UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.

Transcription

1 UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS CALGARY, ALBERTA May, 214 c Maksym Terychnyi 214

2 Absrac Using a Lévy process we generalize formulas in Bo e al. (21) o he Esscher ransform parameers for he log-normal disribuion which ensures he maringale condiion holds for he discouned foreign exchange rae. We also derive similar resuls, bu in he case when he dynamics of he FX rae is driven by a general Meron jump-diffusion process. Using hese values of he parameers we find a risk-neural measure and provide new formulas for he disribuion of jumps, he mean jump size, and he Poisson process inensiy wih respec o his measure. The formulas for a European call foreign exchange opion are also derived. We apply hese formulas o he case of he log-double exponenial and exponenial disribuion of jumps. We provide numerical simulaions for he European call foreign exchange opion prices wih differen parameers. i

3 Acknowledgemens I am happy o ake his opporuniy o express my graiude o my research supervisors Professor Anaoliy Swishchuk and Professor Rober Ellio. They inroduced me o he new (for me) area of Mahemaical Finance, gave many valuable remarks, and made suggesions for improvemens. I am graeful for heir aenion o my hesis preparaion and echnical commens. ii

4 Table of Conens Absrac i Acknowledgemens ii Table of Conens iii Lis of Figures iv Lis of Symbols v 1 Inroducion Lieraure review Lévy Processes Lévy processes. Basic definiions and heorems Characerisic funcions of Lévy processes Iô s formula wih jumps and is applicaions Girsanov s heorem wih jumps Currency derivaives Types of currency derivaives Forward conracs Currency fuures Currency opions Pure currency conracs Currency opion pricing. Main resuls Currency opion pricing for general Lévy processes Currency opion pricing for log-double exponenial processes Currency opion pricing for log-normal processes Currency opion pricing for Meron jump-diffusion processes Currency opion pricing for exponenial processes Numerical Resuls Log-double exponenial disribuion Exponenial disribuion Bibliography Apendix iii

5 Lis of Figures and Illusraions 5.1 Double-exponenial disribuion (green) vs. normal disribuion (red), mean=, dev= S = 1, T =.5, θ 1 = 1, θ 2 = 1, p =.5, mean normal =, sigma normal = S = 1, T = 1., θ 1 = 1, θ 2 = 1, p =.5, mean normal =, sigma normal = S = 1, T = 1.2, θ 1 = 1, θ 2 = 1, p =.5, mean normal =, sigma normal = S = 1, T =.5, θ 1 = 5, θ 2 = 1, p =.5, mean normal =, sigma normal = S = 1, T = 1., θ 1 = 5, θ 2 = 1, p =.5, mean normal =, sigma normal = S = 1, T = 1.2, θ 1 = 5, θ 2 = 1, p =.5, mean normal =, sigma normal = S = 1, T =.5, θ 2 = 1, p = Opion price of European Call: S = 1, T =.5, p = Opion price of European Call: θ = Opion price of European Call: θ = Opion price of European Call: θ = iv

6 Lis of Symbols, Abbreviaions and Nomenclaure Symbol Ch. Def. FX rae Prop. U of C SDE w.r.. Definiion Chaper Definiion foreign exchange rae Proposiion Universiy of Calgary sochasic differenial equaion wih respec o v

7 Chaper 1 Inroducion The srucure of he hesis is as follows: Chaper 1 gives a brief inroducion o he problem of FX currency opion pricing wih regime-swiching parameers. Chaper 2 provides an overview of resuls in he area under consideraion. In paricular, we give deails of models for pricing foreign currency opions incorporaing sochasic ineres raes, based on Meron s (1976, [3]) sochasic ineres rae model for pricing equiy opions. We discuss pricing coningen claims on foreign currencies under sochasic ineres raes using he Heah e al. (1987, [17]) model of erm srucure (Amin e al., 1991, [3]). A model for cross-currency derivaives, such as PRDC (Power-Reverse Dual-Currency) swaps wih calibraion o currency opions, developed by Pierbarg ( 25, [35]), is also described in our overview. We give deails of an approximaion formula for he valuaion of currency opions under a jump-diffusion sochasic volailiy processes for spo exchange raes in a sochasic ineres rae environmen as proposed by Takahashi e al. (26, [44]). A coninuous ime Markov chain which deermines he values of parameers in a modified Cox- Ingersoll-Ross model was proposed by Goue e al. (211, [11]) o sudy he dynamics of foreign exchange raes. In our research we use a hree sae Markov chain. We presen a wo-facor Markov modulaed sochasic volailiy model wih he firs sochasic volailiy componen driven by a log-normal diffusion process and he second independen sochasic volailiy componen driven by a coninuous-ime Markov chain, as proposed by Siu e al. (28, [41]). Finally, a Markov modulaed jump-diffusion model (including a compound Poisson process) for currency opion pricing as described in Bo e al. (21, [8]). We should also menion ha opion pricing is discussed in he Masers Thesis Hedging Canadian Oil. 1

8 An Applicaion of Currency Translaed Opions by Cliff Kichen (see [26], 21). A he end of Chaper 2 we describe a problem which will be solved in our hesis: generalize resuls in [8] o he case when he dynamics of he FX rae are driven by a general Lévy process ([34]). In Chaper 3 we sae all heorems and definiions necessary for our research. In 3.1 we give definiions for a general Lévy process and he mos common cases of such processes: Brownian moion and he compound Poisson process. In 3.2 we give some formulas for characerisic funcions of paricular cases of Lévy processes ha will be used o derive a main resul. In he nex wo secions 3.3, 3.4 we give deails of basic compuaional heorems used in he hesis: Io s formula wih jumps, Girsanov heorem wih jumps. In Chaper 4 we provide an overview of currency derivaives based on Bjork (1998, [6]). In 4.1 differen ypes of currency derivaives wih several examples are proposed. Basic foreign currency opion pricing formulas are given in 4.2. In Chaper 5 we formulae main resuls of our research: 1) In Secion 5.1 we generalize formulas in [8] for he Esscher ransform parameers which ensure ha he maringale condiion for he discouned foreign exchange rae is saisfied for a general Lévy process (see (5.3)). Using hese values of he parameers, (see (5.39), (5.4)), we proceed o a risk-neural measure and provide new formulas for he disribuion of jumps, he mean jump size, (see (5.2)), and he Poisson process inensiy wih respec o his measure, (see (5.19)). A he end of 5.1 pricing formulas for he European call foreign exchange opion are given (I is similar o hose in [8], bu he mean jump size and he Poisson process inensiy wih respec o he new risk-neural measure are differen). 2) In secion 5.2 we apply he formulas (5.19), (5.2), (5.39), (5.4) o he case of log-double exponenial processes (see (5.51)) for jumps (see (5.58)-(5.63)). 3) In Secion 5.4 we derive resuls similar o 5.1 when he ampliude of he jumps is driven by a Meron jump-diffusion process, (see (5.88)-(5.9), (5.18)-(5.19)). 2

9 4) In Secion 5.5 we apply formulas (5.88)-(5.9), (5.18)-(5.19) o a paricular case of an exponenial disribuion, (see (5.11)) of jumps (see (5.113)-(5.115)). In Chaper 6 we provide numerical simulaions of he European call foreign exchange opion prices for differen parameers, (in case of log-double exponenial and exponenial disribuions of jumps): S/K, where S is an iniial spo FX rae, K is a srike FX rae for a mauriy ime T ; parameers θ 1, θ 2 arise in he log-double exponenial disribuion ec. The resuls of Chapers 5-6 are acceped for publicaion in he Journal Insurance: Mahemaics and Economics (see [42]) and in he Journal of Mahemaical Finance (see [43]). In he Appendix codes for Malab funcions used in numerical simulaions of opion prices are provided. 3

10 Chaper 2 Lieraure review Unil he early 199s he exising academic lieraure on he pricing of foreign currency opions could be divided ino wo caegories. In he firs, boh domesic and foreign ineres raes are assumed o be consan whereas he spo exchange rae is assumed o be sochasic. See, e.g., Jarrow e al. (1981, [3]). The second class of models for pricing foreign currency opions incorporaed sochasic ineres raes, and were based on Meron s 1973, [31]) sochasic ineres rae model for pricing equiy opions. Le us menion he main poins of he model in [31]. Le he sock price S (or spo FX rae in our case) follow he dynamics described by he following equaion: ds S = (α λk)d + σdw + (Z 1)dN. (2.1) Here W is a Brownian moion, α λk is a drif, (Z s 1)dN s is a compound Poisson process(see Def. 3.3 in Ch. 3 for deail); k is a mean jump size if a jump occurs. In oher words: if he Poisson even does no occur, and ds S = (α λk)d + σdw, (2.2) ds S = (α λk)d + σdw + Z 1, (2.3) if a jump occurs. The soluion pah for spo FX rae S will be coninuous mos of he ime wih jumps of differen signs and ampliudes occurring a discree poins in ime. The opion price Π(S, ) can be wrien as a wice coninuously differeniable funcion of he sock price and ime: namely, Π = F (S, ). If he sock price yields he dynamics 4

11 defined in (2.1)-(2.3), hen he opion reurn dynamics can be wrien in a similar form as dπ Π = (α π λk π )d + σ π dw + (Z π 1)dN, (2.4) where α π is he insananeous expeced reurn on he opion Π ; (σ π ) 2 is he insananeous variance of he reurn, condiional on he Poisson even no occurring; (Z π 1)dN is an independen compound Poisson process wih parameer λ; k π = E(Z π 1). Using Iô s heorem (see Ch. 3, 3.4, for deails) we obain: α π = 1/2σ2 S 2 F ss (S, ) + (α λk)sf s (S, ) + F + λe(f (SZ, ) F (S, )), (2.5) F (S, ) where F ss (S, ), F s (S, ), F (S, ) are parial derivaives. σ π = F s(s, )σs, (2.6) F (S, ) Le g(s, T ) be he insananeous expeced rae of reurn on he opion when he curren sock price is S and he opion expiry dae is T. Then he opion price F (S, ) yields he following equaion (see (2.5) and [3] for more deails): 1/2σ 2 S 2 F ss (S, T ) + (α λk)sf s (S, T ) F (S, T ) g(s, T )F + λe(f (SZ, T ) F (S, T )) =, (2.7) wih he following boundary condiions: F (, T ) =, F (S, ) = max(, S K), (2.8) where K is a srike price. We shall consider a porfolio sraegy which consiss of he sock(fx rae in our case), he opion, and he riskless asse wih reurn r. Assume, ha we have zero-bea porfolio. From [55]: A Zero-bea porfolio is a porfolio consruced o have zero sysemaic risk or, in oher words, a bea of zero. A zero-bea porfolio would have he same expeced reurn as he risk-free rae. Such a porfolio would have zero correlaion wih marke movemens, given ha is expeced reurn equals he risk-free rae, a low rae 5

12 of reurn. In oher words he jump componen of he socks reurn will represen nonsysemaic risk (see again [3]). Under his assumpion and using (2.5)-(2.7) we obain he following equaion for currency opion price: 1/2σ 2 S 2 F ss (S, T ) + (r λk)sf s (S, T ) F (S, T ) rf + λe(f (SZ, T ) F (S, T )) = (2.9) wih he same boundary condiions (2.8). Pricing equaion (2.9) is similar o (2.7), bu (2.9) does no depend on eiher α or g(s, T ). Insead, as in he sandard Black-Scholes formula, he dependence on he ineres rae, r appears. Noe, ha he jump componen influences he equilibrium opion price in spie of he fac ha i represens non-sysemaic risk. The soluion o (2.9) for he opion price when he curren sock price is S and he expiry ime is T can be calculaed using he formula (see again [3]): e λt (λt ) n F (S, T ) = E [ B(SZe λkt, T ; K, σ 2, r) ]. (2.1) n! n= Here B(SZe λkt, T ; K, σ 2, r) is he Black-Scholes opion pricing formula for he no-jump case (see [7] and Ch. 4, 4.2 for deail); E is an expecaion operaor over he disribuion of Z, Z is he disribuion of jumps (see (2.1), (2.3). A similar model can be found in Grabbe (1983, [23]), Adams e al. (1987, [1]). Unforunaely, his pricing approach did no include a full-fledged erm srucure model ino he valuaion framework. Moreover, he Forex marke is more complicaed can no be adequaely described by he model wih he assumpion of a zero-bea porfolio. Amin e al. (1991, [3]), o our bes knowledge, were he firs o sar discussing and building a general framework o price coningen claims on foreign currencies under sochasic ineres raes using he Heah e al. (1987, [17]) model of erm srucure. We shall inroduce he assumpions underlying he economy as in [3]. The auhors specify he evoluion of forward domesic and foreign ineres raes, (see he deails of forward agreemens in 6

