Pricing the American Option Using Itô s Formula and Optimal Stopping Theory

Transcription

1 U.U.D.M. Projec Repor 2014:3 Pricing he American Opion Uing Iô Formula and Opimal Sopping Theory Jona Bergröm Examenarbee i maemaik, 15 hp Handledare och examinaor: Erik Ekröm Januari 2014 Deparmen of Mahemaic Uppala Univeriy

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3 PRICING THE AMERICAN OPTION USING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY JONAS BERGSTRÖM Abrac. In hi hei he goal i o arrive a reul concerning he value of American opion and a formula for he perpeual American pu opion. For he ochaic dynamic of he underlying ae I look a wo cae. The fir i he andard Black-Schole model and he econd allow for he ae o jump o zero i.e defaul. To achieve he goal aed above he fir couple of ecion inroduce ome baic concep in probabiliy uch a procee and informaion. Before inroducing Iô formula hi paper conain a no oo rigorou inroducion of ochaic differenial equaion and ochaic inegraion. Then he Black-Schole model i inroduced followed by a ecion abou opimal opping heory in order o arrive a he American opion. Acknowledgemen. I would like o hank my upervior Erik Ekröm for hi uppor. Hi inpu ha been eenial for he progre and rucure of hi hei. 1. Inroducion Thi ecion conain he definiion of one of he mo imporan building block in coninuou probabiliy namely Wiener proce. Thi i followed by definiion of informaion and maringale. Definiion 1.1. A Wiener proce, W = {W ; 0}, aring from W 0 = 0 i a coninuou ime ochaic proce aking value in R. W ha independen incremen i.e W v W u and W W are independen whenever u v W + W N(0, ) Definiion 1.2. The ymbol F W denoe he informaion generaed by he proce W = {W ; 0} for [0, ]. If Y i a ochaic proce. Y F W, hen Y i aid o be adaped o he filraion F W. Thi imply mean ha Y can be oberved a ime. For example le Y := e r W. Then Y F W. Thi i becaue, given he rajecory of W beween 0 and, he value of Y can be deermined. If we define Y := e r W +ɛ, ɛ > 0 hen Y / F W ince W +ɛ exi in he fuure beyond our informaion a ime. 1

4 2 JONAS BERGSTRÖM Definiion 1.3. (Maringale) The proce X F W if i called a Maringale E[ X ] < E[X F W ] = E[X ], X F W i called a Submaringale if E[X F W ] E[X ],. X F W i called a Supermaringale if E[X F W ] E[X ],. 2. Sochaic Differenial Equaion Le X be a ochaic proce ha reemble he value of an ae. Wha i a reaonable way o mahemaically conruc price evoluion in coninuou ime? Tha i, wha can be aid abou dx = X +d X? One can aume ha X hould change proporionally wih he incremen of ime, d. Driven by he ae fundamenal value and marke expecaion, d, hould be amplified by a deerminiic funcion of he ae. Call hi funcion u(x ). Thu one arrive a dx = u(x )d. To make hi model more realiic one alo add a non-deerminiic erm dw = W +d dw N(0, d) ha i amplified by a deerminiic funcion σ ha depend on he ame variable a u. The reul i wha i called a ochaic differenial equaion (SDE) ha decribe he local dynamic of a ochaic proce in coninuou ime { dx = u(x )d + σ(x )dw X 0 = x were he oluion o hi yem i X = x + 0 u(x ) d + 3. ochaic inegral 0 σ(x ) dw Thi ecion i devoed o give a good inerpreaion of 0 f() dw. If one aume ha f i a imple funcion over [0, ], meaning ha [0, ] can be pli in maller inerval were f i equal o ome conan on repecive inerval, hen one can formulae he ochaic inegral a where 0 = 0 <... < n =. 0 n 1 f() dw = f( k )(W k+1 W k ) k=0

