Lecure Noes o Finansiella Deriva (5B1575) VT 22 Harald Lang, KTH Maemaik Noe 1: No Arbirage Pricing Le us consider a wo period marke model. A conrac is defined by a sochasic payoff X a bounded sochasic variable a a fuure ime T ( mauriy ) and a price of ha conrac p which is paid oday. We assume ha one conrac is a bank accoun, a conrac which gives a secure payoff of 1 a mauriy, and whose price oday p = e rt, where ypically r >. Of course, r is inerpreed as he ineres rae from now o mauriy. The marke consiss of a se of conracs, and we assume ha we can compose arbirary porfolios of conracs, i.e., if we have conracs 1,..., n wih payoffs X 1,..., X n and prices p 1,..., p n hen we can compose a porfolio consising of λ i unis of conrac i, i = 1,..., n where λ 1,..., λ n are arbirary real numbers. I.e., we assume ha we can ake a shor posiion in any conrac, and ignore any divisibiliy problems. The oal cos of such a porfolio is of course n 1 λ ip i and he oal payoff is n 1 λ ix i. We call also such porfolios conracs, so he se of conracs, defined as pairs (p, X), consiue a linear space. We assume ha he marke is arbirage free in he following sense: 1. There is no conrac (p, X) such ha X almos surely (a.s.) and p <. 2. There is no conrac (p, X) such ha X a.s., E [X] > and p =. I is easy o see ha condiion 1 in paricular implies he law of one price: if X 1 = X 2 a.s., hen p 1 = p 2. We denoe by p(x) he unique price of a conrac whose payoff is X. Now we define an operaor E [X] on all payoffs of conracs by E [X] def = e rt p(x). Now we esablish he following properies of his operaor: Properies of operaor E. 1. E is linear; i.e., E [αx + βy ] = αe [X] + βe [Y ], where α and β are real numbers and X and Y are payoffs of conracs. 2. E is posiive; i.e., if X a.s., hen E [X]. 3. E [1] = 1. 4. For any payoff X such ha X a.s. E [X] = E [X] =. 1
Proof 1. The law of one price implies ha p(αx + βy ) = αp(x) + βp(y ). Now muliply boh sides by e rt o ge he required relaion. 2. Assume ha X a.s. Then, by arbirage condiion 1, p(x), and hence E [X] = e rt p(x). 3. We have by definiion p(1) = e rt, hence E [1] = e rt p(1) = 1. 4. Assume ha X a.s. If E [X] = hen p(x) = and hence E [X] mus be =, else E [X] would be > which would be an arbirage ype 2. If, on he oher hand, E [X] =, hen X = almos surely, and since we don have arbirage ype 1, p(x). Bu since also X = almos surely, p( X), i.e., p(x), so in fac p(x) =. Bu hen i follows ha E [X] = e rt p(x) =, and he proof is complee. Recall ha he operaor E is defined only on sochasic variables ha are payoffs of conracs. Bu assume for a momen ha any bounded sochasic variable X is he payoff of a conrac on he arbirage-free marke under sudy. Then we could define a probabiliy measure P by P (A) def = E [1 A ] for any (measurable) se A, where 1 A denoes he indicaor funcion of A. I is easy o verify, using he properies of E, ha he axioms for a probabiliy measure is saisfied. The value E [X] will hen be he expeced value of X wih respec o he measure P. Propery 4 of he operaor E implies ha P (A) = P (A) =, where P denoes he original probabiliy measure, i.e., he wo probabiliy measures P and P are equivalen. This means, by he Radon-Nikodym heorem, ha here is a sochasic variable Z > such ha E [Z] = 1 and E [X] = E [ZX] for any bounded sochasic variable X. For his reason, we will here call he operaor E [X] he equivalen risk-neural expeced value of X, even if no all bounded sochasic variables are payoffs of conracs. We will come back o he issue of he exisence of an equivalen probabiliy measure laer. We have hus proved ha if he marke is arbirage free, hen here exiss an equivalen expecaion E such ha p = e rt E [X] for any conrac (p, X). Now we prove he opposie, which is quie easy: If here exiss an an equivalen expecaion E, i.e., an operaor E saisfying p = e rt E [X] and properies 1 4, hen he marke is arbirage free. Indeed, if X almos surely, hen by propery 2, E [X], hence p = e rt E [X], so we can no have an arbirage ype 1. Nor can we have an arbirage ype 2, since if X a.s. and E [X] >, hen, by propery 2, E [X], and by propery 4, E [X] and hus p = e rt E [X] >. In summary: Theorem 1. A wo period marke, wih a bank accoun giving he fixed ineres rae r, is arbirage free if and only if here is an operaor E, which we call an equivalen risk-neural expecaions operaor, which saisfies properies 1 4 and p = e rt E [X]. 2