13 Ch.4, 4.1.1) in he following way: hey consider four sources of uncerainy across he wo economies, (domesic and foreign), defined by four independen sandard Brownian moions (W 1, W 2, W 3, W 4 ; [, τ]) on a probabiliy space (Ω, F, P). The domesic forward ineres rae r d (, T ), (he counry s forward ineres rae conraced a ime for insananeous borrowing and lending a ime T wih T τ) changes over ime according o he following sochasic differenial equaion (see Assumpion 1, [3]): dr d (, T ) = α d (, T, ω)d + 2 i=1 σ di (, T, r d (, ))dw i (2.11) for all ω Ω, T τ. Noe, ha he same wo independen Brownian moions W 1, W 2 drive he domesic forward ineres rae. These random shocks can be explained as a shor-run and long-run facor changing differen mauriy ranges of he erm srucure. The domesic bond price in unis of he domesic currency can be wrien (see [3]): { T } B d (, T ) = exp r d (, u)du. (2.12) The dynamics of he domesic bond price process is given by he equaion: where db d (, T ) = [r d (, ) + b d (, T )]B d (, T )d + 2 i=1 α di (, T )B d (, T )dw i, (2.13) T α di = σ di (, u, r d (, u))du, i = 1, 2, (2.14) T b d (, T ) = α d (, u, ω)du [ T 2 σ di (, u, r d (, u))du]. (2.15) A similar assumpion holds for he foreign forward ineres rae dynamics (see Assumpion 2, [3]): i=1 dr f (, T ) = α f (, T, ω)d + 2 i=1 σ fi (, T, r f (, ))dw i (2.16) 7

14 for all ω Ω, T τ; r f (, T ) is a forward ineres rae conraced a ime for insananeous borrowing and lending a ime T wih T τ. The connecion beween he wo markes is he spo exchange rae. The spo rae of exchange (in unis of he domesic currency per foreign currency) is governed by he following SDE (see see Assumpion 3, [3]): ds = µ S d + Here µ is a drif par, δ di, i = 1,, 4 are he volailiy coefficiens. 4 i=1 δ di S dw i. (2.17) The spo exchange rae S depends on he same hree Brownian moions W 1, W 2, W 3 running he domesic and foreign forward ineres raes. As a resul, here exis correlaions beween he spo exchange rae S, he domesic r d (, T ), and he foreign ineres raes r f (, T ). One independen Brownian moion W 4 is added o govern he exchange rae dynamics. To price opions from he domesic perspecive, we need o conver all of he securiies o he domesic currency. We assume ha a domesic invesor has his holdings of foreign currency only in he form of foreign bonds B f, or unis of he foreign money marke accoun B f. Boh holdings are convered o domesic currency according o he formulas: B f = B f S, (2.18) B f = B f S. (2.19) The sochasic processes describing hese securiies, denominaed in he domesic currency, are given by (see Appendix, [3]): [ d B f = B f (µ d () + r f (, ))d + 4 i=1 δ di ()dw i ] (2.2) db f = B f [ µ f()d + 4 i=1 8 (α fi (, T ) + δ di ())dw i ] (2.21)

15 where µ f() = r f (, ) + b f (, T ) + µ d () + 4 δ di ()α fi (, T ). (2.22) The pricing formula for he European Call opion, wih exercise price K and mauriy ime T, for his model is also given in [3](see p ): [ ( 4 C(, T, K) = E max, B f (, T )S exp i=1 i=1 T (α fi (, T ) + δ di ())dw i, T ) 1/2 (α fi (, T ) + δ di ()) 2 d KB d (, T ) ( 2 T 2 T ] exp α di (, T )dw i, 1/2 αdi(, 2 T )d). (2.23) i=1 i=1 Here E is he mahemaical expecaion wih respec o he risk-neural measure P. (Deails abou risk-neural pricing can be found in Bjork [6] and Shreve [4],V.2). The derivaion of formulas for risk-neural currency opion pricing for several paricular cases and general Lévy process (see Ch. 3, 3.1) are in he main resuls of our research (see Ch.5). Melino e al. (1991, [29]) examined he foreign exchange rae process, (under a deerminisic ineres rae), underlying he observed opion prices, and Rumsey (1991, [38]) considered cross-currency opions. Mikkelsen (21, [33]) considered cross-currency opions wih marke models of ineres raes and deerminisic volailiies of spo exchange raes by simulaion. The auhor uses a cross-currency arbirage-free LIBOR marke model, (clmm), for pricing opions, (see [33], [56]), ha are evaluaed from cross-marke dynamics. In [33] he marke consiss of wo economies, each of which has is own domesic bond marke. An invesor can inves in wo fixed income markes: his home currency marke and a foreign currency denominaed marke. To ensure arbirage free pricing of cross-currency derivaives he spo foreign exchange rae and arbirage free foreign exchange rae dynamics are inroduced. I was proved ha if LIBORs are modeled as log-normal variables all forward foreign exchange raes canno be modeled as log-normal wihou breaking he arbirage-free assumpion. Schlogl (22, [39]) exended marke models o a cross-currency framework. He did no 9

16 include sochasic volailiies ino he model and focused on cross currency derivaives such as differenial swaps and opions on he swaps as applicaions. He did no consider currency opions. Pierbarg (25, [35]) developed a model for cross-currency derivaives, such as PRDC (Power-Reverse Dual-Currency) swaps, wih calibraion o currency opions; he used neiher marke models nor sochasic volailiy models. An economy consiss of wo currencies as before (see [33]): domesic and foreign. Le P be he domesic risk-neural measure. Le P i (, T ), i = d, f be he prices, in heir respecive currencies, of he domesic and foreign zero-coupon discoun bonds. Also le r i (), i = d, f, be he shor raes in he wo currencies. Le S be he spo FX rae. The forward FX rae (he break-even rae for a forward FX ransacion) F (, T ) saisfies he condiion(see [35]): F (, T ) = P f(, T ) P d (, T ) S (2.24) following from no-arbirage argumens. The following model is considered in [35]: dp d (, T ) P d (, T ) = r d()d + σ d (, T )dw d, (2.25) dp f (, T ) P f (, T ) = r f()d ρ fs σ f (, T )γ(, S )d + σ f (, T )dw f, (2.26) ds S = (r d () r f ())d + γ(, S )dw S, (2.27) where (W d, W f, W S ) is a hree dimensional Brownian moion under P wih he correlaion marix: Gaussian dynamics for he raes are given by: T { σ i (, T ) = σ i () exp 1 ρ df ρ ds ρ df 1 ρ fs. (2.28) ρ ds ρ fs 1 1 s } χ i (u)du ds, i = d, f, (2.29)

17 where σ d (), σ f (), χ d (), χ f () are deerminisic funcions. A price of a call opion on he FX rae wih srike K and mauriy T is ( { T } ) C(T, K) = E exp r d (s)ds max(, S T K). (2.3) The forward FX rae F (, T ) has he following dynamics (see [35], Prop. 5.1): df (, T ) F (, T ) = σ f(, T )dw T f () σ d (, T )dw T d () + γ(, F (, T )D(, T ))dw T S (). (2.31) Here ( W T d (), W T f (), W T S ()) is a hree-dimensional Brownian moion under he domesic T -forward measure. Moreover, here exiss a Brownian moion dw F () under he domesic T -forward measure P T, such ha: where df (, T ) F (, T ) = Λ(, F (, T )D(, T ))dw F (), (2.32) Λ(, x) = (a() + b()γ(, x) + γ 2 (, x)) 1/2, a() = (σ f (, T )) 2 + (σ d (, T ))2 2ρ ds σ f (, T )σ d (, T ), b() = 2ρ fs σ f (, T ) 2ρ ds σ d (, T ), D(, T ) = P d(, T ) P f (, T ). (2.33) In [35] a Markovian represenaion for he foreign FX rae is used. Define: c(, T, K) = P d (, )E T (F (, T ) K) +, (2.34) where: (x) + = max(, x). T > is a he fixed selemen dae of he FX forward rae. We wish o find a funcion Λ(, x) such ha, in he model defined by he equaion: df (, T ) F (, T ) = Λ(, F (, T )D(, T ))dw F (), (2.35) he values of European opions {c(, T, K)} for all, K, < T, K < are equal o he values of he same opions in he iniial model (2.25)- (2.28) (or (2.31)-(2.34)). The answer is given in [35] (see Theorem 6.1). 11

18 The local volailiy funcion Λ(, x), for which he values of all European opions {c(, T, K)},K in he model (2.35) are he same as in he model (2.2), is given by Λ(, x) = E T ( Λ 2 (, F (, T )D(, T )) F (, T ) = x ). (2.36) Here E T is a condiional mahemaical expecaion wih respec o measure P T (see (2.31). In he case of European opion pricing, he dynamics of he forward FX rae F (, T ) under he measure P T are given in Corollary 6.2([35]): where df (, T ) F (, T ) = Λ(, F (, T )D(, T ))dw F (), (2.37) Λ(, x) = (a() + b() γ(, x) + γ 2 (, x)) 1/2. (2.38) The formulas for γ(, x), r() can also be found in his corollary. V. Pierbarg also derives approximaion formulas for he value of opions on he FX rae for a given expiry T and srike K, (see [35], Theorem 7.2). To value opions on he FX rae wih mauriy T, he forward FX rae can be approximaed by he soluion of he following sochasic differenial equaion: df (, T ) = Λ(, F (, T ))(δ F F (, T ) + (1 δ F )F (, T ))dw F (). (2.39) Here δ F = 1 + T b()η() + 2 γ(, F (, T ))η() w() d, (2.4) 2 Λ 2 (, F (, T )) η() = γ(, F (, T ))(1 + r())(β() 1), (2.41) w() = T u() u()d, (2.42) u() = Λ 2 (, F (, T )) Λ 2 (s, F (, T ))ds. (2.43) 12

19 The value c(t, K) of he European call opion on he FX rae wih mauriy T and srike K is equal o: c(t, K) = P d (, T )B( F (, T ), k + 1 δ F F (, T ), σ F δ F, T ), (2.44) δ F δ F δ F = ( 1 T T Λ 2 (, F (, T )d) 1/2. (2.45) Here B(F, K, σ, T ) is he Black-Schloes formula for a call opion wih forward price F, srike K, (see [6], chaper 17), volailiy σ, and ime o mauriy T. A similar model and approximaions for pricing FX opions are given in [26] by Lech Grzelak and Kees Ooserlee (21). In Garman e al. (1983, [6]) and Grabbe (1983, [23]), foreign exchange opion valuaion formulas are derived under he assumpion ha he exchange rae follows a diffusion process wih coninuous sample pahs. Takahashi e al. (26, [44]) proposed a new approximaion formula for he valuaion of currency opions under jump-diffusion sochasic volailiy processes for spo exchange raes in a sochasic ineres raes environmen. In paricular, hey applied marke models developed by Brace e al (1998,[9]), Jamshidian (1997, [2]) and Milersen e al (1997, [32]) o model he erm srucure of ineres raes. As an applicaion, Takahashi e al. applied he derived formula o he calibraion of volailiy smiles in he JPY/USD currency opion marke. In [44] he auhors suppose, ha he variance process of he spo FX rae is governed by he following dynamics: dv = ξ(η V )d + θ V υ d W. (2.46) Here W is a d-dimensional Brownian moion under he domesic risk neural measure, ξ, η and θ are posiive scalar parameers and υ is a d-dimensional consan vecor wih υ = 1 o represen he correlaions beween he variance and oher facors. The condiion 2ξη > θ 2 ensures ha V remains posiive if V =. This sochasic volailiy model is inroduced by 13

20 Heson in [18](1993). Le F (, T ) denoe a forward exchange rae wih mauriy T a ime. Suppose he dynamics of he process F (, T ) are defined by he equaion: where σ F df (, T ) F (, T ) = σ F dw d, (2.47) is a deerminisic d-dimensional vecor-funcion, and W d is a d-dimensional Brownian moion, (see [44], formulas 4-6). In he case of Merons jump-diffusion ([3], 1976) equaion (2.47) can be presened in he following form: d(, T ) (, T ) = σ F dw d + Z dn λkd, (2.48) where Z dn is a compound Poisson process wih inensiy λ and jump size Z. (See again Ch. 3, Def.3.3 or [34] for deails). A Fourier ransform mehod for currency call opion pricing is used in [44]: c(s, K, T ) B d (, T ) 1 2π + e iku Φ X (u i) Φ (u i) du + c (S, K, T ) iu(1 + iu) B d (, T ). (2.49) Here c(s, K, T ) is an opion price wih srike price K, mauriy ime T, and S is an iniial spo exchange rae, B d (, T ) is a price of zero coupon bonds wih mauriy T in domesic currency; { } Φ (u) = exp σ2 T 2 (u2 + iu). (2.5) Φ X (u) is a firs order approximaion of he characerisic funcion of he process (see [44], formulas 9-15): X = log F (, T ) F (, T ) ; (2.51) c (S, K, T ) = B d (, T )(F N(d + ) KN(d )), (2.52) d ± = k ± 1 2 σ2 T σ T, (2.53) σ - is a consan depending on T, S, domesic and foreign forward ineres raes (see [44], 3); N(x) is a sandard normal cumulaive disribuion funcion. The firs erm on he 14

