Saddlepoint method for pricing European options under Markov-switching Heston's stochastic volatility model

Zhang, Mengzhe; Chan, Leunglung A icle Saddlepoin me hod o p icing Eu opean op ions unde Ma ko -swi ching Hes on's s ochas ic ola ili y model Jou nal o Risk and Financial Managemen P o ided in Coope a ion wi h: MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel Sugges ed Ci a ion: Zhang, Mengzhe; Chan, Leunglung (2022) : Saddlepoin me hod o p icing Eu opean op ions unde Ma ko -swi ching Hes on's s ochas ic ola ili y model, Jou nal o Risk and Financial Managemen , ISSN 1911-8074, MDPI, Basel, Vol. 15, Iss. 9, pp. 1-9, h ps://doi.o g/10.3390/j m15090396 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/274917 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ Ci a ion: Zhang, Mengzhe, and Leunglung Chan. 2022. Saddlepoin Me hod o P icing Eu opean Op ions unde Ma ko -Swi ching Hes on’s S ochas ic Vola ili y Model. Jou nal o Risk and Financial Managemen 15: 396. h ps:// doi.o g/10.3390/j m15090396 Academic Edi o : Thanasis S engos Recei ed: 26 July 2022 Accep ed: 26 Augus 2022 Published: 6 Sep embe 2022 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). Jou nal o Risk and Financial Managemen A icle Saddlepoin Me hod o P icing Eu opean Op ions unde Ma ko -Swi ching Hes on’s S ochas ic Vola ili y Model Mengzhe Zhang 1and Leunglung Chan 2,* 1Psychome ics and Analy ics B anch, NSW Educa ion S anda ds Au ho i y, Sydney, NSW 2001, Aus alia 2School o Ma hema ics and S a is ics, Uni e si y o New Sou h Wales, Sydney, NSW 2052, Aus alia *Co espondence: [email protected] Abs ac : This pape e alua es he p ices o Eu opean-s yle op ions when dynamics o he unde lying asse is assumed o ollow a Ma ko -swi ching Hes on’s s ochas ic ola ili y model. Unde his amewo k, he expec ed e u n and he long- e m mean o he a iance o he unde lying asse ely on s a es o he economy modeled by a con inuous- ime Ma ko chain. The e is e idence ha he Ma ko -swi ching Hes on’s s ochas ic ola ili y model pe o ms well in cap u ing majo e en s a ec ing p ice dynamics. Howe e , due o he na u e o he model, analy ic solu ions o he p ices o op ions o o he inancial de i a i es do no exis . By means o he saddlepoin me hod, an analy ic app oxima ion o Eu opean-s yle op ion p ice is p esen ed. The saddlepoin me hod gi es an e ec i e app oxima ion o op ion p ices unde he Ma ko -swi ching Hes on’s s ochas ic ola ili y model. Keywo ds: Eu opean-s yle op ions; Ma ko -swi ching Hes on’s s ochas ic ola ili y model; saddlepoin me hod; Ma ko chain 1. In oduc ion I is well documen ed ha he s ochas ic ola ili y (SV) models ake he ola ili y smile e ec in o accoun in he eal ma ke s. Needless o say, he SV models a e mo e ealis ic han he s anda d Black–Scholes model. Ne e heless, Vo (2009) conduc ed an empi ical s udy on a Ma ko -swi ching s ochas ic ola ili y model and showed ha he beha iou o c ude oil p ice is well explained unde he Ma ko -swi ching SV model. A signi ican egime-swi ching e ec on he oil ma ke s is well documen ed and hei nume ical esul s e eal ha he o ecas ing powe o he SV model wi h Ma ko swi ching ou pe o ms i s non-swi ching coun e pa . The empi ical s udy also inds ha he SV model