5 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY3 For a non-imple f one creae a equence of imple funcion, f n wih cerain properie. 0 f() dw = lim n 0 f n ()dw. Propoiion 3.1. For any proce f, wih condiion hen E[f 2 ] < f(τ) i adaped o F W τ E[ f(τ)dw τ F W ] = 0 Proof. In hi proof one aume ha f i imple becaue he full proof i ouide of he cope of hi hei. From he law of ieraed expecaion i i rue, for <, ha E[E[ f(τ)dw τ F W ] F W ] = E[ f(τ)dw τ F W ] Now looking a he lef-hand ide inide he fir expecaion i follow ha E[ n 1 f(τ)dw τ F W ] = E[f(τ k )(W τk+1 W τk ) F W ] k=0 where = τ 0 <... < τ n =. Since f(τ k ) depend on he value of he proce W from o τ k (which are all given by F W, for k. 0 k n). Becaue of independen incremen W τk+1 W τk doe no depend on he inerval [, τ k ] and hence i independen of f(τ k ). Then i follow ha E[f(τ k )(W τk+1 W τk ) F W ] = E[f(τ k ) F W ]E[W τk+1 W τk F W ] = 0 E[ f(τ)dw τ F W ] = 0 E[ f(τ)dw τ F W ] = E[0 F W ] = 0 Theorem 3.2. Aume ha dx = u(x )d + σ(x )dw. If u = 0 P-a. X i a maringale. Proof. dx = u(x )d + σ(x )dw have he oluion X = X + u(x τ ) dτ + σ(x τ )dw τ,.

6 4 JONAS BERGSTRÖM Taking expeced value yield E[X F W ] = E[X F W ] + E[ = X + E[ u(x τ ) dτ F W ] u(x τ ) dτ F W ] + E[ σ(x τ )dw τ F W ] = X 4. Iô Formula Iô formula help o give a decripion of he local dynamic of a ochaic proce ha i a funcion of an underlying proce wih a given ochaic differenial. Theorem 4.1. (Iô Formula) Aume ha he ochaic proce X = {X ; 0} ha he differenial dx = u(x )d + σ(x )dw where u,σ F W. Define a new ochaic proce f(, X ), where f i aumed o be mooh. Given he muliplicaion able (d) 2 = 0 d dw = 0 (dw ) 2 = d he ochaic differenial for f become (1) df = f f d + x dx f 2 x 2 (dx ) 2 or equivalenly df = ( f + u f x σ2 2 f f )d + σ x2 x dw Proof. To prove ha (1) hold for all we look a he aylor expanion around he fixed poin (, x). f(+h, x+k) = f(, x)+h f f (, x)+k x (, x)+1 2 (h2 2 f 2 +2hk 2 f x +k2 2 f 2 )+I I = 1 3! (h + k x )3 f( + h, x + k) where 0 < < 1. Now le h d and k dx o ge f(+d, x+dx ) = f(, x)+ f d+ f x dx f 2 2 (d)2 + 2 f x d dx f 2 x 2 (dx ) 2 +I

7 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY5 Since (d) 2 = 0, d dx = d(ud + σdw ) = 0 and he fac ha ( d + x dx ) 3 = 0 i follow ha df = f( + d, x + dx ) f(, x) = f Now we obain f d + x dx f 2 x 2 (dx ) 2 (dx ) 2 = u 2 (d) 2 + 2uσd dw + σ 2 (dw ) 2 = σ 2 d df = ( f + u f x σ2 2 f f )d + σ x2 x dw Theorem 4.2. (Feynman-Ka c) Aume ha F olve he boundary value problem { F + u F x σ2 2 F rf = 0 x 2 F (T, ) = Φ(). Alo aume ha E[ σ(x ) F x (, X )e r 2 ] < and σ(x ) F x e r i adaped o F W. Aume ha X i he oluion o { dx = u(x )d + σ(x )dw Then i follow ha where X = x F (, x) = e r(t ) E.x [Φ(X T )] E,x [. ] = E[. X = x]. Proof. To prove hi define a new ochaic proce and ue Iô formula. f(, X ) = e r F (, X ) df = e r ( rf + F + u F x σ2 2 F F )d + σe r x2 x dw where u = u(x ), σ = σ(x ), F = F (, X ) ec. The oluion o hi SDE i f(t, X T ) = f(, X )+ T e r ( rf + F +u F x +1 2 σ2 2 F x 2 )d+ where he inegrand are evaluaed a (, X ). Since F olve he PDE above i follow ha f(t, X T ) = f(, X )+ T T r F σe x dw e rt F (T, X T ) = e r F (, x)+ r F σe x dw T σe r F x dw