21 righ-hand side of he formula (2.49) is he difference beween he call price of his model and he Black-Scholes call price(see [6], Chaper 17). Also, Ahn e al. (27, [2]) derived explici formulas for European foreign exchange call and pu opions values when he exchange rae dynamics are governed by jump-diffusion processes. Hamilon (1988) was he firs o invesigae he erm srucure of ineres raes by raional expecaions economeric analysis of changes in regime. Goue e al. (211, [11]) sudied foreign exchange rae using a modified Cox-Ingersoll-Ross model under a Hamilon ype Markov regime swiching framework, where all parameers depend on he value of a coninuous ime Markov chain. Definiion 2.1 ([11], Def. 2.1) Le (X ) [;T ] be a coninuous ime Markov chain on finie space S := {1,, K}. Wrie F X := {σ(x s ); s } for he naural filraion generaed by he coninuous ime Markov chain X. The generaor marix of X will be denoed by Π X and given by Π X ij, i j; i, j S (2.54) Π X ii = j i Π X ij (2.55) oherwise. In our numerical simulaions we shall use a slighly differen marix Π X : Π X ij, i, j S, j S Π X ij = 1 for any i S. (2.56) Noe, ha Π X ij represens probabiliy of he jump from sae i o sae j. The definiion of a CIR (Cox-Ingersoll-Ross) process wih parameers values depending on he value of a coninuous ime Markov chain is given in [11]: Definiion 2.2 ([11], Def. 2.2) Le, for all [; T ], (X) be a coninuous ime Markov chain on a finie space S := {e 1, e 2,, e K } defined as in Def We will call a Regime swiching CIR (in shor, RS-CIR) he process (r ) which is he soluion of he SDE given by 15

22 dr = (α(x ) β(x )r )d + σ(x ) r dw. (2.57) Here for all [; T ], he funcions α(x ), β(x ), and σ(x ) are funcions such ha for 1 i K α(i) {α(1),, α(k)}, β(i) {β(1),, β(k)}, and σ(i) {σ(1),, σ(k)}. For all j {1,, K}: α(j) >, 2α(j) σ(j) 2. In [11] Goue e al. invesigaed hree sae Markov swiching regimes. One regime represens normal economic dynamics, he second one is for crisis and he las one means good economy. They also sudy wheher more regimes capure beer he economic and financial dynamics or no. The advanages and disadvanages of models wih more han hree regimes are discussed and suppored by numerical simulaions. Three sae regime swiching models are imporan in our research, because hey adequaely describe movemens of he foreign currency exchange rae in he Forex marke. In Forex we have hree disincive rends: up, down, sideways. Zhou e al. (212, [45]) considered an accessible implemenaion of ineres rae models wih regime-swiching. Siu e al. (28, [41]) considered pricing currency opions under a wo-facor Markov modulaed sochasic volailiy model wih he firs sochasic volailiy componen driven by a log-normal diffusion process and he second independen sochasic volailiy componen driven by a coninuous-ime Markov chain: ds S = µ d + V dw 1 + σ dw 2, dv V = α d + βdw υ, (2.58) Cov(dW υ, dw 1 ) = ρd. 16

23 Here S is a spo FX rae; W 1 = {W 1 } T, W 2 = {W 2 } T are wo independen Brownian moions on (Ω, F, P ); (Ω, F, P ) is a complee probabiliy space; T [, ) is a ime index se; W υ = {W υ } T is a sandard Brownian moion on (Ω, F, P ) correlaed wih W 1 wih coefficien of correlaion ρ. In [41] he auhors define a coninuous-ime, finie-sae Markov chain X = {X } T on (Ω, F, P ) wih sae space S = {e 1, e 2,, e n } (similar o [11], Def. 2.1, 2.2). They ake he sae space S for X o be he se of uni vecors (e 1,, e n ) R n wih probabiliy marix Π X similarly o (2.54)-(2.56). The parameers σ, µ are modelled using his finie sae Markov chain: µ :=< µ, X >, µ R n ; σ :=< σ, X >, σ R n. (2.59) Insananeous marke ineres raes {r d } T, {r f } T of he domesic and foreign money marke accouns are also modeled using he finie sae Markov chain in [41] by: r d :=< r d, X >, r d R n ; r f :=< r f, X >, r f R n. (2.6) Le B d := {B d } T, B f := {B f } T denoe he domesic and foreign money marke accouns. Then heir dynamics are given by he following equaions (see [41]): { } B d = exp rudu d, { } B f = exp rudu f. (2.61) Siu e al. consider he European-syle and American-syle currency opion pricing (see [41], 4, 5) and provide numerical simulaions for opions of boh ypes (see [41], 6). Bo e al. (21, [8]) deals wih a Markov-modulaed jump-diffusion (modeled by compound Poisson process) for currency opion pricing. They use he same finie-sae Markov 17

24 chain as in [41] o model coefficiens in he equaion for FX rae dynamics: ds S = (α λk)d + σdw + (e Z 1)dN. (2.62) Here W is a Brownian moion, α λk is a drif, (e Z 1)dN is a compound Poisson process (see Def. 3.3 in Ch. 3 for deail); k is a mean jump size if a jump occurs. The ampliude of he jumps Z is log-normally disribued. The European call currency opion pricing formulas are also derived (see [8], formulas 2: 2.18, 2.19; 3: ). A random Esscher ransform (see [8], 2, formula 2.5; [13], [15] for he deailed heory of he Esscher ransform) is used o deermine a risk-neural measure, o find opion pricing formulas and o evaluae he Esscher ransform parameers which ensure ha he discouned spo FX rae is a maringale. (See [8], 2, Theorem 2.1). Heson (1993, [18]) found a closed-form soluion for opions having sochasic volailiy wih applicaions o he valuaion of currency opions. Bo e al. (21, [8]) used his model of sochasic volailiy and considered he following model of a FX marke: ds = S (α kλ )d + S V dw + S (e Z 1)dN, (2.63) dv = (γ βv )d + σ υ V W 1 (2.64) W 1 = ρw ρ 2 W 2. (2.65) Here γ, σ υ, β >, ρ < 1; W and W 2 are wo independen Brownian moions; S is a spo FX rae, α kλ is a drif; (e Z 1)dN is a compound Poisson process; V is a sochasic volailiy. They also provided numerical simulaions for he model in [18]. We noe ha currency derivaives for he domesic and foreign equiy marke and for he exchange rae beween he domesic currency and a fixed foreign currency wih consan ineres raes are discussed in Bjork (1998, [6]). We also menion ha currency conversion 18

25 for forward and swap prices wih consan domesic and foreign ineres raes is discussed in Benh e al. (28, [5]). The main goal of our research is o generalize he resuls in [8] o he case when he dynamics of he FX rae are driven by a general Lévy process (See Ch. 3, 3.1 for he basic definiions of Lévy processes, or [34] for a complee overview). In oher words, jumps in our model are no necessarily log-normally disribued. In paricular, we shall invesigae logdouble exponenial and exponenial disribuions of jumps and provide numerical simulaions for European-call opions which depend on a wide range of parameers. (See Ch. 5, 6). 19

26 Chaper 3 Lévy Processes 3.1 Lévy processes. Basic definiions and heorems We shall give a definiion of a general Lévy Process. Le (Ω, F, P ) be a filered probabiliy space wih F = F T, where (F ) [,T ] is he filraion on his space. Definiion 3.1. ([34]) An adaped, real valued sochasic process L = (L ) T wih L = is called a Lévy process if he following condiions are saisfied: (L1): L has independen incremens, i.e. L L s is independen of F s for any < s < < T. (L2): L has saionary incremens, i.e. for any < s, < T he disribuion of L +s L does no depend on. (L3): L is sochasically coninuous, i.e. for every T and ɛ > : limp ( L L s > ɛ) =. s Examples of Lévy process arising in Mahemaical Finance are (linear) drif, a deerminisic process, Brownian moion, non-deerminisic process wih coninuous sample pahs, and he Poisson and compound Poisson processes. Definiion 3.2. ([34]) Le (Ω, F, P ) be a probabiliy space. For each ω Ω, suppose here is a coninuous funcion W ( > ) ha saisfies W = and ha depends on ω. Then W ( ) is a Brownian moion if for all = < 1 < < m he incremens W 1 = W 1 W, W 2 W 1,..., W m W m 1 are independen and each of hese incremens is normally disribued wih E[W i+1 W i ] =, Var[W i+1 W i ] = i+1 i. 2

27 A Brownian moion model requires an assumpion of perfecly divisible asses and a fricionless marke, (i.e. here are no axes and no ransacion coss occur eiher for buying or selling).however, asse prices in a real marke can have jumps which can be described by a compound Poisson process. Definiion 3.3. ([34]) A compound Poisson process is a coninuous-ime (random) sochasic process wih jumps. The jumps arrive randomly according o a Poisson process and he size of he jumps is also random, wih a specified probabiliy disribuion ν. A compound Poisson process, defined by a rae λ > and jump size disribuion ν, is a process N P = Z i, (3.1) i=1 where Z i is an idenically disribued random variable, (N ) T is Poisson process wih a rae λ. Definiion 3.4. ([34]) A compound compensaed Poisson process has he form N P = Z i E[Z] i=1 λ s ds. (3.2) Noe, ha a compound compensaed Poisson process is a maringale (see [34]). The sum of a (linear or non-linear, depending on ime ) drif, a Brownian moion and a compound Poisson process is again a Lévy process. I is ofen called a Lévy jump-diffusion process. Noe here exis jump-diffusion processes which are no Lévy processes. This holds, when he disribuion of jumps gives non converging o zero probabiliy of jumps wih a big ampliude, and hen he condiion (L3) does no hold. We shall give several examples of hese. We now deermine he characerisic funcions of all hree ypes of Lévy processes menioned above. 21

28 3.2 Characerisic funcions of Lévy processes The characerisic funcion of Brownian moion is given a (see for Example [34] for he derivaion of all he resuls in his chaper): where σ is he volailiy of a marke. E [e u ] { 1 σsdws = exp 2 u2 process, defined by a rae λ > and jump size disribuion ν, is: } σsds 2, (3.3) The characerisic funcion of compound Poisson [ E e u N ] { } k=1 Z k = exp λ s (e ux 1)ν(dx)ds. (3.4) R The characerisic funcion of compound compensaed Poisson process, defined by a rae λ > and jump size disribuion ν: [ E e u( N k=1 Z k E[Z] λsds)] { } = exp λ s (e ux 1 ux)ν(dx)ds. (3.5) R We shall assume ha he process L = (L ) T is a Lévy jump-diffusion, i.e. a Brownian moion plus a compensaed compound Poisson process. The pahs of his process can be described by L = µ s ds + N σ s dw s + Z k E[Z] k=1 λ s ds. (3.6) Noe ha since Brownian moion and compound compensaed Poisson processes are maringales, a Lévy process L y is maringale if and only if µ =. Using expressions for characerisic funcions (3.2)-(3.5) we obain he characerisic funcion of he Lévy process L : E [ e ul] { = exp u s µ s ds u2 σsds 2 + λ s R } (e ux 1 ux)ν(dx)ds. (3.7) In he case of compound Poisson Process(no compensaed): E [ e ul] { = exp u s µ s ds u2 σsds λ s R } (e ux 1)ν(dx)ds. (3.8)

29 In he sequel, we shall consider jumps driven by a compound Poisson Process o describe he spo FX rae movemens of currency markes. We also consider sufficien condiions for Lévy process of such a ype o be a maringale. The following decomposiion of he characerisic funcion of general Lévy process is rue (see [34], p. 12 ): Theorem 3.1. Consider a riple (µ, σ, ν) where µ R, σ R > and ν is a measure saisfying ν() = and (1 R x 2 )ν(dx) <. Then, here exiss a probabiliy space (Ω, F, P ) on which four independen Lévy processes exis, L 1, L 2, L 3, L 4 where L 1 is a drif, L 2 is a Brownian moion, L 3 is a compound Poisson process and L 4 is a square inegrable (pure jump) maringale wih an a.s. counable number of jumps of magniude less han 1 on each finie ime inerval. Taking L = L 1 + L 2 + L 3 + L 4, we have ha here exiss a probabiliy space on which a Lévy process L = (L ) T wih characerisic funcion E [ e ul] { s = exp u µ s ds + 1 } 2 u2 σsds 2 + λ s (e ux 1 ux1 x 1 )ν(dx)ds for all u R, is defined. R (3.9) To prove he Theorem 3.1 we can use Iô s formula wih jumps. (see he nex chaper). 3.3 Iô s formula wih jumps and is applicaions In our research shall assume, ha he asse price is described by he following sochasic differenial equaion: ds = S ( µ d + σ dw + (e Z 1)dN ). (3.1) Here µ is a drif, σ is a volailiy, (e Z 1)dN is he compound Poisson process (N is he Poisson process wih inensiy λ >, e Z 1 are he jump sizes, Z has arbirary disribuion ν). The parameers σ, λ are modulaed by a finie sae Markov chain (ξ ) T in he following way σ = σ, ξ, σ R n +, 23

