Pricing American currency options in a jump diffusion model

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1 Pricing American currency opions in a jump diffusion model Marc Chesney and M. Jeanblanc July 1, 23 This paper has benefied from he helpful commens of N. Bellamy, J. Beroin, P.Collin Dufresne, R. Ellio, L. Gauhier, M. Yor and X. Zhang. We hank Michael Suchanecki for accurae remarks on a firs version. Absrac In his aricle he problem of he American opion valuaion in a jump diffusion seing is ackeld. The perpeual case is firs considered. Wihou possible disconinuiies (i.e. wih negaive jumps in he call case), known resuls concerning he currency opion value as well he exercise boundary are obained wih a maringale approach. Wih possible disconinuiies of he underlying process a he exercise boundary (i.e. wih posiive jumps in he call case) original resuls are derived by relying on firs passage ime and overshoo associaed wih a Lévy process. For finie life American currency calls, he formula given by Baes (1991) or Zhang (1995), is rederived in he conex of a negaive jump size. I is basically an exension of he Barone-Adesi and Whaley (1987) approach. This formula is esed. I is shown ha Baes (1996) model generaes good resuls when he process is coninuous a he exercise boundary. However, wih possible disconinuiies resuls generaed by Baes (1996) model are less accurae. Keywords : American opions, perpeual opions, exercise boundary, incomplee markes, jump diffusion model, Laplace ransform, sopping imes, Lévy exponen, overshoo. 1 Inroducion Several aricles have already focused on he valuaion of European opions when he underlying value follows a jump diffusion process. Meron (1976) was he firs o obain a closed form soluion. More recenly Duffie e al. (2) also considered his problem in a larger seing. The problem of he American opion valuaion, specially useful for real opions, is more complex. I was ackled by Sco (1997) and Baes (1996) in a more general conex (jumps, sochasic volailiy,...). The former auhor obained an accurae European opion value by using he Fourier ransform, bu he early exercise premium was obained by a sraighforward applicaion of he Barone-Adesi and Whaley (1987) formula, Deparemen finance e Economie, Groupe HEC, 7835 Jouy en Josas France, Equipe d analyse e probabiliés, Universié d Evry Val d Essonne, Rue du Père Jarlan 9125 Evry Cedex, France. 1

2 in which he possible jumps are no aken ino accoun. The laer auhor also obained in his conex a European opion formula. In order o derive he early exercise premium, he relies on Baes (1991) in which possible jumps are now aken ino accouns. Basically he obained an exension of he Barone- Adesi and Whaley (1987) approach in a jump diffusion seing. Pham (1997) considered he American pu opion valuaion in a jump diffusion model (Meron s assumpions) and relaed his opimal sopping problem o a parabolic inegrodifferenial free boundary problem. By exending he Riesz decomposiion obained by Carr, Jarrow and Myneni (1992) in a diffusion model, he derived a decomposiion of he American pu price as he sum of is corresponding European price and he early exercise premium. Bu he laer erm ress on he idenificaion of he exercise boundary. In he same conex, Zhang (1994) relies on variaional inequaliies and shows how o use numerical mehods (finie difference mehods) in order o price he American pu. Zhang (1995) ses his pricing problem ino a free boundary problem, and by using he Mac Millan s (1983) approximaion, she obains a price for he perpeual pu,and an approximaion of he finie mauriy pu price. These resuls are obained only when jumps are posiive, i.e. wihou disconinuiies of he underlying process a he exercise boundary. Masroeni and Mazeu (1995 and 1996) obain an exension of Zhang (1994) resuls in a mulidimensional sae space. Mulinacci (1996), Mordecki (1999) and (22) and Gerber and Shiu (1998) (and (1999) only in he presence of jumps) also considered he American opion pricing problem. By relying on a maringale approach, we also exend in our sudy, he Barone-Adesi and Whaley (1987) approach and obain approximaions of he American premium and of he criical sock price (see Bunch and Johnson (2) wihou jumps) which fi he assumpions concerning he dynamics of he underlying value (basically Baes Model). We show ha his exension generaes good resuls when jumps are negaive (for currency call). However, when jumps are posiive, he qualiy of he resuls depends on he size of he jumps. Indeed he use of he Barone-Adesi and Whaley exension is more delicae, because of possible disconinuiies in he process a he exercise boundary. Therefore, a new approach is developed in his conex for perpeual opions. We firs consider he valuaion of a perpeual currency call opion, and obain he exercise boundary in wo cases: only negaive jumps and hen only posiive ones. In he firs case, analyical soluions for he price are based on he compuaion of he Laplace ransform of he firs passage ime of he process a he exercise boundary, which is obained using of he sopping heorem and of Lévy s exponen. In he second case, he valuaion problem is more difficul o ackle, he process being possibly disconinuous a he criical boundary. The overshoo a he exercise boundary is inroduced. Our original resuls are derived by relying on Beroin s book (1996) on Lévy s processes. Indeed, Laplace ransforms involving he firs passage ime and he overshoo a he exercise boundary are used. As in Zhang (1995) or Baes (1991), we hen show how an approximaion of he finie mauriy opion price can be obained by relying on he perpeual mauriy case. When he jump is posiive, he approximaion is less accurae because of possible disconinuiies a he exercise boundary. This aricle is organized as follows. In secion 2 he perpeual opion is considered. In secion 3, an approximaion of he finie mauriy American opion value is derived. In secion 4, he accuracy of he approximaion formula is esed, by comparing i o a numerical approach. 2 The valuaion of he perpeual opion Le S, σ 2, r, δ, λ, φ, T, K, C A (S, T ), P A (S, T ), C E (S, T ), P E (S, T ) denoe he underlying spo foreign exchange rae, he variance of he foreign currency rae of reurn, he domesic risk-free rae and he foreign ineres rae (or he rae of dividend for an index opion), he inensiy and he size of he jump, he mauriy, he srike price of he opion, he American and European call and pu prices, respecively. Le us assume perfec markes, consan r, δ, σ, λ, and φ, a jump diffusion process for he currency rae and a zero price of he marke jump risk, following Meron (1976). We assume ha he dynamics of he risk adjused process (S, ) are defined as follows: ds /S = (r δ λφ)d + σdw + φdn = (r δ)d + σdw + φdm (1) 2

