Financial Mathematics: A Comprehensive Treatment (Campolieti-Makarov)

Transcription

1 Financial Mahemaics: A Comprehensive Treamen Campoliei-Makarov Soluions o seleced exercises Alan Marc Wason July 9, 6 Conens Chaper 4: Primer on Derivaive Securiies Chaper : One-dimensional Brownian moion and relaed processes. 4 3 Chaper : Inroducion o coninuous-ime sochasic calculus 3 4 Chaper : Risk-Neural pricing in he Black-Scholes Economy: one underlying sock 5 5 Chaper 3: Risk-Neural pricing in a Muli-Asse Economy 37 6 Chaper 4: American opions 5 7 Chaper 5: Ineres rae modeling and derivaive pricing 53

2 Chaper 4: Primer on Derivaive Securiies Exercise.. CM4, Exercise 4.7 A European binary opion is a so-called all-or-nohing claim on an underlying asse. For example, one share of a cash-or-nohing binary call has a payoff of exacly one dollar if he asse price ends up above he srike and zero oherwise, i.e. he payoff funcion is ΛS S K S. Similarly, a cash-or-nohing binary pu has payoff S<K. Assume ha he asse price process {S} is a geomeric Brownian moion. a Derive he Black-Scholes exac pricing formulas for boh he binary call CS, T and he pu P S, T. b Give he relaionship beween he binary call and pu price when boh opions have he same srike K and mauriy T. c Derive he exac formula for he Greek dela of he binary call and pu: c C S p P S. and Soluion. a By he risk-neural pricing formula we have C, S e rt Ẽ ΛST S S e Λ rt σ r Se T +σ T Z S S e rt Ẽ Z ln σ K T S r σ T e rt P Z ln σ SK r T + σ T e rt P Z α e rt N α where in he las line we denoed α ln σ S T K + r σ T. b The pu-call pariy relaionship for European opions reads rt C, S P, S S Ke whereby P, S C, S S + Ke rt e rt N α + K S c... Exercise.. CM4, Exercise 4.8 Consider he European-syle opion wih payoff ΛS K S K S and assume he geomeric Brownian moion model for he sock price process {S}.

3 a Find he opion value by decomposing he payoff in erms of binary opions wih appropriae indicaor funcions. b Derive formulas for he following sensiiviies of he opion value: V S, Γ V S Θ V and Soluion. a The payoff can be decomposed in erms of binary opions b... ΛS K S K S S K S K so using he pricing formula from he previous exercise we obain V, S e rt ẼΛST S S e rt Ẽ K ST K S S e rt Ẽ ST K S S e rt Ẽ ST K S S C K, S C K, S e rt N αk N αk where αk i σ ln S T K i + r σ T. 3

4 Chaper : One-dimensional Brownian moion and relaed processes. Exercise.. CM4, Exercise. a Show ha he PDF of a sum of wo coninuous random variables X and Y is given by he convoluion of he PDF s f X and f Y : f X+Y x f X xf Y z x dx b Use he resul in a o show ha a sum of wo independen sandard normal random variables resuls in a normal random variable and find he PDF of such a sum. c Assuming ha X N µ, σ and X N µ, σ are correlaed wih CorrX, X ρ, find he mean and variance of ax + a X for a, a R. Soluion. a We sar by compuing he CDF of he sum: denoing by f X,Y x, y he join densiy funcion of X and Y we have P X + Y z P X z Y and differeniaing wih respec o z we obain f X+Y z d dz P X + Y z f X xf Y z x dx z Y f X,Y z y, y dy f X,Y x, y dx dy f X z yf Y y dx where in we used independence of X and Y and in we changed variables o x z y. b We compue he sum of wo sandard normal variables by convolving heir densiy funcions: if X, Y N,, hen f X u f Y u π e u / so f X+Y z e x / e z x / dx π /4 π e z e u / du / π e z /4 π e z e x z dx which shows ha X + Y is a normal variable wih mean and variance. Exercise.. CM4, Exercise.3 Suppose ha X X, X, X 3 be a 3-dimensional 4 Gaussian random vecor wih mean zero and covariance marix Σ X. Se Y 3 + X X + X 3 and Z X X 3. 4

