New Pricing Framework: Options and Bonds

Transcription

1 arxiv: v [q-fin.pr] 14 Oc 014 New Pricing Framework: Opions and Bonds Nick Laskin TopQuark Inc. Torono, ON, M6P P Absrac A unified analyical pricing framework wih involvemen of he sho noise random process has been inroduced and elaboraed. Two exacly solvable new models have been developed. The firs model has been designed o value opions. I is assumed ha asse price sochasic dynamics follows a Geomeric Sho Noise moion. A new arbirage-free inegro-differenial opion pricing equaion has been found and solved. The pu-call pariy has been proved and he Greeks have been calculaed. Three addiional new Greeks associaed wih marke model parameers have been inroduced and evaluaed. I has been shown ha in diffusion approximaion he developed opion pricing model incorporaes he well-known Black- Scholes equaion and is soluion. The sochasic dynamic origin of he Black-Scholes volailiy has been uncovered. The new opion pricing model has been generalized based on asse price dynamics modeled by he superposiion of Geomeric Brownian moion and Geomeric Sho Noise. A generalized arbiragefree inegro-differenial opion pricing equaion has been obained and solved. Based on his soluion new generalized Greeks have been inroduced and calculaed. To model sochasic dynamics of a shor erm ineres rae, he second model has been inroduced and developed based on Langevin ype equaion wih sho noise. I has been found ha he model address: nlaskin@rock .com 1

2 provides affine erm srucure. A new bond pricing formula has been obained. I has been shown ha in diffusion approximaion he developed bond pricing formula goes ino he well-known Vasiček soluion. The sochasic dynamic origin of he long-erm mean and insananeous volailiy of he Vasiček model has been uncovered. A generalized bond pricing model has been inroduced and developed based on shor erm ineres rae sochasic dynamics modeled by superposiion of a sandard Wiener process and sho noise. Despie he non-gaussianiy of probabiliy disribuions involved, all newly elaboraed models have he same degree of analyical racabiliy as he Black Scholes model and he Vasiček model. This circumsance allows one o derive simple exac formulae o value opions and bonds. Key words: Financial derivaives fundamenals, Sho noise, Opion pricing equaion, Green funcion, Pu-call pariy, Greeks, Black- Scholes equaion, Shor erm ineres rae, Vasiček model, Affine erm srucure, Bond pricing formula. Conens 1 Inroducion 4 Asse price sochasic dynamics 6.1 Geomeric Sho Noise moion Characerisic funcional of sho noise Opion pricing equaion and is soluions A new arbirage-free inegro-differenial pricing equaion Exac soluion o he inegro-differenial equaion Call opion Pu opion Pu-Call pariy 16 5 Greeks Common Greeks New Greeks Gaussian jumps model Greek κ

3 5..3 Greek µ Greek ǫ Black-Scholes equaion Diffusion approximaion Opion pricing equaion The soluion o Black-Scholes equaion Funcion L 1 (l) in diffusion approximaion Funcion L (l) in diffusion approximaion Soluion o opion pricing equaion in diffusion approximaion Greeks in diffusion approximaion Greek vega Black-Scholes Greeks Generalized opion pricing framework Geomeric Brownian moion and Geomeric Sho Noise Generalized arbirage-free inegro-differenial opion pricing equaion Generalized Greeks Shor erm ineres rae dynamics Langevin equaion wih sho noise Bond price Affine erm srucure The erm srucure equaion Vasiček model Diffusion approximaion Long-erm mean and insananeous volailiy Mean and variance Generalized shor erm ineres rae model Vasiček model wih sho noise Bond price: generalized formula Generalized erm srucure equaion Conclusion 55 3

4 11 Appendix A: Green s funcion mehod o solve he Black- Scholes equaion 57 1 Appendix B: Greek Θ C in diffusion approximaion 60 1 Inroducion The aim of his paper is o inroduce and elaborae a new unified analyical framework o value opions and bonds. The opions pricing approach is based on asse price dynamics ha has been modelled by he sochasic differenial equaion wih involvemen of sho noise. I resuls in Geomeric Sho Noise moion of asse price. A new arbirage-free inegro-differenial opion pricing equaion has been developed and solved. New exac formulas o value European call and pu opions have been obained. The pu-call pariy has been proved. The Greeks have been calculaed based on he soluion of he opion pricing equaion. Three new Greeks associaed wih he marke model parameers have been inroduced and evaluaed. I has been shown ha he developed opion pricing framework incorporaes he well-known Black-Scholes equaion [1]. The Black- Scholes equaion and is soluions emerge from our inegro-differenial opion pricing equaion in he special case which we call diffusion approximaion. The Geomeric Sho Noise model in diffusion approximaion explains he sochasic dynamic origin of volailiy in he Black-Scholes model. The bonds pricing analyical approach is based on he Langevin ype sochasic differenial equaion wih sho noise o model a shor erm ineres rae dynamics. I resuls in non-gaussian random moion of shor erm ineres rae. A bond pricing formula has been obained and i has been shown ha he model provides affine erm srucure. The new bond pricing formula incorporaes he well-known Vasiček soluion []. The Vasiček soluion comes ou from our bond pricing formula in diffusion approximaion. The sochasic dynamic origin of he Vasiček long-erm mean and insananeous volailiy has been uncovered. The paper s main resuls are presened by Eqs.(1), (), (16), (), (43), (47), (10)-(14), (13), (145), (150), (157), (159), (178), (18) and (191). New formulae o evaluae he common Greeks for call and pu opions have been obained based on an exac soluion of he inegro-differenial opion pricing equaion. The formulae are lised in Table 1 and Table. 4

5 The paper is organized as follows. In Sec. Geomeric Sho Noise moion has been inroduced and applied o model asse price dynamics. A new arbirage-free inegro-differenial equaion o value opions is obained in Sec.3. I has been shown ha Green s funcion mehod is an effecive mahemaical ool o solve his equaion. The exac analyical soluions o his equaion have been found for European call and pu opions. The pu-call pariy has been proved in Sec.4. The Greeks, including hree newly inroduced Greeks, have been calculaed in Sec.5. Three new Greeks are sensiiviies associaed wih he marke parameers involved ino he definiion of he sho noise process. The Gaussian model for asse price jumps has been considered o find he formulae for hree new Greeks. The diffusion approximaion of he inegro-differenial opion pricing equaion has been defined and elaboraed in Sec.6. I has been shown ha he well-known Black-Scholes equaion and is volailiy come ou from he inegro-differenial pricing equaion in diffusion approximaion. I has been shown as well ha he soluion o he Black-Scholes equaion sraighforwardly follows from he exac soluion o he inegro-differenial pricing equaion. The well-known Black-Scholes Greeks for European call opions have been replicaed from he new common Greeks in diffusion approximaion. The Black-Scholes Greeks for European call opions are summarized in Table 3. The generalized opion pricing framework has been presened in Sec.7. The generalizaion comes from he idea o accommodae a superposiion of Geomeric Brownian moion and Geomeric Sho Noise in he equaion for asse price dynamics. The oucome of implemening his idea is a generalized pricing equaion. New formulae o value European call and pu opions have been obained as soluions o he generalized pricing equaion. The special limi cases of hose formulas have been developed and discussed. The generalized Greeks have been inroduced and calculaed. Table 4 displays new formulas for he generalized Greeks. In Sec.8 shor erm ineres rae has been modeled by he Langevin sochasic differenial equaion wih sho noise. A new bond pricing formula has been found, and i has been shown ha he new model provides affine erm srucure. The bond pricing formula is he soluion o he inegrodifferenial erm srucure equaion. The well-known Vasiček model for shor erm ineres rae wih is long-erm mean and insananeous volailiy comes 5

6 ou from our model in diffusion approximaion. I has been shown as well ha he Vasiček bond pricing formula comes ou from he new bond pricing formula in diffusion approximaion. I has o be emphasized ha our model is non-gaussian, while he Vasiček model is Gaussian, ha is, probabiliy disribuions of shor erm ineres rae are differen for our model and he Vasiček model. I is ineresing, ha despie his difference boh models possess exacly he same mean and variance. A generalized bond pricing model has been inroduced and developed in Sec.9 based on shor erm ineres rae sochasic dynamics modeled by superposiion of a sandard Wiener process and sho noise. A new bond pricing formula has been found and i has been shown ha he generalized model provides affine erm srucure. The generalized erm srucure equaion has been found and solved. I has o be emphasized ha despie he non-gaussianiy of probabiliy disribuions involved, all newly developed quaniaive models o value opions and bonds have he same degree of analyical racabiliy as he Black Scholes model [1] and he Vasiček model []. Analyical racabiliy allows one o obain new exac simple formulas o value opions and bonds. The paper s main resuls and findings have been summarized and discussed in he Conclusion. Appendix A develops Green s funcion mehod o solve he Black-Scholes equaion for European call opions. The well-known soluion has been obained sraighforwardly wihou preliminary ransformaions o conver he Black-Scholes equaion o he hea equaion. To our bes knowledge, he implemened Green s funcion approach has no ye been presened anywhere. Appendix B is a walkhrough o show ha in diffusion approximaion he equaion for he new Greek Θ C goes ino he well-known formula for he Black-Scholes Greek hea. Asse price sochasic dynamics.1 Geomeric Sho Noise moion I is supposed ha asse price S() follows he sochasic differenial equaion ds = SF()d, (1) 6