3 where (W, ) is a Wiener process, (N, ) a Poisson process wih inensiy λ and M he compensaed maringale M = N λ. In order ha S remains non-negaive, he coefficien φ mus be chosen sricly greaer han 1. The jumps of S occur when he Poisson process jumps and S = S S = S (1 + φ) N. Hence, he jump is sricly posiive (S S ) for φ >, and he jump is sricly negaive if 1 < φ <. The soluion of (1) is S = xe X where X = (r δ λφ σ2 2 ) + σw + ln(1 + φ)n. The independence of he incremens of X implies ha (exp(kx g(k)), ) is a maringale, where g(k) is he Lévy exponen defined by g(k) = bk σ2 k(k 1) + λ((1 + φ) k 1 kφ) wih b = r δ. (See e.g. Beroin (1996) for deails) The funcion k g(k) is convex, goes o plus infiniy as k goes o infiniy, and g() =, herefore he equaion g(k) = u admis, for u wo soluions of opposie sign. We denoe by g 1 (u) he posiive soluion and by g 1,n (u) he negaive one. Noe ha for δ, he equaliy g(1) = r δ < r leads o g 1 (r) > 1. Jump diffusion models induce incompleeness of he marke. Therefore here is an infiniy of viable prices for opions corresponding o an infiniy of equivalen maringale measures ha can be characerized by idenifying he jump risk marke price. By assuming ha he jump risk is non priced, a unique American pricing formula can be derived (see Meron in he conex of European Opions). When mauriy ends o infiniy, he exercise boundary converges o a ye unknown value. The call opion value is given by : C A (x) = C A (x, ) = sup E((S τ K)e rτ ) (2) τ where τ runs over sopping imes, or, w.l.g. over hiing imes wih L S = x. 2.1 Negaive jumps T L = inf { : S L} Firs le us assume ha he jumps are negaive, i.e., 1 < φ. The exchange rae process is herefore coninuous a he exercise boundary, i.e., S TL = L and [ C A (x) = max (L K)E(e rt L ) ]. L x In appendix A, he Laplace ransform of T L is derived : for L x ( E(e rt L x ρ ) = L) where ρ = g 1 (r) is greaer han 1. Hence: C A (x) = max L x [ ( x ρ ] (L K) L) We can hus derive he value of he exercise boundary as he value where he supremum in he righ hand side of he las equaion is obained : L c = (3) ρ ρ 1 K (4) The opion value is herefore, for L c x ( ) ρ x C A (x) = (L c K) (5) L c 3