5 a Find he probabiliy disribuion of Y. b Find he probabiliy disribuion of he vecor Y, Z. Soluion. We sar by recalling he following facs: i If X X,..., X n is an n-dimensional vecor of random variables wih covariance marix Σ X namely, Σ X ij CovX i, X j and Y MX where M is an m n marix and Y Y,..., Y m is an m-dimensional vecor of random variables, hen Σ Y MΣ X M ii If Y b + MX, where Y is an m-dimensional vecor of random variables, b is a consan m-dimensional vecor and X is an n-dimensional vecor of random variables wih mean σ X and covariance Σ X, hen Y has mean vecor and covariance marix given by µ Y b + Mµ X, Σ Y MΣ X M iii An n-dimensional random vecor has he n-dimensional mulivariae normal disribuion if and only if every combinaion n i c ix i is a normal random variable. In paricular, i follows from ii ha if X N µ X, Σ X and Y b + MX, hen Y N b + Mµ X, MΣ X M iv If Z has he sandard n-dimensional normal disribuion, Z N, I n and A is a symmeric posiive definie marix wih Cholesky facorizaion A U U, hen X U Z has an n-dimensional normal disribuion wih covariance marix Σ X A, so X N, A, We now solve he exercise.. Y has a normal disribuion by iii. If b,,, hen Y N + b, b Σ X b wih so Y N,. b Σ X b 4 3. Noe ha Y Z + no A { }} { so by iii he vecor W : Y, Z has a bivariae normal disribuion wih mean µ W, + A, and covariance marix Σ W AΣ X A. 8 5 X X X 3

6 Exercise.3. CM4, Exercise.8 For a Brownian moion {W } and is naural filraion, calculae E s W 3 for s. Soluion. Mehod. Taking he differenial of W 3 and using I s lemma we see ha dw 3 3W dw + 3W d since W u dw u is a square-inegrable maringale, i follows ha ha so is W 3 3 W u du, whence E s W 3 E s W 3 3 W u du + E s 3 W u du s W 3 s 3 W u du + 3E s W u du As for he las expecaion we have E s W u du so in conclusion s E s W u du + s W u du + sw s s W u du s E s W 3 W 3 s + 3 sw s Mehod. We can also compue he expecaion direcly: E s W 3 E s W W s + W s 3 E s W W s 3 Now noe ha W u du + s E s W u du + 3E s W W s W s + 3E s W W sw s + E s W 3 s E s W W s 3 E s W W s W s W se s W W s W s W se W W s sw s E s W W sw s W se s W W s E s W 3 s W 3 s In he firs equaion we used ha a normal variable has hird cenral momen in fac i has odd cenral momens, as he remark below shows. In he second equaion, we used he fac ha W s is F s -measurable for he firs equaliy so i can be aken ou of he expecaion and also he fac ha W W s is F s -independen for he second equaliy, so ha he condiional expecaion E s is in fac uncondiional. In he hird equaion we simply used ha W s is F s -measurable o ake i ou of he expecaion. 6

7 Remark. Cenral momens of a normal random variable. The n-h cenral momen is by definiion ˆm n E X EX n. Noice ha for he normal disribuion EX µ, and ha Y X µ also follows a normal disribuion, wih zero mean and he same variance σ as X. Therefore, finding he cenral momen of X is equivalen o finding he raw momen of Y. ˆm n E X EX n E X µ n yx µ σ n πσ y n e y σ dy yσu π u n e u du x µ n e x µ σ dx πσ πσ σ n u n e u σdu The laer inegral is zero for odd n as i is he inegral of an odd funcion over a real line. So consider u w u n e u du u n e u du π π w n e w dw n π w π w n / e w dw n Γ n + π where Γx sands for he Euler s Gamma funcion. Using is properies we obain ˆm n σ n n!! ˆm n+ Exercise.4. CM4, Exercise.9 Find he disribuion of W + + W n for n N. Soluion. A sum of normal random variables is again normal, so i suffices o compue is mean and variance. Clearly E W + + W n since each W k N, k. As for he variance, recall ha n VarW + + W n and for i < j k VarW k + i<j CovW i, W j CovW i, W j EW iw j EW i W j W i + EW i VarW i i We hence observe ha VarW + W + + VarW + W + W Var W k k n Var W k k. Var n W k + n + k 7 n n Var n W k + n k