7 wih random force F() modeled by he sho noise process (see, Eq.(6) in [3]) F() = n η k ϕ( k ), () k=1 where random jumps η k of asse price are saisically independen and disribued wih probabiliy densiy funcion p(η), random ime poins k, which are arrival imes of asse price jumps, are uniformly disribued on ime inerval [,T], so ha heir oal number n obeys he Poisson law wih parameer λ, and deerminisic funcion ϕ() is he response funcion. I is supposed ha defined by Eq.() sho noise process F() describes he influence of differen flucuaing facors on asse price dynamics. A single sho noise pulse η k ϕ( k ) describes he influence of a piece of informaion which has become available a random momen k on he asse price a a laer ime. Ampliude η k responds o he magniude of he asse price pulseϕ( k ). Ampliudesη k arerandomsaisicallyindependen variables, subjec o he marke informaion available. We assume as well ha each pulse has he same funcional form or, in oher words, one general response funcion ϕ() can be used o describe he asse price dynamics. We call he sochasic dynamics inroduced by Eqs.(1) and () Geomeric Sho Noise moion.. Characerisic funcional of sho noise By definiion he characerisic funcional Φ[α(τ)] of random force F() is T Φ[α(τ)] = exp i dτα(τ)f(τ), (3) where α(τ) is an arbirary sufficienly smooh funcion and... sands for he average over all randomness involved ino he random force F(). The characerisic funcional conains all informaion abou saisical momens of random force F(). For example, he mean value of F() can be calculaed as funcional derivaive < F() >= 1 i δ δα() Φ[α(τ)] α(τ)=0, (4) 7

8 whilehecorrelaionfuncion< F( 1 )F( ) >isgivenbyhesecondorder funcional derivaive < F( 1 )F( ) >= 1 i δ δα( 1 )δα( ) Φ[α(τ)] α(τ)=0. (5) To evaluae he characerisic funcional Φ[α(τ)] we need o define probabilisic characerisics of each of hree sources of randomness involved in Eq.(). Therefore, assuming ha hese hree sources of randomness are independen of each oher, we have hree saisically independen averaging procedures [3]: 1. Averaging over uniformly disribued ime poins k on he inerval [,T],... T = n k=1 1 T T d k... (6). Averaging over random asse price jumps, which are saisically independen and disribued wih probabiliy densiy funcion p(η),... η = n k=1 3. Averaging over random number n of price jumps, dη k p(η k )... (7)... n = e λ(t ) (λ(t )) n..., (8) n! n=0 which is in fac he averaging wih he Poisson probabiliy densiy funcion, and λ is he rae of arrival of price jumps, i.e. he number of jumps per uni ime. Now we are in posiion o calculae Φ[α(τ)]. Firs, by performing seps #1 and # we obain T n Φ[α(τ)] = exp i dτα(τ) η k ϕ(τ k ) (9) k=1 k,η 8

9 = 1 T T d Then, le us do sep #3, dηp(η) exp iη T dτα(τ)ϕ(τ ) n. Φ[α(τ)] = 1 T T d dηp(η) exp iη T n dτα(τ)ϕ(τ ) n T = e λ(t ) (λ(t )) n 1 [ n! T n=0 d (10) T dηp(η) exp iη dτα(τ)ϕ(τ ) ]n T T = exp λ d dηp(η) exp{iη dτα(τ)ϕ(τ )} 1. Hence, we found he equaion for he characerisic funcional Φ[α(τ)] defined by Eq.(3) T Φ[α(τ)] = exp λ d dηp(η) exp{iη Furher, we chose, as an example, he response funcion T dτα(τ)ϕ(τ )} 1. (11) ϕ() = δ(), (1) where δ() is dela-funcion. In his case we have for he characerisic funcional Φ[α(τ)] T Φ[α(τ)] = exp λ dτ dηp(η)(e iηα(τ) 1). (13) 9

10 Now, having Eq.(13) and definiions (4) and (5) one can easily obain he mean < F() > < F() >= λ dηp(η)η, (14) and he correlaion funcion < F( 1 )F( ) > < F( 1 )F( ) >= λ dηp(η)η δ( 1 )+ < F( 1 ) >< F( ) >. (15) 3 Opion pricing equaion and is soluions 3.1 A new arbirage-free inegro-differenial pricing equaion Having he characerisic funcional Φ[α(τ)] given by Eq.(13) and assuming a fricionless and no-arbirage marke, a consan risk-free ineres rae r, and asse price dynamics governed by he Geomeric Sho Noise moion given by Eq.(1), we inroduce a new arbirage-free inegro-differenial opion pricing equaion C(x, ) +(r q) C(x,) x (16) +λ where { dηp(η) C(x+η,) C(x,) (e η 1) C(x,) } rc(x,) = 0, x x = ln S K, (17) and C(x,) is he value of a European call opion 1 on dividen-paying asse, S is asse price governed by Eq.(1), K is he srike price, r is he risk- 1 An opion which gives he owner he righ, bu no he obligaion, o buy an asse, a a specified price (srike price K), by a predeermined dae (mauriy ime T). 10

11 free ineres rae, q is coninuously paid dividend yield, which is a consan, p(η) is he probabiliy densiy funcion involved ino Eq.(). The erminal condiion (or payoff funcion) for a European call opion is C(S,T) = max(s K,0), (18) where T is an opion mauriy ime. If we ake ino accoun Eq.(17), hen we can wrie Eq.(18) as C(x,T) = Kmax(e x 1,0). (19) Thus, he new generalized opion pricing framework has been inroduced by Eqs.(16) and (19). To value a European pu opion P(x,) we have he same equaion as Eq.(16) while he erminal condiion is P(S,T) = max(k S,0). (0) Wih help of Eq.(17) he erminal condiion (0) for a European pu opion becomes P(x,T) = Kmax(1 e x,0). (1) If we go from C(x,) o C(S,), where x and S are relaed o each oher hrough Eq.(17), hen we can wrie an opion pricing equaion (16) in he form C(S, ) +(r q)s C(S,) S () +λ { dηp(η) C(Se η,) C(S,) (e η 1)S C(S,) } rc(s,) = 0, S wih he erminal condiion given eiher by Eq.(18) or by Eq.(0). When valuing an opion, i is common pracice o consider ime o expiry T insead of ime. Taking ino accoun ha An opion which gives he owner he righ, bu no he obligaion, o sell an asse, a a srike price K, on he mauriy dae T. 11

12 we can presen Eq.(16) in he form +λ (T ), (3) C(x,T ) C(x,T ) +(r q) (T ) x { } dηp(η) C(x+η,T ) C(x,T ) (e η C(x,T ) 1) x rc(x,t ) = 0, while he erminal condiion (19) for a call opion becomes (4) C(x,T ) =T = C(x,0) = Kmax(e x 1,0), (5) and he erminal condiion (1) for a pu opion becomes P(x,T ) =T = P(x,0) = Kmax(1 e x,0). (6) 3. Exac soluion o he inegro-differenial equaion 3..1 Call opion To solve Eq.(4) subjec o erminal condiion given by (5) we will use he Green s funcion mehod. By definiion, Green s funcion G(x x,t ) saisfies he inegrodifferenial equaion +λ G(x x,t ) (T ) +(r q) G(x x,t ) x dηp(η){g(x+η x,t ) G(x x,t ) (7) (e η 1) G(x x,t ) } rg(x x,t ) = 0, x 1

13 and he erminal condiion G(x x,t ) =T = G(x x,0) = δ(x x ), (8) where T is he ime o mauriy. Having Green s funcion G(x x,t ), we wrie he soluion o Eq.(4) wih condiion (5) in he form C(x,T ) = K dx G(x x,t )max(e x 1,0), (9) and he soluion o Eq.(4) wih condiion (6) in he form P(x,T ) = K dx G(x x,t )max(1 e x,0). (30) Green s funcion inroduced by Eqs.(7) and (8) can be found by he Fourier ransform mehod. Green s funcion G(x x,t ) reads G(x x,t ) = 1 dke ik(x x ) G(k,T ), (31) where G(k,T ) is he Fourier ransform of Green s funcion defined by G(k,T ) = In erms of G(k,T ), Eq.(7) akes he form dxe ikx G(x,T ). (3) +λ G(k,T ) (T ) = [ r +ik(r q) (33) dηp(η) { e ikη 1 ik(e η 1) } ]G(k,T ), while he erminal condiion (8) becomes G(k,T ) =T = G(k,0) = 1. (34) 13

14 The soluion o he problem defined by Eqs.(33) and (34) is G(k,T ) = e r(t ) (35) exp [ik(r q)+λ dηp(η) { e ikη 1 ik(e η 1) } ](T ), Then Eq.(31) gives us Green s funcion G(x x,t ) G(x x,t ) = e r(t ) dke ik(x x ) (36) exp [ik(r q)+λ dηp(η) { e ikη 1 ik(e η 1) } ](T ). Subsiuing Eq.(36) ino Eq.(9) yields for he value of a European call opion C(x,T ) C(x,T ) = Ke r(t ) exp [ik(r q)+λ dx dke ik(x x ) dηp(η) { e ikη 1 ik(e η 1) } ](T ) (37) max(e x 1,0). Furher, changing inegraion variable x o z z = x x +(r q λς)(t ), (38) and inroducing parameer l defined by l = x+(r q λς)(t ), (39) 14

15 yield r(t ) el C(x,T ) = Ke Ke r(t ) l dz l dz where ς and ξ(k) have been inroduced by ς = dke ikz z exp{λ(t )ξ(k)} (40) dke ikz exp{λ(t )ξ(k)}, dηp(η)(e η 1), (41) and ξ(k) = dηp(η)(e ikη 1), (4) wih p(η) being he probabiliy densiy funcion of he magniude of asse price jumps. I is easy o see ha Eq.(40) can be wrien in he form C(S,T ) = Se q(t ) L 1 (l) Ke r(t ) L (l), (43) where funcions L 1 (l) and L (l) are defined as L 1 (l) = e λς(t ) L (l) = 1 l l dz dz dke ikz z exp{λ(t )ξ(k)}, (44) dke ikz exp{λ(t )ξ(k)}, (45) wih ς and ξ(k) given by Eqs.(41) and (4). Thus, we found he new equaion (43) o value a European call opion when he sochasic dynamics of asse price is governed by Eq.(1). 15