4 and x K for x < L c. Wihou jumps (φ = ), we obain he known formula for he price of a perpeual call in a Black and Scholes framework.. Indeed in his case, g is a polynomial of degree 2, and wih µ = ( r δ σ 2 /2 ) /σ. ρ = µ + µ 2 + 2r σ (6) 2.2 Posiive jumps Equaion (2) can be rewrien as follows in erms of he process X : where C A (x) = sup E(e rt (l) (xe X T (l) K)). l T (l) = inf{, X ln(l/x) = l} = T L. The jump size being now posiive, he process S could be disconinuous a he exercise boundary and herefore S TL (resp. X T (l) ) is no longer equal o L (resp. l) wih probabiliy 1. The overshoo κ is defined in erms of X by: X T (l) = l + κ(l). Le us inroduce a funcion f as follows: for L x i.e.: f(x, L) = E(e rt L (S TL K)) = E(e rt (l) (xe X T (l) K)) = E(e rt (l) (Le κ(l) K)) f(x, u) = E(e rt (ln(u/x)) (xe ln(u/x)+κ(ln(u/x)) K)) By definiion of he perpeual exercise boundary L c : hence: Le us define: = ue(e rt (ln(u/x))+κ(ln(u/x)) ) KE(e rt (ln(u/x)) ). (7) Φ(q, u) = f(x, L c ) = sup f(x, L) L x f u (x, L c) =, x < L c. (8) u 1 x e q ln( u x ) f(x, u)dx. (9) By he change of variables y = ln u x, we obain Φ(q, u) = + e qy f(ue y, u)dy i.e., by relying on equaion (7): Φ(q, u) = uα(q, r) Kβ(q, r) wih: α(q, r) = β(q, r) = + + e qy E(e rt (y)+κ(y) )dy e qy E(e rt (y) )dy. Therefore: Φ (q, u) = α(q, r). u Furhermore, equaions (8) and (9) lead o: Φ u (q, L c) = f(l c, L c ) q Φ(q, L c ) L c L c 4

5 hence: α(q, r) = L c K L c q L c (L c α(q, r) Kβ(q, r)). Now he funcions α and β are known in erms of an auxiliary funcion h (see appendix A) : α(q, r) = hence from an easy compuaion we obain : The funcion h is given by : h(r, q) h(r, 1) h(r, q) h(r, ), β(q, r) = (q + 1)h(r, q) qh(r, q) L c = h(u, k) = (1) h(r, ) h(r, 1) K (11) u ĝ(k) ĝ 1 (u) k where ĝ(k) = g( k). Now by relying on equaions (11) and (12), he exercise boundary is given by: L c = r ĝ() ĝ 1 (r) ĝ 1 (r) + 1 r ĝ( 1) K Using ha ĝ() =, ĝ( 1) = g(1) = r δ, and ĝ 1 (r) = g 1,n (r), we obain L c = r δ Proposiion 1 The exercise boundary for a perpeual call is (12) g 1,n (r) 1 g 1,n K (13) (r) L c = r δ g 1,n (r) 1 g 1,n K (r) where g 1,n (r) is he negaive roo of g(k) = r I is sraighforward o check ha wihou jump (φ = ) boh formula (4) and (13) coincide. Indeed, in his case, le us check ha r g 1,n (r) 1 δ g 1,n (r) = g 1 (r) g 1 (r) 1 (14) where g is he Lévy exponen in he case φ =, so ha g 1 (r) is he posiive roo of bk+ 1 2 σ2 k(k 1) = r and g 1,n (r) is he negaive roo. Usual relaions beween he sum, he produc of roos and he coefficiens lead o he resul. The case of a pu opion can be solved using he symmerical relaionship beween he American call and pu boundaries (see appendix E, or Mordecki (22)): L p (K, r, δ, φ, λ)l c (K, δ, r, φ, λ(1 + φ)) = K2 1 + φ From (13), he exercise boundary L p of he perpeual pu in a jump diffusion seing wih consan negaive jumps can be obained: L p = K 2 L c (K, δ, r, φ 1+φ, λ(1 + φ)) = K r δ where he funcion γ is he Lévy exponen of he process Y : γ 1,n (δ) γ 1,n (δ) 1 (15) Y = (δ r ˆλ ˆφ σ2 2 ) + σw + ln(1 + ˆφ)N. 5

