Currency Derivatives under a Minimal Market Model with Random Scaling

Transcription

1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 54 March 25 Currency Derivaives under a Minimal Marke Model wih Random Scaling David Heah and Eckhard Plaen ISSN

2 Currency Derivaives under a Minimal Marke Model wih Random Scaling David Heah and Eckhard Plaen March 29, 25 Absrac. This paper uses an alernaive, parsimonious sochasic volailiy model o describe he dynamics of a currency marke for he pricing and hedging of derivaives. Time ransformed squared Bessel processes are he basic driving facors of he minimal marke model. The ime ransformaion is characerized by a random scaling, which provides for realisic exchange rae dynamics. The pricing of sandard European opions is sudied. In paricular, i is shown ha he model produces implied volailiy surfaces ha are ypically observed in real markes. 99 Mahemaics Subjec Classificaion: primary 9A2; secondary 6G3, 62P2. JEL Classificaion: G, G3, D52 Key words and phrases: currency derivaives, sochasic volailiy, random scaling, minimal marke model. Universiy of Technology Sydney, School of Finance & Economics and Deparmen of Mahemaical Sciences, PO Box 23, Broadway, NSW, 27, Ausralia

3 Inroducion The well-known Black-Scholes model (BSM) uses geomeric Brownian moions o model he dynamics of he underlying securiies. This leads o a convenien and compac asse pricing model ha is characerized by deerminisic volailiies. Furhermore, he BSM admis an equivalen risk neural maringale measure so ha he sandard risk neural pricing mehodology can be direcly applied. In currency markes he volailiies of exchange raes are no deerminisic, as has been observed in a wide range of sudies, see, for insance, Malz (997). This is also refleced in deviaions of observed derivaive prices from he prices prediced under he BSM. Therefore, i is essenial o provide reliable prices and hedging prescripions for a range of derivaive securiies. One major line of research uses one-facor local volailiy funcion models, see, for insance, Derman & Kani (994) and Dupire (992), o capure he effec of sochasic volailiy in an exchange rae. However, for more advanced applicaions such a model remains somewha resriced because i allows only one facor o drive he exchange rae dynamics. A muli-facor model is essenial if one aims o model he currency marke as a consisen family of inerrelaed cross currency exchange raes. Using inraday exchange rae daa, heir dynamics have been analyzed in Breymann, Kelly & Plaen (25). This sudy suggess ha an exchange rae should be modeled by a raio of wo ime ransformed squared Bessel processes. In Plaen (2) he minimal marke model (MMM) models exchange raes as such raios of squared Bessel processes in a parsimonious and consisen manner. Since he resuling currency model is a generalizaion of he MMM, i does no allow for he exisence of an equivalen risk neural maringale measure, as poined ou in Heah & Plaen (22a, 22b). Therefore, o obain a consisen arbirage free pricing framework we apply he fair pricing concep of he benchmark approach developed in Plaen (22). More precisely, we model he dynamics of differen denominaions of he growh opimal porfolio (GOP), which is he porfolio ha maximizes expeced logarihmic uiliy. An exchange rae hen equals he raio of wo currency denominaions of he GOP. This paper documens he fac ha realisic implied volailiy surfaces for sandard European opions on exchange raes can be generaed naurally from he proposed MMM wih random scaling. The paper is organized as follows. In Secion 2 a general currency marke is described. Using he benchmark approach, Secion 3 inroduces he fair pricing of derivaives. A generalizaion of he MMM wih random scaling is proposed in Secion 4. The corresponding pricing funcionals for a wide class of coningen claims are derived in Secion 5. European call and pu opions on exchange raes are described in Secion 6, where a new momen expansion procedure is employed o compue implied volailiy surfaces. Finally, in Secion 7 a series of numerical experimens are described. These documen he ype of realisic implied volailiy 2