30 λ = λ, ξ, λ R n +. Using Iô s formula wih jumps, solve (3.1). We sae his heorem firs, because i will be used several imes in he sequel. Theorem 3.2. ([36], p , [46]). Le a(x, ), b(x, ) be adaped sochasic processes and f(,x) be a funcion for which parial derivaives f, f x, f xx are defined and coninuous. Le also dx = a(x, )d + b(x, )dw + Y dn. (3.11) Then he following holds df(x, ) = [f + a(x, )f x b2 (x, )f xx ]d + b(x, )f x dw (3.12) +[f(x + Y, ) f(x, )]dn. The soluion for (3.1) has he following form S = S e L, (S is a spo FX rae a ime = ), where L is given by he formula: L = (µ s 1/2σ 2 s)ds + i=1 N σ s dw s + Z i. (3.13) We can consider jumps in (3.1) of a differen form Z 1 insead of e Z 1. Then (3.1) akes he form: ( ) ds = S µ d + σ dw + (Z 1)dN. (3.14) In his case, o apply Iô s formula we mus require Z : Z >. Then we obain he following soluion for (3.14) S = S e L, where L is given by he formula: L = (µ s 1/2σ 2 s)ds + i=1 N σ s dw s + log Z i. (3.15) Noe ha for mos well-known disribuions, (normal, double exponenial disribuion of Z, ec), L is no a Lévy process, since log Z i for small Z. However, he probabiliy of jumps wih even size is a posiive consan, depending on he ype of disribuion. We shall consider he soluion of he differenial equaion (3.14) in a separae secion laer. (see Ch. 5.4). 24

31 3.4 Girsanov s heorem wih jumps The primary goal of our research is o derive formulas for currency opion pricing. For his purpose we need o ransfer from he iniial probabiliy measure wih Poisson process inensiy λ and densiy funcion of jumps ν o a new risk neural measure wih new λ, ν. Noe ha his measure is no unique. To calculae hese new quaniies we shall use Girsanov s heorem wih jumps (see [36], p ; [4], p ). Theorem 3.3. Le W, T be a Brownian moion on a probabiliy space (Ω, F, P ) and le F, T be a filraion for his Brownian moion. Le Θ, T be an adaped process; λ, ν are new Poisson process inensiy and densiy funcions for jumps respecively. Define Z (1) { = exp Θ s dw s 1/2 } Θ 2 sds, (3.16) Z (2) { } = exp (λ s λ s)ds Z = Z (1) Z (2) N i=1 λ sν (Z i ) ds (3.17) λ s ν(z i ) W = W + Θ s ds + Z E[Z] λ sds (3.18) and assume ha E T Θ2 sz 2 s ds <. Se Z = Z(T ). Then EZ = 1 and under he probabiliy measure P given by P (A) = A Z ωdp ω, foralla F, he process W, T is a maringale. Moreover, W + Θ sds is a sandard Brownian moion, Z E[Z] λ sds is maringale wih respec o new measure P. When λ s = λ s, Theorem 3.3 coincides wih he usual Girsanov heorem for he Brownian moion. 25

32 Chaper 4 Currency derivaives 4.1 Types of currency derivaives A foreign exchange derivaive is a financial derivaive whose payoff depends on he foreign exchange rae(s) of wo (or more) currencies. In his secion we shall consider hree main ypes of foreign exchange derivaives: foreign exchange forwards, currency fuures, currency opions (foreign exchange opions) Forward conracs Following Bjork, (see [6], p. 12) a forward conrac on X, made a, is an agreemen which sipulaes ha he holder of he conrac pays he deerminisic amoun K a he delivery dae T, and receives he sochasic amoun X a T. Nohing is paid or received a he ime, when he conrac is made. Noe ha forward price K is deermined a ime. Forward conracs are agreemens o buy or sell a a cerain ime in he fuure, he mauriy ime, an asse a a price sipulaed in advance. Forwards are a common insrumen used o hedge currency risk, when he invesor is expecing o receive or pay a cerain amoun of money expressed in foreign currency in he near fuure. Forward conracs are binding conracs, (conrary o opions) and, herefore, boh paries are obliged o exercise he conrac condiions and buy (sell) he asse a agreed price. There are hree ypes of prices associaed wih a forward conrac. The firs is a forward price F, ha is he price of one uni of he underlying asse o be delivered a a specific ime in he fuure, ( = T ). The second K is a delivery price, fixed in advance in a conrac.the hird is he value of a forward conrac f(), T. The value of F is chosen in such a way ha f() = (K = F a he iniial momen). There is no iniial price o ener such agreemens, excep 26

33 for bid-ask spread, (see [51]). Afer he iniial ime he value of f() may vary depending on variaions of he spo price of he underlying asse, he prevailing ineres raes, ec (see [19], Chaper 5,6; [27], Chaper 1 for deails of ypes of currency derivaives). The forward price F, = can be calculaed from he formula([19], Chaper 5, p. 17): F = S e (r q)t. (4.1) Here S is he spo price of he underlying asse, q is a rae of reurn of invesmen, r is he risk free rae, and T is mauriy ime. A each momen of ime, unil mauriy, he value of a long forward conrac f(), T (for buy) is calculaed by he following formula ([19], Chaper 5, p. 18): f() = (F K)e (r q)t. (4.2) Here F, K are he curren forward price and delivery price (a ime T ) respecively (see [27], p. 273). Because F changes wih ime, he price of a forward conrac may have boh posiive and negaive values, (see (4.2)). I is imporan for banks, and he finance indusry in general, o value he agreemen each day. In case of forward conracs on foreign currencies he underlying asse is he exchange rae, or a cerain number of unis of a foreign currency. Le us wrie S for he spo exchange rae, (price of 1 uni of foreign currency (EURO for example) in domesic currency, (dollars for example)), and F as he forward price, boh expressed in unis of he base currency per uni of foreign currency. Holding he currency gives he invesor an ineres raes profi a he risk-free rae prevailing in he respecive counry. Denoe by r d as he domesic risk-free rae and r f as he risk-free rae in he foreign counry. The forward price is hen similar o (4.1), (see [19], Chaper 5, p. 113): F = S e (r d r f )T. (4.3) Relaion (4.3) is he well-known ineres rae pariy relaionship from inernaional finance. If condiion (4.3) does no hold, here are arbirage opporuniies. If F > S e (r d r f )T, a profi could be obained by: 27

34 a) borrowing F = S e r f T in domesic currency a rae r d for ime T ; b) buying e r f T of he foreign currency and inves his a he rae r f ; c) shor selling a forward conrac on one uni of he foreign currency. A ime T, he arbirageur will ge one uni of foreign currency from he deposi, which he sells a he forward price F. From his resul, he is able o repay he loan S e (r d r f )T and sill obain a ne profi of F S e (r d r f )T (see [51]). We now lis he forward rading seps for he Forex marke (see [54]): 1) Choose he buy currency and he sell currency. The exchange rae appears auomaically and is called he Spo Rae. 2) Choose he forward dae. The Forward Poins and he Forward Rae appears auomaically. 3) Choose he amoun of he conrac and he amoun you wish o risk. The Sop-Loss rae appears. 4) Read he Response Message. This ells you if you have enough in your accoun o make he deal 5) Finish he deal by pressing he Accep buon. Then your deal is open and running Currency fuures Following Bjork, (see [6], p. 13) a fuures conrac is very close o he corresponding forward conrac in he sense ha i is sill a conrac for he delivery of X a T. The difference is ha all he paymens, from he holder of he conrac o he underwrier, are no longer made a T. Le us denoe he fuures price by F (; T, X); he paymens are delivered coninuously over ime, such ha he holder of he conrac over he ime inerval [s, ] receives he amoun F (; T, X) F (s; T, X) from he underwrier. Finally he holder will receive X, and pay F (T ; T, X), a he delivery dae T. By definiion, he (spo) price (a any ime) of he enire fuures conrac equals zero. Thus he cos of enering or leaving a fuures conrac is zero, and he only conracual obligaion is he paymen sream described above. 28

35 Fuures are similar o forward agreemens in erms of he end resul. A fuure conrac represens a binding obligaion o buy or sell a paricular asse a a specified price a a sipulaed dae. Fuures have, however, specific feaures. For insance heir sandardizaion and heir payoff procedure disinguish hem from forwards. For a new conrac several erms need o be specified by he exchange house prior o any ransacion, (see [51]): 1) he underlying asse: wheher i is a commodiy such as corn, or a financial insrumen such as a foreign exchange rae; 2) he size of he conrac: he amoun of he asse which will be delivered; 3) he delivery dae and delivery arrangemens; 4) he quoed price. A he mauriy ime he price of he fuures conrac converges o he spo price of he underlying asse. If his does no hold, arbirage opporuniies would arise, hus forcing, from he acion of agens in he marke, he price of he fuures o go up or down. We now explain he main differences beween Forward and Fuures ineres raes conracs for he EURO/USD currency pair. The mos popular ineres rae fuures conrac in he Unied Saes is he 3 monhs Eurodollar ineres rae fuures conrac, (see [19], [52]) raded on he Chicago Mercanile Exchange. The Eurodollar currency fuures conrac is similar o a forward rae agreemen because i locks in an ineres rae for a fuure period. For shor mauriies (less han 1 year) he wo conracs seem o be he same and he Eurodollar fuures ineres rae can be reaed he same as he corresponding forward ineres rae. For longer mauriies he differences beween conracs are very imporan. Le us compare a Eurodollar fuures conrac on an ineres rae for he period of ime [T 1, T 2 ] wih a Forward agreemen for he same period. The Eurodollar fuures conrac is seled daily, (he difference in he prior agreed-upon price and he daily fuures price is seled daily). The final selemen is a ime T 1 and reflecs he realized ineres rae beween imes T 1, T 2. On he conrary, he forward agreemen is no seled daily and he final selemen, reflecing he changes in 29

36 ineres raes beween T 1, T 2 is made a ime T 2. So, he main difference would be on he cash-flows processes associaed wih forwards, (which occur only a delivery dae) and wih fuures, which occur every rading day (see [19], Chaper 4, Secion, 4.7; Chaper 6, Secion 6.3 for deail). This is called mark o marke. There are several main differences beween forward agreemens and fuures conracs (see [19]): Forwards Privae conrac No sandardized Limied range of delivery daes Seled a mauriy dae Final cash selemen Fuures Traded on an exchange Sandardized Wide range of delivery daes Seled daily Closed ou before maure Currency opions In our research our main emphasis is on currency opions and pricing hem. In conras o forwards, (fuures) conracs, an opion is he righ, bu no he obligaion, o buy (or sell) an asse under specified erms (see [27], Chaper 12 for deails). In he currency marke (he Forex marke for example) i means he righ o buy (or sell) some amoun of foreign currency a a fixed momen of ime in fuure a a previously sipulaed exchange rae. A foreign exchange opion, (commonly shorened o jus FX opion), is a derivaive where he owner has he righ bu no he obligaion o exchange money denominaed in one currency ino anoher currency a a pre-agreed exchange rae on a specified dae (see [53]). The opion ha gives you a righ o purchase somehing is called call opion, o sell somehing is a pu opion. There are wo main ypes of opions: European opions and American opions. A holder of American opion can exercise he opion a any ime before he expiraion dae. An owner of European opion can exercise only a he expiraion ime. Opions have an advanage over forward (fuures) conracs: while proecing agains a downside risk (if he rend goes agains you, he invesor does no exercise he opion), hey 3

37 do no sop he invesor from profiing from unexpeced upward movemens of he foreign exchange raes. In oher words, he invesor who buys a call opion anicipaes ha asse price will increase, for example, in he Forex marke ha he EURO/USD exchange rae will go up, so ha a he expiraion dae he/she can buy he asse a he srike price K and sell i a he spo price S. His/her gain is hen max(,s-k). If he spo price decreases o less han he srike price K, he opion is no exercised. The person who buys a pu opion expecs ha asse price will decrease (or for example in Forex marke EURO/USD exchange rae goes down), so ha a he expiraion dae he/she can sell he asse a he srike price K and buy i a he spo price S. His/her gain is hen max(,k-s). If he spo price increases o more han he srike price K, he opion is no exercised. 4.2 Pure currency conracs Suppose here are wo currencies: he domesic currency (say US dollars), and he foreign currency (EUROs). Denoe he spo exchange rae a ime by S. By definiion S is quoed as: Spo exchange rae S = Unis of domesic currency Uni of foreign currency. We assume now ha he domesic and foreign ineres raes r d, r f are deerminisic consans; B d, B f are he riskless asse prices in domesic and foreign currencies respecively. We model he spo exchange raeusing geomeric Brownian moion. Then we have he following dynamics for S : ds = S α S d + S σ S dw. (4.4) Here W is a Brownian moion under he objecive probabiliy measure P ; α S, σ S are deer- 31

