= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

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1 Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ S,, and wih ineres raes r = r which are assumed o be no consan, bu sochasic. Tha is, under he risk neural measure, we ignore dividends ds S = r d + σ S, db S 19.1 wih ineres raes given by a mean revering Ornsein-Uhlenbeck or Vasicek process, dr = r r d + σ r db r 19. wih a consan ineres rae volailiy σ r. This model is in paricular relevan for pricing longer daed opions wih mauriies of 5, 10 or 0 years. The Brownian moions db S and db r may have a non vanishing correlaion, The soluions o 19.1 and 19. are given by > s and db S db r = ρ d 19.3 S = S s e s ru σ S,u /du+ s σ S,udB S u 19.4 r = e s r s + r1 e s + σ r e s eu db r u 19.5 For he model 19.1,19., here is a closed form soluion for plain vanilla calls and pus. For a consan ime independen sock volailiy σ S, = σ S, a derivaion can be found in [1]. For ime dependen volailiy he derivaion is similar. There is he following 153

2 154 Chaper 19 Theorem 19.1: Le, under he risk neural measure, he sock and ineres rae processes S, r be given by 19.1,19. and le g be given by Then he following saemens hold: g := 1 e 19.6 a Le H = HS T, r T be some non exoic payoff wih mauriy T. Then: V S, r,, T := E [e ] T r sds HS T, r T S, r = P, T + v,t φ, r,t φ, ξ e φ +ξ where R H v,t S e P,T r,t φ, ξ := r + e T r r σ r g T dφ dξ π σ r g T [ ρ,t φ + 1 ρ,t 1/ ξ ] 19.8 ρ,t := ρ σ S,u e T u du + σr g T v,t g T 19.9 v,t is given in b below and P, T is he price of a zero bond in he Vasicek model, P, T := E [ e ] r sds r = A, T e gt r r A, T = e σ r g T [T ]} σ r 4 gt b The price of a plain vanilla european call/pu is given by ɛ = +1 for call, ɛ = 1 for pu V S,, T = ɛ S Nɛd + P, T KNɛd } 19.1 where d ± = v,t = [ ] log S/K ± v P,T,T / v,t σ S,u + ρ σ r σ S,u g T u + σ r g T u } du Remarks: i Since ineres raes have dimension 1/ime, he IR volailiy σ r has dimension 1/ime 3. The sock volailiy has dimension 1/ ime and has dimension 1/ime. Thus v,t in above is indeed a dimensionless quaniy as i should be.

3 Chaper ii For consan sock volailiy σ S,u = σ S =: σ, v,t = σ T + ρ σ r σ = σ 1 T + ρ σ σ r T 1 e or, wih τ := T, 1 e T u du + σ r T 1 e T + T + σr T 1 e T + 1 e T 1 e T v = vτ = σ τ + ρ σ σ r τ gτ + σ r τ gτ+g τ and ρ,t = ρτ = ρ σ g τ + σr g τ vτ g τ Proof of Theorem: We have o compue he expecaion V S, r,, T = E [e = E [e r sds H ] r sds HS T, r T S, r S e ru σ S,u /du e ] T σ S,u dbu S, rt S, r wih saisfy he SDE s S T = S e ru σ S,u /du+ σ S,u dbu S r T = e T r + r1 e T + σ r e T T e u db r u wih correlaed Brownian moions We have ds S = r d + σ S, db S 19.0 dr = r r d + σ r db r 19.1 db S db r = ρ d 19. r u du = 1 e T r + r T 1 e T + σr 1 e T v db r v =: µ,t + σ r g T v db r v 19.3

4 156 Chaper 19 where we inroduced he noaion µ,t = 1 e T r + r T 1 e T 19.4 g = 1 e 19.5 Inroducing furher η,t = 1 σ S,u du 19.6 and db r db 19.7 db S = ρ db + 1 ρ dz 19.8 wih wo independen Brownian moions db and dz, he expecaion reads V S, r,, T = e µ,t e σr g T u db u 19.9 H S e µ,t η,t +σ r g T udb u + T σ S,u ρdb u+ 1 ρ dz u, r T dw BdW Z where dw B = dw B u } <u T denoes he Wiener measure wih respec o B. To eliminae he facor e σr g T u db u in he firs line of 19.9, we make he subsiuion of variables d B u = db u + σ r g T udu or, for s, T ], and leave Z unchanged. Then, by Girsanov s Theorem such ha e σr B s = B s + σ r s g T u du g T u db u 1 σ r g T u du dw B = dw B V S, r,, T = e µ,t e 1 σ r g T u du 19.3 H S e µ,t η,t +σ T r g T u[d B u σ rg T u du] σ e S,u ρ[d B u σ rg T u du]+ 1 ρ dz u }, r T dw BdW Z = e µ,t + 1 σ r H g T u du S e µ,t η,t σ r g T u du ρσ r σ S,u g T u du e ρσ S,u +σ rg T u}d B u+ 1 ρ σ S,u dz u, r T dw BdW Z