An Applicaion o Fuures and Forwards Le us divide up he ime spell from now o mauriy T in small ime spells now =, 1,..., n = T. A conrac is any porfolio composed now, whose payoff is realized a mauriy T, however, we allow re-allocaing he porfolio a each ime k. A re-allocaion here means ha we sell some asses and buy some asses such ha he ne cos is zero. No ne cash flow hus akes place a oher imes han now and mauriy. Consider firs a forward conrac on some underlying asse whose spo price a mauriy is S T. A forward conrac is a conrac whose price (value) oday is zero, and whose payoff a mauriy is S T G where G is he forward price. Hence, we have he equaliy = e rt E [S T G ], hence = E [S T G ] = E [S T ] G, since G is a consan, so E [G ] = G E [1] = G. The forward price is hus he same as he risk-neural expeced spo price: G = E [S T ]. Nex consider a fuures conrac on he same underlying asse. In his case he marking o marke gives a cash-flow F k F k 1 in each period k, where F k is he fuures price noed a k. Consider he following conrac: a ime k 1 we ener a long posiion on a fuures conrac, and a ime k we collec he cash flow F k F k 1 (which may be negaive), and close ou he conrac (for insance by aking an opposie shor posiion.) The dividend F k F k 1 is deposied (or borrowed) in a bank accoun unil mauriy. The final payoff a mauriy is hus (F k F k 1 )e r(t k). Since he cos of his conrac is zero, we have = e rt E [(F k F k 1 )e r(t k) ], i.e., = E [F k F k 1 ], i.e., E [F k ] = E [F k 1 ]. Thus we have F = E [F ] = E [F 1 ] =... = E [F T ] = E [S T ] so he fuures price F is also equal o he risk-neural expeced spo price: G = E [S T ]. So, we have proved: Theorem 2. If he ineres rae is deerminisically consan, he fuures price and he forward price coincide. 3
Noe 2: The No Arbirage Theorem In lecure noe 1 we proved he exisence of an equivalen expecaions operaor E which is defined on he sochasic payoffs of conracs. Under reasonable condiions i is rue ha here is a probabiliy measure P, defined on he given sample space and sigma algebra of measurable ses, such ha P is equivalen o he original probabiliy measure P and E is he expecaions operaor wih respec o P. We will now prove ha his is rue under he much simplifying condiion ha he sigma-algebra of measurable ses is finiely generaed, i.e., here is a finie number of subses A 1,..., A m such ha he sigma algebra consiss of arbirary unions of such ses. This assumpion is no unrealisic, bu for mahemaical reasons oo resricive. For he proof we need a Lemma: Proof Assume ha V is a subspace of R n wih he alernaing sign propery: every non zero vecor of V has boh sricly negaive and sricly posiive enries. Then here is a vecor λ wih all enries sricly posiive which is orhogonal o V wih respec o he naural inner produc on R n. Le K be he subse of R n K = {ū R n u 1 +... + u n = 1 and u i for all i}. Obviously K and V have no vecor in common. Le λ be he vecor of shores Euclidean lengh such ha λ = k v for some vecors k and v in K and V respecively. The fac ha such a vecor exiss needs a proof, bu we leve ha ou. We wrie λ = k v where k K and v V. Now noe ha for any [, 1] and any k K, v V, he vecor k + (1 ) k K and v + (1 ) v V, hence ( k + (1 ) k ) ( v + (1 ) v ) as a funcion of on [, 1] has a minimum a =, by definiion of k and v, i.e., ( k v) + (1 ) λ 2 = 2 k v 2 + 2(1 )( k v) λ + (1 ) 2 λ 2 has minimum a = which implies ha he derivaive w.r.. a = is. This gives ( k v) λ λ 2 or, equivalenly k λ λ 2 v λ for all v V and k K. Bu since V is a linear space, i follows ha we mus have v λ = for all v V. I remains o prove ha λ has sricly posiive enries. Bu we have k λ λ 2 for all k K, in paricular we can ake k = (1,,..., ) which shows ha he firs enry of λ is > and so on. This complees he proof of he lemma. Proof of he main saemen 1 A payoff X has consan values on each of he ses A i, le us denoe i X(A i ). m This means ha X(ω) = X(A i )1 Ai (ω) where 1 A denoes he indicaor funcion of he se A. Some of he ses A i may have zero probabiliy; for noaional convenience le P(A i ) > for i = 1,..., n and P(A i ) = for i = n + 1,..., m. We associae each conrac wih he n + 1-vecor ( p, X(A 1 ),..., X(A n ) ), i.e., we ignore X:s values on he zero ses. The se of such vecors consiue a 4
linear subspace of R n+1 and has he alernaing sign propery. Indeed; assume ha a leas one X(A j ) is posiive and none is negaive; hen X a.s. and E [X] >, hence p > by arbirage condiion 2, so he firs enry of he vecor is negaive, and we have a leas one negaive and a leas one posiive enry. If a leas one X(A j ) is negaive and none is posiive, we can look a he negaive of he conrac ( p, X(A 1 ),..., X(A n ) ) and he siuaion is brough back o he previous case. Finally, if all X(A j ):s are =, hen p = by arbirage condiion 1. Hence, by he lemma, here are posiive numbers λ,..., λ n such ha λ p + λ 1 X(A 1 ) +... + λ n X(A n ) = for all conracs. We can now wrie, for any conrac p = n 1 λ i λ X(A i ) = d n q i X(A i ) (1) 1 