8 6 JONAS BERGSTRÖM T F (, x) = e r(t ) r( ) F F (T, X T ) σe x dw Taking expeced value yield F (, x) = e r(t ) E[Φ(X T )] 5. Ae Dynamic in he Black-Schole model Wihin he framework of he Black-Schole(BS) model here exi wo ae: A ock and a bond Bond. Bond have he following dynamic in he BS-model: Noe ha hi i equivalen o db = rb d db = rb B = B 0 e r d and ha he local rae of reurn i equal o: db B d = r 5.2. Sock. Sock are aid o have he following dynamic in he BS-model: dx = ux d + σx dw. Here he conan u and σ are, repecively, he local mean of reurn and he volailiy of X. In conra o a bond he local rae of reurn on a ock i ochaic: dx X d = u + σ dw d Theorem 5.1. The oluion o he equaion { dx = rx d + σx dw X 0 = x 0 i given by X = x 0 exp((r 1 2 σ2 ) + σw ) and in addiion we have E[X ] = x 0 e r Noe: Here he drif erm of X i r raher han u. Thi i becaue of o called rik neural valuaion which i explained in heorem 6.2. proof Ue Iô formula on f(, W ) = x 0 exp((r 1 2 σ2 ) + σw ). f(, x) = x 0 exp((r 1 2 σ2 ) + σx)

9 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY7 f = (r 1 2 σ2 )f(, x), f x = σf(, x), 2 f x 2 = σ2 f(, x) df = f(, x)((r 1 2 σ2 )d + σdw σ2 (dw ) 2 ) From he muliplicaion able in heorem (4) i follow ha df = f(, x)(rd + σdw ) dx = rx d + σx dw To prove he econd claim rewrie he expreion above a X = x 0 + Now ake expeced value o ge 0 rx τ dτ + E[X ] = x 0 + r 0 0 σx τ dw τ E[X τ ] dτ m() := E[X ] m() = x 0 + r 0 m(τ) dτ dm d = rm() m() = er m(0) = e r x 0 6. Opion and Opion Pricing An opion i a conrac derived from ome underlying ae which give he holder of he conrac he righ (bu no he obligaion) o buy or ell he underlying ae for a deermined amoun, called he exercie price. The opion i called European if he conrac can only be ued a a pecific ime in he fuure, called mauriy. If he opion can be exercied a any ime beween oday and mauriy hen he opion i called American. The righ o buy i called a call opion and he righ o ell i called a pu opion. Definiion 6.1. A coningen claim, χ, i a random variable ha i adaped o FT X where T i called he mauriy, i.e a ime T he payoff of χ can be deermined by looking a proce X = {X ; 0}. A claim i aid o be imple if χ = Φ(X T ), where Φ i called a conracfuncion. The call opion and he pu opion are wo imple coningen claim who ha conrac funcion max{x K, 0} and max{k x, 0} repecively, where K i he rike price. To price an opion one fir aume ha he marke where financial ae are raded i complee, meaning ha every payoff rucure of a conrac can be replicaed uing bond and ock. Anoher imporan aumpion i he abence of arbirage opporuniie in he marke. Thi aumpion lead o he concep of a o called rik free meaure when aking expeced value of ae price in he BS-model.

10 8 JONAS BERGSTRÖM Theorem 6.2. Le X reemble a ock price wih dynamic dx = ux d + σx dw. Aume ha X i raded a a complee marke ha i free of arbirage. If one ake expeced value, X mu have he following dynamic dx = rx d + σx dw in order o make he marke free of arbirage. meaure. Thi i called a rik free The heorem ay ha if, while aking an expeced meaure, he drif erm i anyhing oher hen r here will exi an arbirage opporuniy. Proof. f(, X()) := e r X where X ha he following dynamic when aking expeced value dx = ûx d + σx dw. In hi proof one can aume ha û > r. By Iô formula one arrive a df = ( f + ûx f x σ2 X 2 which ha he oluion e r X = e rτ X τ + ( f + ûx f x σ2 X 2 τ 2 f f )d + σ x2 x dw 2 f x 2 )d + By replacing X by he deerminiic funcion x i follow ha f(, x) := e r x, f = re r x, f x = e r, 2 f x 2 = 0. Inering hi ino he inegrand in he expreion above yield e r X = e rτ X τ + and by aking expecaion i follow ha (2) E[e r( τ) X F τ ] = X τ + (û r) τ (û r)xe r d + τ τ τ σe r dw σ f x dw xe r( τ) d > X τ. τ The lef hand ide can be een a he value of a conrac a ime τ ha give X a ime. The expreion ay ha hi conrac i greaer han X τ even hough hey have he ame payoff rucure. Now i exi an arbirage opporuniy in he marke. By horening he conrac a ime τ, one can immediaely buy X τ while ill having capial a one dipoal. A ime one cloe he hor by elling he ock a value X and pay for he value of he conrac a ime wich i X hence one make a rik free profi. If however we had he rik free meaure i.e û = r hen (2) become E[e r( τ) X F τ ] = X τ