4 surfaces ha can be obained under he MMM. 2 Currency Benchmark Model Consider he evoluion of he savings accouns of d + currencies, d {, 2,...}. To be mahemaically precise, hese are modeled on a filered probabiliy space (Ω, A T, A, P ), where he filraion A = (A ) [,T ] fulfills he usual condiions, see Karazas & Shreve (998), wih A being rivial. Here A describes he informaion ha is available a ime [, T ]. The measure P is he real world probabiliy measure. Le r j denoe he jh shor rae a ime [, T ], j {,,..., d}. This is he shores forward rae for he jh currency. In his paper we assume, for simpliciy, ha r j = {r j, [, T ]} is a nonnegaive deerminisic funcion of ime. Noe however ha he shor rae can be made sochasic wihou changing significanly he resuls of his paper. Using hese definiions r denoes he domesic shor rae a ime. The jh savings accoun process B j = {B j, [, T ]} of he jh currency, when denominaed in his currency, is given by he differenial equaion db j = B j r j d (2.) for [, T ] wih B j =. The locally riskless savings accoun can be inerpreed as he limi in probabiliy of a sequence of rollover shor erm bond accouns wih imes o mauriy converging o zero. The (i, j)h exchange rae X i,j denoes he price of one uni of he jh currency a ime [, T ], when measured in unis of he ih currency i, j {,,..., d}. Viewed from he domesic marke, he quaniy X,j denoes he exchange rae for he jh currency. Wihou loss of generaliy, we inerpre he h currency as he domesic currency. I is assumed ha all exchange rae processes are coninuous. We inroduce d + primary securiy accoun processes, which are chosen o be he savings accouns. The jh savings accoun value a ime, when denominaed in unis of he domesic currency, is hen given by he expression for [, T ] and j {,,..., d}. S,j = X,j B j, (2.2) We now inroduce porfolios of primary securiy accouns, where S = {S = (S,, S,,..., S,d ), [, T ]} denoes he vecor process of primary securiy accoun prices when expressed in unis of he domesic currency. We call a predicable, S-inegrable sochasic process δ = {δ = (δ,..., δ d ), [, T ]} a sraegy. The quaniy δ j (, ) denoes he number of unis of he jh primary securiy accoun ha are held a ime according o he sraegy δ. In 3

5 addiion, le S (δ) = {S (δ), [, T ]} be he value process of he corresponding porfolio, where S (δ) is denominaed in unis of he domesic currency, ha is S (δ) = j= δ j S,j, (2.3) almos surely, for [, T ]. We say ha a sraegy δ is self-financing if ds (δ) = j= δ j ds,j (2.4) for [, T ]. Here he sochasic differenials are defined as Iô differenials, see Proer (24). In pracical erms, he self-financing propery means ha all changes in he value of he porfolio are due o gains or losses from rade. We assume hroughou he paper ha all porfolios and sraegies are self-financing and omi herefore his aribue from now on. Of paricular imporance in our analysis will be he growh opimal porfolio (GOP). This is he porfolio ha maximizes expeced logarihmic uiliy from erminal wealh, see Kelly (956), Long (99) or Karazas & Shreve (998). I has been shown in Plaen (22) ha he given coninuous complee marke has a unique GOP. According o he above menioned lieraure, he value D a ime of he GOP, when denominaed in unis of he domesic currency, saisfies he sochasic differenial equaion (SDE) dd = D ( r d + θ,k ( θ,k ) ) d + dw k (2.5) for [, T ] wih D >. Here he processes W k = {W k, [, T ]}, k {, 2,..., d}, denoe independen sandard Wiener processes. Noe ha he volailiies θ,k, k {, 2,..., d}, of he GOP are he corresponding domesic marke prices of risk and he risk premium of he GOP is he sum of he squares of hese marke prices of risk. I has been shown in Plaen (24a, 24b, 25) ha he GOP D can be approximaed by a well diversified accumulaion world sock index and can herefore be inerpreed as a marke index. Also Breymann, Kelly & Plaen (24) have found in a comparison of inraday world sock indices ha he marke capializaion weighed world sock index is a good proxy for he GOP. We assume ha a oal reurn world sock index (WSI) can be considered as a good approximaion for he GOP. To illusrae hese ideas we show in Figure he values of a WSI in Briish Pound and US dollar over he period from 97 unil 24. This index has been consruced for illusraive purposes by using marke capializaion and sock index daa as available from Thomson Financial. Noe ha for hese currency denominaions he index is on average growing over ime. 4

6 4 35 GBP USD ime Figure : WSI denominaed in GBP and USD. Le us denoe by D i he denominaion of he GOP in unis of he ih currency, i {,,..., d}. I is sraighforward o conclude by symmery ha as in (2.5) he GOP D, i when denominaed in unis of he ih currency, mus saisfy he SDE dd i = D i ( r i d + θ i,k ( θ i,k ) ) d + dw k (2.6) for [, T ] and i {,,..., d}. Here θ i,k is he marke price of risk of he ih currency denominaion wih respec o he kh Wiener process W k a ime. We assume ha he ih denominaion of he GOP remains sricly posiive, ha is, D i > almos surely for all [, T ] and i {,,..., d}. Furhermore, he volailiy processes θ j,k are assumed o be predicable and such ha T j= ( ) θ j,k 2 s ds < almos surely. Obviously, since he GOP is unique, he (i, j)h exchange rae a ime can be expressed by he raio X i,j = Di D j (2.7) for [, T ] and i, j {,,..., d}. Using he Iô formula ogeher wih (2.6) and (2.7) i can be seen ha dx i,j = X i,j (( r i r j + θ i,k ( θ i,k ) ) θ j,k d + ( θ i,k ) ) θ j,k dw k (2.8) 5