38 minisic consans. The dynamics of B d, B f are given by db d = r d B d d, (4.5) db f = r f B f d. (4.6) A T-claim ( a financial derivaive) is any sochasic variable Z = Φ(S(T )), where Φ is a some given deerminisic funcion. A European call opion is an example of T-claim and allows he holder o exercise he opion, (i.e., o buy), only on he opion expiraion dae T. I has he following value a ime T: Z = max(s T K, ). (4.7) The foreign currency plays he same role as a domesic sock wih a coninuous dividend. We shall prove his below. We shall use he following assumpion: all markes are fricionless and liquid. All holdings of he foreign currency are invesed in he foreign riskless asse, i.e. hey will evolve according o he dynamics (4.6), (see [6], Chaper 17). Using he sandard heory of derivaives we have he following risk neural valuaion formula where Q is a risk-neural maringale measure. The soluion o he SDE (4.4) is Π(, Z) = e r d(t ) E Q,S [Φ(S(T ))], (4.8) S = S e (α S 1/2σ 2 S )+σ SW. (4.9) The possibiliy of buying he foreign currency and invesing i a he foreign shor rae of ineres, is equivalen o he possibiliy of invesing in a domesic asse wih a price process 32

39 B f = B f S, where he dynamics of B f are given by d B f = B f (α + r f )d + B f σ dw. (4.1) Using he Girsanov heorem wih θ = α S + r f r d σ S he risk neural Q-dynamics of B f are given by d B f = r d Bf d + B f σ S dw, (4.11) where W is a is a Q-Wiener process. Using (4.5) and (4.11), Iô s formula (see Theorem 3.2) and he fac ha S = B f B f we have he following dynamics for S under Q: ds = S (r d r f ) + S σsdw. (4.12) Theorem 4.1 (see [6], Chaper 17). The arbirage free price Π(, Φ) for he T-claim Z = Φ(S T ) is given by Π(, Φ) = F (, S ) where F (, s) = e r d(t ) E Q,s[Φ(S T )], (4.13) where Q-dynamics of S are given by (4.12) and S = s is he iniial condiion. This resul follows direcly from he fac ha S T e r d(t ) is a maringale (see Girsanov s heorem, Theorem 3.3 ). Alernaively, F (, s) can be obained as he soluion o he boundary value problem wih he boundary condiion F (T, s) = Φ(s). F + s(r d r f ) F s s2 σs 2 2 F s r df = ; (4.14) 2 33

40 As a resul, foreign currency can be reaed exacly as a sock wih a coninuous dividend. So, he price of he European call, Z = max[s T K, ], on he foreign currency, is given by he modified Black-Scholes formula (see [6], Chaper 17): F (, s) = se r f (T ) K[d 1 ] e r d(t ) K[d 2 ], (4.15) where d 1 (, x) = 1 σ S T { ( x ) ( log + r d r f + 1 ) } K 2 σ2 S (T ), (4.16) d 2 (, s) = d 1 (, s) σ S T. (4.17) In his secion he marke also includes a domesic equiy wih price S d, and a foreign equiy wih a price S f. We shall model he equiy dynamics using geomeric Brownian moion. Now, here are hree risky asses in our marke: S, S f, S d. As a resul a hreedimensional Brownian moion is used o model he marke: The dynamic model of he enire economy, under he objecive measure P, is as follows (see [6], Chaper 17): ds = S α S d + S σ S dw, (4.18) ds d = S d α d d + S d σ d dw, (4.19) ds f = S f α f d + S f σ f dw, (4.2) db d = r d B d d (4.21) db f = r f B f d. (4.22) 34

41 Here W = [W 1 W 2 W 3 ] T is a a hree-dimensional Wiener process, consising of hree independen Brownian moions; σ is a 3 3 marix of he following form: σ = σ S σ d = σ S1 σ S2 σ S3 σ d1 σ d2 σ d3 (4.23) σ f σ f1 σ f2 σ f3 and is inverible. There are he following ypes of T -conracs (see [6], Chaper 17): 1) A European foreign equiy call, sruck in he foreign currency, i.e. an opion o buy one uni of he foreign equiy a he srike price of K unis of he foreign currency. The value of his claim a he dae of expiraion is, expressed in he foreign currency, is Z f = max[s f T K, ]. (4.24) Expressed in erms of he domesic currency he value of he claim a T is: Z d = S T max[s f T K, ]. (4.25) 2) A European foreign equiy call, sruck in domesic currency, i.e. a European opion o buy one uni of he foreign equiy a ime T, by paying K unis of he domesic currency. Expressed in domesic erms he price of his claim is given by Z d = max[s T S f T K, ]. (4.26) 3) An exchange opion, which gives he righ o exchange one uni of he domesic equiy for one uni of he foreign equiy. The corresponding claim, expressed in erms of he domesic currency, is In he general case, a T-claim has he following form Z d = max[s T S f T Sd T, ]. (4.27) ( ) Z = Φ S T, ST d, S f T, 35

42 where Z is measured in he domesic currency. Similarly o (4.13) we shall use he following risk-neural valuaion formula: F (, s, s d, s f ) = e r d(t ) E Q,s,s d,s f [Φ(S T )], (4.28) where S d = s d, S f = s f is an iniial condiion. The marke defined in (4.18)-(4.22) is equivalen o a marke consising of he componens S d, S f, B f, B d, where B f = B f S, (4.29) S f = S f S. (4.3) Using (4.29), (4.3) we can derive from (4.18)-(4.22) he following equivalen equaions modeling he FX marke (see [6],Chaper 17): ds d = S d α d d + S d σ d dw, (4.31) d S f = S f (α f + α S + σ f σ T S )d + S f (σ f + σ S )dw, (4.32) d B f = B f (α S + r f )d + B f σ S dw (4.33) db d = r d B d d. (4.34) S d, S f, B f can be inerpreed as he prices of domesically raded asses. Similarly o (4.1), (4.11) we can proceed o he risk-neural measure in (4.31)-(4.34), (see [6], Chaper 17): ds d = S d r d d + S d σ d dw, (4.35) 36

43 d S f = S f r d d + S f (σ f + σ S )dw, (4.36) d B f = B f r d d + B f σ S dw, (4.37) ds = S (r d r f )d + S σ S dw, (4.38) ds f = S f (r f σ f σ T s )d + S f σ f dw. (4.39) Theorem 4.2 (see [6], Chaper 17). The arbirage free price Π(, Φ) for he T -claim Z = Φ(S T, S d T, S f T ) is given by Π(, Φ) = F (, s, s d, s f ) where F (, s, s d, s f ) = e r d(t ) E Q [Φ(S,s,s d,s f T, ST d, S f T )], (4.4) where he Q-dynamics are given by (4.35)-(4.39). Alernaively F (, s, s d, s f ) can be obained as a soluion o he boundary value problem F + s(r d r f ) F s + F sd r d s + s F f r d d s + { f 1 s 2 σ S 2 2 F 2 s + 2 (sd ) 2 σ d 2 2 F (s d ) + ( s f ) ( ) 2 σ 2 f 2 + σ S 2 + 2σ f σs T 2 F s d sσ d σ T S 2 F s d s + s f s(σ f σ T S + σ S 2 ) 2 F s f s + s d sf (σ d σ T f + σ d σ T S ) 2 F ( s f ) 2 } + s d s f r df = ; (4.41) wih a boundary condiion F (T, s, s d, s f ) = Φ(s, s d, s f ). Theorem 4.3 (see [6], Chaper 17). The arbirage free price Π(, Φ) for he T -claim Z = Φ(S T, S d T, S f T ) 37

44 is given by Π(, Φ) = F (, s, s d, s f ) where F (, s, s d, s f ) = e r d(t ) E Q [Φ(S,s,s d,s f T, ST d, S f T )], (4.42) where he Q-dynamics are given by (4.35)-(4.39). Alernaively F (, s, s d, s f ) can be obained as a soluion of he boundary value problem F + s(r d r f ) F s + F sd r d s + d sf (r f σ f σs T ) F s + { } f 1 s 2 σ S 2 2 F 2 s + 2 (sd ) 2 σ d 2 2 F (s d ) + 2 (sf ) 2 2 F + (s f ) 2 s d sσ d σ T S 2 F s d s + sf sσ f σ T S 2 F s f s + sd s f σ d σ T f wih a boundary condiion F (T, s, s d, s f ) = Φ(s, s d, s f ). 2 F s d s f r df = ; (4.43) 38

45 Chaper 5 Currency opion pricing. Main resuls 5.1 Currency opion pricing for general Lévy processes Le (Ω, F, P) be a complee probabiliy space wih a probabiliy measure P. Consider a coninuous-ime, finie-sae Markov chain ξ = {ξ } T on (Ω, F, P) wih a sae space S, he se of uni vecors (e 1,, e n ) R n wih a rae marix Π 1. The dynamics of he chain are given by: ξ = ξ + Πξ u du + M R n, (5.1) where M = {M, } is a R n -valued maringale wih respec o (F ξ ) T, he P- augmenaion of he naural filraion (F ) T, generaed by he Markov chain ξ. Consider a Markov-modulaed Meron jump-diffusion which models he dynamics of he spo FX rae, given by he following sochasic differenial equaion (in he sequel SDE, see [8]): ds = S ( µ d + σ dw + (e Z 1)dN ). (5.2) Here µ is drif parameer; W is a Brownian moion, σ is he volailiy; N is a Poisson Process wih inensiy λ, e Z 1 is he ampliude of he jumps, given he jump arrival ime. The disribuion of Z has a densiy ν(x), x R. The parameers µ, σ, λ are modeled using he finie sae Markov chain: µ :=< µ, ξ >, µ R n +; σ :=< σ, ξ >, σ R n +; λ :=< λ, ξ >, λ R n +. (5.3) 1 In our numerical simulaions we consider hree-sae Markov chain and calculae elemens in Π using Forex marke EURO/USD currency pair 39

46 The soluion of (5.74) is S = S e L, (where S is he spo FX rae a ime = ). Here L is given by he formula: L = (µ s 1/2σ 2 s)ds + σ s dw s + Z s dn s. (5.4) There is more han one equivalen maringale measure for his marke driven by a Markovmodulaed jump-diffusion model. We shall define he regime-swiching generalized Esscher ransform o deermine a specific equivalen maringale measure. Using Iô s formula we can derive a sochasic differenial equaion for he discouned spo FX rae. To define he discouned spo FX rae we need o inroduce domesic and foreign riskless ineres raes for bonds in he domesic and foreign currency. The domesic and foreign ineres raes (r d ) T, (r f ) T are defined using he Markov chain (ξ ) T (see [8]): r d = r d, ξ, r d R n +, r f = r f, ξ, r f R n +. The discouned spo FX rae is: ( ) S d = exp (rs d rs f )ds S, T. (5.5) Using (5.76), he differeniaion formula, see Ellio e al. (1982, [12]) and he sochasic differenial equaion for he spo FX rae (5.74) we find he sochasic differenial equaion for he discouned discouned spo FX rae: ds d = S d (r d r f + µ )d + S d σ dw + S d (e Z 1)dN. (5.6) To derive he main resuls consider he log spo FX rae Using he differeniaion formula: Y = log ( S S ). 4

47 Y = C + J, where C, J are he coninuous and jump par of Y. They are given in (5.78), (5.79): C = ( ) r d s rs f + µ s ds + σ s dw s, (5.7) J = Z s dn s. (5.8) Le (F Y ) T denoe he P-augmenaion of he naural filraion (F ) T, generaed by Y. For each [, T ] se H = F Y F ξ T. Le us also define wo families of regime swiching parameers (θ c s) T, (θ J s ) T : θ m =< θ m, ξ >, θ m = (θ m 1,..., θ m n ) R n, m = {c, J}. Define a random Esscher ransform Q θc,θ J P on H using hese families of parameers (θ c s) T, (θ J s ) T (see [8], [13], [15] for deails): [ E exp L θc,θ J,θJ dqθc = dp =: (5.9) H θc sdc s + ) θj s dj s ( θc sdc s + ) ]. θj s dj s F ξ ( exp The explici formula for he densiy L θc,θ J of he Esscher ransform is given in he following heorem. A similar saemen is proven for he log-normal disribuion in [8]. The formula below can be obained by anoher approach, considered by Ellio and Osakwe ([14]). Theorem 5.1. For T he densiy L θc,θ J of Esscher ransform defined in (5.9) is given by L θc,θ J ( = exp θsσ c s dw s 1/2 ( λ s ( exp θs J Z s dn s R ) (θsσ c s ) 2 ds (5.1) ) e θj s x ν(dx) 1 ) ds. 41