11 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY9 which i a correc pricing of uch a conrac. Hence no arbirage opporuniy exi. Noe: The la expreion in he proof ay ha anding a ime τ he value of a ock i equal o he dicouned expeced value of he ock given all informaion available a ime τ. Thi i alo in line wih he o called efficien-marke hypohei. 7. The black-schole equaion To price a coningen claim one fir make a couple of financial and mahemaical aumpion. Fir we aume ha he marke i free of arbirage opporuniie and ha i i complee. Anoher financial aumpion i ha every porfolio coniing of bond and ock are o called elf financed meaning ha every new rade of acquiiion of ae mu be financed by elling par of he porfolio. No exogenou inflow or ouflow of capial i allowed. Here we are alo dealing wih imple coningen claim meaning ha he conrac only depend on he value of he ae a mauriy, T i.e. χ = Φ(X T ). Now one define a ochaic proce V (, S ) a he value of he coningen claim χ a ime < T and add he mahemaical aumpion ha V C 1,2. Now if one ue Iô formula and inroduce dynamic for porfolio and hen ue cerain hedging poiion (ee [1]) and he above aumpion one can derive ha (3) { V + rx V x σ2 X 2 2 V rv = 0 x 2 V (T, X T ) = Φ(X T ) which according o Feynman-Ka c ha he oluion V (, X ) = e r(t ) E[Φ(X T )]. If one replace he ochaic variable wih deerminiic one (3) become { V V (, x) + rx x (, x) σ2 x 2 2 V (, x) rv (, x) = 0 x 2 V (T, x) = Φ(x) which i called he Black-Schole equaion. 8. Opimal Sopping Theory Thi ecion conain a implified examinaion of opimal opping heory. For a more rigorou analyi of opimal opping heory ee [3]. Definiion 8.1. (Sopping ime) A random variable τ : Ω [0, ] i called a Markov ime if {τ } F 0. A Markov ime i called a opping ime if τ <.

12 10 JONAS BERGSTRÖM Noe: {τ } F mean ha we can deermine, anding a ime if he opping even ha occurred or no Example. Le X be a proce in dicree ime which i adaped o F. Le τ := inf{n 0; X n A} i.e τ meaure he ime when he proce ake value in A for he fir ime. Here τ i a opping ime becaue: {τ n} = {w Ω; τ(w) n} for ome probabiliy pace Ω and {w Ω; τ(w) n} = {w Ω; X (w) A, for ome n} n = {X A}. =1 Since X i adaped {X A} F F n {τ n} F n 8.2. Opimal opping raegy in coninuou ime. Aume one i holding a coningen claim a ime ha expire a a fuure ime T. The value of he coningen claim i aumed o depend on a ock, X, wih he rik-neural dynamic dx = rx d + σx dw Wihin he ime inerval [, T ] one ha he opion o eiher exercie or coninue o hold he coningen claim. Sanding a a fixed poin in ime,, he value of exerciing he coningen claim a ha ime i Φ(X ) were Φ i a conrac funcion. Here, depending on he price level of X and he naure of Φ, migh no be he opimal ime o op i.e exerciing he coningen claim. To analyze hi problem furher one define V (, x) = up E,x [e r(τ ) Φ(τ, X τ )]. τ T So anding a (, x), were x = X, V (, x) i by definiion he opimal value one can achieve from exerciing he coningen claim a ome ime in he fuure beween and T. Le ˆτ [, T ] be he opping ime ha aifie V (, x) = E,x [e r(ˆτ ) Φ(X τ )]. Then ˆτ i he opimal opping ime when anding a (, x). Now he queion i o find cerain condiion ha V (, x) mu aify. Le be he malle elemen in [, T ]. V (, X ) = Φ(X ). If V (, X ) = Φ(X ) i mean ha he opimal value one can receive i when exerciing