5 Chaper wih r T given by r T = e T r + r1 e T T + σ r e T u db r u = e T r + r1 e T T + σ r e T u d B u σ r g T udu = e T r + r1 e T σ r g T T + σ r e T u d B u =: q,t + Y,T where we abbreviaed and we used he relaion q,t = e T r + r1 e T σ r g T Y,T = σ r e T u d B u e T u g T udu = e T u e T u du = 1 1 e T = 1 1 e T T 1 e T + e = 1 g T The random variables X,T := ρσ S,u + σ r g T u} d B u + 1 ρ T σ S,u dz u Y,T = σ r e T u d B u are Gaussian wih mean 0 and variances v,t := V[X,T ] = [ρσs,u + σ r g T u ] } + 1 ρ σs,u du w,t := V[Y,T ] = σr e T u du = σr g T CovX,T, Y,T = ρσr σ S,u e T u + σrg T ue T u} du = ρσ r where we used again in he las line. Thus, inroducing F,T := µ,t η,t σ r σ S,u e T u du + σ r g T g T u du ρσ r σ S,u g T u du 19.4

6 158 Chaper 19 and recalling r T = q,t + Y,T, we obain V S, r,, T = e µ,t + 1 σ r g T u du R H S e F,T +X,T, q,t + Y,T e 1 X,T,Y,T C 1 X,T,T Y,T dx de C,T dy,t,t = e µ,t + 1 σ r g T u du H S e F,T + v,t x, q,t + w,t y e 1 x,yr 1,T x y de dx dy R,T π R π where C,T = V[X,T ] CovX,T, Y,T CovX,T, Y,T V[Y,T ] denoes he covariance marix of X,T and Y,T and R,T he corresponding correlaion marix, wih R,T = 1 ρ,t ρ,t = CovX,T, Y,T V[X,T ]V[Y,T ] = ρσ r ρ,t 1 σ S,u e T u du + σ r g T v,t σ r g T = ρ σ S,u e T u du + σr g T v,t g T Transforming o independen Gaussians, V = e µ,t + 1 σ r g T u du S e F,T + v,t x, q,t + w,t ρ,t x + 1 ρ,t 1/ y e x +y R H dx dy π For he special case H 1, and he las equaion reduce o V = P, T = E [ e r udu r ] = e µ,t + 1 σ r g T u du 19.49

7 Chaper Wih ha, we obain where we inroduced e F,T = e µ,t σr g T u du ρσ T r σ S,u g T u du η,t = e 1 σ r m,t := 1 σ r g T u du ρσ r σ S,u g T u du η,t / P, T =: e m,t /P, T From he definiion of v,t in v,t = = such ha, recalling 19.6, g T u du ρσ r σ S,u g T u du η,t [ρσs,u + σ r g T u ] } + 1 ρ σs,u du σ S,u + ρσ r σ S,u g T u + σrg T u } du 19.5 m,t = v,t Thus, now for a general H again, coninuing wih 19.48, S V S,, T = P, T H P,T em,t + v,t x, x,t x, y e x +y R = P, T + v,t x, x,t x, y e x +y R H v,t S e P,T dx dy π dx dy π where x,t x, y = q,t + w,t ρ,t x + 1 ρ,t 1/ y = r + e T r r σ r g T +σ r g T [ ρ,t x + 1 ρ,t 1/ y ] In paricular, for a plain vanilla call HS T = S T K +, he same compuaion as in he Black-Scholes case see chaper 6 gives } S V Call S, r,, T = P, T Nd P,T + KNd = S Nd + P, T KNd where [ d ± = log S /K P,T ] ± v,t v,t 19.57