7 for [, T ] wih iniial value X i,j = Di D j > and i, j {,,..., d}. Furhermore, by (2.2), (2.) and (2.7), he value of he jh savings accoun, when denominaed in unis of he domesic currency, is given by he formula S,j = D D j B j (2.9) for [, T ] and j {,,..., d}. By applicaion of he Iô formula we obain from (2.9), (2.) and (2.6) for S,j he SDE ( ( ) ( ) ) ds,j = S,j r d + θ,k θ j,k θ,k d + dw k (2.) for [, T ] wih iniial value S,j > and j {,,..., d}. To avoid redundan primary securiy accouns we assume ha he domesic volailiy marix b = [b j,k wih j, kh domesic volailiy ] d j, b j,k = θ,k θ j,k (2.) is inverible, see Plaen (22). We call he above model a currency benchmark model, where he GOP plays he role of he benchmark. 3 Benchmarked Securiies and Fair Pricing Under he benchmark approach, we call any price ha is expressed in unis of he GOP a benchmarked price. Thus, he benchmarked jh savings accoun value a ime is obained by he raio Ŝ (j) Ŝ (j) = S,j D (3.) for [, T ] and j {,,..., d}. By applicaion of he Iô formula we obain from (3.), (2.) and (2.5) for he benchmarked jh savings accoun Ŝ(j) he SDE dŝ(j) = Ŝ(j) θ j,k dw k (3.2) for [, T ], j {,,..., d}. Noe ha he SDE (3.2) for Ŝ(j) is drifless and Ŝ(j) is herefore a nonnegaive (A, P )-local maringale, see Proer (24), 6

8 j {,,..., d}. Similarly, by applicaion of he Iô formula i follows from (2.4), (2.9) and (2.6) ha any benchmarked porfolio Ŝ (δ) = S(δ) D (3.3) is also drifless, ha is dŝ(δ) = j= δ j dŝ(j) (3.4) for [, T ]. This means ha benchmarked porfolio processes are (A, P )- local maringales. Consequenly, any nonnegaive benchmarked porfolio is an (A, P )-supermaringale, see Proer (24). By using his propery i is shown in Plaen (22) ha from zero iniial capial one canno generae, by using any nonnegaive self-financing porfolio, some sricly posiive wealh wih sricly posiive probabiliy. In his sense he above benchmark model is arbirage free. The Radon-Nikodym derivaive Λ (j) = {Λ (j), [, T ]}, wih Λ (j) = Ŝ(j) Ŝ (j) (3.5) for [, T ] and j {,,..., d}, for he candidae risk neural measure of he jh currency denominaion equals up o a consan facor he jh benchmarked savings accoun. Obviously, by (3.2) he process Λ (j) is an (A, P )-local maringale, j {,,..., d}. Mos of he lieraure on currency derivaive pricing makes he assumpion ha here exis equivalen risk neural maringale measures for all currency denominaions. This means ha one assumes ha he Radon-Nikodym derivaives Λ (), Λ (),..., Λ (d) of he corresponding candidae risk neural measures are for all currency denominaions maringales. To see wheher his assumpion makes pracical sense we plo in Figure 2 he Radon-Nikodym derivaive for he US dollar denominaion, where we inerpre again he WSI as GOP. Here he US shor rae has been se o abou 5%, which is above is average of 4.%, see Dimson, Marsh & Saunon (22). I is obvious ha he Radon-Nikodym derivaive process shows a sysemaic decline over ime. This can be observed for all major currencies. In addiion, i is widely recognized ha he world sock porfolio will ouperform in he long run he savings accoun of any currency, see Dimson, Marsh & Saunon (22). This provides furher evidence ha he Radon-Nikodym derivaive, given in (3.5), should sysemaically decline in he long run. Such a sysemaic decline is no ypical for rajecories of maringales, however, i is ypical for rajecories of nonnegaive sric supermaringales, as is suggesed by he MMM ha we will inroduce laer. The sysemaic decline of he graphs in Figure 2 can be aken as a warning ha one should avoid he sandard risk neural pricing mehodology for realisic currency marke models. 7

9 ime Figure 2: Radon-Nikodym derivaive for US currency denominaion. Since we do no rely on he sandard risk neural pricing mehodology we need a more general pricing concep. For he pricing of derivaives we apply he concep of fair pricing, as inroduced in Plaen (22). A price process is called fair if, when expressed in unis of he GOP, forms an (A, P )-maringale. This means, benchmarked fair prices are maringales. However noe ha savings accouns do no need o be fair. Also in a risk neural world, where an equivalen risk neural maringale measure exiss, one can show ha risk neural prices of derivaives are fair. For a mauriy dae T (, ) le H T denoe an A T -measurable, nonnegaive coningen claim, which is expressed in unis of he domesic currency and has finie expeced benchmarked value, ha is ( ) HT E A < (3.6) D T for all [, T ]. The benchmarked, fair price û HT () a ime of his coningen claim is given by he condiional expecaion ( ) HT û HT () = E A (3.7) for [, T ]. This specificaion makes he benchmarked price process û HT = {û HT (), [, T ]} an (A, P )-maringale ha maches he coningen claim H T a he mauriy dae T. Any nonnegaive porfolio ha replicaes he coningen claim is according o (3.4) and (3.3) an (A, P )-supermaringale. I has herefore an iniial value greaer or equal o he fair value. The fair value describes in he given currency benchmark model he minimal price ha allows one o replicae he coningen claim by a nonnegaive porfolio, see Plaen (22). D T 8