16 3.. Pu opion Subsiuion of Eq.(36) ino Eq.(30) yields for he value of a European pu opion by P(x,T ) = Ke r(t ) exp [ik(r q)+λ This equaion can be presened as dx dke ik(x x ) dηp(η) { e ikη 1 ik(e η 1) } ](T ) (46) max(1 e x,0). P(S,T ) = Ke r(t ) L (l) Se q(t ) L 1 (l), (47) where l is given by Eq.(39) while funcions L 1 (l) and L (l) are inroduced L 1 (l) = e λς(t ) L (l) = 1 l l dz dz dke ikz z exp{λ(t )ξ(k)}, (48) dke ikz exp{λ(t )ξ(k))}, (49) wih ς and ξ(k) defined by Eqs.(41) and (4). Thus, we obained he new equaion (47) o value a European pu opion when he sochasic dynamics of asse price is governed by Eq.(1). 4 Pu-Call pariy The pu-call pariy is a fundamenal relaionship beween he values of European call and pu opions, boh wih he same srike price K and ime o expiry T. This relaionship is a manifesaion of he no-arbirage 16

17 principle. The pu-call pariy equaion is model independen and has he form C(S,T ) P(S,T ) = Se q(t ) Ke r(t ). (50) To prove his relaionship we use Eqs.(43) and (47) o obain C(S,T ) P(S,T ) = (51) = Se q(t ) (L 1 (l)+l 1 (l)) Ke r(t ) (L (l)+l (l)), From he definiions of funcions L given by Eqs.(44) and (45), and funcions L given by Eqs.(48) and (49) we have and L 1 (l)+l 1 (l) = 1, (5) L (l)+l (l) = 1. (53) Le s show, for example, ha equaion (53) holds. Indeed, wih help of Eqs.(45) and (49) we wrie for Eq.(53) = L (l)+l (l) = 1 ( dz dke ikz exp{λ(t )ξ(k)}) (54) dkδ(k)exp{λ(t )ξ(k)} = exp{λ(t )ξ(0)} = 1, here we used ξ(k = 0) = 0 as i follows immediaely from Eq.(4). A similar consideraion can be provided o prove Eq.(5). By subsiuing Eqs.(5) and (53) ino Eq.(51) we complee he proof of pu-call pariy equaion (50). 5 Greeks The purpose of his Secion is o calculae he Greeks based onequaion (43). 17

18 In opion pricing fundamenals and opion rading, Greek leers are used o define sensiiviies (Greeks) of he opion value in respec o a change in eiher underlying price (i.e., asse price) or parameers (i.e., risk-free rae, ime o mauriy, ec.). The Greeks can be considered as effecive ools o measure and manage he risk in an opion posiion. The mos common Greeks are he firs order derivaives: dela, rho, psi and hea as well as gamma, a second-order derivaive of he opion value over underlying price. To simplify calculaions of he Greeks le s noe ha he following equaions hold Se q(t ) L 1(l) l = Ke r(t ) L (l), (55) l Se q(t ) L 1(l) = Ke r(t ) L (l), (56) l l where funcions L 1 (l), L (l), L 1 (l), L (l) are given by Eqs.(44), (45), (48) and (49). 5.1 Common Greeks The Greek dela for a call opion C is given by C(S,T ) C = = e q(t ) L 1 (l). (57) S The Greek rho ρ C for a call opion is C(S,T ) ρ C = = (T )Ke r(t ) L (l). (58) r The Greek psi ψ C for a call opion is he firs parial derivaive of opion value C(S,T ) wih respec o he dividend rae q, ψ C = C(S,T ) q The Greek hea Θ C for a call opion is defined by = (T )Se q(t ) L 1 (l). (59) Thus, we have C(S,T ) Θ C = (T ) = C(S,T ). 18

19 Θ C = qse q(t ) L 1 (l) rke r(t ) L (l)+se q(t ) λςl 1 (l) λse q(t )e λς(t ) + λke r(t ) l l dz dze z dke ikz ξ(k)exp{λ(t )ξ(k)} (60) dke ikz ξ(k)exp{λ(t )ξ(k)}. Finally, he second order sensiiviy gamma Γ C for a call opion is Γ C = C(S,T ) = e q(t ) L 1(l) l S l S = e q(t ) L 1 (l). (61) S l Funcions L 1 (l) and L (l) in he formulas above are given by Eqs.(44) and (45). Table 1 summarizes he common Greeks for a call opion. Call Dela, C = C(S,T ) e q(t ) L S 1 (l) Gamma, Γ C = C (S,T ) e q(t ) L 1 (l) = Ke r(t ) L (l) S S l S l Rho, ρ C = C(S,T ) r Psi, ψ C = C(S,T ) q Thea, Θ C = C(S,T ) (T )Ke r(t ) L (l) (T )Se q(t ) L 1 (l) qse q(t ) L 1 (l) rke r(t ) L (l) +λςse q(t ) L 1 (l) l dze z dke ikz ξ(k)e λ(t )ξ(k) λse q(t ) e λς(t ) + λke r(t ) l dz dke ikz ξ(k)e λ(t )ξ(k) Table 1. Common Greeks (Call opion) Common Greeks forapu opioncanbeeasily foundby using hepu-call pariy equaion (50). The Greek dela for a pu opion P is 19

20 P(S,T ) P = = e q(t ) L 1 (l) = C e q(t ), (6) S where C is he Greek dela for a call opion given by Eq.(57). The Greek rho ρ P for a pu opion is ρ P = P(S,T ) r = (T )Ke r(t ) L (l) = ρ C (T )Ke r(t ), (63) where ρ C is he Greek rho for a call opion given by Eq.(58). The Greek psi ψ P for a pu opion is he firs parial derivaive of opion value P(S,T ) wih respec o he dividend rae q, ψ P = P(S,T ) q = (T )Se q(t ) L 1 (l) = ψ C +(T )Se q(t ), (64) where ψ C is he Greek psi for a call opion given by Eq.(59). The Greek hea Θ P for a pu opion is defined by P(S,T ) Θ P = (T ) = P(S,T ). Thus, from he pu-call pariy equaion (50) we have Θ P = Θ C qse q(t ) +rke r(t ), (65) where Θ C is he Greek hea for a call opion given by Eq.(60). Finally, he second order sensiiviy gamma Γ P for a pu opion is Γ P = P(S,T ) S = e q(t ) S L 1 (l) l = e q(t ) S L 1 (l) l where Γ C is he Greek gamma for a call opion given by Eq.(59). Hence, we conclude ha = Γ C, (66) Γ P = Γ C. (67) Funcions L 1 (l), L 1 (l), and L (l) in he formulas above are given by Eqs.(44), (48) and (49). 0

21 Table summarizes he common Greeks for a pu opion. Pu Dela, P = P(S,T ) e q(t ) L S 1 (l) = C e q(t ) Gamma, Γ P = P (S,T ) S Rho, ρ P = P(S,T ) r Psi, ψ P = P(S,T ) q Thea, Θ P = P(S,T ) Table. Common Greeks (Pu opion) 5. New Greeks 5..1 Gaussian jumps model L 1 (l) l L 1 (l) l - e q(t ) = e q(t ) = Γ S S C (T )Ke r(t ) L (l) = ρ C (T )Ke r(t ) (T )Se q(t ) L 1 (l) = ψ C +(T )Se q(t ) Θ C qse q(t ) +rke r(t ) The new Greeks are opion sensiiviies associaed wih he marke parameers involved in he definiion of sho noise process F()(see, Eq.()). Besides parameer λ, which is he rae of asse price jumps arrival, here can be oher marke parameers associaed wih probabiliy densiy funcion p(η) involved ino he definiion of sho noise F(). As soon as we specify he probabiliy densiy funcion p(η), we will ge he marke parameers associaed wih i. Having hese parameers, we can inroduce a few new Greeks. A his poin we assume, as an example, ha he probabiliy densiy funcion of asse price jump magniudes is a normal disribuion p(η) = 1 (η ν) exp{ δ δ }, (68) where he marke parameers ν and δ are he mean and variance of asse price jump magniudes. The firs order derivaives of a call opion wih respec o parameers λ, ν and δ will bring hree new Greeks, which do no exis in he Black-Scholes opion pricing framework. I follows immediaely from pu-call pariy equaion (50) ha all hree new Greeks are he same for call and pu opions. Thus, we need o calculae hese Greeks for call opion only. 5.. Greek κ We inroduce he noaion κ C for he derivaive of call opion wih respec o λ, which is he rae of asse price jumps arrival. Hence, he definiion of 1

22 he Greek kappa κ C is C(S,T ) κ C =. (69) λ The Greek κ C is a new Greek ha does no exis in he Black-Scholes framework, because of he absence of marke parameer λ. To find he Greek kappa we differeniae Eq.(43) wih respec o λ. The resul is κ C T = ςse q(t ) L 1 (l) +Se q(t )e λς(t ) Ke r(t ) l l dz dz dke ikz z ξ(k)exp{λ(t )ξ(k)} (70) dke ikz ξ(k)exp{λ(t )ξ(k)}. Comparing Eqs.(60) and (70) le s obain he relaionship beween Θ C and κ C or κ C = T λ (qse q(t ) L 1 (l) rke r(t ) L (l) Θ C ), (71) Θ C = qse q(t ) L 1 (l) rke r(t ) L (l) λκ C T. (7) Thus, we discovered a new fundamenal relaionship beween common Greek Θ C and newly inroduced Greek κ C. I has already been menioned ha he kappa for a pu opion κ P is he same as kappa for a call opion κ C, κ P = κ C, (73) because of he pu-call pariy equaion (50). The fundamenal relaionship beween he common Greek Θ P and newly inroduced κ P has he form