where d = n 1 λ i/λ and q i = λ i /(dλ ) for i = 1,..., n. Noe ha n 1 q i = 1, so if we define q j = for j = n+1,..., m, we can hus inerpre q i as new arificial probabiliies q i = P (A i ) of he evens A i, i = 1,..., m. Now (1) can be wrien p = d E [X] where E denoes expecaion wih respec o he P -measure. Noe also ha P (A i ) = P(A i ) =, so he wo measures are equivalen. The Radon- Nikodym derivaive of P wih respec o P, dp dp = q i P(A i ) on he ses A i, i = 1..., n and may be defined as any number on A j for j = n + 1,..., m which we may choose posiive so as o ge a sricly posiive Radon-Nikodym derivaive. If here is a bank accoun wih a fixed ineres rae r, he discoun facor d is idenified as d = e rt. Q.E.D. 5
Noe 3: Change of numéraire and maringale pricing Now assume ha here are several poins in ime,..., n = T when rading can ake place. We consider firs conracs (recall ha porfolios are also conracs) which pay a sochasic payoff X a ime T, and look a he marke price p k a k for k = 1...T. These prices are sochasic variables which are realized a k. [In a more fancy language: we have a filraion of sigma algebras... F F +1... represening he informaion available a ime..., + 1..., and he price process {p } is adaped o his filraion.] We assume ha he marke is arbirage free, and in view of earlier noes, ha here is an equivalen probabiliy measure P wih expecaions operaor E such ha p = d E [X] for all conracs. Here d is he discoun facor, which is equal o e ρt if here is a zero coupon bond wih mauring a T and ρ is he corresponding zero rae. We now assume ha here is a money marke accoun, a bank accoun (or bond) which pays a (coninuous) ineres rae he shor ineres rae r beween wo consecuive ime periods 1 and. The shor ineres rae r is a sochasic variable which is realized in period 1, i.e., when I inves an amoun a ime 1 I know he ineres I will receive in he nex period, bu he shor rae a laer ime periods are sochasic. In oher words, he shor rae process r is predicable. A rading sraegy is a plan elling us wha o inves a each period of ime, depending on he sae of he world as of ha ime. For insance, a rading sraegy migh say ha if even A occurs a ime, hen go long x in paper X and go shor y in paper Y. A self financing rading sraegy is one where afer he iniial invesmen, all furher rading is self financing unil he end dae, i.e., in each of he periods 1, 2,..., T 1, he ne invesmen is zero. Thus, wha we have called conracs are in fac porfolios following a self financing rading sraegy. Now we inroduce a new measure by a change of numéraire: Firs, noe ha E [d exp(σ1 T r i )] = 1, for an invesmen of 1 in he bank accoun will evenually end up wih a wealh of exp(σ T 1 r i ). Now define a new equivalen measure P by d P = d exp(σ T 1 r i ) dp. This means ha for any bounded sochasic variable Y, Ẽ [Y ] = d E [exp(σ T 1 r i )Y ], hence, if (p, X) is any conrac, we can le Y = exp( Σ T 1 r i )X o ge Ẽ [exp( Σ1 T r i )X] = d E [X] = p: p = Ẽ [exp( Σ T 1 r i )X] for any conrac. Wha we have done is called a change of numéraire: he measure P has he zero coupon bond as numéraire, whereas P has he money marke accoun (shor rae) as a numéraire. Now consider a conrac whose price in each period is a sochasic variable p, =, 1,..., T which is realized in period. Consider he following rading sraegy: If he even A has occurred in period, hen buy e r 1+...+r unis of he corresponding porfolio a he price p per uni, and finance he purchase by 6
borrowing a he shor rae r +1 and roll he deb forward by renewing he deb in each period a he shor ineres rae. Then sell he securiies a ime s and inves he amoun a he shor rae and roll forward unil ime T. This means ha a ime T he ne value is p e r 1+...+r T + p s e r 1+...+r +r s+1 +...+r T if he even A did occur a ime, and oherwise. We may wrie his as ( p e r 1+...+r T + p s e r 1+...+r +r s+1 +...+r T )1 A where 1 A is he indicaor funcion of he even A. Since here is no iniial invesmen and all payoffs ake place a ime T, we have Ẽ [( p + p s e r +1... r s ) 1 A ] =. Le Ẽ denoe expecaion condiional on all informaion realized a (i.e., condiional on he sigma algebra F ). Employing he ieraed expecaions formula and using he fac ha A is realized a, we ge = Ẽ [ Ẽ [( p + p s e r +1... r s ) 1 A ] ] = Ẽ [ 1 A Ẽ [ p + p s e r +1... r s ] ] and since his is rue for any A realized a, we conclude ha Ẽ [ p + p s e r +1... r s ] = Since in paricular p is realized a ime, we can re-wrie his as in paricular, p = Ẽ [p s e r +1... r s ] p = Ẽ [p +1 e r +1 ] This says ha if he marke is arbirage free, hen here is an equivalen probabiliy measure P such ha he discouned price process {p exp( Σ 1 r i )} of any conrac (i.e., any porfolio under self financing re-allocaion) is a maringale. For his reason, we call Ẽ he Equivalen Maringale Measure. I is rivial o see ha he opposie is rue: If here is a probabiliy measure under which he discouned price process of any conrac is a maringale, hen here are no arbirage possibiliies. Forward and fuures prices. As an example, we will describe he fuures and forward prices of a securiy in erms of he Maringale measure. In order o ease noaion somewha we inroduce R(s, ) def = e r s+1+...