10 The fair price u HT () of he coningen claim H T a ime, when expressed in unis of he domesic currency, is hen according o (3.3) obained by he fair pricing formula u HT () = D û HT () (3.8) for [, T ]. An analogous formula holds for each currency denominaion. If here exiss an equivalen risk neural maringale measure P wih Radon- Nikodym derivaive Λ () T = d P dp AT = B T D DT a ime, hen he fair pricing formula (3.8) coincides wih he classical risk neural pricing formula, as can be seen by he relaions ( D u HT () = E D T B T B B BT ) ( Λ H T A = E T Λ B BT ) ( B H T A = Ẽ B T H T A ). Here Ẽ denoes expecaion under he domesic equivalen risk neural maringale measure, see Plaen (22). However, as indicaed previously, a realisic currency benchmark model is unlikely o have an equivalen risk neural maringale measure. This is why we will use he fair pricing formula (3.8). 4 Minimal Marke Model wih Random Scaling The model ha we consider now is a generalizaion of he minimal marke model (MMM) suggesed in Plaen (2, 22). The MMM generaes sochasic volailiies ha involve ransformaions of squared Bessel processes. As discussed in Plaen (2) and Breymann, Kelly & Plaen (24), he MMM capures a number of imporan sylized empirical facs observed for indices and exchange raes. Under he generalized version of he MMM ha we are going o consider, he jh denominaion D j of he GOP a ime [, T ] is modeled by he equaion D j = (Z j ) ν j 2 B j, (4.) where Z j = {Z j, [, T ]} is a ime ransformed squared Bessel process of dimension ν j > 2, which saisfies he SDE dz j = ν j 4 (q j,k ) 2 γ d + Z j γ q j,k dw k (4.2) for [, T ] wih Z j = (D) j 2 ν j 2 >, j {,,..., d}. We know, see Revuz & Yor (999), ha he above ime ransformed squared Bessel process says almos surely sricly posiive. The (j, k)h scaling level q j,k = {q j,k, [, T ]} will be specified below, as well as he scaling process γ = {γ, [, T ]}. In general, one could inroduce a separae scaling for each squared Bessel process. However, for simpliciy we consider here only a single common scaling process γ. The scaling 9

11 levels q j,k can, for insance, be modeled as coninuous ime Markov chains. For simpliciy, we assume here ha hese are consans. By applicaion of he Iô formula i follows from (4.) and (4.2) ha he jh denominaion D j of he GOP a ime saisfies he SDE (2.5) wih (j, k)h volailiy, ha is j, kh marke price of risk, ( θ j,k νj ) = 2 q j,k γ Z j (4.3) for [, T ], j {,,..., d}, k {, 2,..., d}. This volailiy is sochasic since i is proporional o he inverse of he square roo of he jh squared Bessel process Z j. We allow he scaling γ o be random o generae globally for he currency marke random ime ransformaions for he squared Bessel processes involved. One can say ha he random scaling models he rading aciviy in he currency marke. To model his in deail, we assume ha he random scaling γ saisfies he SDE dγ = a(, γ ) d + b(, γ ) ϱ k dw k + (ϱ k ) 2 d W (4.4) for [, T ] wih γ. Here W = { W, [, T ]} is an independen sandard Wiener process ha does no drive he rading noise of he marke. Furhermore, he kh scaling correlaion ϱ k is assumed o be a consan. The drif coefficien funcion a(, ) and he diffusion coefficien funcion b(, ) are given funcions o be chosen such ha he SDE (4.4) has a unique srong soluion and he scaling γ remains almos surely nonnegaive for all [, T ]. The above formulaion of he random scaling is sill raher general. Below we will specify furher deails based on empirical evidence. We hen obain a Markovian muli-facor currency benchmark model. Curren numerical mehods applied o derivaive pricing problems work reasonably well only up o wo or hree dimensions. This is also he dimensionaliy of he model considered here. Numerically, we are herefore on he borderline of wha is achievable wih curren algorihms and echniques. As we will see below, when pricing currency opions, he derivaive prices are srongly influenced by he randomness in he scaling. To be more specific, we inroduce he marke aciviy process m = {m, [, T ]}, see Breymann, Kelly & Plaen (24, 25) and Heah & Plaen (24), given by m = γ ξ (4.5) wih he exponenial funcion { ξ = ξ exp } η s ds (4.6)