wi h Ma ko swi ching is able o cap u e la ge shocks o he oil ma ke s well. These ad an ages o empi ical indings gi e us a mo i a ion o p icing op ions unde SV models wi h Ma ko swi ching. This pape shows how o p ice a Eu opean-s yle op ion in Ma ko -swi ching Hes on’s s ochas ic ola ili y model. Unde he Ma ko -swi ching Hes on’s s ochas ic ola ili y model, he key pa ame e s o he model ollow Ma ko chains which swi ch om a s a e o ano he s a e as ime e ol es. The Ma ko -swi ching ( egime-swi ching) beha iou could e lec he dynamic p e e ences and ime- a ying belie s o ade s o he changing le els o economy ac i i ies. In gene al, he Ma ko chain is unobse able and he chain is modeled by a s ochas ic di e en ial equa ion (SDE). Zhu e al. (2012) de i e an explici Eu opean op ion p icing o mulas in a wo-s a e egime by sol ing a couple o pa ial di e en ial equa ions (PDEs) using he Fou ie ans o m me hod. Chan and Zhu (2021b) ob ain a closed- o m o mulas o Lookback op ions unde a Ma ko -swi ching Wiene p ocess. Chan and Zhu (2015) de i e a closed- o m o mula o Ame ican con e ible bonds in a Black–Scholes–Me on’s model wi h egime swi ching. Chan and Zhu (2021a) ob ain an analy ic app oach o p icing Ame ican op ions wi h Ma ko swi ching. Ellio e al. J. Risk Financial Manag. 2022,15, 396. h ps://doi.o g/10.3390/j m15090396 h ps://www.mdpi.com/jou nal/j m J. Risk Financial Manag. 2022,15, 396 2 o 9 (2013) p ice op ions unde he CEV model wi h Ma ko swi ching. Chan and Zhu (2015) conside a homo opy analysis me hod o p ice ba ie op ions unde he Ma ko -swi ching Wiene p ocess. Ellio e al. (2014) use a quad a ic app oxima ion app oach o p ice ba ie op ions unde he Ma ko swi ching model. Bollen (1998) calcula es a Eu opean op ion p ice unde he Ma ko swi ching model ia a binomial ee. Ha dy (2001) p oposes a ecu si e algo i hm o p ice Eu opean op ions unde he Ma ko swi ching model. Duan e al. (2002) use a la ice-app oxima ion me hod o p ice Eu opean-s yle and Ame ican- s yle op ions unde he Ma ko swi ching model. Boyle and D a iam (2007) use a ini e di e ence scheme o p ice exo ic op ions unde he Ma ko swi ching model. Li e al. (2012) de i e he bounds o exo ic op ion p ice using semide ini e p og amming (SDP) unde he Ma ko swi ching model. Lu and Pu i (2020) e alua e Ma ko -swi ching Ame ican op ion ia Laplace ans o m. Egami and Ke khish ili (2020) simpli y an op imal s opping p oblem o a wo-s a e Ma ko -swi ching model o a pai o no-swi ching op imal s opping p oblems. Ellio and Lian (2013) in es iga e he p icing o a iance swaps unde a Ma ko -swi ching s ochas ic ola ili y model. The saddlepoin me hod has been widely used in app oxima ing he p obabili y dis ibu ion o he pa ial sum o independen andom a iables since i was p oposed by Daniels in 1954. In his pape , ins ead o app oxima ing dis ibu ions, we use he Lugannani and Rice (LR) o mula (Daniels 1987) o app oxima e ail p obabili ies o loga i hm o he unde lying asse . In o de o ob ain he p ice dynamics unde a ma ingale measu e, we adop an Essche ans o m (Ellio e al. 2005). Once we ob ain he closed- o m cumulan gene a ing unc ions (CGFs) o ln(S ) unde he ma ingale measu e Q and he physical p obabili y measu e P , espec i ely, hen he