6 Here N is a Poisson process wih inensiy ˆλ = λ(1 + φ) and ˆφ = φ/(1 + φ). Hence, he Lévy exponen of Y is γ(k) = (δ r)k σ2 k(k 1) + λ((1 + φ) 1 k 1 + φ(k 1)) = g(1 k) r + δ The perpeual exercise boundary for he pu can also be obained by relying on he procedure used for he call (See Appendix B for deails): L p = r δ g 1 (r) + 1 g 1 K (16) (r) Boh formulae are he same if which reduces o γ 1,n (δ) γ 1,n (δ) 1 = g 1 (r) + 1 g 1 (r) γ 1,n (δ) + g 1 (r) = 1 his las equaliy is now obvious from γ(k) = g(1 k) r + δ. Now ha he perpeual exercise boundary for call and pus are known, opion prices can be derived. 3 American and European In a jump-diffusion seing, a decomposiion of he American opion price ino he European price and he American premium can also be obained. We denoe by L c (T ) he opimal boundary We do no know how o compue his boundary, however, we shall ge he price of he perpeual call. Proposiion 2 Le us assume ha in he risk neural economy he dynamics of he currency price are given by equaion (1) wih consan coefficiens. The price of he American Currency Call saisfies he following decomposiion (where θ sands for he ime o mauriy T and x for he curren value of he underlying, i.e. S ): wih: C A (x, θ) = C E (x, θ) + δx rk + θ n= e + θ n= e (r+λ)v (λv)n (δ+λ)v (λv)n N (d 1 (L c (θ v), n; v))dv N (d 2 (L c (θ v), n; v))dv (17) d 1 (z, n; v) = ln(x/z) + (r δ λφ + σ2 /2)v + n ln(1 + φ) σ v d 2 (z, n; v) = d 1 (z, n; v) σ v and he perpeual American call opion is herefore given by: C A (x) = δx where L c is given by equaion (13). + + e n= + + rk n= (δ+λ)v (λv)n e (r+λ)v (λv)n N (d 1 (L c, n; v))dv N (d 2 (L c, n; v))dv (18) 6

7 Proof: Le and T be fixed and apply Iô s lemma o he funcion: (u, x) e r(u ) C A (x, T u) on he inerval [, T ]: e r(t ) C A (S T, ) = C A (S, T ) + T +σ + T where he differenial generaor L is defined by: T e r(u ) L(C A )(S u, T u)du e r(u ) S u C A x (S u, T u)dw u e r(u ) [C A ((1 + φ)s u, T u) C A (S u, T u)] dn u (19) L(f)(x, ) = σ2 2 x2 2 f f (x, ) + (r δ λφ)x f (x, ) + (x, ) rf(x, ) x2 x In he coninuaion region he American call price saisfies he P.D.E. given in appendix C and herefore L(C A )(x, T u) is equal o λ(c A ((1 + φ)x, T u) C A (x, T u)). In he sopping region he American call is equal o is inrinsic value, hence: E( L(C A )(x, T u) + λ(c A ((1 + φ)x, T u) C A (x, T u))1 x<lc(t u) = +(rk (δ + λφ)x)1 x Lc(T u) The second inegral in he righ hand side of equaion (19) is a maringale; moreover using ha T h u dn u F ) = E( T h u λdu F ) for any predicable process h, aking condiional expecaion wih respec o F of boh members of (19), we ge afer some rivial simplificaions C A (S, T ) = C E (S, T ) hence (See deails in Appendix D): C A (S, T ) = C E (S, T ) + rk n= λ n n= T δλ n T T e r(u ) E((rK δs u )1 Su>L c(t u) F )du (u ) n e (δ+λ)(u ) N (d 1 (L c (T u), n, u ))du (u ) n e (r+λ)(u ) N (d 2 (L c (T u), n, u ))du and equaion (17) is obained. When he mauriy T ends o infiniy, equaion (18) is obained. Along he same lines a decomposiion for he pu can be obained. Proposiion 3 Le us assume ha in he risk neural economy he dynamics of he currency price are given by equaion (1) wih consan coefficiens. The price of he American currency pu saisfies he following decomposiion: P A (x, θ) = P E (x, θ) + rk δx + θ n= e + θ n= e (δ+λ)v (λv)n (r+λ)v (λv)n N ( d 2 (L p (θ v), n; v))dv N ( d 1 (L p (θ v), n; v))dv (2) wih d 1, d 2 are defined in he previous proposiion and he perpeual American pu opion is herefore given by: P A (x) = rk δx + + e n= + + n= (r+λ)u (λu)n e (δ+λ)u (λu)n 7 N ( d 2 (L p, n, u))du N ( d 1 (L p, n, u))du