13 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY 11 he coningen claim a ime which i o ay ha ˆτ =. Thi lead o a beer definiion of ˆτ namely ˆτ = inf{ T ; V (, X ) = Φ(X )}. So he opimal raegy i o hold he coningen claim unil ome ime ˆτ when V (ˆτ, Xˆτ ) = Φ(Xˆτ ). Thi lead o ha V (, X ) > Φ(X ) < ˆτ which lead o he inequaliy V (τ, X τ ) Φ(X τ ), τ [, T ]. To reach new concluion on he behavior of V (, x) we ue Iô formula on he proce fe r V (, X ) d(e r V (, X )) = e r ( r + L)V (, X )d + e r σ V x dw where L = + rx x σ2 X 2 The oluion o hi SDE i 2. x 2 e r(+h) V (+h, X +h ) = e r V (, x)+ where L = + rx x σ2 X 2 2 x 2. +h e r ( r+l)v (, X ) d+ Muliplying boh ide wih e r and aking expecaion yield (4) e rh E x, [V ( + h, X +h )] = V (, x) + E,x [ +h +h e r( ) ( r + L)V d]. Now anding a (, x), wha happen o V if i i no opimal o op i.e V (, x) > Φ(, x). Thi mean ha he opimal value i in he fuure. So inead of exerciing he coningen claim a one hold i ill + h. Here we alo le h end o zero, o minimize lo of informaion of V beween and + h. Thi mean ha he value of V mu be he dicouned expeced value of hi fuure V. Inering hi ino (4) give E,x [ V (, x) = e rh E,x [V ( + h, X +h )] +h e r( ) ( r + L)V (, X )d] = 0. If one divide hi expreion wih h and coninue o le h end o zero i will reul in he inegrand evaluaed a. Thi mean ha LV (, x) = rv (, x). Alernaively if we had V (, x) = Φ(, x) i would no be opimal o coninue o hold he coningen claim ill +h. Thi lead o he ric inequaliy V (, x) > e rh E,x [V ( + h, X +h )]. e r σ V x dw

14 12 JONAS BERGSTRÖM Inering hi ino (4) yield LV (, x) < rv (, x). Propoiion 8.2. Aume ha V i enough differeniable and V (, x) = up τ T E,x [e r(τ ) Φ(τ, X τ )]. Define he region C := {(, x); V (, x) > Φ(, x)} hen he following hold: V (T, x) = Φ(T, x) LV (, x) = rv (, x) (, x) C LV (, x) < rv (, x) (, x) / C where L = + rx x σ2 x 2 2. x 2 The opimal opping ime anding a (, x) i ˆτ = inf{ ; V (, X ) = Φ(, X )} 9. The American Pu Opion When he opporuniy of exerciing an opion a any ime before mauriy exi hing ge complicaed. Here he conrac funcion, in conra o he European opion, can no depend olely on he price of he ae a mauriy. When mauriy i finie here doe no exi an analyic formula for he pricing of an American pu opion. However, in he cae of he call opion, i doe exi. Propoiion 9.1. The price of an American call opion wih finie mauriy coincide wih i European counerpar. Proof. Thi i proved by howing ha he opimal opping ime for he opion equal mauriy. The opimal opping problem i Define a new proce up E,x [e r(τ ) max(x τ K, 0)]. τ T Z = e r( ) max(x K, 0) = e r max(e r X e r K, 0) and prove ha Z i a ubmaringale. Thi i done in wo ep. Sep 1: Y := e r X e r K. When increae, o doe e r K hence, even hough i i a deerminiic funcion, e r K i a ubmaringale. e r X were X ha he dynamic under he rik neural valuaion i.e dx = rx d + σx dw i, following from heorem 6.2, a maringale. So i follow ha Y i a ubmaringale i.e E[Y F ] E[Y ],. Sep 2: γ(y) = max{y, 0} i a convex funcion of y, hence by Jenen inequaliy E[γ(Y ) F ] γ(y ),