8 So and hence Var n k W k n W k N k n k k, nn + n + 6 nn + n + 6 Exercise.5. CM4, Exercise. Le a,, a n R and < < < n. Find he disribuion of n i a iw i. Noe ha he choice a k n, k n, leads o Asian opions. Soluion. Arguing as in he previous exercise, we conclude ha n n n Var a i W i Var a i W i + a n n + a n a k k i i k so by inducion n Var a i W i i n i a i i + a i a k k i k Exercise.6. CM4, Exercise. Suppose ha he processes {X} and {Y } are respecively given by X x o + µ x + σ x W and Y y o + µ y + σ y W, where x, y, µ x, µ y, σ x >, σ y > are real consans. Find he covariance CovX, Y s, for s,. Soluion. Exercise.7. CM4, Exercise. Consider he process X x + µ + σw, where x, µ and σ are real consans. Show ha x + µ K EmaxX K, 9 x + µ KN σ + σ x + µ K n σ Soluion. Exercise.8. CM4, Exercise.3 By direcly calculaing parial derivaives, verify ha he ransiion PDF p ; x e x / π of sandard Bronwian moion saisfies he diffusion equaion u, x u, x x 8

9 Soluion. We simply compue he derivaives p, x π 3 p x, x e x π e x + x x π e x p x, x e x x π π e x whereby clearly p, x x p, x Exercise.9. CM4, Exercise.4 Find he ransiion PDF of {W n } for n N. Soluion. Exercise.. CM4, Exercise.5 Find he ransiion PDF of {S n }, where S S e µ+σw,, S >, for n N. Soluion. Exercise.. CM4, Exercise.9 Show ha he mean and covariance funcions of he Brownian bridge from a o b on, T are, respecively for s,, T. m a + b a T, cs, s s T Soluion. Recall ha if X X,T W T W T is he Brownian bridge from o, en X,T a b a + b a T + X so m X a b,t c X a b,t a + b a T + m X, s c X, s 9

10 and i suffices o compue m X and c X, s. m X EX E W T W T, c X, s E XXs E W T W T W s st W T EW W s s T EW W T T s s T EW sw T + s T W T Exercise.. CM4, Exercise. Consider he GBM; process S S e µ+σw,, S >. The respecive sampled maximum and minimum of his process are defined by M S sup Su, m S inf Su u u and he firs hiing ime o a level B > is defined by Derive expressions for he following: T S B inf{ : S B}. a P M S y, S s, >, < x y <, S y. b P M S y, >, S y <. c P T S B, >, S < B. Soluion. a Le X µ + σw. Denoing M sup s W s, We know ha f M X,Xw, y e µ +µy f M,W w, y f M,W w, y w y π e w y / Noe ha S s if and only if X ln s S so P M S y, S s P M X ln ys, X ln ss and no P M X m, X x P M X m, X x m x e µ f M X,Xw, y dy dw m x µy w y e e w y / dy dw π And if you have enough paience you can ype he compuaion of he laer inegral...

11 b As above noe ha P M S y P M X ln y so i suffices o inegrae he join S densiy f M X,Xw, y e µ +µy w y e w y / π over he region < y w, w ln y S, namely P M X ln ys ln y S w m m y f M X,Xw, y dy dw f M X,Xw, y dw dy + where he second equaliy follows from Fubini s heorem. m f M X,Xw, y dw dy c To compue ha P T S B, simply noe ha and we compued he laer in par b. Remark.. We recall briefly how o obain P T S B P M S B P M S B f M X,Xw, y e µ +µy f M,W w, y 3 f M,W w, y w y π e w y / where, recall X µ + σw, M X sup s Xs and M sup s W, where W is a sandard Bronwian moion. The join densiy of Bronwian moion and is maximum 4 follows from he reflecion principle P W w, T W m P W m w Noe on he one hand ha for he LHS we have P T W m, W w P M m, W w As for he RHS, since W N, we have P W m w m w w m π e u / du Equaing boh and differeniaing wih respec o m and w yields 4. f M,W x, y dy dx To show 3, assume for simpliciy ha σ. I hen suffices o perform a change of measure ha renders he Brownian moion drifless. Assuming ha he marke price of risk is γ and leing Ŵ W + γ we ge dx µ γd + dŵ 4

12 which is drifless if γ µ. The change of measure is given by he Radon-Nikodym process d ρ ˆP exp µ dp σ ds µ ds e µ µw e µ µŵ Hence P M X m, X x dp E I M X m,x x Ê d ˆP Ê ρ I M X m,x x e µ Ê e µŵ I M Ŵ m,ŵ x e µ E e µw I M m,w x e µ m x e µy f M,W w, y dy dw I M X m,x x where in we used ha X Ŵ and in we used he fac ha he random variables M Ŵ and Ŵ under measure ˆP are he same as M and W under measure P. This shows 3.