8 160 Chaper 19 and his proves he heorem. Calibraion of he Model o European Call Opions We assume ha he ineres rae process has already been calibraed o he iniial yield curve by using a suiable shif funcion r 0 d r 0 + ϕ}d wih a piecewise consan ϕ as well as o swapions and caples. Now we wan o fi he model o ATM call opions on S wih he mauriies T 1, T,..., T n. Le σ imp T i be he implied volailiies of hose call opions. We assume ha he insananeous sock volailiy σ S, =: σ is piecewise consan on he inervalls [0, T 1, [T 1, T,...,[T n 1, T n ]. From he Theorem 19.1 above we find ha he implied volailiy σ BSVas,imp squared, imes mauriy of a plain vanilla call in he BS-Vasicek model wih mauriy T is given by v 0,T v T = σbsvas,impt T = 0 σu + ρσ r σug T u + σrg T u } du For piecewise consan σu u [Ti 1,T i =: σ i he above equaion becomes we wrie emporarily σ r insead of σ r for he ineres rae vol, o avoid confusion wih he equiy σ i s v Ti = = i j=1 i j=1 j T j 1 σu + ρσ r σug T i u + σ r g T i u } du σj T j T j 1 + ρσ r Tj σ j T j 1 g T i udu + σ r } T j T j 1 g T i u du wih T 0 := 0. Now he calibraion is done by requiring σ BSVas,imp T i = σ model imp T i! = σ marke imp T i σ imp T i for all i = 1,,..., n. The righ hand side of is aken from marke daa, and he lef hand side is given by he expression Thus we are led o he following sysem of quadraic equaions for σ i : For all i = 1,,..., n σ impt i = i j=1 σj ρσ + σ r Tj j T j T j 1 g T i udu + σr } Tj T j T j 1 g T i u Tj du T i 19.61

9 Chaper where we pu T j = T j T j 1. From his se, σ 1, σ,..., σ n can be deermined inducively saring wih σ 1 : For i = 1, he above equaion simplifies o σimpt 1 = σ1 ρσ + σ r T1 1 T 1 T 0 g T 1 udu + σr } T1 T 1 T 0 g T 1 u du 19.6 which gives σ 1 = ρ σr 1 T 1 T 0 ± g T 1 udu σimp T 1 + ρ σ r T1 T 1 T 0 g T 1 udu σ r T 1 1 T 0 g T 1 u du We have 1 s 1 s 1 1 e s g s udu = 1 s g s u du = e s s + 1 e s s For small x := s < 1 we may Taylor expand o obain 1 e x = 1 1 x+x / x 3 /6+Ox 4 x x = 1 x + x + 6 Ox Thus, 1 s g s udu = x + x + 6 Ox3 x + x + 6 Ox3 = 1 = s 1 g s s u du = 1 + O s [ 1 x + ] x x + x + 3 Ox3 = 1 x 3 + Ox3 = s 3 + O s Realisic daa from IR calibraion are around 10 percen and σ r around 0.5 o 1 percen, a leas in pre-financial-crisis imes 10%, σ r 1% Then, since σ imp T is around 0 o 40 percen, for mauriies less han, say, 5 years such ha T i 0.5 o jusify a Taylor expansion, σ impt 1 σr T 1 1 T 0 g T 1 u du σ impt 1 σ r T 1 /3 σ impt 1 T 1/

10 16 Chaper 19 should be posiive. If his is no he case σ 1 = 0 may be a reasonable choice. Now suppose ha σ 1,..., σ i 1 have already been deermined. Then σ i is obained from σimpt i = σi ρσ + σ r Ti i T i T i 1 g T i udu + σr } Ti T i T i 1 g T i u T du i T i + s i 1 σ 1,..., σ i where we pu s i 1 = i 1 σj ρσ + σ r Tj j T j T j 1 g T i udu + σr } Tj T j T j 1 g T i u Tj du T i 19.7 j=1 If one would subsiue in he above expression T i by T i 1, one would obain he expression for σ impt i 1. Therefore we abreviae Then, σ i = ρ σr + T i i T i σ imp,i T i σ imp,i := σ impt i s i 1 σ 1,..., σ i T i 1 g T i udu ρ σ r Ti T i T i 1 g T i udu σ r Ti T i T i 1 g T i u du if he above expression is a posiive real number and σ i = 0 oherwise. Finally, 1 s 1 s s g T udu = 1 1 e T e s s g T u du = 1 1 e T e T s T s + e T e T s s s

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