+r. Firs consider a zero coupon bond wih par value 1 mauring a ime T. The price P (, T ) of his bond a ime is P (, T ) = Ẽ [R(, T ) 1 1]. Now consider a forward conrac wrien oday ( = ), and le G be he forward price and S T he sochasic price of he underlying asse a ime T. Since 7
he price of he conrac oday is zero, we mus have = Ẽ [R(, T ) 1 (S T G )], i.e., G = P (, T ) 1 Ẽ [R(, T ) 1 S T ] Now consider a fuures conrac on he same underlying asse. In his case, he amoun F F 1 is paid o he long holder of he conrac a ime = 1,..., T. Here F is he fuures price a ime. If we go long one such conrac a ime and close i ou (go shor one conrac) a he end of ime period + 1, hen he amoun invesed in period is zero, and he payoff in period + 1 is F +1 F. Hence = Ẽ [e r +1 (F +1 F )]. Bu r +1 is known a ime (recall ha r is predicable) so i can be regarded as a consan; hence, Ẽ [F +1 F ] =, or, since also F is is realized in period, F = Ẽ [F +1 ]. This means ha he sequence {F } T is a maringale under he P-measure. Hence F = Ẽ [F T ], i.e., since by definiion F T = S T, F = Ẽ [S T ] Noe ha if he ineres rae R(,T) is deerminisic, hen G P (, T ) = R(, T ) 1, so = F, for hen G = P (, T ) 1 Ẽ [R(, T ) 1 S T ] = P (, T ) 1 P (, T )Ẽ [S T ] = Ẽ [S T ] = F. Oherwise, we can do he following compuaion: G = P (, T ) 1 Ẽ [R(, T ) 1 S T ] = P (, T ) 1( Ẽ [R(, T ) 1 ]Ẽ [S T ] + cov[r(, T ) 1, S T ] ) = F + P (, T ) 1 cov[r(, T ) 1, S T ] So G = F + correcion erm, where he correcion erm is posiive or negaive depending on he sign of he correlaion beween he ineres rae and he spo price of he underlying asse. 8
Noe 4: Black s pricing model Consider a derivaive of a variable whose value is V. Le T be he mauriy dae of he derivaive, G he forward price for a conrac mauring a T, G he value of G a ime zero, e rt he discoun facor from ime zero o T (equivalenly, e rt is he price of a zero coupon bond wih face value 1 mauring a T ). Black s model assumes ha he value of V a ime T, V T, is sochasic, wih a log-normal probabiliy w.r.. he risk-adjused probabiliy measure P, i.e., wih he zero coupon bond as numeraire: V T = A e σ T z, where A is some consan and σ is also a consan referred o as he volailiy of V and z is a sochasic variable which has a normal (,1) disribuion w.r.. he P -measure. Since a forward conrac has a zero price when i is wrien, we have = e rt E [A e σ T z G ] i.e., G = E [A e σ T z ] = A e σ2 T/2 Assume ha he value of he derivaive a mauriy is some funcion Φ(V T ) of V T, e.g., if i is a European call opion, hen Φ(V T ) = max[v T K, ] where K is he srike price. The price of he derivaive a ime mus hen be p = e rt E [Φ(A e σ T z )] Subsiuing G e σ2 T/2 for A from he relaion above yields P = e rt E [Φ(G e σ2 T/2+σ T z )] = e rt 2π Φ(G e σ2 T/2+σ T z ) e z2 /2 dz This is Black s pricing formula. Noe ha he price is independen of he parameer A i is already incorporaed in he price G. Black-Scholes pricing formula Le us use Black s pricing formula o price a European opion on a sock. We use he same noaion and assumpions as in he previous secion, and also ha he underlying sock pays no dividend beween now and mauiriy of he opion. Le he price of he sock oday be S and he forward price G so ha S = e rt E [S T ] and = e rt E [S T G ] from which we infer ha G = Se rt The value of he opion a mauriy is some funcion Φ(S T ) of he sock price, so he curren price p of he opion is p = e rt 2π Φ(Se (r σ2 /2)T +σ T z ) e z2 /2 dz 9
which is he famous Black-Scholes pricing formula for European opions. Noe ha he price of he opion is independen of he expeced growh rae (under he rue probabiliy measure) of he sock value. However, in order o compue he price, we need o know he volailiy σ of he sock price under he risk adjused measure P. If rading is possible in coninuous ime unil mauriy, and if log(s) follows an Iô process, i urns ou ha his volailiy is he same as ha under he rue probabiliy measure P ; his is essenially a consequence of Girsanov s heorem, as we will see laer. 1