13 3 Chaper : Inroducion o coninuous-ime sochasic calculus Exercise 3.. CM4, Exercise.4 Evaluae he following double sochasic inegral s dw u dw s Soluion. The inner inegral is simply s dw u W s W W s and hen W s dw s can be compued by applying I s formula o dfw wih fx x. Indeed, dw s W sdw + ds so inegraing over s and re-arranging we obain W s dw s W Exercise 3.. CM4, Exercise.5 Show ha by using an appropriae I formula. W s dw s 3 W 3 W s ds Soluion. Applying I s formula o compue dfw s, wih fx x 3 we obain dw 3 s 3W sdw s + 3W sds so re-arranging and inegraing over s yields as claimed. W s dw s 3 W 3 W s ds Exercise 3.3. CM4, Exercise.6 Use he I isomery propery o calculae he variances of he following I inegrals. Also explain why he inegrals are well defined. a W s / dw s. b W s + s dw s. c W s + s 3/ dw s. 3

14 Soluion. Recall ha an I inegral Xs dw s is well defined provided ha he inegrand Xs is Fs-measurable which is obviously saisfied in all hree cases and provided ha i saisfies he square inegrabiliy condiion E X s ds E X s ds < and he I isomery propery saes ha Var Xs dw s a By I isomery we have Var W s / dw s b For he inegral W s + s dw s we have ha Var W s + s dw s E W s + s 4 ds EX s ds EW sds E W 4 s + 4W 3 ss + 6W ss + 4W ss 3 + s 4 3s + 6s 3 + s 4 ds where in we used he resul of Exercise.9, o he effec ha EW 3 s and EW 4 s 6 EW s ds 3. This inegral is clearly finie, which shows ha he square inegrabiliy condiion is saisfied. This could have been concluded direcly from, since we are inegraing a coninuous funcion E W s + s 4 over a compac se. Exercise 3.4. CM4, Exercise.8 Using I s formula, show ha he process defined by X : W 4 6 W u du, is a maringale wih respec o a filraion for Brownian moion. Soluion. Noe ha X f, W, wih f, x x 4 6 W u du so by I s formula so X is an I inegral dx 6W d + 4W 3 dw + 6W d 4W 3 dw X 4 W 3 s dw s which is a maringale provided ha he usual square inegrabiliy is saisfied, which rivially is by coninuiy of Brownian moion E W 3 s ds E W 6 s ds < 4

15 Exercise 3.5. CM4, Exercise.9 Use I s formula o show ha for any ineger k EW k kk E W k s ds and use his o derive a formula for all he momens of he sandard normal disribuion. Soluion. By I s formula we have whence and aking expecaions as claimed. dw k kw k dw + kk W k d W k k W k s dw s + EW k kk kk EW k s ds W k s ds Exercise 3.6. CM4, Exercise. Show ha M e / sinw, is a maringale wih respec o a filraion for Brownian moion. Soluion. By I s formula dm e/ sinw + e / cosw dw e/ sinw d e / cosw dw and since M we have M e s/ cosw sdw s which is an I inegral provided ha he square inegrabiliy condiion e s Ecos W sds < is saisfied and his is clearly he case, by coninuiy. Exercise 3.7. CM4, Exercise. Use I s formula o show ha for any non-random, coninuously differeniable funcion f, he following formula of inegraion by pars holds: fs dw s fw f sw s ds Soluion. By I s formula we have dfw f W d + fdw and inegraing we conclude as claimed. fw f s dw s + fs dw s 5

16 Exercise 3.8. CM4, Exercise.5 Le N x be he sandard normal CDF and consider he process X def W N, < T T Express his process as an I process and show ha i is a maringale wih respec o any filraion of BM. Find he limiing value XT lim T X. Soluion. Consider he funcion By I s formula we have f, x N x T x T π e u / du df, W f f, W d + x, W dw + f, W d x The derivaives in quesion are given by f, x e x π f x, x e x T π f, x e x T x π T, x T 3/, T, x T 3/ whence dx e W T dw π T This is a maringale provided ha he square inegrabiliy condiion is saisfied, namely However E T u e E W u T u e uz T u T u T u e W u T u du < e z / dz π π T u u e z + T u dz < for u, T. Is inegral over, is hus also finie being he inegral of a coninuous funcion over a compac inerval. Finally, i is clear ha lim X N W lim T T T 6