12 for [, T ]. Here ξ > is a consan and η is he piecewise consan deerminisic long erm ne growh rae of he marke. If one hinks of he marke aciviy m as flucuaing around one, hen he ne growh rae process η = {η, [, T ]} needs o offse he average growh in he random scaling process γ over he long erm. This is why he raio γ ξ appears in (4.5). The marke aciviy m is designed o model he normalized rading aciviy a ime. By using inraday daa he marke aciviy process for he US dollar denominaion has been analyzed in Breymann, Kelly & Plaen (24). According o hese resuls, a realisic marke aciviy process is obained when he diffusion coefficien funcion in (4.4) is chosen o be muliplicaive wih b(, γ) = β γ. (4.7) for (, γ) [, T ] (, ). In addiion, hese resuls sugges ha he drif coefficien funcion has he form ( ) a(, γ ) = ξ β 2 γ A + γ η (4.8) for [, T ] and γ, where he aciviy volailiy β = {β, [, T ]} is some deerminisic funcion of ime. Using Iô s formula ogeher wih (4.4), (4.5) and (4.6) he SDE for m is given by for [, T ]. dm = β 2 A(m ) d + β m ξ ϱ k dw k + (ϱ k ) 2 d W The paricular choice of he funcion A( ) conrols he ype of feedback which characerizes he random scaling. In Breymann, Kelly & Plaen (24) i was poined ou ha a good choice for he funcion A( ) is of he form ( p A(m) = 2 g ) 2 m m (4.9) for m [, ), wih some speed of adjusmen parameer g and some reference level p. The marke aciviy process can be shown o have he gamma densiy as is saionary densiy wih mean p and variance for parameers g > and g g p >, see Heah & Plaen (24). 5 Pricing Funcions for Coningen Claims Le us now check, wheher he above MMM wih random scaling can provide currency derivaive prices ha are consisen wih hose observed in he marke.

13 For he sudy of currency derivaives le us, for simpliciy, consider he case of wo independen squared Bessel processes Z and Z, saisfying according o (4.2) he SDE dz i = ν i 4 (qi,i+ ) 2 γ d + Z i γ q i,i+ dw i+ (5.) for [, T ] and i {, } wih he consan scaling levels q, and q,2. By (4.3) and (2.5) he corresponding ih denominaion D i of he GOP hen saisfies he SDE (( ( dd i = D i r i νi ) 2 (q i,i+ ) 2 γ Z i ) d + ν i 2 2 ( ) q i,i+ γ 2 dw i+ Z i ) (5.2) for [, T ], i {, }. Now, consider he case where he coningen claim H T, when expressed in unis of he domesic currency, has he form H T = H T (D T, D T, γ T ), (5.3) which is ha of a European currency opion wih mauriy T. Due o he given Markovian srucure, he corresponding benchmarked fair price process û HT, given by (3.7), is such ha û HT () = û(, D, D, γ ). (5.4) Here he funcion û : [, T ] (, ) 3 [, ) is differeniable wih respec o ime on (, T ) and wice differeniable wih respec o he componens (D, D, γ) on (, ) 3. This allows us o apply he Iô formula o obain he maringale represenaion H T (D T, D T, γ T ) D T = û(, D, D, γ ) T T T ν 2 2 ν 2 2 q, γs Z s q,2 γs Z s D s D s b(s, γ s ) û(, D s, D s, γ s ) γ û(, D s, D s, γ s ) D dw s û(, Ds, Ds, γ s ) dw 2 D s 2 ϱ k dws k + 2 (ϱ k ) 2 d W s (5.5) for [, T ). Here he funcion û saisfies by (5.2) and (4.4) he Kolmogorov backward equaion L û(, D, D, γ) = (5.6) 2

14 wih operaor L û(, D, D, γ) = û(, D, D, γ) + D r + ( ν 2 ) 2 (q, ) 2 γ 2 û(, D, D, γ) ( D ) 2 B ν 2 D + D r + ( ν 2 ) 2 (q,2 ) 2 γ 2 ( D ) 2 B ν 2 û(, D, D, γ) D + + ( ν 2 ) 2 (q, ) 2 γ 2 2 ( D ) 2 B ν 2 ( ν 2 ) 2 (q,2 ) 2 γ 2 2 ( D ) 2 B ν 2 + a(, γ) û(, D, D, γ) γ + b(, γ) ϱ D ν b(, γ) ϱ 2 D ν û(, D, D, γ) (D ) 2 2 û(, D, D, γ) (D ) (b(, 2 û(, D, D, γ) γ))2 γ 2 q, ( ) B ν 2 γ 2 û(, D, D, γ) D γ D q,2 γ for (, D, D, γ) (, T ) (, ) 3 wih erminal condiion ( ) B ν 2 2 û(, D, D, γ) (5.7) D γ D û(t, D, D, γ) = H T (D, D, γ) D (5.8) for (D, D, γ) (, ) 3. The above se of equaions describes he pricing funcions for a range of currency derivaives. Some of hese will be sudied below. This PDE formulaion can be exended o include he case of many pah dependen derivaives, such as barrier and binary opions, by appropriae modificaion of he boundary condiions, see Heah & Plaen (996). Noe ha in hese cases he PDE operaor appearing in (5.7) does no change. 6 European Opions on Exchange Raes Le us now apply he benchmarked pricing formula (3.7) for compuing a benchmarked European call opion price on he exchange rae process X, = {X, = 3