Noe 5: Binomial ree as a numerical soluion o Black-Scholes model Assume ha we wan o calculae he price and hedging porfolio for a European opion, using a binomial ree model. The underlying securiy is assumed o eiher go up by a facor e x +σ or down by a facor e x σ in each period. The value of x depend on he exac model employed, and σ is he volailiy of he underlying asse. In doing so, we come up wih a value V of he hedging porfolio in period. In he nex period, period 1, we re-calculae he value of he hedging porfolio, again using he binomial ree. If he securiy price has indeed increased x +σ by he facor e or e x σ, hen he re-balanced porfolio in ha period will cos exacly he same as he currenly held porfolio, i.e., he rading sraegy is self-financing, since we will be using he same binomial ree. A problem arises if his doesn happen, bu he price dynamics has deviaed from ha of he binomial ree model. We hen have o re-calculae he value of he hedging porfolio V 1 (S) in a new binomial ree, saring a he hen curren value S of he underlying securiy. V 1 (S) is hus he cos of he new hedging porfolio we wan o hold in period 1. Le us inroduce some noaion: The known period price of he securiy is S, he sochasic price in period 1 is S. Now, wheher or no S has followed he binomial dynamics, we can calculae he value V 1 (S) of he hedging porfolio in period 1 using he binomial ree model based on he hen curren securiy price S. Le v = V 1 (S) V be he difference beween he cos of he new hedging porfolio in period 1 and he value in period 1 of he porfolio held from he previous period. If he value S of he underlying securiy follows he binomial dynamics exacly, hen v =, for he porfolio sraegy is hen self financing by consrucion. We now assume ha he rue dynamics of S is ha of Black Scholes, i.e., S = S e ν +σ θ where θ is a N(,1)-variable. The value of v will hen depend on he movemen of S: v = v(ν + σ θ), and v = if he dynamics exacly maches ha of he binomial ree model, i.e., if θ = x ν σ ±1. We employ he following resul from calculus: if he funcion f(x) = for wo values x 1 and x 2 of x, hen f(x) = 1 2 f (c)(x x 1 )(x x 2 ) for some c in he smalles inerval conaining x, x 1 and x 2. We conclude ha v(ν + σ θ) = σ2 2 v (ν + σ ξ)(θ 1 x ν σ )(θ + 1 x ν σ ) where ξ is a sochasic variable whose value is somewhere in he smalles inerval conaining x ν σ ± 1 and θ. We wrie his as v = σ2 2 v ()(θ 2 1) + η where η is a sochasic variable wih small norm: η = O( 3/2 ). Noe ha E [θ 2 1] =, so we have 11
v = χ + η where E [χ] = and Var [χ] = O( 2 ). Now we add up all v:s for all periods 1... n: n v i = i=1 n χ i + η i i=1 n χ i + i=1 n η i Noe ha he θ:s for differen periods are assumed o be sochasically independen, hence he χ:s are uncorrelaed. Thus: n v i = i=1 n i=1 Var [χ i] + i=1 n η i = n O( 2 ) + n O( 3/2 ) = O( ). i=1 This proves he main resul of his noe: Binomal ree and Blasc-Scholes dynamics: Under he Black Scholes dynamics assumpion, he binomial ree model will yield a rading sraegy which is no fully self-financing, bu he oal cash flow is of he order of magniude O( ). More precisely, he oal cash flow is a sochasic variable whose norm is O( ). The relaion is: he securiy price in he binomial model eiher goes up by a facor e x +σ or down by he facor e x +σ, where σ is he volailiy of he securiy price, and x can be chosen arbirarily in order o achieve desirable values of he risk adjused probabiliies. We noe in paricular ha, jus as in Black Scholes pricing formula, he growh rae ν of he Black Scholes securiy dynamics doesn ener anywhere in he binomial compuaion. 12
Noe 6: Ho-Lee s Ineres Rae Model in Discree Time Ho Lee s model is a model of he shor ineres rae, and is a no arbirage model; he parameers of he model can be chosen such ha he curren erm srucure is correcly represened. Time is discree:, 1,..., n+1 = T. The ineres rae r k from k 1 o k is assumed o be k r k = θ k + σ k = 1,..., n 2 b j where {b k } are sochasic variables, b k is realized a ime k 1 (fancy language: {b k } is a predicable sochasic process) and σ and θ k are real numbers; i.e., hey are parameers of he model. The probabiliy disribuion of b k is given for he equivalen maringale measure P wih he money marke accoun as a numeraire: b k = 1 wih (risk-adjused) probabiliy.5 and b k = 1 wih probabiliy.5; and hey are hus assumed o be saisically independen. We can now compue he price Z 4 a of a zero coupon bond mauring a 4 wih face value 1: Z 4 = Ẽ [e r 1 r 2 r 3 r4 1] = Ẽ [e θ 1 θ 2 θ 3 θ 4 σ(3b 2 +2b 3 +b 4 ) ] = e θ 1 θ 2 θ 3 θ 4 Ẽ [e 3σb 2 ] Ẽ [e 2σb 3 ] Ẽ [e σb 4 ] = e θ 1 θ 2 θ 3 θ 4 cosh(3σ) cosh(2σ) cosh(σ) And, by he same oken, in general Z k = e θ 1 θ k cosh ( (k 1)σ )... cosh(σ) Combining wih he same expression for Z k 1 we ge Z k = e θ k cosh ( (k 1)σ ) Z k 1 i.e., θ k = f k + ln [ cosh ( (k 1)σ )] (1) [ def Zk 1 ] where f k = ln is he forward rae from k 1 o k. Z k Thus, if he parameers θ k are chosen according o (1), hen he he model reflecs he curren erm srucure. The parameer σ is ypically chosen o be he (esimaed) volailiy of he one period rae. Once we have he parameers of he model, we can price any ineres rae derivaive in a binomial ree. We show he procedure by an example, where we wan o price a European call opion mauring a 2 wih srike price 86 on a zero coupon bond mauring a 4 wih face value 1 when he following parameers are given: θ 1 =.6, θ 2 =.61, θ 3 =.62, θ 4 =.63, σ =.1. We represen he ineres raes in a binomial ree: 1 2 3.6.71.82.93.51.62.73.42.53.33 13