17 Exercise 3.9. CM4, Exercise.6 Suppose ha he processes {X} and {Y } have he log-normal dynamics Show ha he process Z X Y dx X µ X d + σ X dw dy Y µ Y d + σ Y dw is also log-normal, wih dynamics dz Z µ Z d + σ Z dw and deermine he coefficiens µ Z and σ Z in erms of hose of X and Y. Solve he same problem now assuming ha X and Y are governed by wo correlaed Brownian moions W X and W Y, respecively, where CorrW X, W Y ρ, for a given correlaion ρ. Soluion. Exercise 3.. CM4, Exercise.7 Consider he ime-homogeneous diffusion X have a sochasic differenial dx 3X d + XdW, X Find he sochasic differenial for he process Y : X and find he generaor for Y. Soluion. Noe ha Y fx wih f, x x. By I s formula we have dy dfx f x XdX + f x XdXdX X dx dxdx 4X 3/ X 3X d + dw X d 3X d + dw X 3Y d + dw Y Recall ha given an I process is generaor is he operaor X µ, Xd + σ, XdW, G,x µ, x x + σ, x x which acs on funcions f C,. In our case, for he process Y we have G,x Y 3x x x + x 7

18 Exercise 3.. CM4, Exercise.8 Le X W and Y e W. Find he sochasic differenial of Z X Y. Compue he mean and variance of Z. Soluion. By I s formula, he differenials of he wo processes X and Y are dx W + d + W dw, dy e W dw + ew d To compue dz dfx, Y we apply he -dimensional I s formula o he funcion fx, y x y. Since f x y, f y x y, f xx, f xy y, f yy x y 3 we have omiing he dependence X, Y in he firs line dz f x dx + f y dy + f xxdxdx + f xy dxdy + f yydy dy X X dx Y Y dy Y dxdy + Y 3 dy dy e W W + d + W dw W e W ew d + e W dw e W W e W d + W e 3W e W d e W W + W e W + W e W d +W e W W dw and I am very likely o have made a misake somewhere... Exercise 3.. CM4, Exercise.9 Le X be a ime-homogeneous diffusion process solving an SDE dx cx + σdw wih iniial condiion X x R and where c, σ are consans. Consider he process defined by Y X c X s ds σ,. a Represen Y as an I process and show ha i is a maringale wih respec o any filraion for Brownian moion. b Compue he mean and variance of Y for all. Soluion. a By I s formula we have whence dy XdX + σ d cx d σ d Y Y + σ cx d + σxdw cx d σxdw Xs dw s x + σ Xs dw s which is a maringale provided ha he square inegrabiliy condiion EX s ds is saisfied. We verify his in par b. 8

19 b Clearly EY x. In order o compue he variance, we sar by solving he SDE for X using an inegraing facor. Noe ha whence de c X ce c X + ce c Xd + σe c dw σe c dw X xe c + σ e c s dw s The expeced value of X is EX xe c and by I isomery we have VarX σ e c s ds σ c ec s σ c ec Since we also have VarX EX EX we conclude ha σ VarY 4σ EX s ds 4σ c ecs + x e cs ds σ4 e c x + c c Exercise 3.3. CM4, Exercise. Use he I formula o wrie down sochasic differenials for he following processes. a Y exp σw σ. Also find he expecaion and variance of he process X Y s ds. b Z fw where f is coninuously differeniable. Soluion. a By I s formula we have dy e σw σ σ d + σdw + σ σe σw σ dw σy dw If A Y s ds, i is clear ha EA As for he variance we have and EY s ds e σs Ee σw ds VarA EA EA EA EA E Y u du which is compued as on CM4, Page 43. Y v dv 9 e σs e σs ds E Y uy v du dv