15 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY 13 and hu Z i a ubmaringale. Hence E,x [Z T ] E,x [Z τ ], τ T So, following he reul of propoiion 9.1, one focue on he American pu opion. Thu, in hi conex one aume ha he conrac funcion, Φ, ha he form Φ(x) = max{k x, 0} = (K x) +. Wih an American pu opion one hope for a decreae in he value of he underlying ae o ha i i below he rike price K. Le X be he proce ha reemble he value of he underlying ae wih he rik-neural dynamic: dx = rx d+σx dw. One aume ha here i a price level, call i b(),. when X goe below hi level i i opimal o op. For b() o be opimal i mu be maller han K i.e be in he money. Alo by heorem 5.1 he oluion o he SDE of he underlying ae look like X = x exp((r 1 2 σ2 ) + σw ) o i i reaonable o demand ha b() > 0. Theorem 9.2. Aume ha V (, x) = up τ T E,x [e r(τ ) (K X τ ) + ]. If here exi a b() and u(, x) ha aifie: u(t, x) = (K X T ) + 1. u(, x) > (K x) + x > b 2. u(, x) = (K x) + x b 3. Lu(, x) = ru(, x) x > b 4. Lu(, x) < ru(, x) x b 5. u(, x) = 1 x = b 6. x where L = + rx x σ2 x 2 2 x 2, hen i follow ha u(, x) = V (, x) Proof. Ue Iô formula on e r u(, X ). Le τ [, T ] be a opping ime. e rτ u(τ, X τ ) = e r u(, x)+ 2. and 3. ay ha τ e rτ (K X τ ) + e r u(, x)+ 4. and 5. give e r ( r+l)u(, X ) d+ e rτ u(τ, X τ ) e rτ (K X τ ) + τ e rτ (K X τ ) + e r u(, x) + τ e r ( r+l)u(, X ) d+ τ Now aking expeced value on boh ide yield: r u e x (, X ) dw. τ r u e x (, X ) dw. E,x [e rτ (K X τ ) + ] e r u(, x) E[e r(τ ) (K X τ ) + ] u(, x) up E,x [e r(τ ) (τ, X τ ) + ] u(, x) τ T r u e x (, X ) dw.

16 14 JONAS BERGSTRÖM To prove he revere inequaliy ue he ame argumen bu change τ o τ := T τ b, where τ b := inf{ T ; X b}. Then i follow ha e rτ u(τ, X τ ) = e r u(, x)+ τ e r ( r+l)u(, X ) d+ τ Beween and τ X will alway be above b o from 4. i follow ha ( r+l)u(, X ) = 0 e rτ u(τ, X τ ) = e r u(, x)+ 1. and 3. u(τ, X τ ) = (K X τ ) + which lead o e rτ (K X τ ) + = e r u(, x) + τ τ r u e x (, X ) dw r u e x (, X ) dw r u e x (, X ) dw and muliplying boh ide wih e r and aking expeced value yield: u(, x) = E,x [e r(τ ) (K X τ ) + ] up E,x [e r(τ ) (τ, X τ ) + ] τ T u(, x) = V (, x) 10. The Perpeual American Pu Opion When dealing wih an American pu opion ha ha a finie mauriy, unlike i European couner par, no analyic formula for i value i known. If however one reric only o he cae when here i no mauriy or equivalenly when i i equal o infiniy, an analyic formula can be derived. Wih T = he opion i called a perpeual opion. When he opion ha no expiring dae he value of he opion doe no depend on ime bu only on he price level of he underlying ae i.e i derivaive wih repec o ime i zero. The opimal price fronier, b, become a conan. Now one i looking for b and V (x) ha aify he condiion of heorem 9.2, and a a conequence of he heorem, he oluion o he perpeual American pu opion ha been obained. I follow from condiion 4. in heorem 9.2 ha 1 2 σ2 x 2 V (x) + rxv (x) rv (x) = 0 for x > b. Thi ODE can be rewrien a an Cauchy Euler equaion: (5) x 2 V (x) + αxv (x) + βv (x) = 0, α = 2r σ 2, β = α.