8 where L p is given by equaion (15). 4 An approximaion of he opion value Le us rely on he Barone-Adesi and Whaley (1987) approach, and on Baes (1991) aricle. Le us assume ha he jump size is negaive. If he American and European opion values saisfied he same linear P.D.E. (in he coninuaion region), heir difference C, he American premium, mus also saisfy his P.D.E. in he same region. Le us wrie : where: C(S, T ) = yh(s, y) y = 1 e rt and where h is a wo argumen funcion ha has o be deermined. In he coninuaion region h saisfies he following p.d.e. (see also Meron (1976) and appendix C where Io s lemma is applied o he discouned American Call) obained by a change of variables: σ 2 2 x2 2 h + (r δ)x h x2 x rh y h (1 y)r y λ( h λx h((1 + φ)x, y) + h(x, y)) = x Like he auhors, le us now assume ha he erm wih he derivaive of h wih respec o y is negligeable. Wheher or no i is a good approximaion is an empirical issue ha will be considered in he following secion. The following equaion has o be solved : σ 2 2 x2 2 h + (r δ)x h x2 x rh y λ( h λx h((1 + φ)x, y) + h(x, y)) = x The perpeual opion value saisfies almos he same differenial equaion. The only difference resides in he fac ha y is equal o 1 in he perpeual case, and herefore we have r y insead of r in he hird erm of he lef hand-side. In secion I (see equaion 5)) we derived is soluion which is for a negaive jump: h = ηx ρ where η is sill unknown, and by relying on appendix A: ρ = g 1 ( r y ) Bu when S ends o he exercise boundary L c (T ), by coninuiy of he opion value, he following equaion is saisfied : L c (T ) K = C E (L c (T ), T ) + yηl c (T ) ρ (21) and by use of he smooh fi condiion he following equaion is obained: 1 = C E x (L c(t ), T ) + yηρl c (T ) ρ 1 (22) Wihin a jump-diffusion model, his condiion was derived by Zhang (1944) in he conex of variaional inequaliies and by Pham (1995) wih a free boundary formulaion. We hus have a sysem of wo equaions (21) and (22) and wo unknowns η and L c (T ) ha can be solved. L c (T ) is he implici soluion of: L c (T ) = K C E (L c (T ), T ) + (1 C E x (L c(t ), T )) L c(t ) ρ and he approximaion formula is he following : C A (S, T ) = C E (S, T ) + A(S /L c (T )) ρ (23) if: S < L c C A (S, T ) = S K 8

9 oherwise, wih: where (see Meron 1976): A = (1 C E x (L c(t ), T ))) L c(t ) ρ and: C E (S, T ) = + n= e λt (λt ) n n ln(1+φ) (δ+λφ e T )T C BS (S, T, r δ λφ + C BS (S, T, θ, σ) = S N(d 1 ) Ke θt N(d 2 ) n ln(1 + φ), σ) (24) T d 1 = ln(s /K) + (θ + σ 2 2 )T σt d 2 = d 1 σ T The laer erms are given by Black and Scholes (1973). An approximaion for he pu prices, in a conex where he jump size is posiive, can be obained by relying on equaion (??). A decomposiion of he American opion price ino wo componens,namely he European opion price and he American premium was herefore derived. Pham (1995) has exended he Riesz decomposiion obained by Carr- Jarrow- Mynemi (1992) or Jacka (1991) wihin a diffusion model. He obains an early exercise premium which ress on he idenificaion of he exercise boundary. In our paper, as in Zhang (1995), simulaneously approximaions of he premium and of he exercise boundary are derived. These resuls were obained by Baes (1991) in a more general conex in which 1 + φ is a log-normal random variable (for he pu). This means ha his resuls could be used even wih posiive jumps for an American call. However, as shown in appendix C, he differenial equaion which soluion is he American opion approximaion value, akes a specific form jus below he exercise boundary. Unforunaely here is no known soluion o his equaion. Posiive jumps generae possible disconinuiies in he process a he exercise boundary, and herefore he problem is more difficul o solve (see secion I). As shown in following secion, Baes approximaion gives beer resuls when he size of he jump is negaive or small. 5 Simulaions In his secion, formula (23) is esed. Firs we compare he resuls generaed by his formula o hose obained by use of a numerical mehod (explici mehod). Then, we compue wha is called he pseudo American call price (las column). I is he sum of he erm given in equaion (24) (Meron formula), which corresponds o he European price of he opion, and of he American premium given by Barone Adesi and Whaley (1987), wihou jumps (φ = ). We checked wheher good approximaions for he American call prices were obained. In able 1, i urns ou ha formula (23) generaes very accurae resuls. However, he posiive sign of he ineres rae differenial induces an early exercise premium which is negligeable. In his case, he European price gives already a very good approximaion of he American one. In able 2, he ineres rae differenial is negaive and formula (23) seems o generae prey accurae resuls. Furhermore hese resuls are usually beer han hose generaed by he pseudo American valuaion. In able 3 and 4 he jump is posiive, and is size is prey high (1%). We observe ha Baes exension of Barone-Adesi and Whaley formula is prey good when he opion is ou of he money. Oherwise, when he opion is in he money he resuls are no saisfying. Indeed, he probabiliy for he underlying o reach quickly he exercise boundary and o be disconinuous a his level is higher for in he money han for ou he money calls. For in he money calls, i is beer o use he pseudo American call price. When he size of he jump is smaller, (see able 5 and 6), he size of he biais ges also smaller, and he qualiy of he resuls improve. By comparing able 2 and 4, we observe ha ou of he money call opions have higher prices wih posiive jumps han wih negaive ones, he reverse being rue for in he money calls. Posiive 9