saddlepoin equa ion can be sol ed ia any symbolic compu ing package, such as Maple. The emaining p ocedu e o p icing op ions is jus simple algeb aic manipula ions. Glasse man and Kim (2009) use saddlepoin me hods o calcula e a Eu opean call op ion in a jump-di usion amewo k. Zhang and Chan (2016) ob ain saddlepoin app oxima ion o Eu opean call op ions in a Ma ko - swi ching model and ou cu en wo k ex ends hei wo k o a Ma ko -swi ching SV model. The pape is o ganized as ollows. Sec ion 2 e iews Ma ko -swi ching Hes on’s s ochas ic ola ili y model. Sec ion 3discusses he saddlepoin me hod and LR o mula. Sec ion 4de i es he cumulan gene a ing unc ions o he model unde di e en p obabili y measu es. Sec ions 5and 6 e e o nume ical esul s and conclusion, espec i ely. 2. Ma ko -Swi ching Hes on’s S ochas ic Vola ili y Model A e adop ing an Essche ans o m (Ellio e al. 2005), he dynamics o unde lying asse and he ola ili y in he Ma ko -swi ching Hes on’s s ochas ic ola ili y model unde a ma ingale measu e, P, a e assumed o ollow he ollowing SDEs: dS = S d +√σ S dWS , (1) dσ =κ(θ? −σ )d +σ √σ dWσ . (2) He e is he sho a e, θ is he long- e m mean o he a iance, κ is a mean- e e ing speed pa ame e o he a iance, σ is he ola ili y o ola ili y, µ is he mean a e o e u n and θ? :=θ −ρσ (µ − ) . (dWS , dWσ ) is a wo-dimensional B ownian mo ion wi h dWS ,dWσ =ρ . We de ine Y =ln(S ) and apply I o’s o mula o Y o ge a new p ocess (Y,σ) wi h dY = ( −1 2σ )d +√σ S dWS , (3) dσ =κ(θ? −σ )d +σ √σ dWσ . (4) F om Hes on (1993), he alue o a Eu opean call op ion is in o m o C( ,S) = EQexp(− (T− ))(ST−K)+ =S0Q(YT>ln(K)) −Kexp(− (T− ))P(YT>ln(K)) . (5) J. Risk Financial Manag. 2022,15, 396 3 o 9 He e Q is de ined by he measu e change dQ dP=e− (T− )eYT−Y0 , which uses ST as a nu- me ai e asse . The dynamics o (Y,σ) can be w i en as dY = ( +1 2σ )d +√σ S dWS,Q , (6) dσ = (κθ? −(κ−ρσ )σ )d +σ √σ dWσ,Q . (7) He e WS,Q and Wσ,Q a e s anda d B ownian mo ions unde Q wi h a co ela ion pa ame e ρ . To make he pa ame e s depend on he chains, we in oduced he no a ion in he ollowing pa ame e s µ ,θ and whe e µ :=hµ,X i, :=h ,X i, θ :=hθ,X i. He e h.i deno es an inne p oduc . X is a con inuous- ime Ma ko chain. We assume ha he Ma ko chain X is i educible. Wi hou loss o gene ali y, we can iden i y he s a e space o he chain X wi h he ini e se o uni ec o s E:={e1 , e2 , ··· , eN} , whe e ei:= ( 0, ··· , 1, ··· , 0 )0∈ <N . F om Ellio e al. (1994), he semi-ma ingale ep esen a ion o he chain is gi en by X: X =X0+Z 0A0X du +M . (8) He e M is a ma ingale and A0 is he anspose o A . A0:= [λij]i,j=1,2,···,N deno es he in ensi y ma ix o he chain X , λij is he cons an a e o ansi ion o he chain X om s a e ei o s a e ej . Fo mo e de ails abou he Ma ko -swi ching model, see Guo (2001) and Bu ing on and Ellio (2002). 