17 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY 15 Thi i olved by eeking a oluion of he form V (x) = x m which by inering ino (5) give V (x) = C 1 x + C 2 x α. Since V (x) i bounded by K i follow ha C 1 = 0. Condiion 3. and 6. in heorem 9.2 ay repecively ha V (b) = (K b) + and V (b) = 1. V (b) = C 2 b α = (K b) + = K b C 2 = b α (K b) and V (x) = ( b x )α (K b). V (x) = αb α x (1+α) (K b) V (b) = αb α b (1+α) (K b) = 1 which i equivalen o b = αk 1 + α = 2rK σ 2 σ 2 2r + σ 2 2rK = 2r + σ 2. So one arrive a he following formula for an American perpeual pu opion { b α V (x) = x (K b) x > b K x x b where b = 2rK 2r and α = 2r + σ2 σ jump o defaul model Under he BS-model ock price, which have a poiive iniial value, can never drop o zero (hi follow from heorem 5.1). Hiory ugge ha he probabiliy of a o called defaul i non-zero. In hi conex defaul mean ha he financial eniy in queion fail o mee i financial obligaion, uch a inere paymen on heir loan. If hi happen he value of hi eniy will drop o zero wih no chance of a recovery. Here we do no aume ha counrie can defaul. To incorporae defaul one can ue poion procee. Le N() = he number of defaul beween 0 and. where λ i he defaul ineniy. N( + ) N() P o(λ ) Empirical evidence how a poiive correlaion beween corporae bond yield and credi defaul wap (CDS) pread (ee [4]). A CDS pread i he

18 16 JONAS BERGSTRÖM fee ha he buyer of he CDS pay o he iuer. If he chance of defaul increae o doe he fee, for having he inurance. So in hi conex one expand he BS-model by aying ha here exi a ock and wo bond. A corporae bond and a overeign bond. In he normal BS-model λ = 0, ha i o ay boh bond mu have he ame rae of reurn namely r. If however λ > 0 hen i follow ha if inveor hold a corporae bond, hey need o be compenaed for aking a bigger rik in comparion if hey held a governmen bond. Thi lead o aume ha he rae of reurn of a corporae bond i r + λ. When uing a rik neural valuaion he dynamic of a ock price become were dn po(λ d) dx = (r + λ)x + σx dw X dn I i alo aumed ha λ i a decreaing funcion of he underlying ae, X. Uing he above aumpion one can arrive a an ODE for a perpeual American pu opion wih a non zero defaul ineniy, λ, namely (6) 1 2 σ2 x 2 V + (r + λ(x))xv (r + λ(x))v = λ(x)k Theorem Aume ha V 1 olve he homogeneou differenial equaion V (x) + p(x) V (x) + q(x) V (x) = 0. Then anoher linear independen oluion, V 2 ha he form: x 1 y V 2 = V 1 u(x), u(x) = V1 2 exp( p(z) dz) dy. Theorem Aume ha V 1 and V 2 are wo linearly independen oluion o he homogeneou differenial equaion V (x) + p(x) V (x) + q(x) V (x) = 0. The paricular oluion o he differenial equaion ha he form V (x) + p(x) V (x) + q(x) V (x) = β(x) V p = u 1 (x) V 1 (x) + u 2 (x) V 2 (x) where { u 1 V 1 + u 2 V 2 = 0 u 1 V 1 + u 2 V 2 = β(x) Thi yem ha a oluion if [ V1 V de( 2 V 1 V 2 ] ) 0

19 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY 17 Theorem (Cramer Rule) AX = B, A = a a 1n......, X = x 1..., B = b 1... a n1... a nn x n b n x i = de(a i) de(a), A i = a a 1i 1 b 1 a 1i+1... a 1n a n1... a ni 1 b n a ni+1... a nn i.e he i:h column of A i replaced by B. For proof of all hree of he above heorem ee [5]. To make calculaion eaier we wrie (6) a V + α 1 x V α 1 x 2 V = β 2(r + λ(x)) α = σ 2, β = 2Kλ(x) σ 2 x 2. One oluion o he homogenou differenial equaion, V + α 1 x V α 1 x 2 V = 0 i V 1 = x. Theorem 11.1 give he econd linearly independen oluion, namely V 2 = V 1 u(x), where u(x) = x 1 exp( y α V1 2 z dz) dy. To find he paricular oluion, V p, o (6) one ue heorem V p = V 1 u 1 (x) + V 2 u 2 (x) { u1 V 1 + u 2 V 2 = 0 u 1 V 1 + u 2 V 2 = β which can alo be wrien in marix form: [ ] [ ] [ ] V1 V 2 u1 0 = V 1 V 2 u 2 β V V 1 = 1 2 = (V 1 u) = V 1 u + V 1 u d x = u + V 1 dx ( 1 V1 2 exp( = u + V 1 ( 1 x α V1 2 exp( z dz)) = u + 1 V 1 exp( x α z dz). y α dz) dy) z

