae how he mehod performs on he paricular example of he currency riple dollar, euro, yen. We find a very saisfacory agreemen beween our formulas and he marke a one week and one monh mauriies. References [1] D. Breeden and R. Lizenberger. Prices of sae-coningen claims implici in opion prices. J. Bus., 51(4):621 651, 1978. [2] V. Durrleman. From implied o spo volailiies. Technical repor, 2005. [3] N. El Karoui, M. Jeanblanc-Picqué, and S. Shreve. Robusness of he Black and Scholes formula. Mah. Finance, 8(2):93 126, 1998. [4] E. Reiner. Quano mechanics. RISK, 5(10):59 63, 1992. 16
Figure 5: S EURJP Y, C EURJP Y, and M EURJP Y a one week mauriy reconsruced wih full formula (dashed line) and wih parial formula (doed line) agains he observed values (solid line). 17
Figure 6: S EURJP Y, C EURJP Y, and M EURJP Y a one monh mauriy reconsruced wih full formula (dashed line) and wih parial formula (doed line) agains he observed values (solid line). 18
Figure 7: S USDJP Y, C USDJP Y, and M USDJP Y a one week mauriy reconsruced wih full formula (dashed line) and wih parial formula (doed line) agains he observed values (solid line). 19
Figure 8: S USDJP Y, C USDJP Y, and M USDJP Y a one monh mauriy reconsruced wih full formula (dashed line) and wih parial formula (doed line) agains he observed values (solid line). 20
Figure 9: S EURUSD, C EURUSD, and M EURUSD a one week mauriy reconsruced wih full formula (dashed line) and wih parial formula (doed line) agains he observed values (solid line). 21
Figure 10: S EURUSD, C EURUSD, and M EURUSD a one monh mauriy reconsruced wih full formula (dashed line) and wih parial formula (doed line) agains he observed values (solid line). 22
Figure 11: S EURJP Y, C EURJP Y, and M EURJP Y a one week mauriy reconsruced wih full formula (dashed line) and wih parial formula (doed line) agains he observed values (solid line). 23
Figure 12: Daily reurns (op), one week a-he-money implied volailiies (middle), and insananeous covariances beween σ USDJP Y and ρ USDJP Y/EURUSD, and beween σ EURUSD and ρ USDJP Y/EURUSD (boom). 24