48 In addiion,he random Esscher ransform densiy L θc,θ J (see (5.9), (5.1)) is an exponenial (H ) T maringale and saisfies he following SDE: N dl θc,θ J L θc,θ J Proof of Theorem 5.1. = θ c σ dw + (e θj Z 1)dN λ ( R ) e θj x ν(dx) 1 d. (5.11) The compound Poisson Process, driving he jumps (ezs 1) and he Brownian moion W are independen processes. Consequenly: [ ( ) ] E exp θsdc c s + θs J dj s F ξ = [ ( E exp θs(µ c s 1/2σs)ds 2 + Le us calculae: Wrie ) ] [ ( ) ] θsσ c s dw s ) F ξ E exp θs J Z s dn s F ξ. (5.12) [ ( ) ] E exp θs J Z s dn s F ξ. ( ) Γ := exp α s dn s, α s = θs J Z s. Using he differeniaion rule (see [12]) we obain he following represenaion of Γ : Γ = Γ + M J + ],] Γ s R (e αs 1)ν(dx)λ s ds, (5.13) where M J = ],] Γ s (e αs 1)dNs ],] Γ s R (e αs 1)ν(dx)λ s ds is a maringale wih respec o F ξ. Using his fac and (5.83) we obain: [ ( ) ] ( ( ) ) E exp θs J Z s dn s F ξ = exp λ s e θj s x ν(dx) 1. ds R (5.14) We have from he differeniaion rule: where σ is he volailiy of a marke. E [e u ] { 1 σsdws = exp 2 u2 42 } σsds 2, (5.15)

49 Subsiuing (5.14) and (5.15) ino (5.12) we obain: [ ( ) E exp θsdc c s + θs J dj s F ξ ] = ( exp θs(µ c s 1/2σs)ds Subsiuing (5.16) ino he expression for L θc,θ J ) ( ( (θsσ c s ) 2 ds) exp λ s R in (5.9) we obain: ) ) e θj s x ν(dx) 1 ds. (5.16) L θc,θ J ( = exp θs(µ c s 1/2σs)ds 2 + [ ( exp θs(µ c s 1/2σs)ds If we represen L θc,θ J ( exp ( exp θs J Z s dn s (θ c sσ s ) 2 ds) θ c sσ s dw s 1/2 ) ( θsσ c s dw s ) exp θs J Z s dn s ) (5.17) ) ( ( ) 1 exp λ s e θj s x ν(dx) 1 ds)] = λ s ( we obain (5.11). I follows from (5.11) ha L θc,θ J R R ) (θsσ c s ) 2 ds ) ) e θj s x ν(dx) 1 ds. in he form L θc,θ J = e X (see (5.17)) and apply differeniaion rule is a maringale. We shall derive he following condiion for he discouned spo FX rae ((5.76)) o be maringale. These condiions will be used o calculae he risk-neural Esscher ransform parameers (θ c, s ) T, (θ J, ) T and give o he measure Q. Then we shall use hese values s o find he no-arbirage price of European call currency derivaives. Theorem 5.2. Le he random Esscher ransform be defined by (5.9). Then he maringale condiion (for S d, see (5.76)) holds if and only if he Markov modulaed parameers (θ c, θ J, T ) saisfy for all T he condiion: r f r d + µ + θσ c 2 + λ θ,j k θ,j =. (5.18) Here he random Esscher ransform inensiy λ θ,j of he Poisson Process and he main percenage jump size k θ,j are, respecively, given by = λ e θj s x ν(dx), (5.19) λ θ,j R 43

50 k θ,j = R e(θj +1)x ν(dx) R eθj x ν(dx) as long as R eθj x ν(dx) < +, R e(θj +1)x ν(dx) < +. is 1, (5.2) Proof of Theorem 5.2. The maringale condiion for he discouned spo FX rae S d To derive his condiion a Bayes formula is used: E θc,θ J [S d H u ] = S d u, u. (5.21) E θc,θ J [S d H u ] =,θj E[Lθc S d H u ], (5.22) E[L θc,θ J H u ] aking ino accoun ha L θc,θ J is a maringale wih respec o H u, so: [ E L θc,θ J ] H u = L θc,θ J u. (5.23) Using formula (5.76) for he soluion of he SDE for he spo FX rae, we obain an expression for he discouned spo FX rae in he following form: ( S d = Su d exp (rs f rs d + µ s 1/2σs)ds 2 + u Then, using (5.1), (5.24) we can rewrie(5.93) as: [ L θ c,θ J E L θc,θ J u S d u σ s dw s + ] [ ( H u = Su d E exp θsσ c s dw s 1/2 ( exp θs J Z s dn s u u u λ s ( ( exp (rs f rs d + µ s 1/2σs)ds 2 + u R [ ( Su d E exp (θs c + 1)σ s dw s 1/2 u ) ( exp (rs F rs D + µ s + θsσ c s)ds 2 u u u ) ) e θj s x ν(dx) 1 ds σ s dw s + u u Z s dn s ), u. (5.24) ) (θsσ c s ) 2 ds (5.25) Z s dn s ) H u ]= ) ((θs c + 1)σ s ) 2 ds (5.26) ( ( exp λ s e θj s x ν(dx) 1 ) ds u R 44 ] ) H u

51 [ ( ) E exp (θs c + 1)Z s dn s H u ]. u Using he expression for he characerisic funcion of Brownian moion (see (5.15)) we obain: From (5.14) we obain: [ ( E exp (θs c + 1)σ s dw s 1/2 u u ) ((θs c + 1)σ s ) 2 ds H u ]= 1. (5.27) [ ( ) ( ( ) ) E exp (θs c + 1)Z s dn s H u ]= exp λ s e (θj s +1)x ν(dx) 1 ds. (5.28) u R Subsiuing (5.27), (5.28) ino (5.26) we obain finally: ( exp [ L θ c,θ ] ( J ) E S d L θc,θ J H u = Su d exp (rs f rs d + µ s + θsσ c s)ds 2 (5.29) u u ( λ s e θj s x ν(dx) 1 ) ) ( ( ds exp λ s e (θj s +1)x ν(dx) 1 ) ) ds. R R u From (5.29) we obain he maringale condiion for discouned spo FX rae: r f r d + µ + θ c σ 2 + λ [ R u e (θj s +1)x ν(dx) R ] e θj s x ν(dx) =. (5.3) We now prove, ha under he Esscher ransform he new Poisson process inensiy and mean jump size are given by (5.19), (5.2). Noe ha L J = Z s dn s is he jump par of Lévy process in he formula (5.75) for he soluion of SDE for spo FX rae. We have: ] E Q [e LJ = Ω ( ) exp Z s dn s L θc,,θ J, (ω)dp (ω), (5.31) where P is he iniial probabiliy measure and Q is he new risk-neural measure. Subsiuing he densiy of Esscher ransform (5.1) ino (5.31) we have (see also [14]): ] [ ( E Q [e LJ = E P exp θsσ c s dw s 1/2 ( ) )] [ ( λ s e θj s x ν(dx) 1 ds E P exp R 45 ) (θsσ c s ) 2 ds (5.32) (θ J s + 1)Z s dn s ) ].

52 Using (5.14) we obain: ( ) ( ( ) ) E P [exp (θs J + 1)Z s dn s ]= exp λ s e (θj s +1)x ν(dx) 1 ds. (5.33) R Subsiue (5.33) ino (5.32) and aking ino accoun he characerisic funcion of Brownian moion (see (5.15)) we obain: E Q [e LJ ] ( ( = exp λ s e θj s x ν(dx) R [ R e(θj s +1)x ν(dx) R eθj s x ν(dx) Reurning o he iniial measure P, bu wih differen λ θ,j E λ, ν [e LJ ] ) ) 1] ds. (5.34), k θ,j,we have: ( ( = exp λ θ,j s e x ν(dx) ) ) 1 ds. (5.35) R Formula (5.19) for he new inensiy λ θ,j of Poisson process follows direcly from (5.34), (5.35). The new densiy of jumps ν is defined from (5.35) by he following formula: R e(θj +1)x ν(dx) R eθj x ν(dx) = R e x ν(dx). (5.36) We now calculae he new mean jump size given he jump arrival wih respec o he new measure Q: k θ,j = (e Z(ω) 1)d ν(ω) = (e x 1) ν(dx) = e x ν(dx) 1 = Ω R R where he new measure ν(dx) is defined by he formula (5.36). R e(θj +1)x ν(dx) 1, (5.37) R eθj x ν(dx) We can rewrie he maringale condiion (5.3) for he discouned spo FX rae in he following form: where λ θ,j, k θ,j r f r d + µ + θσ c 2 + λ θ,j k θ,j =, (5.38) are given by (5.19), (5.2) respecively. If we pu k θ,j =, we obain he following formulas for he regime swiching Esscher ransform parameers yielding he maringale condiion (5.38): θ c, = rd r f µ, (5.39) σ 2 46

53 θ J, : R e(θj, +1)x ν(dx) R eθj, x ν(dx) = 1. (5.4) In he nex secion we shall apply hese formulas (5.39), (5.4) o he log-double exponenial disribuion of jumps. We now proceed o he general formulas for European calls (see [8], [3]). For he European call currency opions wih a srike price K and he ime of expiraion T he price a ime zero is given by: [ Π (S, K, T, ξ) = E θc,,θ J, e ] T (rd s rf s )ds (S T K) + F ξ. (5.41) Le J i (, T ) denoe he occupaion ime of ξ in sae e i over he period [, T ], < T. We inroduce several new quaniies ha will be used in fuure calculaions: R,T = 1 T T (r d s r f s )ds = 1 T n (ri d r f i )J i(, T ), (5.42) i=1 where J i (, T ) := T < ξ s, e i > ds; U,T = 1 T σ T sds 2 = 1 T n σi 2 J i (, T ); (5.43) i=1 λ θ,t = 1 T T λ θ J,T = 1 T n i=1 (1 + k θ J s )λ θ J s ds = 1 T λ θ J i J i (, T ); (5.44) n i=1 (1 + k θ J i )λ θ J i J i (, T ); (5.45) V 2,T,m = U,T + mσ2 J T, (5.46) where σ 2 J is he variance of he disribuion of he jumps. R,T,m = R,T 1 T T λ θ J s k θ J s ds + 1 T T log(1 + k θ J T s ) ds = (5.47) 47

54 R,T 1 T n i=1 λ θ J i k θ J i + m T n log(1 + k θ J i=1 T i ) J i (, T ), where m is he number of jumps in he inerval [, T ], n is he number of saes of he Markov chain ξ. Noe, ha in our consideraions all hese general formulas (5.42)-(5.47) menioned above are simplified by he fac ha: k θ J = wih respec o he new risk-neural measure Q wih Esscher ransform parameers given by (5.39), (5.4). From he pricing formula in Meron (1976, [3]) le us define (see [8]) Π (S, K, T ; R,T, U,T, λ θ,t ) = m= BS (S, K, T, V 2,T,m, R,T,m ) e T λθ,j,t (T λ θ m!,t )m (5.48) where BS (S, K, T, V 2,T,m, R,T,m) is he sandard Black-Scholes price formula (see [6]) wih iniial spo FX rae S, srike price K, risk-free rae r, volailiy square σ 2 and ime T o mauriy. Then, he European syle call opion pricing formula akes he form (see [8]): Π (S, K, T ) = Π (S, K, T ; R,T, U,T, λ Θ,J,T ) (5.49) [,] n ψ(j 1, J 2,..., J n )dj 1...dJ n, where ψ(j 1, J 2,..., J n ) is he join probabiliy disribuion densiy for he occupaion ime, which is deermined by he following characerisic funcion (See [14]): E [ exp { u, J(, T ) }] = exp{(π + diag(u))(t )} E[ξ ], 1, (5.5) where 1 R n is a vecor of ones, u = (u 1,..., u n ) is a vecor of ransform variables, J(, T ) := {J 1 (, T ),..., J n (, T )}. 5.2 Currency opion pricing for log-double exponenial processes The log-double exponenial disribuion for Z (e Z 1 are he jumps in (5.74)), plays a fundamenal role in mahemaical finance, describing he spo FX rae movemens over long 48

55 period of ime. I is defined by he following formula of he densiy funcion: ν(x) = pθ 1 e θ 1x +(1 p)θ 2 e θ 2x. (5.51) x x< The mean value of his disribuion is: The variance of his disribuion is: var(θ 1, θ 2, p) = 2p θ 2 1 mean(θ 1, θ 2, p) = p θ 1 1 p θ 2. (5.52) + 2(1 p) θ 2 2 ( p 1 p ) 2. (5.53) θ 1 θ 2 The double-exponenial disribuion was firs invesigaed by Kou in [25]. He also gave economic reasons o use such a ype of disribuion in Mahemaical Finance. The double exponenial disribuion has wo ineresing properies: he lepokuric feaure (see [22], 3; [25] ) and he memoryless propery (he probabiliy disribuion of X is memoryless if for any non-negaive real numbers and s, we have Pr(X > + s X > ) = Pr(X > s), see for example [48]). The las propery is inheried from he exponenial disribuion. A saisical disribuion has he lepokuric feaure if here is a higher peak (higher kurosis) han in a normal disribuion. This high peak and he corresponding fa ails mean he disribuion is more concenraed around he mean han a normal disribuion, and i has a smaller sandard deviaion. See deails for fa-ail disribuions and heir applicaions o Mahemaical Finance in [47]. A lepokuric disribuion means ha small changes have less probabiliy because hisorical values are cenered by he mean. However, his also means ha large flucuaions have greaer probabiliies wihin he fa ails. In [28] his disribuion is applied o model asse pricing under fuzzy environmens. Several advanages of models including log-double exponenial disribued jumps over oher models described by Lévy processes (see [25]) include: 1) The model describes properly some imporan empirical resuls from sock and foreign currency markes. The double exponenial jump-diffusion model is able o reflec he lepokuric feaure of he reurn disribuion. Moreover, he empirical ess performed in [37] 49