20 b By I s formula we have dz d fw f W d + fdw Exercise 3.4. CM4, Exercise. A ime-homogeneous diffusion process X has sochasic differenial dx X XdW. Assuming ha < X <, show ha he process Y ln has a consan diffusion coefficien. X X Soluion. Noe ha Y fx wih fx ln x x. Clearly f x x x. By I s formula we have dy f XdX + f XX X d dw + Od so ha he diffusion erm is consan and equal o. Since f x x, he drif erm will x x be given by f XX X X. Exercise 3.5. CM4, Exercise.3 Le X s dw s, where <. Provide he sochasic differenial equaion for X and check your answer by solving he SDE obained subjec o he iniial condiion X. Soluion. Noe ha X f, Y where f, y y and Y s dw s. By I s formula we have dx f, Y d + f y, Y dy + f yy, Y dy dy Y d + dy dw s s dw s s X d + dw dw d + d + dw Exercise 3.6. CM4, Exercise.5 Le gy be a given funcion of y, and suppose ha fx is a soluion of f x gfx. Show ha X fw is a soluion of he SDE dx gxg Xd + gxdw

21 Soluion. Since f x gfx, clearly f x g fxf s. By I s formula we have ha as claimed. dx dfw f W dw + f W d gfw dw + g fw f W gxdw + g XgXd Exercise 3.7. CM4, Exercise.6 Use Exercise.5 o solve he following non-linear SDE dx X 3 d + X dw, X x Soluion. By Exercise.5, he soluion is given by X fw, where { f x gfx fx, f x Separaing variables one immediaely obains which is singular a x x. fx x xx Exercise 3.8. CM4, Exercise.3 Consider he boundary value problem for he hea equaion V + V x, V T, x fx Show ha he soluion is given by V, x πt fye y x T dy Soluion. This is essenially idenical o CM4, Example.. By he Feynman-Kac heorem CM4, Theorem.7, if a sochasic process {X} saisfies he SDE hen he C, funcion V, x solving he PDE admis a represenaion dx µ, Xd + σ, XdW, 5 V + σ, x V + µ, x V x x, T, x R+, V T, x fx provided ha E fxt <. V, x E fxt X x 6

22 In our case, µ, x and σ, x so he SDE 5 becomes dx dw and has a soluion XT X + W T W. The represenaion 6 hen reads V, x E fxt X x E fx + W T W X x E fx + W T W πt T dy fye y x fyϕ x,t y dy where in we used he fac ha EφX φyϕ Xy dy, where ϕ X denoes he PDF of X, and also ha x + W T W N x, T. Exercise 3.9. CM4, Exercise.34 Deermine f, x saisfying he following boundary value problem V + σ x V x + µx f x, T, x R+, ft, x K,K x Soluion. Wrie φx K,K x more generally o begin wih and we will specialize a he end. By he Feynman-Kac heorem CM4, Theorem.7, if a sochasic process {X} saisfies he SDE dx µ, Xd + σ, XdW, 7 hen he C, funcion f, x solving he PDE admis a represenaion f + σ, x f + µ, x f x x, T, x R+, ft, x φx f, x E φxt X x 8 provided ha E φxt <. In his case, he drif and volailiy are µ, x µx and σ, x σx, for real consans µ, σ > and he SDE 7 reads dx µxd + σxsw. This is a linear SDE wih soluion c.f. CM4, Equaion.7 XT X exp The represenaion 8 hen reads µ σ f, x E φxt X x E φ X exp µ σ E φ x exp µ σ φ x exp µ σ T + σw T W T + σw T W X x T + σw T W T + σy e y T dy 9 πt

23 We now specialize o φx K,K x. Noe ha if and only if ln σ K x exp K µ σ T x }{{} no α so ha φ x exp 9 becomes µ σ f, x T + σy α α µ σ T + σy K y ln σ K µ σ T x }{{} no α if and only if y α, α whence he inegral in y e T dy N α N α πt Exercise 3.. CM4, Exercise.36 Assume ha a sock price process {S} saisfies he SDE ds rsd + σsd W wih consans r, σ >, and where { W } is a sandard P -BM. By using Girsanov s heorem, find he explici expression for he Radon-Nikodym derivaive process d ρ ˆP d P er such ha he process defined by Ŝ S, is a ˆP -maringale. Give he SDE saisfied by he sock price S wih respec o he ˆP -BM. Soluion. We sar by compuing he differenial of he process Ŝ. e r dŝ d r er S S d er S ds + e r S 3 σ S d σd σŝ + d W Define Ŵ : W σ. By Girsanov s heorem, Ŵ is a ˆP -BM under he measure defined by he Radon-Nikodym derivaive process d ˆP ρ d P exp σ ds + σ d W s e σ +σ W and he SDE saisfied by he sock price S wih respec o he ˆP -BM is dŝ σdŵ 3