The ineres rae from one period o he nex is obained by going eiher one sep o he righ on he same line, or sep o he righ o he line below; each wih a (risk adjused) probabiliy of.5. We can compue he value a 2 of he bond: since is value a 4 is 1, he value a 3 and 2 is obained recursively backwards: 2 3 84.794 91.119 88.254 92.96 91.856 94.838 96.754 The value of he opion can now also be obained by backward recursion: 1 2 2.39 1.5 3.853 2.254 5.856 The value of he opion is hus 2.39. I is easy o price also American or oher more exoic derivaives in his binomial ree model. 14
Noe 7: The Black-Scholes-Meron Pricing Formula I will derive he Black-Scholes-Meron Pricing Formula, using maringales and Iô calculus however, I m rying o use as lile mumbo-jumbo as possible. This means ha I have o invoke Iô s represenaion formula and some Iô calculus, bu I avoid e.g. Girsanov s heorem and changes of measure. The reason for his is ha I wan o make he idea as ransparen as possible; he cos is ha he compuaions are more echnically involved han proofs relying on change of measures and Girsanov s heorem ha can be found in many ex books. Of course, he proofs are he same, i is jus he presenaion ha differ. A good way o undersand wha is going on is I hope o firs read his presenaion, and hen he ex-book. Le W be a Wiener process, T defined on a probabiliy space (Ω, F, P ) equipped wih a filraion [ ] T F which is generaed by he Wiener = process W. Consider now a bounded sochasic variable X, measurable w.r.. F = F T which is he payoff of a conrac (or a self financing porfolio of conracs) wrien a ime. We will derive an expression for he price p of ha conrac in erms of he prices of wo oher conracs whose prices a ime are B and S respecively. The prices of hese asses are assumed o be defined by he following Iô processes: { ds = µ S d + σ S dw db = r B d Here µ, σ and r are bounded adaped processes and σ >. The asse B can be hough of as a bond, and S an asse whose value is underlying ha of he derivaive X. Firs we inroduce some noaion: B = e Iô s formula now gives λ = µ r σ u = e v = u 1 λ s dw s 1 2 = e (he marke price of risk ) λ2 s ds λ s dw s + 1 2 r s ds B, S = e du = λ u dw λ2 s ds dv = λ 2 v d + λ v dw r s ds S, X = e d S = (µ r ) S d + σ S dw d B = r s ds X Now we employ Iô s represenaion formula on he sochasic variable u T X: u T X = A + h dw 15
for some adaped process h and consan A. Define Y = A + h s dw s We are now in a posiion o creae a self financing porfolio of he asses B and S which replicaes X. The porfolio consiss of ϕ of he asse B and ψ of he asse S, he value of he porfolio a any ime is hus i.e., V = ϕ B + ψ S Ṽ = e r s ds V = ϕ B + ψ S We will chose ϕ and ψ such ha wo condiions are saisfied: firs, Ṽ = v Y, and second: he porfolio is self financing. The firs condiion hen implies ha he porfolio replicaes X, indeed, ṼT = v T Y T = X, so V T = X. Now, by Iô s formula d(v Y ) = (dv )Y + v dy + d v y, Y On he oher hand, = λ v (λ Y + h ) d + v (λ Y + h ) dw ϕ d B + ψ d S = ψ (µ r ) S d + ψ σ S dw So if we choose ψ such ha ψ σ S = v (λ Y + h ) hen d(v Y ) = ϕ d B + ψ d S and if we now choose ϕ such ha ϕ B + ψ S = v Y, hen dṽ = ϕ d B + ψ d S which implies ha he porfolio is self financing, and Ṽ = v Y, so i replicaes X. The value of he derivaive mus hence if here is no arbirage be ha of he replicaing porfolio. If p is he value a ime and p = p e r sd s he discouned value, hen p = v Y = v E [Y T ] = v E [u T X]: Theorem: The discouned price p of he derivaive is given by p = v E [u T X] I is common pracice o inroduce an equivalen probabiliy measure whose Radon- Nikodym derivaive w.r.. he rue measure is u T. Noe ha u T > and E [u T ] = 1, so i is a permissible Radon-Nikodym derivaive. If we denoe expecaion w.r.. his measure by Ê we can wrie he above 16