23 Θ P = qse q(t ) L 1 (l)+rke r(t ) L (l) λκ P T, (74) which can be easily verified wih help of Eqs.(65), (5), (53), (7) and (73). Sraighforward subsiuion of Eq.(43) ino he definiion (69) and calculaion of he derivaive of C(S,T ) wih respec o λ yield he ideniies for κ C wih involvemen of some common Greeks. For example, i is easy o see ha he ideniy holds κ C = S C λ 1 ρ C T λ, (75) where C, and ρ C are defined by Eqs.(57) and (58) Greek µ Le us inroduce he Greek mu µ C as he firs order derivaive of a call opion wih respec o parameer ν, which is he mean of asse price jump magniudes, C(S,T ) µ C =. (76) ν The Greek µ C is a new Greek ha does no exis in he Black-Scholes framework, because of he absence of marke parameer ν. Calculaing he derivaive of C(S,T ) wih respec o ν yields +Se q(t )e λς(t ) Ke r(t ) µ C λ(t ) = ς ν Se q(t ) L 1 (l) l l dz dz dke ikz z ξ(k) ν dke ikz ξ(k) ν exp{λ(t )ξ(k)} (77) exp{λ(t )ξ(k)}. I is easy o see from Eqs.(41) and (4) ha for p(η) given by Eq.(68) we have 3

24 and ς ν = δ eν+ = (ς +1), (78) ξ(k) = e ikν k δ = ik(ξ(k)+1). (79) ν Therefore, Eq.(77) can be rewrien as +Se q(t )e λς(t ) Ke r(t ) µ C λ(t ) = Se q(t ) (ς +1)L 1 (l) l l dz dz Furher, aking ino accoun ha dke ikz z ik(ξ(k)+1)exp{λ(t )ξ(k)} (80) dke ikz ik(ξ(k)+1)exp{λ(t )ξ(k)}. ike ikz = z eikz, and performing inegraion over dz we find +Se q(t )e λς(t ) µ C λ(t ) = ςse q(t ) L 1 (l)+ (81) l dze z dke ikz ξ(k)exp{λ(t )ξ(k)}, which can be considered as he equaion o calculae he Greek µ C. Le us show ha Eq.(81) can be used o find new ideniies wih involvemen of he Greek µ C and some common Greeks. For insance, if we rewrie Eq.(81) in he form ( µ C λ = L1 (l) Se q(t ) λ L ) 1(l) l, (8) l λ 4

25 hen i is easy o see ha he ideniy holds µ C λ = S C λ +ς(t )S Γ C, (83) where he Greek Γ C is defined by Eq.(61). Wih he help of Eq.(75) we can obain anoher ideniy from Eq.(83) κ C = µ C λ 1 ρ C T λ ς(t )S Γ C, (84) which esablishes he relaionship beween newly inroduced Greeks κ C and µ C Greek ǫ A new Greek epsilon ǫ C is inroduced as he derivaive of a call opion wih respec o parameer δ, which is he sandard deviaion of asse price jump magniudes, C(S,T ) ǫ C =. (85) δ The Greek ǫ C is a new Greek ha does no exis in he Black-Scholes framework, because of he absence of marke parameer δ. The derivaive of C(S,T ) wih respec o δ can be expressed as ǫ C λ(t ) = ς δ Se q(t ) L 1 (l) +Se q(t )e λς(t ) Ke r(t ) l l dz dze z dke ikz ξ(k) δ dke ikz ξ(k) δ I follows from Eqs.(41) and (4) ha exp{λ(t )ξ(k)} (86) exp{λ(t )ξ(k)}. ς δ = δ eν+ = δ(ς +1), (87) 5

26 and ξ(k) = e ikν k δ = k δ(ξ(k)+1). (88) δ Furher, aking ino accoun ha we have k e ikz = z eikz, ǫ C δλ(t ) = (ς +1)Se q(t ) L 1 (l) +Se q(t )e λς(t ) l dze z dk z eikz (ξ(k)+1)exp{λ(t )ξ(k)} Ke r(t ) l dz Inegraion by pars over dz yields dk z eikz (ξ(k)+1)exp{λ(t )ξ(k)}. ǫ C λ(t )δ = ςse q(t ) L 1 (l) +Se q(t )e λς(t ) + Ke r(t ) l dze z dke ikz ξ(k)exp{λ(t )ξ(k)} (89) dke ikl (ξ(k)+1)exp{λ(t )ξ(k)}, which can be considered as he equaion o calculae he Greek ǫ C. Comparing he equaion above wih Eq.(83) le s conclude ha he following relaionship holds 6

27 + Ke r(t ) ǫ C λ(t )δ = µ C λ(t ) +S Γ C (90) dke ikl ξ(k)exp{λ(t )ξ(k)}, where Γ C is given by Eq.(61). Sraighforward subsiuion of Eq.(43) ino definiion (85) and calculaion of he derivaive of C(S,T ) wih respec o δ yield he ideniies for ǫ C wih involvemen of common Greeks C and ρ C. For example, i is easy o see ha he ideniy holds ǫ C = S C δ 1 ρ C T δ, where C, and ρ C are defined by Eqs.(57) and (58). 6 Black-Scholes equaion 6.1 Diffusion approximaion We are aiming o show ha he well-known Black-Scholes equaion can be obained from he inegro-differenial opion pricing equaion(16). To ge he Black-Scholes equaion le s consider he marke siuaion when he variance of asse price jump magniudes δ 0, while he arrival rae of price jumps λ in such a way ha he produc λ dηp(η)η remains finie. We call his case diffusion approximaion. Due o he condiion δ 0, he expression under he inegral sign in Eq.(16) can be expanded in η up o he second-order + λ C(x, ) dηp(η)η { C(x,) x +(r q) C(x,) x C(x,) } rc(x,) = 0. x (91) 7

28 To simplify he equaions in downsream consideraion i is convenien o inroduce he noaion σ = λ dηp(η)η. (9) In general, he diffusion approximaion is he case when he mean of asse price jump magniudes ν 0, he variance of asse price jump magniudes δ 0 and he arrival rae of price jumps λ while he producs λ dηp(η)η and λ dηp(η)η remain finie. 6. Opion pricing equaion Wih help Eq.(9), equaion (91) can be rewrien as C(x, ) + σ C(x,) x Then he subsiuion +(r q σ ) C(x,) rc(x,) = 0. (93) x yields S K = ex, (94) C(S, ) or + σ (S S ) C(S,)+(r q σ )S C(S,) rc(s,) = 0 (95) S C(S, ) + σ C(S,) S +(r q)s C(S,) rc(s,) = 0. (96) S S This is he famous Black-Scholes equaion [1], which has o be supplemened wih he erminal condiion given by Eq.(18) for call opion and by Eq.(0) for pu opion. 8

29 Thus, i has been shown ha he new pricing equaion (16) goes ino he Black-Scholes equaion (96) in diffusion approximaion. The parameer σ inroduced by Eq.(9) is called volailiy of asse price S. If we go from ime o he ime o expiry T, hen Eq.(96) can be rewrien as C(S,T ) + σ C(S,T ) (T ) S S +(r q)s while he erminal condiions (18), (0) became C(S,T ) rc(s,t ) = 0, S (97) C(S,T ) =T = C(S,0) = max(s K,0), (98) for a European call opion, and P(S,T ) =T = P(S,T) = max(k S,0). (99) for a European pu opion. 6.3 The soluion o Black-Scholes equaion The sraighforward soluion o Eq.(97) wih he erminal condiion (98) has been presened in Appendix A. Here we show ha soluion(43) o he inegro-differenial pricing equaion (4) goes ino he Black-Scholes soluion in diffusion approximaion. In he diffusion approximaion we have and λς = λ λξ(k) = λ dηp(η)(η + η σ ) = λν +, (100) dηp(η)(ikη k η ) = ikλν k σ, (101) where σ is given by Eq.(9) and ς and ξ(k) are defined by Eqs.(41) and (4). 9

30 6.3.1 Funcion L 1 (l) in diffusion approximaion TofindheequaionforL 1 (l)indiffusionapproximaionwesubsiueeqs.(100) and (101) ino Eq.(44) L 1 (l) diff L (diff) 1 = e (λν+ σ )(T ) x+(r q λν σ )(T ) } dke ikz exp {(ikλν k σ )(T ). Calculaion of he inegral over dk resuls in 1 = } dke ikz exp {(ikλν k σ )(T ) 1 σ (T ) exp { } (z +λν(t )). σ (T ) dze z (10) (103) Subsiuing Eq.(103) ino Eq.(10) and changing inegraion variable z y yield z y = z +λν(t ), L (diff) 1 = 1 σ (T ) (104) x+(r q σ )(T ) Wih help of a new inegraion variable w { dyexp 1 ( ) } y σ T +σ T. y w = y σ T +σ T, we rewrie Eq.(104) in he following way 30

31 L (diff) 1 = 1 Then funcion L (diff) 1 akes he form x+(r q σ )(T ) σ +σ T T dw e w. (105) L (diff) 1 = N(d 1 ), (106) where funcion N(d) is he cumulaive disribuion funcion of he sandard normal disribuion [4] N(d) = 1 d dw e w, (107) and parameer d 1 has been inroduced by d 1 = σ x+(r q )(T ) σ +σ T = T σ x+(r q + σ T )(T ). (108) Thus, i has been shown ha in diffusion approximaion funcion L 1 (l) goes ino funcion N(d 1 ), L 1 (l) diff L (diff) 1 = N(d 1 ). (109) The similar consideraion brings he equaion for funcion L 1 (l) in diffusion approximaion L 1 (l) diff L (diff) 1 = N( d 1 ), (110) wih N(d) and d 1 defined by Eqs.(107) and (108). On a final noe we presen he equaion for L 1 (l)/ l in diffusion approximaion L 1 (l) l diff 1 σ T N (d 1 ), where N (d) = exp( d /)/ is derivaive of N(d) wih respec o d and d 1 is given by Eq.(108). 31