24 Exercise 3.. CM4, Exercise.38 Consider a one-dimensional general diffusion process {X} having a ransiion PDF ps, ; x, y, s <, wih respec o a given probabiliy measure P, for all x, y in he sae space of he process. Assume a change of measure P ˆP is defined by he Radon-Nikodym derivaive process d ρ ˆP h, X, dp Le ˆps, ; x, y denoe he PDF wih respec o he measure ˆP. Show ha he wo ransiion PDF s are relaed by h, y ˆps, ; x, y ps, ; x, y hs, x Soluion. ˆP s, ; x, y ˆP X y Xs x Ê X y Xs x ρ s E ρ X y Xs x hs, Xs E h, X X y Xs x h, u u y ps, ; x, u du hs, x y h, ups, ; x, u du hs, x so by he firs fundamenal heorem of calculus as claimed. ˆps, ; x, y ˆP h, y s, ; x, y ˆps, ; x, y y hs, x 4

25 4 Chaper : Risk-Neural pricing in he Black-Scholes Economy: one underlying sock Exercise 4.. CM4, Exercise.4 Assume he sandard Black-Scholes model in an economy wih consan coninuously compounded ineres rae r and wih sock price process {S} as a GBM wih consan volailiy σ and consan coninuous dividend yield q. Le S S > be he spo a ime < T, where T is he expiry dae. Derive he corresponding arbirage-free ime- pricing formula V, S for a European opion wih he following payoffs: a ΛST N n a ns n T, N, a n R. b ΛST S α T KI ST >K, α R \. Soluion. a Denoe as usual τ T. Clearly N N V, S e rτ Ẽ,S ΛST e rτ Ẽ,S a n S n T e rτ a n Ẽ,S S n T These expecaions are compued in CM4, Equaion.4. Wriing ST Se r q σ τ+σ τz, Z τ W T W N, n n we have: Ẽ,S S n T S Ẽ,S n e nr q σ τ+nσ τz S n e nr q σ τ Ẽ e nσ τz S n e nr q+ σ n τ where in we used CM4, Formula A.. Puing everyhing ogeher we conclude ha b In his case we have V, S e rτ N n a n S n e nr q+ σ n τ V, S e rτ Ẽ,S S α T KI ST >K e rτ Ẽ,S S α T I ST >K Ke rτ Ẽ,S IST >K Wrie as usual ST Se r q σ τ+σ τz, Z τ W T W N, 5

26 The second expecaion above is simply Ẽ,S IST >K P Se r q σ τ+σ τz > K P Z > ln K S r q σ τ σ τ P Z < ln S K + r q σ τ σ τ no N d The firs expecaion is compued in CM4, Equaion.4: Ẽ,S S α T I ST >K S n e nr q σ τ Ẽ,S e σ τz I{Z> ln K S r q σ τ S n e nr q+ σ n τ N σ τ ln S n K + r q + σ τ σ τ } so in conclusion we have: V, S e rτ S n e nr q+ σ n τ N d ln S K + r q σ τ σ τ ln S n K + r q + σ τ σ τ Ke rτ N d Exercise 4.. CM4, Exercise.5 Assume he sandard Black-Scholes model in an economy wih consan coninuously compounded ineres rae r and wih sock price process {S} as a GBM wih consan volailiy σ and consan coninuous dividend yield q. A European call spread has payoff, ST K, ΛST ST K, K < ST < K + ɛ, ɛ, ST K + ɛ where K, ɛ are posiive real consans. a Give a skech of he payoff funcion. b Derive a formula for he opion s presen value V, S and, S V S. c Find lim ɛ V, S and lim ɛ V, S and explain your resuls. a Soluion. 6