15 D, [, T ]} wih srike K and mauriy dae T. Then he benchmarked fair D call opion price ĉ T,K (, D, D, γ ) a ime [, T ] is given by ( (X, ) + ) ĉ T,K (, D, D T K, γ ) = E A = E ( ( D T D T K D T ) + A ) (6.) for [, T ]. The corresponding fair call opion price c T,K (, D, D, γ ), expressed in unis of he domesic currency, see (3.8), akes he form ( ( c T,K (, D, D, γ ) = D E K ) + ) A DT DT (6.2) for [, T ]. The benchmarked pricing funcion ĉ T,K saisfies he PDE (5.6) wih erminal condiion ( ĉ T,K (T, D, D, γ) = D K ) + (6.3) D for (D, D, γ) (, ) 3. Noe ha he PDE (5.7) and (5.8) is a hree-dimensional PDE ha is difficul o approximae using sandard numerical mehods. A valuaion procedure ha is numerically racable and can be widely applied for European syle currency derivaives is as follows: Using (3.7) he fair prices PT (, D, γ ) and PT (, D, D, γ ) for he domesic and foreign zero coupon bonds wih a mauriy dae T, when denominaed in domesic currency, are given by ( ) PT (, D, γ ) = D E A (6.4) and for [, T ], respecively. D T ( PT (, D, D, γ ) = D E D T ) A (6.5) Again using (3.7) he fair price p T,K (, D, D, γ ) for a European pu opion wih srike K and mauriy dae T is given by ( ( K p T,K (, D, D, γ ) = D E ) + ) A DT DT (6.6) for [, T ]. The corresponding pu-call pariy relaion beween European pu and call opions is expressed via he equaion p T,K (, D, D, γ ) = c T,K (, D, D, γ )+K P T (, D, γ ) P T (, D, D, γ ) (6.7) 4

16 for [, T ]. Le us now define he ime ransformaion ϕ by he differenial dϕ() = γ d (6.8) for [, T ] wih ϕ() =. Consider he ime ransformed sochasic process V i = {V i ϕ, ϕ [, ϕ(t )]} given by V i ϕ() = Z i (6.9) for [, T ] and i {, }. Using (5.), (6.8) and (6.9) i can be seen ha where dvϕ() i = ν i 4 (qi,i+ ) 2 γ d + q V i,i+ ϕ() i γ dw i+ = ν i 4 (qi,i+ ) 2 dϕ() + q i,i+ Vϕ() i i+ duϕ(), (6.) du i+ ϕ() = γ dw i+ for [, T ] and i {, }. Using (6.8) he quadraic variaion of U i+ is given by U i+ = γ s ds = ϕ() (6.) for [, T ] and hus U i+ is a Wiener process in ϕ-ime, i {, }. Consequenly, in he new ϕ-ime scale V i is a squared Bessel process of dimension ν i ha does no depend on he random scaling process γ. Using (4.), (6.), (6.8) and (6.9) he benchmarked fair price c T,K (, ϕ(), Vϕ(), V ϕ(), γ ) = ĉ T,K (, D, D, γ ) of a European call can also be expressed in he form (( ) + ) c T,K (, ϕ(), Vϕ(), Vϕ(), γ ) = E BT (V ϕ(t ) ) K ν 2 BT 2 (V ϕ(t 2 ) ) ν 2 2 A for [, T ]. (6.2) Consider he filraions (A ) and (A 2 ) generaed by he independen Wiener processes W and W 2, respecively, ogeher wih he filraion (Ã) generaed by he Wiener process W. The filraion (A ) appearing in (6.2) can be defined as he join filraion of (A ), (A 2 ) and (Ã). Tha is, A = A A 2 Ã for [, T ]. Le (A ) be he join filraion given by A = A A 2 ÃT 5