20 18 JONAS BERGSTRÖM We now apply heorem A = [ V1 V 2 V 1 V 2 ] [ 0 V2, A 1 = β V 2 ] [ V1 0, A 2 = V 1 β ] de(a 1 ) = βv 2, de(a 2 ) = βv 1, Theorem 11.3 give de(a) = V 1 V 2 V 1 V 2 = V 1 (u + 1 x α exp( V 1 z dz)) V 2 x α = V 2 + exp( z dz) V 2 x α = exp( z dz). x u α x 1 = βv 2 exp( z dz), α u 2 = βv 1 exp( z dz) x y α x y u 1 = βv 2 exp( z dz) dy, u α 2 = βv 1 exp( dz) dy. z x y α x y V p = V 1 βv 2 exp( z dz) dy + V α 2 βv 1 exp( dz) dy z If λ i aumed o be conan he value of V 1 and V 2 are a follow. V 1 = x, x 1 V 2 = x x 1 = x y y 2 exp( α y 1 dz) dy z exp( αlog(y)) dy 2 x 1 = x dy yα+2 = 1 (α + 1)x α

21 PRICING THE AMERICAN OPTIONUSING ITÔ S FORMULA AND OPTIMAL STOPPING THEORY 19 Se β = γ x 2, γ = 2Kλ σ 2 and iner V 1 and V 2 ino V p yeild V p = xγ x 1 α + 1 = γ α + 1 = γ α y 2 dy γ α(α + 1) = 2Kλ σ 2 σ 2 2(r + λ) Kλ = (r + λ) So he oluion o (6) where λ i conan i V = C 1 x + C 2 x α + γ x (α + 1)x α y α 1 dy Kλ (r + λ). Now we would like V o aify he condiion in heorem 9.2. V (x) K (i) V (b) = K b (ii) (b) = 1 (iii) V x (i) C 1 = 0, (ii) V (b) = C 2 b α + Kλ (r + λ) = K b. Thi lead o ha C 2 = b α (K b Kλ (r + λ) ). V (x) = ( b x )α (K b Kλ (r + λ) ) + Kλ (r + λ) (iii) V = αb α x (α+1) (K b Kλ x x=b (r + λ) ) x=b = 1 which i equivalen o α b (K b Kλ (r + λ) ) = 1 b = α α + 1 (K Kλ (r + λ) ) α = 2(r + λ) σ 2 α α + 1 = = 2(r + λ) σ 2 σ 2 2(r + λ) + σ 2 2(r + λ) 2(r + λ) + σ 2

22 20 JONAS BERGSTRÖM b = K 2(r + λ) 2(r + λ) + σ 2 Kλ (r + λ) = rk r + λ rk r + λ = 2rK 2(r + λ) + σ 2. Now one ha arrived a he following formula for an perpeual American pu opion on a ock wih defaul ineniy λ namely where b = V (x) = 2rK 2(r+λ)+σ 2 { ( b x )α (K b and α = 2(r+λ) σ 2. Kλ Kλ (r+λ) (r+λ) ) + K x, Reference if x > b if x b [1] Björk, T. Arbirage heory in coninuou ime. Third ediion. Oxford univeriy pre [2] Sefanica, D. A primer for he mahmaic of financial engineering. Second ediion. FE Pre, New York [3] Pekir, G. and Shiryaev, A. Opimal opping and free-boundary problem. Birkhäuer Verlag, Bael-Boon-Berlin [4] Carr, P. and Wu, L. A imple robu link beween american pu and credi proecion. [5] Simmon,G. and Kranz, S. Differenial equaion: Theory, echnique and pracice. McGraw-Hill companie. Inernaional ediion