56 Figure 5.1: Double-exponenial disribuion (green) vs. normal disribuion (red), mean=, dev=2 sugges ha he log-double exponenial jump-diffusion model fis sock daa beer han he log-normal jump-diffusion model. 2) The model gives explici soluions for convenience of compuaions. 3) The model has an economical inerpreaion. 4) I has been suggesed from exensive empirical sudies ha markes end o have boh overreacion and underreacion o good or bad news. The jump par of he model can be considered as he marke response o ouside news. In he absence of ouside news he asse price (or spo FX rae) changes in ime as a geomeric Brownian moion. Good or bad news (ouer srikes for he FX marke in our case) arrive according o a Poisson process, and he spo FX rae changes in response according o he jump size disribuion. Since he double exponenial disribuion has boh a high peak and heavy ails, i can be applied o model boh he overreacion (describing he heavy ails) and underreacion (describing he high peak) o ouside news. 5) The log double exponenial model is self-consisen. In Mahemaical Finance, i means ha a model is arbirage-free. The family of regime swiching Esscher ransform parameers is defined by (5.39), (5.4). 5

57 Le us define θ J,, (he firs parameer θ c, has he same formula as in general case) by: ( e (θj s +1)x pθ 1 e θ 1x +(1 p)θ 2 e θ 2x )dx = (5.54) x x< R R ( e θj s x pθ 1 e θ 1x +(1 p)θ 2 e θ 2x )dx. x x< We require an addiional resricion for he convergence of he inegrals in (5.54): θ 2 < θ J < θ 1. (5.55) Then (5.54) can be rewrien in he following form: pθ 1 θ 1 θ J 1 + (1 p)θ 2 θ 2 + θ J + 1 = pθ 1 θ 1 θ J + (1 p)θ 2. (5.56) θ 2 + θ J Solving (5.56) we arrive a he quadraic equaion: (θ J ) 2 (pθ 1 (1 p)θ 2 ) + θ J (pθ 1 + 2θ 1 θ 2 (1 p)θ 2 )+ (5.57) pθ 1 θ pθ 2 θ 2 1 θ 2 θ θ 1 θ 2 =. If pθ 1 (1 p)θ 2 we have wo soluions and one of hem saisfies resricion (5.55): θ J = pθ 1 + 2θ 1 θ 2 (1 p)θ 2 ± (5.58) 2(pθ 1 (1 p)θ 2 ) (pθ1 + 2θ 1 θ 2 (1 p)θ 2 ) 2 4(pθ 1 (1 p)θ 2 )(pθ 1 θ2 2 + pθ 2 θ1 2 θ 2 θ1 2 + θ 1 θ 2 ). 2(pθ 1 (1 p)θ 2 ) Then he Poisson process inensiy (see (5.19)) is: λ θ,j = λ ( pθ1 θ 1 θ J The new mean jump size (see (5.2)) is: + (1 p)θ ) 2. (5.59) θ 2 + θ J k θ,j = (5.6) as in he general case. 51

58 When we proceed o a new risk-neural measure Q we have a new densiy of jumps ν ν(x) = pθ 1 e θ 1x +(1 p)θ 2 e θ 2x. (5.61) x x< The new probabiliy p can be calculaed using (5.36): pθ 1 θ 1 θ J 1 + (1 p)θ 2 θ 2 +θ J +1 pθ 1 θ 1 θ J + (1 p)θ 2 θ 2 +θ J From (5.62) we obain an explici formula for p: = pθ 1 θ (1 pθ 2) θ (5.62) p = pθ 1 θ 1 θ J 1 + (1 p)θ 2 θ 2 +θ J +1 pθ 1 θ 1 θ J + (1 p)θ 2 θ 2 +θ J θ 1 θ 1 1 θ 2 θ 2 +1 θ 2 θ (5.63) 5.3 Currency opion pricing for log-normal processes Log-normal disribuion of jumps wih µ J he mean, σ J he deviaion (see [49]), and is applicaions o currency opion pricing was invesigaed in [8]. More deails of hese disribuions and oher disribuions applicable for he Forex marke can be found in [5]. We give here a skech of resuls from [8] o compare hem wih he case of he log-double exponenial disribuion of jumps discussed in his aricle. The main goal of our paper is a generalizaion of his resul for arbirary Lévy processes. The resuls, provided in [8] are as follows: Theorem 5.3. For T he densiy L θc,θ J of he Esscher ransform defined in (5.9) is given by L θc,θ J ( = exp θsσ c s dw s 1/2 ( exp θs J Z s dn s ) λ s (e θj s µ J +1/2(θs J σ J ) 2 1 ) (θsσ c s ) 2 ds (5.64) ) ds, where µ J, σ J are he mean value and deviaion of jumps, respecively. In addiion, he random Esscher ransform densiy L θc,θ J, (see (5.9), (5.1)), is an exponenial (H ) T maringale and saisfies he following SDE dl θc,θ J L θc,θ J ) = θσ c dw + (e θj Z 1)dN λ (e θj µ J +1/2(θ J σ J ) 2 1 d. (5.65) 52

59 Theorem 5.4. Le he random Esscher ransform be defined by (5.9). Then he maringale condiion (for S d, see (5.76)) holds if and only if he Markov modulaed parameers (θ c, θ J, T ) saisfy for all T he condiion: r f i rd i + µ i + θi c σi 2 + λ θ,j i k θ,j i = for all i, 1 i Λ (5.66) where he random Esscher ransform inensiy λ θ,j i percenage jump size k θ,j i are respecively given by of he Poisson Process and he mean λ θ,j i = λ i e θj i µ J +1/2(θ J i σ J ) 2, (5.67) k θ,j i = e µ J +1/2σ 2 J +θj i σ2 J 1 for all i. (5.68) The regime swiching parameers saisfying he maringale condiion (5.66) are given by he following formulas: θ c, i is he same as in (5.39), θ J, i = µ J + 1/2σ 2 J σ 2 J for all i. (5.69) Wih such a value of a parameer θ J, i : k θ,j i =, λ θ,j i = λ i ( µ J 2σ 2 J ) + σ2 J 8 for all i. (5.7) Noe, ha hese formulas (5.67)-(5.7) follow direcly from our formulas for he case of general Lévy process, (see (5.19),(5.2), (5.4)). In paricular, he fac ha k θ,j i subsiuing (5.4) ino he expression for k θ,j i As R e θj, i x ν(dx) = R e(θj, i +1)x ν(dx) R eθj, i x ν(dx) 1 2πσ 2 J e θj, R we obain from (5.71) he following equaliy: in (5.2). From (5.4) we derive: i x e (x µ J ) 2 2σ J 2 = by = 1 (5.71) dx = e 1 2 (σ J θ J, i ) 2 +θ J, i µ J (5.72) e 1 2 (σ J (θ J, i +1)) 2 +(θ J, i +1)µ J = e 1 2 (σ J θ J, i ) 2 +θ J, i µ J. (5.73) 53

60 The expression for he value of he Esscher ransform parameer θ J, i in (5.69) follows immediaely from (5.73). Insering his value of θ J, i obain he formula (5.7). ino he expression for λ θ,j i in (5.19) we In he numerical simulaions, we assume ha he hidden Markov chain has hree saes: up, down, side-way, and he corresponding rae marix is calculaed using real Forex daa for he hireen-year period: from January 3, 2 o November 213. To calculae all probabiliies we use he Malab scrip (see he Appendix). 5.4 Currency opion pricing for Meron jump-diffusion processes Consider a Markov-modulaed Meron jump-diffusion which models he dynamics of he spo FX rae, given by he following sochasic differenial equaion (in he sequel SDE, see [8]): ds = S ( µ d + σ dw + (Z 1)dN ), Z >. (5.74) Here µ is he drif parameer; W is a Brownian moion, σ is he volailiy; N is a Poisson Process wih inensiy λ, Z 1 is he ampliude of he jumps, given he jump arrival ime. The disribuion of Z has a densiy ν(x), x R. The soluion of (5.74) is S = S e L, (where S is he spo FX rae a ime = ). Here L is given by he formula: L = (µ s 1/2σ 2 s)ds + σ s dw s + log Z s dn s. (5.75) Noe, ha for he mos of well-known disribuions (normal, exponenial disribuion of Z, ec) L is no a Lévy process (see definiion of Lévy process in [34], he condiion L3), since log Z for small Z, bu probabiliy of jumps wih even ampliude is a posiive consan, depending on a ype of disribuion. We call he process (5.75) as Meron jump-diffusion process (see [3], Secion 2, formulas 2, 3) There is more han one equivalen maringale measure for his marke driven by a Markovmodulaed jump-diffusion model. We shall define he regime-swiching generalized Esscher 54

61 ransform o deermine a specific equivalen maringale measure. Using Io s formula we can derive a sochasic differenial equaion for he discouned spo FX rae. To define he discouned spo FX rae we need o inroduce domesic and foreign riskless ineres raes for bonds in he domesic and foreign currency. The discouned spo FX rae is: ( ) S d = exp (rs d rs f )ds S, T. (5.76) Using (5.76), he differeniaion formula, see Ellio e al. (1982, [12]) and he sochasic differenial equaion for he spo FX rae (5.74) we find he sochasic differenial equaion for he discouned discouned spo FX rae: ds d = S d (r d r f + µ )d + S d σ dw + S d (Z 1)dN. (5.77) To derive he main resuls consider he log spo FX rae Using he differeniaion formula: Y = log ( S S ). Y = C + J, where C, J are he coninuous and diffusion par of Y. They are given in (5.78), (5.79): C = ( ) r d s rs f + µ s ds + σ s dw s, (5.78) J = The similar o he Theorem 5.1 saemen is rue in his case. log Z s dn s. (5.79) Theorem 5.5. For T densiy L θc,θ J of Esscher ransform defined in (5.9) is given by L θc,θ J ( = exp θsσ c s dw s 1/2 55 ) (θsσ c s ) 2 ds (5.8)

62 ( exp θs J log Z s dn s In addiion, he random Esscher ransform densiy L θc,θ J (see (5.9), (5.8)) is an exponenial (H ) T maringale and admis he following SDE dl θc,θ J L θc,θ J ( ) ) λ s x θj s ν(dx) 1 ds. R + ( ) = θσ c dw + (Z θj 1)dN λ x θj ν(dx) 1 d. (5.81) R + Proof of Theorem 5.5. The compound Poisson Process, driving jumps N (Z i 1), and he Brownian moion W are independen processes. As a resul: [ ( E exp θsdc c s + θ J s dj s ) F ξ ] = [ ( E exp θs(µ c s 1/2σs)ds 2 + Le us calculae: Wrie ) ] [ ( ) ] θsσ c s dw s ) F ξ E exp θs J log Z s dn s F ξ. (5.82) [ ( ) ] E exp θs J log Z s dn s F ξ. ( ) Γ := exp α s dn s, α s = θs J log Z s. Using he differeniaion rule (see [12]) we obain he following represenaion of Γ : where M J = Γ = Γ + M J + ],] ],] Γ s Γ s (e αs 1)dNs R (e αs 1)ν(dx)λ s ds, (5.83) ],] Γ s R (e αs 1)ν(dx)λ s ds is a maringale wih respec o F ξ. Using his fac and (5.83) we obain: [ ( ) ] ( ( ) E exp θs J log Z s dn s F ξ = exp λ s x θj s ν(dx) 1 )ds. (5.84) R 56