32 6.3. Funcion L (l) in diffusion approximaion TofindheequaionforL (l)indiffusionapproximaionwesubsiueeqs.(100) and (101) ino Eq.(45) L (l) diff L (diff) = 1 x+(r q λν σ )(T ) } dke ikz exp {(ikλν k σ )(T ), dz (111) Following he consideraion provided while we calculaed L (diff) 1 we find by L (diff) = N(d ), (11) where funcion N(d) is defined by Eq.(107) and d has been inroduced d = σ x+(r q )(T ) σ T = d 1 σ T. (113) Thus, i has been shown ha in diffusion approximaion funcion L (l) goes ino funcion N(d ). L (l) diff L (diff) = N(d ). In a similar way he equaion for L (l) can be obained L (l) diff L (diff) = N( d ), (114) wih N(d) and d defined by Eqs.(107) and (113). On a final noe we presen he equaion for L (l)/ l in diffusion approximaion L (l) l diff 1 σ T N (d ), where N (d) = exp( d /)/ is derivaive of N(d) wih respec o d, and d is given by Eq.(113). 3

33 6.3.3 Soluion o opion pricing equaion in diffusion approximaion Using Eqs.(106) and (11) we can wrie Eq.(43) in diffusion approximaion in he form C BS (S,T ) = Se q(t ) N(d 1 ) Ke r(t ) N(d ), (115) where C BS (S,T ) sands for he value of a European call opion in he Black-Scholes model. Using Eqs.(110) and (114) we can wrie Eq.(47) in diffusion approximaion in he form P BS (S,T ) = Ke r(t ) N( d ) Se q(t ) N( d 1 ), (116) where P BS (S,T ) sands for he value of a European pu opion in he Black-Scholes model. Hence, we see ha in diffusion approximaion, soluions (43) and (47) go ino he Black-Scholes soluions for call and pu opions. The key feaure of diffusion approximaion is he Black-Scholes volailiy defined by Eq.(9). In oher words, Eq.(9) sheds ligh ino he sochasic dynamic origin of he Black-Scholes volailiy, which emerges naurally in he diffusion approximaion. 6.4 Greeks in diffusion approximaion Greek vega In he Black-Scholes world here is a volailiy σ. Hence, we can consider he derivaive of call and pu opions wih respec o σ. This is he Greek vega 3. For insance, for a European call opion we have vega υ C which is υ C = C(S,T ) σ = Se q(t ) T N (d 1 ) = Ke r(t ) T N (d ), where N (d) sands for derivaive of N(d) wih respec o d, (117) 3 Acually, here isnosuch Greekname asvegaandυ doesno belongo Greeksymbols. 33

34 N (d) = 1 exp( d /), (118) parameers d 1 and d are defined by Eqs.(108) and (113). I follows from he pu-call pariy law (50), which holds for he Black- Scholes soluions, ha P(S,T ) C(S,T ) υ P = = = υ C, (119) σ σ ha is, he Greeks vega for call and pu opions are he same Black-Scholes Greeks I follows from Eqs.(109), (110), (11) and (114) ha in diffusion approximaion all common Greeks presened in Tables 1 and go ino he well-known Black-Scholes Greeks. Table 3 summarizes he Black-Scholes Greeks for a call opion. Call (Black-Scholes pricing equaion) Dela, C = C(S,T ) e q(t ) N(d S 1 ) Gamma, Γ C = C (S,T ) S Rho, ρ C = C(S,T ) e q(t ) Sσ T N (d 1 ) = Ke r(t ) S σ T N (d ) r (T )Ke r(t ) N(d ) Psi, ψ C = C(S,T ) q (T )Se q(t ) N(d 1 ) Thea, Θ C = C(S,T ) qse q(t ) N(d 1 ) rke r(t ) N(d ) σse q(t ) N (d 1 ) T Vega υ C = C(S,T ) σ Se q(t ) T N (d 1 ) = Ke r(t ) T N (d ) Table 3. Black-Scholes Greeks (Call opion) 7 Generalized opion pricing framework 7.1 Geomeric Brownian moion and Geomeric Sho Noise The new opion pricing framework presened in Sec.3 can be easily generalized o accommodae a superposiion of Geomeric Brownian moion and a Geomeric Sho Noise moion. Indeed, if besides he Geomeric Sho Noise moion we have he Geomeric Brownian moion, hen Eq.(1) becomes 34

35 ds = µsd+σsdw +SF()d, (10) where µ and σ are consans belonging o he Brownian moion process, W is a sandard Wiener process, and he sho noise process F() has been inroduced by Eq.(). 7. Generalized arbirage-free inegro-differenial opion pricing equaion If asse price follows Eq.(10) hen he generalized arbirage-free inegrodifferenial opion pricing equaion has a form C(x, ) + σ C(x,) x +(r q σ ) C(x,) x (11) +λ { dηp(η) C(x+η,) C(x,) (e η 1) C(x,) } rc(x,) = 0, x here x is given by Eq.(17), C(x,) is he value of a European call opion on dividen-paying asse, S is asse price governed by Eq.(10), K is he srike price, r is he risk-free ineres rae, q is coninuously paid dividend yield, which is a consan, p(η) is he probabiliy densiy funcion involved ino Eq.(). If we go from C(x,) o C(S,) where x and S are relaed o each oher by Eq.(17), hen we can wrie he generalized opion pricing equaion in he form C(S, ) + σ C(S,) S +(r q)s C(S,) S S (1) +λ { dηp(η) C(Se η,) C(S,) (e η 1)S C(S,) } rc(s,) = 0, S wih he erminal condiion given eiher by Eq.(18) or by Eq.(0). 35

36 In he case when λ = 0, Eq.(1) becomes he Black-Scholes equaion (see, Eq.(96)). In he case when σ = 0, Eq.(1) becomes he inegrodifferenial opion pricing equaion (see, Eq.()). If we go from ime o he ime o expiry T, hen Eq.(1) can be rewrien as +λ C(S,T ) + σ C(S,T ) C(S,T ) (T ) S +(r q)s S S { } dηp(η) C(Se η,t ) C(S,T ) (e η C(S,T ) 1)S S rc(s,t ) = 0, (13) wih he erminal condiion (98) for a European call opion and he erminal condiion (99) for a European pu opion. The soluion o Eq.(13) wih he erminal condiion (98) is C(S,T ) = Se q(t ) L 1 (l σ ) Ke r(t ) L (l σ ), (14) where new funcions L 1 (l σ ) and L (l σ ) are inroduced by L 1 (l σ ) = e( σ and λς)(t ) L (l σ ) = 1 l σ dz l σ dz } dke ikz z exp {[ σ k +λξ(k)](t ), (15) } dke ikz exp {[ σ k +λξ(k)](t ), (16) wih ς and ξ(k) given by Eqs.(41) and (4), and parameer l σ defined by l σ = x+(r q σ λς)(t ). (17) 36

37 Generalized opion pricing formula (14) for a European call opion follows from asse price sochasic dynamics (10) modeled by superposiion of Geomeric Brownian moion and a Geomeric Sho Noise. Le us consider wo limi cases of Eq.(14). 1. In he limi case when σ = 0 we have L 1 (l σ ) σ=0 = L 1 (l), (18) L (l σ ) σ=0 = L (l), (19) and Eq.(14) goes ino he opion pricing equaion (43) wih l defined by Eq.(39).. In he limi case when λ = 0 we have L 1 (l σ ) λ=0 = N(d 1 ), (130) L (l σ ) λ=0 = N(d ), (131) and Eq.(14) goes ino he Black-Scholes opion pricing formula (115) wih d 1 and d defined by Eqs.(108) and (113). Hence, Eq.(14) describes he impac of he inerplay beween Gaussian Geomeric Brownian moion and non-gaussian Geomeric Sho Noise on he value of a European call opion. On a final noe, le us presen he generalized opion pricing formula for a European pu opion. The soluion o Eq.(13) wih erminal condiion (99) is P(S,T ) = Ke r(t ) L (l σ ) Se q(t ) L 1 (l σ ), (13) where new funcions L 1 (l) and L (l) are inroduced by L 1 (l σ ) = e( σ and λς)(t ) l σ dz } dke ikz z exp {[ σ k +λξ(k)](t ), (133) 37

38 L (l σ ) = 1 l σ dz and he parameer l σ is defined by Eq.(17). Le us consider wo limi cases of Eq.(13). 1. In he limi case when σ = 0 we have } dke ikz exp {[ σ k +λξ(k)](t ), (134) L 1 (l σ ) σ=0 = L 1 (l), (135) L (l σ ) σ=0 = L (l), (136) and Eq.(13) goes ino he opion pricing equaion (47) wih l defined by Eq.(39).. In he limi case when λ = 0 we have L 1 (l σ ) λ=0 = N( d 1 ), (137) L (l σ ) λ=0 = N( d ), (138) and Eq.(13) goes ino he Balck-Scholes opion pricing formula (116) wih d 1 and d defined by Eqs.(108) and (113). Hence, Eq.(13) describes he impac of he inerplay beween Gaussian Geomeric Brownian moion and non-gaussian Geomeric Sho Noise on he value of a European pu opion. I is easy o see ha he value of call opion given by Eq.(14) and he value of pu opion given by Eq.(13) saisfy he fundamenal pu-call relaionship (50). 7.3 Generalized Greeks Here we presen he equaions for common generalized Greeks in he case when he value of a European call opion is given by Eq.(14). The generalized Greek dela for a call opion C is C = C(S,T ) S 38 = e q(t ) L 1 (l σ ). (139)

39 The generalized Greek rho ρ C for a call opion is C(S,T ) ρ C = = (T )Ke r(t ) L (l σ ). (140) r The generalized Greek psi ψ C for a call opion is ψ C = C(S,T ) q The generalized Greek hea Θ C for a call opion is = (T )Se q(t ) L 1 (l σ ). (141) Θ C = C(S,T ) = qse q(t ) L 1 (l σ ) rke r(t ) L (l σ )+Se q(t ) ( σ +λς)l 1(l σ ) Se q(t )e( σ λς)(t ) (14) l dze z dke ikz ( σ k } +λξ(k))exp {[ σ k +λξ(k)](t ) + λke r(t ) l dz dke ikz ( σ k {[ +λξ(k))exp σ k } +λξ(k)](t ). The second order sensiiviy generalized gamma Γ C for a call opion is Γ C = C(S,T ) = e q(t ) L 1(l σ ) l σ S l σ S = e q(t ) L 1 (l σ ). (143) S l σ Finally, he generalized Greek vega υ C for a call opion is υ C = C(S,T ) σ = σ(t )Se q(t ) L 1 (l σ ) 39