3. Saddlepoin Me hods The saddlepoin app oxima ion was p oposed by Daniels in 1954 and is used o app oxima e he p obabili y densi y unc ion o he sum ¯ Y=∑n 1Yk/n , whe e Y0 ks a e iden ically independen dis ibu ed andom a iables. Assume he cumulan gene a ing unc ion G(z)o ¯ Yis known, he p obabili y densi y n(¯ y)o ¯ Yis in he o m o n(¯ y) = n 2πiZτ+i∞ τ−i∞exp nG(z)−z¯ ydz, (9) o any τ∈ {y∈ < :|G(¯ y)|<∞} . Th ough he s eepes descen me hod, Daniels (1954) ga e an asymp o ics o he in eg al (9). The app oxima ion ˆ n(¯ y) could be compu ed by selec ing he pa h o pass h ough he saddle poin ¯ z such ha G0(¯ z)−¯ y= 0. The modulus o he in eg and a ains a maximum a ¯ z as he unc ion G(z)−z¯ y has a minimum a ¯ z . While he in eg and does no make a con ibu ion apa om a neighbou hood o he saddlepoin . Consequen ly, highe o de e ms in he expansion can be igno ed wi hou losing much p ecision. Daniels (1987) ob ained a p obabili y app oxima ion o mula based on he Lugannani– Rice (LR) o mula. Compa ed o hose o he Edgewo h expansion, he e o o he LR o mula is almos uni o mly dis ibu ed o e he whole ange o he expec a ion ¯ y . The LR o mula is gi en by P(¯ Y>¯ y) = 1−Φ(√nˆ w) + φ(√nˆ w)b0 n1/2 +b1 n3/2 +o(n−3/2), (10) whe e b0= 1 /ˆ u− 1 /ˆ w , b1= (λ4/ 8 − 5 λ2 3/ 24 )/ˆ u−λ3/( 2 ˆ u2)− 1 /ˆ u3+ 1 /ˆ w3 , ˆ w=sgn(¯ z)p2(¯ zy −G(¯ z)) , ˆ u=¯ zpG00(¯ z) , λ3=G(3)(¯ z)/G00(¯ z)3/2 and λ4=G(4)(¯ z)/G00(¯ z)4/2. J. Risk Financial Manag. 2022,15, 396 4 o 9 He e Φ , φ a e he cumula i e dis ibu ion unc ion and he p obabili y densi y o he s anda d no mal dis ibu ion, espec i ely. 4. P icing Eu opean Op ions in Ma ko -Swi ching Hes on’s Model I he unde lying asse ollows dynamics (1) and (2), he p ice o a Eu opean call op ion a ime ze o is in he o m o C( ,S) = EQexp(− (T− ))(ST−K)+ =S0Q(YT>ln(K)) −Kexp(− (T− ))P(YT>ln(K)) . (11) The CGF o YT unde P is deno ed by eG(z,Y, ,σ)=E[ezYT] . Deno e FS j , Fσ j and FX by he na u al il a ions gene a ed by he B ownian mo ions WS , Wσ and he Ma ko chain X , up o ime , espec i ely, we ha e FS 1 :=ˆ σ{WS u|u≤ }, Fσ 1 :=ˆ σ{Wσ u|u≤ }, FX :=ˆ σ{Xu|u≤ } and FS 2 :=ˆ σ{WS,Q u|u≤ }, Fσ 2 :=ˆ σ{Wσ,Q u|u≤ }. He e ˆ σ is he smalles σ - ield. To ob ain he CGF we calcula e he expec a ion by enla ging he il a ion, whe e we assume FX T is known. In such a case, all pa ame e s depending on he Ma ko chain X would degene a e o de e minis ic unc ions o ime. Fo ins ance, gi en FX T, he CGF is gi en by he ollowing lemma. Lemma 1. I he unde lying asse ollows he dynamics (1) and (2), and FX T is gi en, hen he CGF o he s ochas ic a iable YT=ln(ST)is gi en by j(z,Y, ,σ)|FX T) = EQhezYT|FS j ∨Fσ j ∨FX Ti =eF(z, ,T)+H(z, ,T)σ+zY, (12) whe e F(z, ,T)and H(z, ,T)a e gi en by F(z, ,T) = ZT h z +κθ?H(z, ,T),Xsids, H(z, ,T) = bj −ρσ z+d σ2 [1−ed(T− ) 1−ged(T− )], g=bj−ρσ z+d bj−ρσ z−d, d=q(ρσ z−bj)2−σ2 (2ujz +z2), b1=κ,b2=κ−ρσ , u1=−1 2,u2=1 2. P oo . When he p ice dynamics ollow (1) and (2), Hes on (1993) demons a es ha he alue o any asse U(S,σ, )mus sa is y he pa ial di e en ial equa ion (PDE) J. Risk Financial Manag. 2022,15, 396 5 o 9 1 2σS2∂2U ∂S2+ρσσ S∂2U ∂S∂σ +1 2σ2 σ∂2 ∂σ2+ S ∂U ∂S+κ(θ?( )−σ)∂U ∂σ − U +∂U ∂ =0. (13) Thus we guess he solu ion o he model has he o m C(S,σ, ) = SP2−Ke− TP1. (14) Bo h o P1 and P2 mus sa is y he o iginal PDE (13). Subs i u ing he p oposed solu ion (14) in o he o iginal PDE (13) shows ha P1and P2mus sa is y he PDEs 1 2σ∂2Pj ∂Y2+ρσσ S∂2Pj ∂Y∂σ +1 2σ2 σ∂2Pj ∂σ2+ ( +ujσ)∂U ∂Y+ (κθ? −bjσ)∂U ∂σ +∂U ∂ =0, (15) o j= 1, 2 and whe e b1=κ , b2=κ−ρσ , u1=−1 2 , u2=1 2 . Pj could be ea ed as he condi ional p obabili y ha he op ion expi es in- he-money, Pj(Y,σ,T, ln[K]) = P [YT≥ln[K]|Y ,σ ,X ]. (16) We can sol e PDE (15) by i s guessing ha he a ine o m solu ion migh be j(z,Y, ,σ)|FX T) = eF(z, ,T)+H(z, ,T)σ +zY . (17) The unc ions F(z, ,T)and H(z, ,T)can be ound by sol ing wo Ricca i ODEs: −∂F ∂ = z+κθ? F(18) −∂H ∂ =1 2z2+ujz+ (ρσ z−bj)H+1 2σ2 H2(19) wi h ini ial condi ions F(z, ,T) = 0 and H(z,T,T) = 0. The solu ions a e F(z, ,T) = ZT h z +κθ?H(z, ,T),Xsids, H(z, ,T) = bj −ρσ z+d σ2 [1−ed(T− ) 1−ged(T− )], g=bj−ρσ z+d bj−ρσ z−d, d=q(ρσ z−bj)2−σ2 (2ujz +z2), b1=κ,b2=κ−ρσ , u1=−1 2,u2=1 2. Lemma 2. I he unde lying asse ollows he dynamics (1) and (2), hen he MGF o he s ochas ic a iable y( ,T) = RT σ(s)ds is gi en by j(z,Y, ,σ) = hΦ(z, ,T)X , Idi×eH(z, ,T)σ +zY , (20) J. Risk Financial Manag. 2022,15, 396 6 o 9 whe e Φ(z, ,T) = exp(A0(T− ) + diag[ z(T− ) +κθ? σ2 ((bj−ρσ z−d)(T− )−2 ln[1−ed(T− ) 1−ged(T− )])]), H(z, ,T) = bj −ρσ z+d σ2 [1−ed(T− ) 1−ged(T− )], g=bj−ρσ z+d bj−ρσ z−d, d=q(ρσ z−bj)2−σ2 (2ujz +z2), b1=κ,b2=κ−ρσ , u1=−1 2,u2=1 2. P oo . In o de o de i e he uncondi ional MGF, we need o calcula e he expec a ion wi hou condi ioning, whe e θ? elies on he chain o X p ocess up o ime T . Consequen ly, we w i e j(z,Y, ,σ) = EezYT|FS ∨Fσ ∨FX  =E[EezYT|FS ∨Fσ ∨FX T|FS ∨Fσ ∨FX ] =E[eF(z, ,T)+H(z, ,T)σ +zY |FS ∨Fσ ∨FX ] =E[RT h z +κθ?H(z, ,T),Xsids|FS ∨Fσ ∨FX ]×eH(z, ,T)σ +zY . (21) By he P oposi ion 3.2 in Ellio and Lian (2013), we ha e E[ZT h z +κθ?H(z, ,T),Xsids|FS ∨Fσ ∨FX ] = hΦ(z, ,T)X , Idi, (22) whe e Φ(z, ,T)is an N-by-N<- alued ma ix gi en by Φ(z, ,T) = exp ZT A0+diag[ z +κθ?H(z, ,T)]ds) =exp(A0(T− ) + diag[ z(T− ) +κθ? σ2 ((bj−ρσ z−d)(T− )−2 ln[1−ed(T− ) 1−ged(T− )])](23) Id= (1, 1, . . . 1)∈ <Nand A0deno es he anspose o A. So, j(z,Y, ,σ) = hΦ(z, ,T)X , Idi×eH(z, ,T)σ +zY . (24) A las Gj(z , Y , , σ) = ln[ j(z , Y , , σ)] = lnhΦ(z, ,T)X , Idi+H(z , , T)σ +zY . Using his CGF, we can apply he saddlepoin app oxima ion o mula p esen ed in he p e ious sec ion o calcula e op ion p ice. C(S,σ, ) = SP2−Ke− TP1. (25) 5. Nume ical Examples In his sec ion, we gi e some examples o p ice Eu opean-s yle call op ions unde Ma ko -swi ching Hes on’s s ochas ic ola ili y model using he saddlepoin me hod. Wi hou he loss o gene ali y, we conside wo- egime case: he i s egime ( X = 1) e e s o he ‘Booms’ s a e o economy and he second egime ( X = 2) e e s o he ‘Recessions’ J. Risk Financial Manag. 2022,15, 396 7 o 9 s a e o economy. The pa ame e s se ing in ou example a e S0 = 100, κ = 2, ρ = − 0.2, σ = 0.2, = [0.03; 0.03], µ= [0.08; 0.04], and θ= [0.009; 0.004]. The ansi ion a e ma ix o he Ma ko chain A is [ − 1, 1; 1, − 1]. Assuming ha he cu en s a e is X0= 1 and ha he cu en ola ili y le el is V0= 0.04. The saddlepoin can be ound by sol ing he saddlepoin equa ion, G0 j(z)−ln[K] = 0, h ough symbolic compu ing so wa es like Ma hema ica o Maple. The esul s o a wo- egime case a e documen ed in Table 1. He e MCi deno es Mon e-Ca lo esul s based on cu en s a e i . We se sample pa hs o Mon e-Ca lo simula ion o 20,000 and he compu a ion is pe o med using Ma lab R2010b p og ams as he benchma k. SA 1 i and SA 2 i e e s o he ASP me hod based on he cu en s a e i wi h i s o de and second o de , espec i ely. The esul s show ha ‘ SA 2 i ’ would de