10 jumps, even if hey have a negaive effec on he risk adjused drif have a srong posiive effec on he probabiliy of exercise for ou he money calls. For hese opions he posiive jump effec is more imporan han he negaive drif effec. For in he money calls, he probabiliy of exercise is already high (wih a negaive jump) and herefore a posiive jump doesn have a srong posiive effec on he probabiliy of exercise. However, due o possible disconinuiies in he process a he exercise boundary, wih posiive jumps, exercise boundaries are smaller wih posiive jumps han wih negaive ones. This implies ha in he money call values are higher wih negaive jumps han wih posiive ones. 6 Conclusion In his paper, original resuls concerning he pricing of perpeual American currency opions in a jump diffusion framework are obained. I has been shown ha he sign of he jump size is a relevan parameer. Wihou disconinuouies a he exercise boundary, known resuls are obained. Wih possible disconinuouies, an overshoo is inroduced and new resuls are derived. For finie life American currency calls, he formula given by Baes (1991) or Zhang (1995), is rederived in he conex of a negaive size jump. I is basically an exension of he Barone-Adesi and Whaley approach (1987). This formula is esed and gives good resuls. However, if he exchange rae can be disconinuous a he exercise boundary (posiive jumps for he underlying value in he call case), he pricing problem, is more difficul o ackle, and one should be very cauious in applying a Barone-Adesi exension. Appendix A Pecherskii and Rogozin resul Le us define T (l) he firs passage ime of he process X a l = ln(l/s ). By applying he opional sopping heorem and by relying on he lef coninuiy of he process a he sopping ime T (l), we obain he following formula : E(exp( g(k)t (l))) = exp( kl) By invering he Levy exponen we obain he Laplace ransform : E(exp( ut (l))) = exp( lg 1 (u)). We recall a resul of Pecherskii and Rogozin which can be found in Beroin : for every riple of posiive numbers (α, β, q), e ql E(e αt (l) βκ(l) h(α, q) h(α, β) )dx = (25) (q β)h(α, q) where ( ) h(α, β) = exp d 1 (e e α βx )P (X dx) (26) Le ĝ he Lévy exponen of he dual process of X, i.e. ˆX = X. h(b, k) = b ĝ(k) ĝ 1 (b) k From he definiion, he Laplace exponen of he dual process is ĝ(k) = g( k). Appendix B For a pu, we can inroduce, he funcion ψ in place of he funcion f and he funcion Ψ in place of Φ ψ(x, u) = E(Ke rt (ln(u/x) xe rt (ln(u/x)+κ(l) ) dx Ψ(q, u) = u x eq ln(u/x) ψ(x, u) ( ) = dye qy KE(e r ˆT (y) ) ue(e r ˆT (y) ˆκ(y) ) 1 (27)

11 where he ha refers o ˆX = X, so ha ˆT (y) = T ( y), ˆκ(y) = κ( y). Then, he same kind of compuaion leads o Appendix C L p = r δ K ĝ 1,n (r) 1 ĝ 1,n (r) = r δ K g 1 (r) + 1 g 1 (r) By applying generalized Io s lemma o he discouned American call price, on he inerval [, T ], he following equaion is obained : e r C A (S, T ) = C A (S, T ) + T e ru C A s (S u, T u)du T r e ru C A (S u, T u)du. + T e ru C A x (S u, T u)s u (r δ λφ)du + T e ru C A x (S u, T u)s u σdw u + σ2 2 T e ru 2 C A x (S 2 u, T u)sudu 2 + T e ru (C A ((1 + φ)s u, T u) C A (S u, T u))dn u In he risk adjused economy, he discouned American call price is a maringale in he coninuaion region. ( v exp( ru) C A x (S u, T u)s u σdw u, v T ) is also a maringale. Therefore, he drif erm is equal o zero. Hence, in he coninuaion region, he American call value saisfies he following differenial equaion: σ 2 2 x2 2 C A x 2 (x, T u) + (r δ λφ)x C A x (x, T u) rc A(x, T u) + C A (x, T u) s + λ(c A ((1 + φ)x, T u) C A (x, T u)) = Lc(T ) Now, if he jump is posiive, and if S belongs o he inerval :[ 1+φ, L c(t )], he value of he American call saisfies he following differenial equaion : σ 2 2 x2 2 C A x 2 (x, T u) + (r δ λφ)x C A x (x, T u) rc A(x, T u) + C A (x, T u) s + λ((1 + φ)x K C A (x, T u)) = because in his case he value of he American opion afer he jump is equal o he inrinsic value. Appendix D Le us compue E(1 Su>L c(t u) F ), he compuaion shall be he same for E(S u 1 Su>L c(t u) F ). From S u = S exp((r δ λφ σ 2 /2)(u ) + σ(w u W ) + (N u N ) ln(1 + φ)), = S exp(ν(u ) + σ(w u W ) + (N u N ) ln(1 + φ)) = S Z where Z is independen of S, we obain E(1 Su>L c(t u) F ) = Υ(S ) where Υ(x) = Q (x exp(ν(u ) + σ(w u W + (N u N ) ln(1 + φ)) > L c (T u)) = Q (ν(u ) + σw u + N u ln(1 + φ))) > ln(l c (T u)) ln x) = Q(N u = n)q (ν(u ) + σw u + n ln(1 + φ) > ln(l c (T u)) ln x). n= Then, he resul follows. Appendix E The pu-call symmery formulae are well known in he case of coninuous processes (See e.g. Deemple (21)). Mordecki (21) esablishes, from he Wiener-Hopf decomposiion a general symmery relaionship for Lévy processes. Here, we presen a simple proof in our case. The price of he currency is S = S e (r δ) e σw 1 2 σ2 e λφ+n ln(1+φ) = S e (r δ) Z 11