40 Se q(t ) σ(t )e ( σ λς)(t ) (144) l σ dz + Ke r(t ) σ(t ) } dke ikz z k exp {[ σ k +λξ(k)](t ) l σ dz } dke ikz k exp {[ σ k +λξ(k)](t ). Funcions L 1 (l σ ) and L (l σ ) in he formulas above are given by Eqs.(15) and (16). Table 4 summarizes he common generalized Greeks for a call opion. 40

41 Call Dela, C = C(S,T ) e q(t ) L S 1 (l σ ) Gamma, Γ C = C (S,T ) e q(t ) L 1 (l σ) S S l σ Rho, ρ C = C(S,T ) = Ke r(t ) S L (l σ) l σ r (T )Ke r(t ) L (l σ ) Psi, ψ C = C(S,T ) q (T )Se q(t ) L 1 (l σ ) Thea, Θ C = C(S,T ) qse q(t ) L 1 (l σ ) rke r(t ) L (l σ ) +(σ /+λς)se q(t ) L 1 (l σ ) Se q(t ) e ( σ λς)(t ) l dke ikz ( σ k +λξ(k))e{[ σ + λke r(t ) l dz dke ikz ( σ k +λξ(k))e{[ σ dze z Vega, υ C = C(S,T ) σ(t )Se q(t ) L σ 1 (l σ ) l σ Se q(t ) σ(t )e ( σ λς)(t ) k +λξ(k)](t )} k +λξ(k)](t )} dz dke ikz z k e {[ σ k +λξ(k)](t )} + Ke r(t ) σ(t ) l σ dz dke ikz k e {[ σ k +λξ(k)](t )} Table 4. Common generalized Greeks (Call opion) Le us consider wo limi cases of Eqs.(139)-(144). 1. In he limi case when σ = 0 he generalized Greek vega defined by Eq.(144) does no exis, while he generalized Greeks defined by Eqs.(139)- 41

42 (143) go ino he common Greeks inroduced by Eqs.(57)-(61) due o Eqs.(18) and (19).. In he limi case when λ = 0 he generalized Greeks defined by Eqs.(139)-(144) go ino he well known Black-Scholes Greeks due o Eqs.(130) and (131). The Black-Scholes Greeks are presened in Table 3. The common generalized Greeks for a pu opion can be found by using he pu-call pariy equaion (50). Hence, new equaions (139)-(144) describe he impac of inerplay beween Gaussian Geomeric Brownian moion and non-gaussian Geomeric Sho Noise on he common generalized Greeks. 8 Shor erm ineres rae dynamics 8.1 Langevin equaion wih sho noise We inroduce a new sochasic model o describe shor erm ineres rae dynamics, dr = ard+f()d, (145) which is a sochasic differenial equaion of he Langevin ype wih he sho noise F() given by Eq.() and he parameer a (a > 0) being he speed a which r() goes o is equilibrium level a. The shor erm ineres rae r() governed by Eq.(145) is a non-gaussian random process. I is easy o see from Eq.(145) ha if a he ime momen he shor erm ineres rae is r(), hen laer on, a he ime momen s, s >, i will be r(s) = e a(s ) r()+ s dτe a(s τ) F(τ) (146) or r(s) = e as r 0 + s dτe a(s τ) F(τ), (147) wih r 0 = r() =0 if we choose =

43 8. Bond price 8..1 Affine erm srucure The price a ime of a zero-coupon bond P(,T,r) which pays one currency uni a mauriy T, (0 T), is P(,T,r) =< exp{ subjec o erminal condiion T dsr(s)} > F, (148) P(T,T,r) = 1, (149) here r(s) is given by Eq.(146), r = r() is he rae a ime, and <... > F sands for he average wih respec o all randomness involved ino he random force F(). Furher, subsiuion of r(s) from Eq.(146) ino Eq.(148) and sraighforward inegraion over ds yields P(,T,r) = exp{a(,t) B(,T)r()}, (150) where he following noaions have been inroduced and B(,T) = T dse a(s ) = 1 e a(t ), (151) a exp{a(,t)} =< exp{ T dsb(s,t)f(s)} > F. (15) Therefore, we conclude ha he new model inroduced by Eq.(145) provides affine erm srucure. To calculae he average in Eq.(15), we use Eqs.(3) and (13) o obain T exp{a(,t)} = exp{λ r ds dηp r (η)(e ηb(s,t) 1)}, (153) 43

44 from which i follows direcly ha A(,T) = λ r T ds dηp r (η)(e ηb(s,t) 1). (154) where B(s,T) is defined by Eq.(151), λ r is he number of shor erm ineresraejumpsperuniime, andp r (η)isheprobabiliydensiyfuncion of jump magniudes. We assume ha probabiliy densiy funcion p r (η) of shor erm ineres rae jump magniudes is a normal disribuion p r (η) = 1 exp{ (η ν r) }, (155) δr where he marke parameers ν r and δ r are he mean and variance of shor erm ineres rae jump magniudes. The calculaion of he inegral over η in Eq.(154) gives he resul δ r dηp r (η)(e ηb(s,t) 1) = exp{ ν r B(s,T)+ δ r B (s,t)} 1. (156) Therefore, we have for A(,T) A(,T) = λ r T { } ds exp{ ν r B(s,T)+ δ r B (s,t)} 1. (157) By inroducing a new inegraion variable y = B(s,T), we can express A(,T) in he form A(,T) = λ r B(,T) 0 dy 1 ay exp{ ν ry + δ r y 1}. (158) Thus, based on shor erm ineres rae dynamics inroduced by Eq.(145), weobainedformula(150)opriceazerocouponbond, whereb(,t)isgiven by Eq.(151), and A(,T) is given by Eqs.(157) or (158). 44

45 8.. The erm srucure equaion Defined by Eq.(148) he price of zero-coupon bond P(,T,r) solves he following inegro-differenial equaion +λ r P(,T,r) ar P(,T,r) r dηp r (η){p(,t,r+η) P(,T,r)} = rp(,t,r), (159) subjec o erminal condiion (149), wih λ r being he number of shor erm ineres rae jumps per uni ime, p r (η) being he probabiliy densiy funcion of jump magniudes, and parameer a being he speed a which he shor erm ineres rae goes o is equilibrium level a. Following Vasiček [] erminology, we call Eq.(159) he erm srucure equaion. To find he soluion o Eq.(159) subjec o erminal condiion (149) le s noe ha he affine erm srucure means ha he bond price admis soluion of he form given by Eq.(150). Subsiuion of Eq.(150) ino Eq.(159) gives he wo ordinary differenial equaions o obain A(, T) and B(,T) da(,t) d +λ r dηp r (η){e ηb(s,t) 1} = 0, (160) db(,t) ab(,t)+1 = 0, (161) d subjec o erminal condiions a = T and A(T,T) = 0, (16) B(T,T) = 0. (163) To solve he sysem of equaions (160) and (161) le s firs solve Eq.(161). By separaing he variables in Eq.(161) we wrie 45

46 and db(,t) ab(,t) 1 = d, 1 ln ab(,t) 1 = +C, a where C is an arbirary consan of inegraion. Hence, we have B(,T) = 1 a (1±eaC e a ). The value of consan C and he sign can be fixed by imposing erminal condiion (163), and finally, we come o he soluion for B(,T) given by Eq.(151). Furher, Eq.(160) can be solved by sraighforward inegraion. Indeed, we have A(T,T) A(,T) = λ r T ds dηp r (η){e ηb(s,t) 1}. I is obvious ha afer aking ino accoun erminal condiion (16), we immediaely come o he soluion for A(,T) given by Eq.(154). Thus, we proved ha he price of zero-coupon bond P(,T,r) defined by Eq.(148) solves he new inegro-differenial erm srucure equaion (159). 8.3 Vasiček model Diffusion approximaion To obain he Vasiček bond pricing formula from Eqs.(150), (151) and (154), we consider he diffusion approximaion when ν r 0, δ r 0 and λ r while he producs λ r dηp r (η)η and λ r dηp r (η)η remain finie. I follows from Eq.(154) ha A diff (,T) in his approximaion is A(,T) diff A diff (,T) (164) 46

47 = λ r dηp r (η){ ηb(s,t)+ η B (s,t)} = λ r ν r B(s,T)+ σ r B (s,t), where B(s,T) is given by Eq.(151), parameer ν r is ν r = dηp r (η)η, (165) and parameer σ r is σ r = λ r dηp r (η)η. (166) 8.3. Long-erm mean and insananeous volailiy Subsiuing B(s, T) given by Eq.(151) ino Eq.(164) and evaluaing he inegral over ds yield A diff (,T) = λ rν r (T B(,T)) (167) a ) + σ r (T B(,T)+ 1 e a(t ). a a Le us inroduce he noaion Then we have b = λ rν r a. (168) A diff (,T) A Vasiček (,T) = (b σ r a )[B(,T) (T )] σ r 4a B (,T). (169) A bond pricing equaion (150) in diffusion approximaion has a form P diff (,T) = exp{a Vasiček (,T) B(,T)r()}. (170) 47