17 for [, T ]. Therefore, A conains all of he informaion regarding he full evoluion of he random scaling process γ over he ime inerval [, T ]. The condiional expecaion (6.2) for he benchmarked fair price of a European call can now be rewrien in he form c T,K (, ϕ(), V ϕ(), V ϕ(), γ ) = E ( c ϕ(t ),T,K(, ϕ(), V ϕ(), V 2 ϕ()) A ), (6.3) where (( ) + c ϕ(t ),T,K(, ϕ(), Vϕ(), Vϕ()) 2 = E BT (V ϕ(t ) ) K ν 2 BT 2 (V ϕ(t 2 ) ) ν 2 2 for [, T ]. A ) (6.4) For simpliciy, we will use he model formulaion wih ϱ = ϱ 2 =. Then for fixed values ϕ() and ϕ(t ) he funcion c ϕ(t ),T,K (, ϕ(), V ϕ(), V ϕ() 2 ) can be calculaed using he known ransiion densiies of he squared Bessel processes V and V 2. To approximae (6.3) we will use he following useful momen expansion E ( c ϕ(t ),T,K(, ϕ(), Vϕ(), Vϕ()) ) 2 A c τ,t,k (, ϕ(), Vϕ(), Vϕ()) 2 τ=e(ϕ(t ) A ) ( τ 2 c τ,t,k(, ϕ(), Vϕ(), Vϕ()) 2 (ϕ(t ( )) ) E ) E ϕ(t ) 2 A A τ=e(ϕ(t ) A) (6.5) for [, T ]. This expression requires an esimae of he condiional variance E((ϕ(T ) E(ϕ(T ) A )) 2 A ). Forunaely, his can be convenienly approximaed using a one-dimensional coupled sysem of PDEs. For his purpose we define a funcion τ : [, T ] (, ) 2 (, ) by τ (, γ, ϕ()) = E ( ϕ(t ) A ) (6.6) for [, T ]. Using he Kolmogorov backward equaion, (4.4) and (6.8), his funcion saisfies he PDE τ (, γ, ϕ) + a(, γ) τ (, γ, ϕ) γ for (, γ, ϕ) (, T ) (, ) 2 wih erminal condiion for (γ, ϕ) (, ) 2. The soluion o his wo-dimensional PDE is given by + 2 b2 (, γ) 2 τ (, γ, ϕ) + γ τ (, γ, ϕ) γ 2 ϕ = (6.7) τ (, γ, ϕ) = ϕ (6.8) τ (, γ, ϕ) = τ, (, γ) + ϕ τ, (, γ), 6

18 where τ, : [, T ] (, ) (, ) and τ, : [, T ] (, ) (, ) are funcions ha saisfy he coupled sysem of PDEs and τ, (, γ) + a(, γ) τ,(, γ) γ τ, (, γ) + a(, γ) τ,(, γ) γ for (, γ) (, T ) (, ) wih erminal condiion and + 2 b2 (, γ) 2 τ, (, γ) γ 2 + γ τ, (, γ) = (6.9) + 2 b2 (, γ) 2 τ, (, γ) γ 2 = (6.2) τ, (T, γ) = (6.2) τ, (T, γ) = (6.22) for γ (, ). A similar valuaion procedure can be used o approximae he funcion τ 2 : [, T ] (, ) 2 (, ) given by τ 2 (, γ, ϕ()) = E ( ϕ 2 (T ) A ) (6.23) for [, T ]. The funcions τ and τ 2 given in (6.6) and (6.23), respecively, are needed o approximae he condiional variance appearing in he expansion (6.5). For all of he numerical resuls described in his secion he above expansion mehod was successfully used. 7 Numerical Resuls To illusrae he ype of pricing effecs ha can be obained, Figure 3 shows a erm srucure of implied volailiies as a funcion of he srike K and mauriy dae T generaed for he above MMM. The defaul parameers used were ν = ν = 4., scaling levels q, = q,2 =.2, shor raes r = r =.5 iniial values D = D =. and mauriy dae T =.. For he random scaling process γ he defaul parameers used were ϱ = 3, g = 2, η =.48 and β =.6 wih iniial values ξ =. and γ =.. For he numerical resuls shown here, implied volailiies were calculaed by using boh he domesic and foreign zero coupon bonds o infer he discoun facor used in he Black-Scholes formula for exchange rae call and pu opions. This ensures ha he implied volailiies calculaed for a call and pu opion wih he same srike and mauriy dae are he same. Inspecion of Figure 3 shows ha for he defaul parameers used he a-hemoney implied volailiies increase gradually as he mauriy dae increases. The implied volailiy smile curvaure, eviden in Figure 3, is due o he scaling process γ. This ype of shape for he erm srucure of implied volailiies is very ypical for observed currency opions. 7

19 T.4 K.2.2 Figure 3: Implied call volailiies as funcion of srike K and ime q2 K.2.5 Figure 4: Implied call volailiies as funcion of srike K and scaling level q,2 wih q, + q,2 = For a fixed mauriy dae T =.25, Figure 4 shows implied volailiies for European call opions as a funcion of he scaling level q,2. Here he normalizaion condiion q, + q,2 =.4 is imposed o ensure ha he a-he-money implied volailiies are approximaely he same for differen values of q,2. Noe ha by changing he magniude of q,2 relaive o q, he implied volailiy curve can be changed from a predominaely negaively skewed smile o a predominanly posiively skewed smile. For a range of sochasic volailiy models his is generally only obained by using differen values for he degree of correlaion beween he driving Wiener processes for he models. Here i is obained more naurally and direcly by using he scaling levels q, and q,2. For a fixed mauriy dae T =.25, Figure 5 displays implied volailiies as a funcion of he srike K and he dimension ν 2 wih ν = ν 2. I can be seen 8