ini ely ou pe o m ‘ SA 1 i ’ o all ange o ma u i y. Table 2demons a es op ion p ices wi h di e en s ikes unde Ma ko -swi ching Hes on’s s ochas ic ola ili y model as well as i s non-swi ching Hes on’s coun e pa . In a non-swi ching Hes on’s model, we assume all pa ame e s’ alue equals he pa ame e s’ alue in he i s egime o he Ma ko -swi ching Hes on’s model. The nume ical esul s a e shown in Figu e 1and i clea ly shows ha ou app oxima ion is qui e p ecise. We also no ice ha he op ion p ices calcula ed om he Ma ko -swi ching Hes on’s model a e cheape han hei non- swi ching coun e pa s o e all ange o s ike p ices gi en in he Table 2. This is because pa ame e s’ alue in a egime-swi ching Hes on’s model is smalle . As θ1<θ2 and µ1<µ2 , so 1 TRθ1d >1 TRθ d and 1 TRµ1d >1 TRµ d . Figu e 2is he plo o he implied ola ili y. The ma u i y ime is ixed a T = 1 and lea es o he pa ame e s unchanged. When ρ is ze o we ge a ola ili y smile and he implied ola ili y s a s o inc ease a e he s ike p ice is g ea e han he ini ial s ock p ice. A nega i e ρ has an ob ious impac on he shape o he ola ili y cu e: he ola ili y smile becomes a smi k. I seems ha he changes in he ini ial egime only has an impac on he alues o implied ola ili y, bu no he shape o ola ili y cu e. Table 1. Call op ion p ices (N = 2). Fi s O de Sec O de T(yea ) MC1MC2SA11SA12SA21SA22 0.1 8.5315 8.2114 8.4556 8.0806 8.5043 8.1738 0.2 8.4257 7.8141 8.0043 7.3541 8.3253 7.7503 0.5 8.0846 7.1332 7.2920 6.2731 7.9300 7.0056 1 7.7494 6.7888 6.8765 5.8592 7.5898 6.6773 Table 2. Call op ion p ices when T = 1, N = 2. wi h Regime-Swi ching wi h Regime-Swi ching wi hou Regime-Swi ching KMC1SA21Fi s S a e 70 32.0532 32.1250 32.3157 80 22.8978 22.6910 23.3394 90 14.0194 14.0200 15.4829 100 7.1860 7.1398 9.3352 110 3.0066 2.7466 5.1183 120 1.0630 0.9057 2.5812 130 0.3573 0.2758 1.2193 140 0.1071 0.0829 0.1175 J. Risk Financial Manag. 2022,15, 396 8 o 9 70 80 90 100 110 120 130 140 0 5 10 15 20 25 30 35 S ike P ice K Op ion P ices P ice wi h egime swi ching (MC) P ice wi h egime swihcing (SA2) P ice wi hou egime swi ching Figu e 1. A pic u e o a Table 2. 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18 K/S0 Implied ola ili y Ini ial Regime = Regime 1 ho = −0.2 ho = 0 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 K/S0 Implied ola ili y Ini ial Regime = Regime 2 ho = −0.2 ho = 0 Figu e 2. Vola ili y Smile (T = 1). 6. Conclusions This pape s udies a saddlepoin app oxima ion app oach o p icing op ions in a Ma ko -swi ching Hes on’s s ochas ic ola ili y model. Ou app oach equi es o de i e a closed- o m exp ession o cumulan gene a ing unc ion, which needs o sol e pa ial di e en ial equa ions. The nume ical esul s in Sec ion 5examine he accu acy o ou me hod and a wo-s a e case has been conside ed. The esul s show ha saddlepoin me hod gi es qui e accu acy o he gi en ange o ma u i ies. Howe e , i he closed- o m cumulan gene a ing unc ion does no exis , hen one needs o ely on a nume ical me hod o ind a saddle poin . In his si ua ion, he p ocedu e becomes slowe . In he u u e, i is wo h conside ing mul i-s a e Ma ko chain case and de eloping an algo i hm o bea he cu se o dimensionali y. Fu he mo e, p icing Ame ican op ion unde