Corollary: The discouned price p of he derivaive is given by p = Ê [ X] Proof: Noe firs ha u is a maringale, since du = λu dw. Now, for any even A realized a and E{1 A E [ Xu T ]} = E{E [1 A XuT ]} = E[1 A XuT ] E{1 A u Ê ( X)} = E{1 A E (u T ) Ê ( X)} = E{E [1 A XuT ]} = E[1 A XuT ] Since A is any even realized a his shows ha E [ Xu T ] = u Ê ( X), which proves he corollary. I.e., he discouned prices { p } T =1 is a maringale under he new probabiliy measure; he equivalen maringale measure. The Black-Scholes Pricing Formula for a European Opion Assume ha µ, σ and r are consans. The price p of a European opion on he underlying asse S T, giving he reurn F (S T ) a ime T is hen p = e rt 2π ( F S e (r 1 2 σ2 )T +σ ) T x e x2 /2 dx Proof: We define W =. Now u T = e λw T 1 2 λ2t. Hence p = E [e λw T 1 2 λ2t e rt F ( S e (µ σ2 /2)T +σw T ) ] = e rt 2π = e rt 2π ( e λ T z 1 2 λ2t F S e (µ 1 2 σ2 )T +σ ) T z e z2 /2 dz ( F S e (µ 1 2 σ2 )T +σ ) T z e 1 2 (z+λ T ) 2 dz = [make change of variable z + λ T = x in inegral]... ( = e rt F S e (r 1 2 σ2 )T +σ ) T x e x2 /2 dx 2π Quad Era Demonsrandum 17
Noe 8: Ho-Lee s Ineres Rae Model in Coninuous Time This is a coninuaion on Lecure Noe 5. Le r() denoe he sochasic insananeous shor rae a ime as seen from now. The Ho-Lee model in coninuous ime is r() = θ() + σw (), W () = where W () is a sandard Wiener process (Brownian moion) under he maringale measure ( he risk adjused probabiliy measure ) wih he money marke accoun (shor rae) as numéraire. The funcion θ() is deerminisic, and chosen so ha he model correcly reflecs he curren erm srucure. Le Z() denoe he curren price of a zero coupon bond mauring a wih par value 1. We denoe by Ẽ expecaion w.r.. he maringale measure. We have Z() = Ẽ [e r(s) ds ] so we sar by compuing he inegral. Using inegraion by pars of Iô calculus, we ge and hence r(s) ds = θ(s) ds + σ W (s) ds = θ(s) ds + σ ( ) Ṽar r(s) ds = σ 2 ( s) 2 ds = σ2 3 3 ( s) dw (s) σ2 a+ Since he E[z] = e 2 if z N(a, σ 2 ) we ge Z() = Ẽ [e from which follows r(s) ds ] = e θ(s) ds Ẽ [e σ θ() = d d ( s) dw (s) ] = e θ(s) ds+ σ2 6 3 ln Z() + σ2 2 2 (1) Equaion (1) defines he funcion θ such ha he model is calibraed wih he curren erm srucure. Noe ha he erm d ln Z() = f() is defined as he d shor forward rae, he he ineres rae for a infiniesimally shor forward rae agreemen (FRA) mauring a. 18
Forward and fuures prices of a zero coupon bond As an example we will compue he forward price and he fuures price of a zero coupon bond using Ho-Lee s model. The forward price G of a zero coupon bond wih par value 1 o be bough a and mauring a T > is, in any model, G = Z(T ) Z(). We now proceed o compue he fuures price F. If p denoes he (sochasic) price of he bond a, We have and p = Ẽ [e T F = Ẽ [p ] r(s) ds ] where Ẽ denoes expecaion condiional on he informaion available a ime. Combining hese and using he law of ieraed expecaions we ge F = Ẽ [e T r(s) ds ] = e T θ(s) ds Ẽ [e σ T W (s) ds ] Employ inegraion by pars of Iô calculus on he las inegral o ge F = e T = e ( d ds = Z(T ) Z() e σ 2 θ(s) ds Ẽ [e W (s) ds ] ln Z(s) σ2 2 s2 ) ds Ẽ [e σ 6 (T 3 3) e σ2 2 = Z(T ) Z() e σ 2 2 2 (T ) = G e σ2 (T s) dw (s) σ(t )W () ] (T s)2 ds+ σ2 2 (T )2 2 2 (T ) We see ha, as anicipaed, he fuures price is lower han he forward price (he bond price is negaively correlaed wih he ineres rae), wih a conversion facor e σ2 2 2 (T ). 19
Noe 9: Ho-Lee s Ineres Rae Model in Coninuous Time, coninued This is a coninuaion of Lecure Noe 8. The aim is o compue he price of a European opion on a zero coupon bond. Le Z(, T ) be he price a ime of a zero coupon bond mauring a T wih face value 1. In order o compue he expression for Z(, T ) we firs noe ha Hence r(s) ds = = = ln θ(s) ds + σ W (s) ds ( d σ2 ln Z(s) + ds 2 s2 ) ds + σ ( Z() ) + σ2 Z(T ) 6 (T 3 3 ) + σ Z(, T ) = Ẽ [e T = Z(T ) Z() e σ 2 r(s) ds ] W (s) d 6 (T 3 3) e σ(t )W () Ẽ [ e σ = Z(T ) Z() e σ 2 2 (T )T σ(t )W () e (T s) dw (s) + σ(t )W () (T s) dw (s)] We are now in a posiion o wrie down a formula for he price of a European opion on a zero coupon bond mauring a T (he bond) mauring a ime (he opion). Le Φ ( Z(, T ) ) be he payoff of he opion a mauriy. The price of ha opion oday is hen p = Ẽ [ Φ ( Z(, T ) ) e r(s) ds] = Z()Ẽ [ Φ ( Z(, T ) ) e σ2 6 3 σ ( s) dw (s)] The problem is o find an analyical expression for his expecaion. A he end of Lecure Noe 7 we found he Black-Scholes-Meron pricing formula for a European opion on a sock. In ha case, we were able o find an inegral expression for he relevan expeced value, since only one inegral of dw appeared in he expression, namely W () = dw (s). I was hen an easy ask o wrie down an inegral formula, and clean i up by making a change of variables in he inegral. This change of variables was acually a derivaion of a special case of Girsanov s heorem. The curren siuaion is somewha more complicaed, for wo differen inegrals of dw appear: firs W () = dw (s) appears in he expression for Z(, T ), and he inegral ( s) dw (s) appears in he exponen semming from he discouning. The remedy is again a change of variables, bu infiniely many such, and ha process is he conen of Girsanov s heorem. 2