20 dim K Figure 5: Implied call volailiies as funcion of srike K and dimension ν 2 wih ν = ν from Figure 5 ha a lower dimension produces more curvaure for he implied volailiy smile. For higher dimensions some degree of curvaure is sill presen. This is due o he naural curvaure produced by he random scaling process γ bea.4 K.2.2 Figure 6: Implied call volailiies as funcion of srike K and scaling volailiy β. Finally, Figure 6 shows implied volailiies as a funcion of he srike K and scaling volailiy β. Noe he dramaic increase in curvaure if he scaling process γ is made more volaile. This figure shows he impac of using a random scaling process and is effec on he curvaure of implied volailiies. The above figures illusrae how he curvaure and skewness of he implied volailiies of exchange rae opions change depending on he parameer choice for he MMM. From a pracical poin of view i is clear ha wih realisic parameer choices one can capure using he above parsimonious model he ypical implied volailiy surfaces of exchange rae opions. As shown in Heah & Plaen (24) he corresponding 9.4

21 index version of his model easily generaes he negaively skewed implied volailiy surfaces commonly observed for index opions. In addiion his reduced index model produces realisic fair bonds and corresponding forward rae curves. Conclusion This paper proposes a minimal marke model wih random scaling o describe he dynamics of a currency marke. This parsimonious and realisic model is based on specifying suiable represenaions for he growh opimal porfolio and is differen denominaions in he domesic and foreign currencies. Time ransformed squared Bessel processes arise for each of he associaed facors. In addiion, he ime ransformaion is iself modeled using a random scaling process. The paricular ime ransformaion has been chosen o represen he effec of having random marke aciviy. The fair pricing concep is applied o price European syle derivaive securiies because he model does no admi an equivalen risk neural maringale measure. Numerical resuls for he proposed hree-facor model have been obained, which show realisic implied volailiy surfaces for European currency opions. A powerful numerical expansion mehod is described ha can be used o efficienly price European syle derivaives. References Breymann, W., L. Kelly, & E. Plaen (24). Inraday empirical analysis and modeling of diversified sock indices. Technical repor, Universiy of Technology, Sydney. QFRC Research Paper 25, o appear in Asia-Pacific Financial Markes 25. Breymann, W., L. Kelly, & E. Plaen (25). An inraday empirical analysis of currencies. (working paper). Derman, E. & I. Kani (994). The volailiy smile and is implied ree. Goldman Sachs Quaniaive Sraegies Research Noes. Dimson, E., P. Marsh, & M. Saunon (22). Triumph of he Opimiss: Years of Global Invesmen Reurns. Princeon Universiy Press. Dupire, B. (992). Arbirage pricing wih sochasic volailiy. In Proceedings of AFFI Conference, Paris. Heah, D. & E. Plaen (996). Valuaion of FX barrier opions under sochasic volailiy. Financial Engineering and he Japanese Markes 3, Heah, D. & E. Plaen (22a). Perfec hedging of index derivaives under a minimal marke model. In. J. Theor. Appl. Finance 5(7),

22 Heah, D. & E. Plaen (22b). Pricing and hedging of index derivaives under an alernaive asse price model wih endogenous sochasic volailiy. In J. Yong (Ed.), Recen Developmens in Mahemaical Finance, pp World Scienific. Heah, D. & E. Plaen (24). Undersanding he implied volailiy surface for opions on a diversified index. Technical repor, Universiy of Technology, Sydney. QFRC Research Paper 28, o appear in Asia-Pacific Financial Markes (). Karazas, I. & S. E. Shreve (998). Mehods of Mahemaical Finance, Volume 39 of Appl. Mah. Springer. Kelly, J. R. (956). A new inerpreaion of informaion rae. Bell Sys. Techn. J. 35, Long, J. B. (99). The numeraire porfolio. J. Financial Economics 26, Malz, A. M. (997). Esimaing he probabiliy disribuion of he fuure exchange rae from opion prices. J. Derivaives 4, Plaen, E. (2). A minimal financial marke model. In Trends in Mahemaics, pp Birkhäuser. Plaen, E. (22). Arbirage in coninuous complee markes. Adv. in Appl. Probab. 34(3), Plaen, E. (24a). A benchmark approach o finance. Technical repor, Universiy of Technology, Sydney. QFRC Research Paper 38, o appear in Mahemaical Finance. Plaen, E. (24b). Modeling he volailiy and expeced value of a diversified world index. In. J. Theor. Appl. Finance 7(4), Plaen, E. (25). On he role of he growh opimal porfolio in finance. Technical repor, Universiy of Technology, Sydney. QFRC Research Paper 44. Proer, P. (24). Sochasic Inegraion and Differenial Equaions (2nd ed.). Springer. Revuz, D. & M. Yor (999). Coninuous Maringales and Brownian Moion (3rd ed.). Springer. 2