48 WerecognizeinEq.(170)heVasičekbondpricingformula[]wihA Vasiček (,T) given by Eq.(169) and B(,T) given by Eq.(151). Hence, we conclude ha b defined by Eq.(168) is long erm mean and σ r inroduced by Eq.(166) is insananeous volailiy of he Vasiček shor erm ineres rae model. The parameer σ r /a appearing in Eq.(169) is someimes called long erm variance. Le us remind ha he Vasiček shor erm ineres rae model has been inroduced by means of he following sochasic differenial equaion [] dr = a(b r)d+σ r dw(), (171) where a is parameer a (a > 0) being he speed a which r() goes o is equilibrium level b a, parameer σ r is insananeous volailiy and W() is a sandard Wiener process. The parameers a, b, and σ r are posiive consans. The parameer a in he Vasiček model has exacly he same meaning as in our model inroduced by Eq.(145). The exisence and meaning of he phenomenological long erm mean b in he Vasiček model is explained in our model by Eq. (168) which emerges naurally in he diffusion approximaion. In oher words, our model provides he quaniaive background o inroduce parameer b and les us conclude ha he long erm mean has sochasic dynamic origin. The origin of insananeous volailiy σ r in he Vasiček model is explained in our model by Eqs.(164) and (166), ha is, our model provides he quaniaive background o inroduce he Vasiček parameer σ r. Thus, we see ha in diffusion approximaion he shor erm ineres rae model defined by Eq.(145) goes exacly ino he well-known Vasiček model []. The long erm mean given by Eq.(168) and insananeous volailiy given by Eq.(166) are presened in erms of λ r and he parameers involved ino probabiliy densiy funcion p r (η) defined by Eq.(155). In oher words, he new sochasic model (145) gives insigh ino sochasic dynamic origin of he Vasiček long erm mean b and a shor erm ineres rae insananeous volailiy σ r. Le us emphasize ha he sochasic dynamics (145) iniiaes a non- Gaussian probabiliy disribuion of shor erm ineres rae in conras o he Vasiček model, which gives he Gaussian disribuion. However, despie he lack of Gaussianiy, he sochasic model (145) has he same degree of analyical racabiliy as he Vasiček model. 48

49 8.3.3 Mean and variance I is ineresing ha he non-gaussian model inroduced by Eq.(145) possesses exacly he same mean and variance as he Vasiček model, which resuls in he Gaussian probabiliy disribuion of shor erm ineres rae. Indeed, i follows from Eq.(147) ha he mean < r(s) > is < r(s) >= e as r 0 + s dτe a(s τ) < F(τ) >, (17) and he variance is 0 Var(r(s)) =< (r(s) < r(s) >) > (173) =< s dτe a(s τ) (F(τ) < F(τ) >) >, 0 where <... > sands for he average defined by Eqs.(6)-(8). Taking ino accoun Eq.(14) and using Eq.(68) we obain for he mean < r(s) >= e as r 0 +b(1 e as ) = b+(r 0 b)e as, (174) where b is given by Eq.(168). To calculae he variance Var(r(s)) le s use Eqs.(15) and (68) o find Var(r(s)) =< (r(s) < r(s) >) >= σ r a (1 e as ), (175) where σ r is given by Eq.(166). The condiional mean < r(s) r() > and variance Var(r(s) r()) are and < r(s) r() >= b+(r() b)e a(s ), (176) Var(r(s) r()) = σ r a (1 e a(s ) ). (177) The equaions for < r(s) r() > and Var(r(s) r()) coincide exacly wih he Vasiček s equaions (see, Eqs.(5) and (6) in []) for condiional mean 49

50 and variance of shor erm ineres rae. I has o be emphasized ha we found an ineresing coincidence beween he means and he variances of he non-gaussian model inroduced by Eq.(145) and he Gaussian Vasiček model. When s, he condiional mean < r(s) r() > goes o he long erm mean b given by Eq.(168), lim < r(s) r() >= b. s The condiional variance Var(r(s) r()) is increasing wih respec o s from zero o he long erm variance, lim Var(r(s) r()) = σ r s a. Hence, he new sochasic model inroduced by Eq.(145) resuls in a seady non-normal probabiliy disribuion for r() wih he mean and he variance b = λ r a dηp r (η)η, σ r a = λ r a dηp r (η)η, where p r (η) is defined by Eq.(155) and λ r is he number of shor erm ineres rae jumps per uni ime. 9 Generalized shor erm ineres rae model 9.1 Vasiček model wih sho noise Now we inroduce a generalized shor erm ineres rae model as a superposiion of he new model defined by Eq.(145) and he Vasiček shor erm ineres rae model given by Eq.(171) dr = a(b r)d+σ r dw()+f()d, (178) 50

51 where all noaions are he same as hey were defined for Eqs.(145) and (171). If we ake ino accoun ha dw() can be presened as dw() = w()d, where w() is he whie noise process, hen Eq.(178) reads dr = a(b r)d+σ r w()d+f()d. I is easy o see from Eq.(178) ha if a he ime momen he shor erm ineres rae is r() hen laer on, a he ime momen s, s >, i will be r(s) = e a(s ) r()+b(1 e a(s ) ) (179) +σ r s dτe a(s τ) w(τ)+ s dτe a(s τ) F(τ), or r(s) = e as r 0 +b(1 e as )+σ r s s dτe a(s τ) w(τ)+ 0 dτe a(s τ) F(τ), (180) wih r 0 = r() =0 if we choose = Bond price: generalized formula The price a ime of a zero-coupon bond P(,T,r) which pays one currency uni a mauriy T, (0 T), is P(,T,r) =< exp{ T dsr(s)} > w, F, P(T,T,r) = 1, (181) where r(s) is given by Eq.(179) and <... > w, F sands for he average over whie noise w() and all randomness involved ino sho noise F(). Furher, subsiuion of r(s) from Eq.(179) ino Eq.(181) and sraighforward inegraion over ds yields 51

52 P(,T,r) = exp{a(,t) B(,T)r()}, (18) where B(,T) has been inroduced by Eq.(151), and exp{a(,t)} = exp{ b(t B(,T))} (183) < exp{ σ r T dsb(s,t)w(s)} > w < exp{ T dsb(s,t)f(s)} > F, because whie noise and sho noise are independen of each oher. Therefore, we conclude ha he generalized model inroduced by Eq.(178) provides affine erm srucure. Furher, we have (see, Eqs.(15), (153) and (157)) < exp{ T dsb(s,t)f(s)} > F (184) T = exp λ r { } ds exp{ ν r B(s,T)+ δ r B (s,t)} 1. While for he average wih respec o he whie noise process 4 we have 4 The characerisic funcional Ψ[α(τ)] of a whie noise process is Ψ[α(τ)] =< exp{i T dτα(τ)w(τ)} > w = exp{ 1 T dτα (τ)}, (185) where α(τ) is an arbirary sufficienly smooh funcion and... w sands for he average over whie noise. T Toprovehis equaion weneed o calculae < exp{i dτα(τ)w(τ)} > w. I can easilybe done if we wrie exp{i T dτα(τ)w(τ)} as a power series expansion and ake ino accoun ha whie noise is a random process which has wo saisical momens only w(τ) w = 0, w(τ 1 )w(τ ) w = δ(τ 1 τ ). 5

53 < exp{ σ r T dsb(s,t)w(s)} > w = exp{ σ r { } σ = exp r a (T ) σ r σ a B(,T) r 4a B (,T). I follows from Eqs.(183)-(186) ha T dsb (s,t)} (186) A(,T) = (b σ r a )[B(,T) (T )] σ r 4a B (,T) (187) +λ r T ds dηp r (η)(e ηb(s,t) 1), which can be wrien as A(,T) = A Vasiček (,T)+A(,T), (188) wih A Vasiček (,T) and A(,T) defined by Eqs.(169) and (157). Thus, based on generalized shor erm ineres rae dynamics inroduced by Eq.(178), we obained he formula (18) for price of a zero coupon bond, where B(,T) is given by Eq.(151) and A(,T) is given by Eq.(187). There are he following limi cases of Eq.(188) A(,T) λ=0 = A Vasiček (,T), (189) which resuls in he Vasiček formula (170) for a bond price, and A(,T) b=0, σ=0 = A(,T), (190) whichresulsinbondpricingformula(150)wihb(,t)givenbyeq.(151) and A(,T) given by Eqs.(157). Hence, he generalized shor erm ineres rae dynamics (178) describes he impac of inerplay beween Gaussian Geomeric Brownian moion and non-gaussian Geomeric Sho Noise on a bond price. 53

54 9..1 Generalized erm srucure equaion Defined by Eq.(181) he price of zero-coupon bond P(,T,r) solves he following inegro-differenial equaion +λ r P(,T,r) +a(b r) P(,T,r) r + σ r P(,T,r) r (191) dηp r (η){p(,t,r+η) P(,T,r)} = rp(,t,r), subjec o erminal condiion (149), wih parameers a, b, and σ r being Vasiček s parameers in Eq.(171), λ r being he number of shor erm ineres rae jumps per uni ime, and p r (η) being he probabiliy densiy funcion of jump magniudes. This is a generalized erm srucure equaion in he case when he shor erm ineres rae solves he sochasic differenial equaion (178). To find he soluion o Eq.(191) subjec o erminal condiion (149) le s noe ha he affine erm srucure means ha he bond price admis soluion of he form given by Eq.(18). Subsiuion of Eq.(18) ino Eq.(191) gives he wo ordinary differenial equaions o obain A(,T) and B(,T). The ordinary differenial equaion o find A(,T) has a form da(,t) +abb(,t)+ σ r d B (,T)+λ r subjec o erminal condiion a = T dηp r (η){e ηb(s,t) 1} = 0, (19) A(T,T) = 0. (193) For B(,T) we have Eq.(161) subjec o erminal condiion given by Eq.(163), hus, we conclude ha B(,T) is given by Eq.(151). Then he soluion o Eq.(19) in erms of B(,T) is A(,T) = T ds abb(,t)+ σ r B (s,t)+λ r 54 dηp r (η)(e ηb(s,t) 1), (194)