Changing measure and employing Girsanov s heorem We will change measure o he one having he zero coupon bond as a numéraire. This is he reverse of wha we did in noe 3 we will go from he money marke accoun as a numéraire o he zero coupon bond as numéraire. We denoe expecaions operaor wih respec o he risk adjused probabiliy measure for conracs mauring a ime wih zero coupon bond as a numeraire by E () (in noe 3 we used he noaion E ), and he relaion beween he wo measures is expressed by Z()E () [Y ] = Ẽ [Y e r(s) ds ] = Z()Ẽ [ Y e σ 2 6 3 σ for any sochasic variabke Y realized a. Hence E () [Y ] = Ẽ [ Y e σ 2 6 3 σ ( s) dw (s)] ( s) dw (s)] Noe ha he facor appearing in he expecaion above is e σ 2 6 3 σ ( s) dw (s) = e σ2 2 ( s)2 ds σ ( s) dw (s) Girsanov s heorem now says: under he new probabiliy measure () def dp = e σ2 2 ( s)2 ds σ ( s) dw (s) d P, he Wiener process W (s) is equal o W () (s) σ s ( u) du = W () (s) σ(s 1 2 s2 ) where W () (s) is a Wiener process under he new P () -measure. Thus, under he P () -measure, he price a ime of a zero coupon bond mauring a T is Hence, we can wrie: where G = Z(T ) Z() Z(, T ) = Z(T ) Z() e σ2 (T )T /2 e σ(t )(W () () σ 2 /2) = Z(T ) Z() e σ2 (T ) 2 /2 σ(t )W () () p = Z()E () [ Φ ( Z(T ) Z() e σ2 (T ) 2 /2 σ(t )W () () )] = Z() 2π Φ ( G e ˆσ2 /2+ˆσ z ) e z2 /2 dz def is he forward price of Z(, T ) and σ = σ(t ). Noe ha his expression for he price of he opion coincides wih ha of Black s pricing formula (see Lecure noe 4) if he volailiy of he underlying asse Z(, T ) is se o σ = σ(t ). 21
Noe 1: Looking Back and some PDE:s In Lecure noe 7 we inroduced a new measure P. We saw ha he discouned price p a ime of a conrac is a maringale under his new measure. The measure is hus he Equivalen Maringale Measure (EMM) when he money marke accoun is he numéraire. Elsewhere we have denoed his measure by a raher han, bu in order o no confuse wih he noaion for discouning, we coninue o use he symbol here. The pricing formula p = Ê [ X] can also be wrien as p = Ê [Xe T r s ds ] in more conformiy wih he noaion used elsewhere in hese noes. The rue probabiliy measure P and he EMM are relaed by he Radon- Nikodym derivaive u T = e 1 2 λ s dw s. By Girsanov s heorem, his implies ha he underlying Wiener process W can be wrien as λ2 s ds dw = dŵ λ ds where Ŵ is a Wiener process under he EMM-measure. In paricular, i means ha ds = µs d + σs dw = µs d + σs(dŵ λ ds) = rs d + σs dŵ This shows ha he drif of he underlying asse S is equal o he risk free rae (which we already knew) and ha he volailiy of he asse inder he EMMmeasure is he same as under he rue measure, in accordance wih wha we claimed earlier (noe 4.) I now follows easily ha he expeced rae of reurn under he EMM measure of he derivaive is also r. Indeed, wih he noaion from noe 7, dp = dv = φ db + ψ ds = φ r B d + ψ (r S d + σ S dŵ) = r V d + ψ σ S dŵ = r p d + ψ σ S dŵ This fac is rue no only under he Black-Scholes model, bu for any conrac. Indeed, under he EMM measure, he discouned price p of any conrac is a maringale. Hence, he acual price p saisfies Since dp = d(e r s ds p ) = r e r s ds p d + e r s ds d p r s ds is realized a and Ê [d p ] = (since p is a maringale), we ge, wih somewha sloppy noaion, Ê [dp ] = r e r s ds p d = r p d 22
The PDE:s Le f(s, ) be he price of a derivaive of he underlying asse S a ime. We assume ha he price of he derivaive only depends of he curren value of he underlying asse. A ime d laer, he price is f(s + ds, + d) = f(s, ) + f (S, ) d + f S (S, ) ds + 1 2 f SS(S, ) ds 2 Taking expecaion w.r.. he EMM-measure gives, since he price rend of he derivaive f also mus be ha of he risk free rae: f + rf d = f + f d + rsf S d + σ2 2 S2 f SS d since he expeced value of ds 2 is σ 2 S 2 d. The calculaion is a somewha informal way of using Iô s formula. Hence we have he Black-Scholes-Meron differenial equaion: rf = f + rsf S + σ2 2 S2 f SS By he very same oken, if f(r, ) is he price of a ineres rae derivarive, depending only on he curren shor rae r, we ge rf = f + µf r + σ2 2 f rr where σ is he volailiy of he shor rae; µ is he rend of he ineres rae under he EMM-measure! This equaion is called he Term Srucure Equaion. Someimes one defines λ = ν µ where ν is ha rue rend of he shor rae, and σ hus have µ = ν λσ, and calls λ he marke price of risk. However, ineres rae models ypically model he dynamics of he ineres w.r.. he EMM-measure direcly, so hey doesn specify wha he marke price of risk is. 23