12 We can wrie ( E(e r (K S ) + ) = E e δ Z ( KS ) ( S ) + = S Ẽ e δ ( KS ) S ) + S and, under Q where d Q F = Z dq F he process S = 1/S follows d S = S ((δ r)d σd W φ 1 + φ d M ) where M = N λ(1 + φ) is a Q-maringale (hence, N is a Q-Poisson process wih inensiy λ(1 + φ)). Hence P (r, δ, x, K; σ, φ, λ) = C(δ, r, K, x; σ, φ, λ(1 + φ)). 1 + φ The same mehod esablishes ha for T L = inf{ : S L} and T L = inf{ : S KS L } ( E(e rt L (K S TL ) + ) = E e δ T L (S S TL ) +). Hence L c (K, r, δ; φ, λ) and L p (S, δ, r; φ, λ(1 + φ)) (where he firs argumen is he srike) saisfy 1 + φ L p = KS L c References and he symmery formula follows. Barone-Adesi, G. and R.E. Whaley, Efficien Analyic Approximaion of American Opion Values, The Journal of Finance, Vol. XLII, n 2, june 1987 Baes, David S., The Crash of 1987 : Wha is expeced. The evidence from opions markes, Journal of Finance Vol n 3, July 1991, pp Baes, David S., Jumps and Sochasic Volailiy : Exchange Rae Process Implici in Deuschmark Opions, The Review of Financial Sudies Vol n 9, pp 69-17, 1996 Beroin, J.: Lévy Processes, Cambridge Universiy Press, 1996 Bunch, D. and H. Johnson: The American Pu Opion and is Criical Sock Price, The Journal of Finance,vol 55, n 5,oc.2. Carr, P., R. Jarrow and R. Myneni: Alernaive Characerizaions of American Pu Opions, Mahemaical Finance 2, 87-15, Deemple J.: American opions symmery properies, in Opion pricing, Ineres raes and risk managemen,jouini, E. and Cvianić, J. and Musiela, M. ediors, Cambridge Universiy Press, 67-14, 21. Duffie, D., J. Pan and K. Singleon: Transform Analysis and Opion Pricing for Affine Jump-Diffusions, Economerica, 68, , 2. Gerber, H.U., and E.S. Shiu: Pricing Perpeual Opions for Jump Processes, Norh American Acuarial Journal, Volume 2, n 3,11-112, Gerber, H.U., and E.S. Shiu: From ruin heory o pricing rese guaranees and perpeual Pu Opions, Insurance: Mahemaics and Economics, 24, 3-14, Gukhal, C.R.: Analyical valuaion of American opions on jump-diffusion processes, Mahemaical Finance, 11, , 21. Mac Millan, L.W.: Analyic approximaion for he American pu opion, Advances in Fuures ans Opion research, 1, ,1986. Masroeni L. and M. Mazeu: An inegro-differenial parabolic variaional inequaliy conneced wih he problem of he American opion pricing, Zeischrif fur Analysis und ihre Anwendungen, 1995 Masroeni, L. and M. Mazeu: Sabiliy for he inegro-differenial variaional inequaliies of he American opion pricing problem, Advances in Mahemaical Sciences and Applicaions,