63 We have from he differeniaion rule: E [e u ] { 1 σsdws = exp 2 u2 } σsds 2 (5.85) where σ is he volailiy of a marke. Subsiuing (5.84) and (5.85) ino (5.82) we obain: [ ( E exp θsdc c s + θ J s dj s ) F ξ ] = ( exp θs(µ c s 1/2σs)ds ) ( ( ) (θsσ c s ) 2 ds) exp λ s x θj s ν(dx) 1 )ds R + (5.86) Subsiuing (5.86) o he expression for L θc,θ J in (5.9) we have: L θc,θ J ( = exp θs(µ c s 1/2σs)ds 2 + [ ( exp θs(µ c s 1/2σs)ds If we presen L θc,θ J (θ c sσ s ) 2 ds) ( exp θsσ c s dw s 1/2 ( λ s ( exp θs J log Z s dn s ) ( θsσ c s dw s ) exp θs J log Z s dn s ) (5.87) ) ( ( )ds)] 1 exp λ s x θj s ν(dx) 1 = R + ) (θsσ c s ) 2 ds ) ) x θj s ν(dx) 1 ds. R + in he form L θc,θ J = e X (see (5.87)) and and apply differeniaion rule we obain SDE (5.81). I follows from (5.81) ha L θc,θ J is a maringale. We shall derive he following condiion for he discouned spo FX rae ((5.76)) o be maringale. These condiions will be used o calculae he risk-neural Esscher ransform parameers (θ c, ) T, (θ J, ) T and give o he measure Q. Then we shall use hese values o find he no-arbirage price of European call currency derivaives. Theorem 5.6. Le he random Esscher ransform be defined by (5.9). Then he maringale condiion (for S d, see (5.76)) holds if and only if Markov modulaed parameers (θ c, θ J, T ) saisfy for all T he condiion: r f r d + µ + θσ c 2 + λ θ,j k θ,j = (5.88) 57

64 where he random Esscher ransform inensiy λ θ,j of he Poisson Process and he main percenage jump size k θ,j are respecively given by λ θ,j = λ x θj s ν(dx), (5.89) R + as long as R + x θj +1 ν(dx) < +. k θ,j = R + x (θj +1) ν(dx) R + x θj ν(dx) 1 (5.9) Proof of Theorem 5.6. The maringale condiion for he discouned spo FX rae S d To derive such a condiion Bayes formula is used: E θc,θ J [S d H u ] = S d u, u. (5.91) E θc,θ J [S d H u ] =,θj E[Lθc S d H u ], (5.92) E[L θc,θ J H u ] aking ino accoun ha L θc,θ J is a maringale wih respec o H u, so: [ E L θc,θ J ] H u = L θc,θ J u. (5.93) Using formula (5.76) for he soluion of he SDE for he spo FX rae, we obain an expression for he discouned spo FX rae in he following form: ( S d = Su d exp (rs f rs d + µ s 1/2σs)ds 2 + u u σ s dw s + Then, using (5.8), (5.94) we can rewrie(5.93) in he following form: [ L θ c,θ ] [ ( J E S d L θc,θ J H u = Su d E exp u u ( exp θs J log Z s dn s u ( exp (rs f rs d + µ s 1/2σs)ds 2 + u u 58 u θ c sσ s dw s 1/2 log Z s dn s ), u. (5.94) ( ) ) λ s x θj s ν(dx) 1 ds R + u σ s dw s + u ) (θsσ c s ) 2 ds (5.95) log Z s dn s ) H u ]=

65 [ ( Su d E exp (θs c + 1)σ s dw s 1/2 ( exp (rs f rs d + µ s + θsσ c s)ds 2 u u ) u ) ((θs c + 1)σ s ) 2 ds (5.96) ( ( exp λ s e θj s x ν(dx) 1 ) ds u R [ ( ) E exp (θs c + 1) log Z s dn s H u ]. u ] ) H u Using expression for characerisic funcion of Brownian moion (see (5.85)) we obain: [ ( ) E exp (θs c + 1)σ s dw s 1/2 ((θs c + 1)σ s ) 2 ds H u ]= 1. (5.97) u Using (5.84) we have: [ ( ) ( ( ) ) E exp (θs c + 1) log Z s dn s H u ]= exp λ s x (θj s +1) ν(dx) 1 ds. (5.98) u R + Subsiuing (5.97), (5.98) ino (5.96) we obain finally: ( exp [ L θ c,θ J E u L θc,θ J u S d H u u ] ( ) = Su d exp (rs f rs d + µ s + θsσ c s)ds 2 (5.99) u ( ( ) ) λ s x θj s ν(dx) 1 ds exp R + u ( λ s x (θj s +1) ν(dx) 1 ) ) ds. R + From (5.99) we ge he maringale condiion for he discouned spo FX rae: r f r d + µ + θ c σ 2 + λ [ x (θj s +1) ν(dx) R + x θj s ν(dx) ]=. (5.1) R + Prove now, ha under he Esscher ransform he new Poisson process inensiy and he mean jump size are given by (5.89), (5.9). Noe ha L J = log Z s dn s is he jump par of Lévy process in he formula (5.75) for he soluion of SDE for spo FX rae. We have: ] ( ) E Q [e LJ = exp log Z s dn s L θc,θ J (ω)dp(ω), (5.11) Ω where P is he iniial probabiliy measure, Q is a new risk-neural measure. Subsiuing he densiy of he Esscher ransform (5.8) ino (5.11) we have: ] [ ( ) E Q [e LJ = E P exp θsσ c s dw s 1/2 (θsσ c s ) 2 ds (5.12) 59

66 Using (5.84) we obain: ( )] [ ( ) λ s x θj s ν(dx) 1 )ds E P exp (θs J + 1) log Z s dn s ]. R + ( ) ( ( ) ) E P [exp (θs J + 1) log Z s dn s ]= exp λ s x (θj s +1) ν(dx) 1 ds. (5.13) R + Puing (5.13) o (5.12) and aking ino accoun characerisic funcion of Brownian moion (see (5.85)) we have: E Q [e LJ ] ( ( [ = exp λ s x θj R s ν(dx) + x (θj s +1) ) ) ν(dx) 1] ds. (5.14) R + R + x θj s ν(dx) Reurn o he iniial measure P, bu wih differen λ θ,j E λ, ν [e LJ ], k θ,j. We obain: ( ( ) ) = exp λ θ,j s x ν(dx) 1 ds. (5.15) R + Formula (5.89) for he new inensiy λ θ,j of he Poisson process follows direcly from (5.14), (5.15). The new densiy of jumps ν is defined from (5.14), (5.15) by he following formula: R + x (θj +1) ν(dx) R + x θj ν(dx) = x ν(dx). R + (5.16) Calculae now he new mean jump size given jump arrival wih respec o he new measure Q: k θ,j = (Z(ω) 1)d ν(ω) = (x 1) ν(dx) = Ω R + R x ν(dx) 1 = + x (θj +1) ν(dx) 1. (5.17) + R + x θj ν(dx) So, we can rewrie maringale condiion for he discouned spo FX rae in he form in (5.88), where λ θ,j, k θ,j are given by (5.89), (5.9) respecively. Using (5.1) we have he following formulas for he families of he regime swiching parameers saisfying he maringale condiion (5.88): θ c, = K + r d r f µ, (5.18) σ 2 6

67 θ J, : x (θ R + J, +1) ν(dx) R + x θ J, ν(dx) = K, (5.19) λ where K is any consan. Noe again, ha he choice for hese parameers is no unique. In he nex secion we shall apply hese formulas (5.18), (5.19) o he exponenial disribuion of jumps. 5.5 Currency opion pricing for exponenial processes Because of he resricion Z s > we can no consider a double-exponenial disribuion of jumps (see [25], [28]) in log Z s dn s. Le us consider exponenial disribuion insead. I is defined by he following formula of densiy funcion: ν(x) = θe θx x. (5.11) The mean value of his disribuion is: mean(θ) = 1 θ. (5.111) The variance of his disribuion is: var(θ) = 1 θ 2. (5.112) The exponenial disribuion like he double-exponenial disribuion has also memorylessness propery. Le us derive he maringale condiion and formulas for he regime-swiching Esscher ransform parameers in case of jumps driven by he exponenial disribuion. Using he maringale condiion for discouned spo FX rae (5.1) we obain: [ ] Γ(θ r f r d + µ + θσ c 2 J + λ + 2) Γ(θJ + 1) =, (5.113) θ θj +1 θ θj where we have such a resricion (and in he sequel): θ J > 1. 61

68 Using (5.89), (5.9) he random Esscher ransform inensiy λ θ,j of he Poisson Process and he main percenage jump size k θ,j are respecively given by λ θ,j Γ(θ J + 1) = λ, (5.114) θ θj k θ,j = θj (5.115) θ Using (5.19) we have he following formula for he families of regime swiching Esscher ransform parameers saisfying maringale condiion (5.113): [ Γ(θ J, ] θ J, + 2) : Γ(θJ, + 1) = K. (5.116) θ θj, +1 θ θj, λ Le us simplify (5.116): θ J, + 1) : Γ(θJ, θ θj, The formula for θ c, in his case is he same as in (5.18). ( θ J, ) = K. (5.117) θ λ Wih respec o o such values of he regime swiching Esscher ransform parameers we have from (5.114), (5.115), (5.117): k J, = K /λ J,. (5.118) When we proceed o a new risk-neural measure Q we have he new θ in (5.11). Using (5.16) we obain: θ = θ θ J + 1. (5.119) From (5.119) we arrive a an ineresing conclusion: θ depends on ime. So, now he disribuion of jumps changes depending on ime (i was no he case before for he log double-exponenial disribuion, where θ was acually a consan, see [8]). So, he compound Poisson Process depends no only on a number of jumps, bu on momens of ime when hey arrive in his case. The same saemen is rue for he mean jump size in (5.115). Bu he pricing formulas (5.41)-(5.49) are applicable o his case as well. 62

69 Chaper 6 Numerical Resuls In he numerical simulaions, we assume ha he hidden Markov chain has hree saes: up, down, side-way, and corresponding probabiliy marix is calculaed using real Forex daa for a hireen-year period: from January 3, 2 o November 213. To calculae all probabiliies we use Malab scrip: Probab_marix_calc1(candles_back_up,candles_back_down, dela_back_up, dela_back_down,candles_up,candles_down, dela_up, dela_down ) (see Appendix for deails). 6.1 Log-double exponenial disribuion In he Figures we shall provide numerical simulaions for he case when he ampliude of jumps is described by a log-double exponenial disribuion. These hree graphs show a dependence of he European-call opion price agains S/K, where S is he iniial spo FX rae, K is he srike FX rae for a differen mauriy ime T in years:.5, 1, 1.2. We use he following funcion in Malab: Draw( S_,T,approx_num,seps_num, ea_1,ea_2,p,mean_normal,sigma_normal) o draw hese graphs 1. The argumens of his funcion are: S is he saring spo FX rae o define firs poin in S/K raio, T is he mauriy ime, approx num describes he number of aemps o calculae he mean for he inegral in he European call opion pricing formula (see Secion 2, (5.49)), seps num denoes he number of ime subinervals 1 Malab scrips for all plos are available upon reques 63

70 o calculae he inegral in (5.49); ea 1, ea 2, p are θ 1, θ 2, p parameers in he logdouble-exponenial disribuion (see Secion 3, (5.51)). mean normal, sigma normal are he mean and deviaion for he log-normal disribuion (see Secion 4). In hese hree graphs: θ 1 = 1, θ 2 = 1, p =.5; mean normal =, sigma normal =.1. Blue line denoes he log-double exponenial, green line denoes he log-normal, red-line denoes he plo wihou jumps. The S/K raio ranges from.8 o 1.25 wih a sep.5; he opion price ranges from o 1 wih a sep.1. The number of ime inervals: num =1. From hese hree plos we conclude ha i is imporan o incorporae jump risk ino he spo FX rae models. Described by Black-Scholes equaions wihou jumps, red line on a plo is significanly below boh blue and green lines which sand for he log-double exponenial and he log-normal disribuions of jumps, respecively. All hree plos have he same mean value and approximaely equal deviaions for boh ypes of jumps: log-normal and log-double exponenial. We invesigae he case when i does no hold (see Figures ). As we can see, he log-double exponenial curve moves up in comparison wih he lognormal and wihou jumps opion prices. If we fix he value of he θ 2 parameer in he log-double exponenial disribuion wih S/K = 1 he corresponding plo is given in Figure 6.7. Figure 6.8 represens a plo of he dependence of he European-call opion price agains values of he parameers θ 1, θ 2 in a log-double exponenial disribuion, again S/K = 1. 64

71 Figure 6.1: S = 1, T =.5, θ 1 = 1, θ 2 = 1, p =.5, mean normal =, sigma normal =.1 Figure 6.2: S = 1, T = 1., θ 1 = 1, θ 2 = 1, p =.5, mean normal =, sigma normal =.1 65

72 Figure 6.3: S = 1, T = 1.2, θ 1 = 1, θ 2 = 1, p =.5, mean normal =, sigma normal =.1 Figure 6.4: S = 1, T =.5, θ 1 = 5, θ 2 = 1, p =.5, mean normal =, sigma normal =.1 66

73 Figure 6.5: S = 1, T = 1., θ 1 = 5, θ 2 = 1, p =.5, mean normal =, sigma normal =.1 Figure 6.6: S = 1, T = 1.2, θ 1 = 5, θ 2 = 1, p =.5, mean normal =, sigma normal =.1 67

74 Figure 6.7: S = 1, T =.5, θ 2 = 1, p =.5 Figure 6.8: Opion price of European Call: S = 1, T =.5, p =.5 68