55 which can be sraighforwardly ransformed ino Eq.(187). Thus, i has been shown ha he price of zero-coupon bond P(,T,r) defined by Eq.(181) solves he generalized erm srucure equaion (191). 10 Conclusion The unified framework consising of wo analyical approaches o value opions and bonds has been inroduced and elaboraed. The opions pricing approach is based on asse price dynamics ha has been modeled by he sochasic differenial equaion wih involvemen of sho noise. I resuls in Geomeric Sho Noise moion of asse price. A new arbirage-free inegro-differenial opion pricing equaion has been developed and solved. New exac formulae o value European call and pu opions have been obained. The pu-call pariy has been proved. Based on he soluion of he opion pricing equaion, he new common Greeks have been calculaed. Three new Greeks associaed wih he marke model parameers have been inroduced and evaluaed. I has been shown ha he developed opion pricing framework incorporaes he well-known Black-Scholes equaion [1]. The Black-Scholes equaion and is soluions emerge from our inegrodifferenial opion pricing equaion in he special case which we call diffusion approximaion. The Geomeric Sho Noise model in diffusion approximaion explains he sochasic dynamic origin of volailiy in he Black-Scholes model. The generalized opion pricing framework has been inroduced and developed based on asse price sochasic dynamics modeled by a superposiion of Geomeric Brownian moion and Geomeric Sho Noise. New formulae o value European call and pu opions have been obained as soluions o he generalized arbirage-free inegro-differenial pricing equaion. Based on hese soluions new generalized Greeks have been inroduced and calculaed. The bonds pricing analyical approach has been developed based on he Langevin ype sochasic differenial equaion wih sho noise o model a shor erm ineres rae dynamics. I resuls in a non-gaussian random moion of shor erm ineres rae. I is ineresing ha despie he lack of normaliy, he new model possesses exacly he same mean and variance as he Vasiček model, which resuls in he Normal disribuion of shor erm ineres rae. A bond pricing formula has been obained and i has been shown ha he model provides affine erm srucure. The new bond pricing formula in- 55

56 corporaes he well-known Vasiček soluion []. The inegro-differenial erm srucure equaion has been obained and i has been shown ha he new bond pricing formula solves his equaion. The well-known Vasiček model for shor erm ineres rae wih is long-erm mean and insananeous volailiy comes ou from our model in diffusion approximaion. The sochasic dynamic origin of he Vasiček long-erm mean and insananeous volailiy has been uncovered. A generalized bond pricing model has been inroduced and developed based on shor erm ineres rae sochasic dynamics modeled by a superposiion of a sandard Wiener process and sho noise. A new bond pricing formulahasbeenfoundandihasbeenshownhahegeneralizedmodelprovides affine erm srucure. The generalized inegro-differenial erm srucure equaion has been obained and solved. I has o be emphasized ha despie he non-gaussianiy of probabiliy disribuions involved, all newly elaboraed quaniaive models o value opions and bonds have he same degree of analyical racabiliy as he Black Scholes model [1] and he Vasiček model []. This circumsance allows one o derive new exac formulas o value opions and bonds. The unified framework can be easily exended o cover valuaion of oher ypes of opions and a variey of financial producs wih opion feaures involved, and o elaborae enhanced shor erm ineres rae models o value ineres rae coningen claims. References [1] F. Black and M. Scholes, The Pricing of Opions and Corporae Liabiliies, The Journal of Poliical Economy 81, (1973), pp [] O. Vasiček, An equilibrium characerizaion of he erm srucure, Journal of Financial Economics 5, (1977), pp [3] N. Laskin, Fracional marke dynamics, Physica A 87, (000), pp [4] M. Abramowiz and I. Segun, eds. (197), Handbook of Mahemaical Funcions wih Formulas, Graphs, and Mahemaical Tables, New York, Dover, formula 6.., page 931, available on-line a hp://people.mah.sfu.ca/ cbm/aands/oc.hm 56

57 11 Appendix A: Green s funcion mehod o solve he Black-Scholes equaion Leusshow howoobainheblack-scholes formulaovalueaeuropeancall opion wihou convering he Black-Scholes equaion ino he hea equaion. In oher words, we are aiming o ge a sraighforward soluion o he Black- Scholes problem given by Eqs.(97) and (98). Using noaion (17), he Black- Scholes equaion (97) wih he erminal condiion given by Eq.(98) can be rewrien in a mahemaically equivalen form C(x,T ) + σ (T ) C(x,T ) x +(r q σ ) ) C(x,T rc(x,t ) = 0, x (195) wih he erminal condiion given by Eq.(5). The Green s funcion mehod is a convenien way o solve he problem given by Eqs.(195) and (5). Green s funcion G BS (x x,t ) of he Black-Scholes equaion saisfies he parial differenial equaion G BS (x x,t ) (T ) = σ G BS (x x,t ) x (196) +(r q σ ) G BS(x x,t ) rg BS (x x,t ) = 0, x and he erminal condiion G BS (x x,t ) =T = G BS (x x,0) = δ(x x ). (197) Having Green s funcion G BS (x x,t ), we can wrie he soluion of Eq.(195) wih erminal condiion Eq.(5) in he form C BS (x,t ) = K dx G BS (x x,t )max(e x 1,0), (198) 57

58 where C BS (S,T ) sands for he value of a European call opion in he Black-Scholes model. Green s funcion inroduced by Eq.(196) wih he erminal condiion (197) can be found by he Fourier ransform mehod. Wih help of definiions (31) and (3), equaion (196) reads G BS (k,t ) (T ) = { σ k } +ik(r q σ ) r G BS (k,t ), (199) and he erminal condiion (197) for G BS (k,t ) is G BS (k,t ) =T = G BS (k,0) = 1. (00) The soluion of he problem defined by Eqs.(199) and (00) is G BS (k,t ) = e r(t ) exp {[ σ k } +ik(r q σ )](T ). (01) SubsiuionofEq.(01)inoEq.(31)givesusheGreen sfunciong BS (x x,t ) of he Black-Scholes equaion (195) = G BS (x x,t ) = e r(t ) exp {[ σ k e r(t ) σ (T ) exp dke ik(x x ) } +ik(r q σ )](T ) { [x x +(r q σ σ (T ) } )(T )]. (0) Then Eq.(198) yields for he value of a European call opion C BS (x,t ) C BS (x,t ) = Ke r(t ) σ (T ) } dx exp { [x x +(r q σ )(T )] σ (T ) max(e x 1,0). 58 (03)

59 By inroducing a new inegraion variable z, x z, z = x x +(r q σ )(T ) σ, T and using Eq.(94) we find for a European call opion C BS (S,T ) C BS (S,T ) = Se q(t ) e σ (T ) d dze z e zσ T (04) Ke r(t ) where parameer d is given by d dzexp } { z, d = ln S σ +(r q )(T ) K σ. (05) T Inroducing inegraion variable y, y = z +σ T in he firs erm of Eq.(04) yields d+σ C BS (S,T ) = Se q(t ) Ke r(t ) d dzexp T } { z. dye y I is easy o see from he equaion above ha we came o he well-known soluion o he Black-Scholes equaion (97) C BS (S,T ) = Se q(t ) N(d 1 ) Ke r(t ) N(d ), where N(d) is he cumulaive disribuion funcion of he sandard normal disribuion given by Eq.(107) and parameer d 1 is d 1 = d +σ T = ln S K wih d defined by Eq.(05). σ +(r q + )(T ) σ, (06) T 59

60 1 Appendix B: Greek Θ C in diffusion approximaion Here we presen a walkhrough o calculae Greek Θ C (see, Eq.(60)) in diffusion approximaion. I follows from Eqs.(41) and (4) ha in diffusion approximaion and λς diff λν + σ, (07) λξ(k) iλkν k σ diff, (08) where parameer σ has been inroduced by Eq.(9). Using Eqs.(07), (08) and subsiuing ike ikz wih e ikz / z we have Θ C diff Θ diff C = qse q(t ) N(d 1 ) rke r(t ) N(d )+Se q(t ) λςn(d 1 ) Se q(t ) e λς(t ) σ (T ) l ( dze z λν ) z + σ z } (z +λν(t )) exp { σ (T ) (09) + Ke r(t ) σ (T ) exp { l dz (λν z + σ (z +λν(t )) σ (T ) }, ) z where Eqs.(109) and (11) have been aken ino accoun. The inegrals involved ino Eq.(09) are l dze z ( λν z + σ ) exp { z 60 } (z +λν(t )) σ (T )

61 = λνe l exp { and l } (l +λν(t l )) +λν dze z exp { σ (T ) + σ ( }) (z +λν(t )) e l exp { z σ z=l (T ) } { + σ dz (λν = σ + σ e l exp l (l +λν(t )) σ (T ) } dze z (z +λν(t )) exp {, σ (T ) z + σ ) exp { z } (z +λν(t )) σ (T ) } (l+λν(t )) = λνexp { σ (T ) ( (z +λν(t )) exp { z σ (T ) }) z=l, } (z +λν(t )) σ (T ) (10) (11) where parameer l is given by Eq.(39). By subsiuing Eqs.(10) and (11) ino Eq.(09) and performing simple algebra we obain Θ diff C = qse q(t ) N(d 1 ) rke r(t ) N(d ) (1) } Se q(t ) e λς(t ) σ (l +λν(t )) (T ) e l exp {. σ (T ) Furher simplificaion of Eq.(1) comes wih he ransformaions of he las erm. Indeed, if we ake ino accoun ha 61

62 and l +λν(t ) = x+(r q σ )(T ), l+λς(t ) = x+(r q)(t ), wih x given by Eq.(17), hen Eq.(1) reads Θ diff C = qse q(t ) N(d 1 ) rke r(t ) N(d ) (13) σse q(t ) T N (d 1 ), where N (d) = exp( d /)/ is derivaive of N(d) wih respec o d, and d 1 is given by Eq.(06). Therefore, we came o he Black-Scholes Greek hea presened in Table 3. 6