13 Meron R.C.: Opion Pricing when Underlying Sock Reurns are Disconinuous, Journal of Financial Economics, 3, , Mordecki, E.: Opimal Sopping for a diffusion wih jumps, Finance and Sochasics, 3, , Mordecki, E.: Opimal Sopping and perpeual Opions for Levy Processes,Finance and Sochasics, 6, , 22. Mulinacci, S.: An approximaion of American opion prices in a jump-diffusion model. Sochasic Process. Appl. 62, 1-17, Pecherskii, E. and Rogozin, B.A., On join disribuions of random variables associaed wih flucuaions of a process wih independen incremens, Theory of probabiliy and Appl., 14, , Pham, H.: Opimal Sopping, Free Boundary and American Opion in a Jump Diffusion Model, Appl. Mah. Opim. 35 n 2, , 1997 Sco, L.O: Pricing Sock Opions in a Jump Diffusion Model wih Sochasic Volailiy and Ineres Raes: Applicaions of Fourier Inversion Mehods, Mahemaical Finance, 7, , 1997 Zhang, X.: Analyse numérique des opions américaines dans un modéle de diffusion des saus, Thèse de docora de l école naionale des Pons e Chaussées, Paris, 1994 Zhang, X.: Formules quasi-explicies pour les opions américaines dans un modèle de diffusion avec saus, Mahemaics and Compuers Simulaion 13

14 TABLE I THEORICAL EUROPEAN AND AMERICAN CALL VALUES Parameers: λ = 1, φ = -.1, K = 1, r-δ =.4 CALL OPTION PRICES EUROPEAN AMERICAN PARAMETERS NO JUMP WITH JUMPS NO JUMP WITH JUMPS Garm. Fin. Meron Fin. Barone Fin. Baes Ap- Kohlh. diff. diff. Adesi diff. prox 1) S meh. meh. meh. r = σ = T = ) r = σ = T = ) r = σ = T = ) r = σ = T =

15 TABLE II THEORICAL EUROPEAN AND AMERICAN CALL VALUES Parameers: λ = 1, φ = -.1, K = 1, r-δ = -.4 CALL OPTION PRICES EUROPEAN AMERICAN PARAMETERS NO JUMP WITH JUMPS NO JUMP WITH JUMPS Garm. Fin. Meron Fin. Barone Fin. Baes Ap- Kohlh. diff. diff. Adesi diff. prox 1) S meh. meh. meh. r = σ = T = ) r = σ = T = ) r = σ = T = ) r = σ = T =

16 TABLE III THEORICAL EUROPEAN AND AMERICAN CALL VALUES Parameers: λ = 1, φ =.1, K = 1, r-δ =.4 CALL OPTION PRICES EUROPEAN AMERICAN PARAMETERS NO JUMP WITH JUMPS NO JUMP WITH JUMPS Garm. Fin. Meron Fin. Barone Fin. Baes Ap- Kohlh. diff. diff. Adesi diff. prox 1) S meh. meh. meh. r = σ = T = ) r = σ = T = ) r = σ = T = ) r = σ = T =

17 TABLE IV THEORICAL EUROPEAN AND AMERICAN CALL VALUES Parameers: λ = 1, φ =.1, K = 1, r-δ = -.4 CALL OPTION PRICES EUROPEAN AMERICAN PARAMETERS NO JUMP WITH JUMPS NO JUMP WITH JUMPS Garm. Fin. Meron Fin. Barone Fin. Baes Ap- Kohlh. diff. diff. Adesi diff. prox 1) S meh. meh. meh. r = σ = T = ) r = σ = T = ) r = σ = T = ) r = σ = T =

18 TABLE V THEORETICAL EUROPEAN AND AMERICAN CALL VALUES Parameers: λ = 1, φ =.6, K = 1, r-δ =-.4 CALL OPTION PRICES EUROPEAN AMERICAN PARAMETERS NO JUMP WITH JUMPS NO JUMP WITH JUMPS Garm. Fin. Meron Fin. Barone Fin. Baes Ap- Kohlh. diff. diff. Adesi diff. prox 1) S meh. meh. meh. r = σ = T = ) r = σ = T = ) r = σ = T = ) r = σ = T =

19 c TABLE VI THEORICAL EUROPEAN AND AMERICAN CALL VALUES Parameers: λ= 1, φ =.2, K = 1, r-δ = -.4 CALL OPTION PRICES EUROPEAN AMERICAN PARAMETERS NO JUMP WITH JUMPS NO JUMP WITH JUMPS Garm. Fin. Meron Fin. Barone Fin. Baes Ap- Kohlh. diff. diff. Adesi diff. prox 1) S meh. meh. meh. r = σ = T = ) r = σ = T = ) r = σ = T = ) r = σ = T =

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