= 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

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.

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.

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 19.7 + σ r g T [ ρ,t φ + 1 ρ,t 1/ ξ ] 19.

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

16 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 19.

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

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.

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

36 The random variables X,T := ρσ S,u + σ r g T u} d B u + 1 ρ T σ S,u dz u 19.37 Y,T = σ r e T u d B u 19.

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

45 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 19.

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

5 m,t = v,t 19.53 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 19.

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

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.

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/

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 19.63 We have 1 s 1 s 1 1 e s g s udu = 1 s g s u du = 1 1 1 e s s + 1 e s s 19.64 19.

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

71 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.

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.