27 b The payoff can be decomposed as a combinaion of call payoffs ΛST ST K + ST K ɛ + so V, S Ẽ,S ST K + Ẽ,S ST K ɛ + e qτ N d + e qτ S K, τ e rτ KN d e qτ S e qτ N d + e qτ S K + ɛ, τ + e rτ KN K, τ d e qτ S K + ɛ, τ As for he dela we have V, S e qτ S N d + e qτ S K, τ N d + e qτ S K + ɛ, τ c Obviously lim ɛ V, S since he erms in he second line of equaion hen equal hose in he firs one and have opposie sign. Noe ha for ɛ he opion payoff is idenically zero. As ɛ, he opion becomes a simple call sruck a K whence lim V, S ɛ e qτ N d + e qτ S K, τ e rτ KN d e qτ S K, τ This can also be seen by aking he limi in equaion and noing ha lim N d + e qτ S ɛ K + ɛ, τ N lim d + e qτ S ɛ K + ɛ, τ N d +, τ N since recall d + x, τ ln x r+ σ τ σ. τ Exercise 4.3. CM4, Exercise.7 Soluion. Exercise 4.4. CM4, Exercise.8 Soluion. Exercise 4.5. CM4, Exercise.9 A so-called pay-laer European opion coss he holder nohing i.e. zero premium o se up a presen ime. The payoff o he holder is ST K +. Moreover, he holder mus pay ou X dollars o he wrier in he case ha ST. Derive an expression fr he fair value of X. Deermine he fair value in he limi of infinie volailiy lim σ Xσ. Assume he sandard Black-Scholes model in an economy wih consan coninuously compounded ineres rae r and wih sock price process {S} as a GBM wih consan volailiy σ and consan coninuous dividend yield q. 7

28 Soluion. The effecive payoff of he pay-laer opion is Λ eff ST ST K + XI {ST K} and we need o deermine he fair value of X such ha he ime- expeced value of he laer is zero, namely Ẽ,S Λ eff ST Ẽ,S ST K + XI {ST K} Clearly Ẽ,S Λ eff ST e rt Ẽ,S ST K + Xe rt Ẽ,S I{ST K} Se qt N d + e qt S K, T Ke rt N d e qt S K, T Xe rt N d e qt S K, T where as usual d ± ln x+r q± σ T σ. T so Imposing Ẽ,SΛ eff ST and solving for X yields Noe ha whence lim σ X. N d + e qt S X Se r qt K, T K N d e qt S K, T lim d ln x + r q ± ±x, τ lim σ T σ σ σ T lim N d +, σ lim N d σ σ T ± lim σ ± Exercise 4.6. CM4, Exercise. Le CS, τ be he Black-Scholes pricing formula of a sandard call opion wih spo S, srike K, fixed ineres rae r, zero sock dividend, consan volailiy σ and ime o mauriy τ >. a Show ha he respecive limiing values of he call price for vanishing and infinie volailiy are given by lim CS, τ S σ Ke rτ +, lim CS, τ S σ + b Give a financial inerpreaion of boh limis. Noe ha he second limi is independen of he srike value K; give a financial inuiion for his fac. Soluion. 8

29 a We performed an idenical compuaion in he previous exercise. The Black-Scholes formula for a call opion is CS, τ SN Also noe ha whence so On he oher hand, d + SK, τ Ke rτ N d SK, τ, d ± x, τ ln x + r ± σ τ σ τ lim d σ τ ± ± lim σ σ lim N d +, σ ± lim N d σ lim CS, τ lim d SN + SK, τ Ke rτ N d SK, τ S σ σ lim d ± S ln S K, τ lim + rτ σ K σ σ τ { +, ln S K + rτ,, ln S K + rτ < Noe furher ha ln S K + rτ if and only if S e rτ. Hence lim CS, τ lim d SN + SK, τ Ke rτ N d SK, τ σ σ I {ln S K +rτ<} + S Ke rτ I {ln S K +rτ } S Ke rτ + b The more volaile he marke is, he more expensive an opion is. In he absence of arbirage, ime- value of a call is bounded above by he spo, so as he volailiy grows indefiniely, he call value converges o is no-arbirage maximum S... Exercise 4.7. CM4, Exercise. Consider he value of a European call opion wrien by an issuer who only has a fracion α < of he underlying asse. Tha is, a expiraion ime T he payoff of his ype of call is given by V T ST K + I {αst ST K} + αst I {αst <ST K} Le C L S, τ; K, α denoe he value of a European call, where τ T is he ime o expiry, K > is he srike, S S is he spo of he underlying. Show ha C L S, τ; K, α CS, K, τ αc K S, α, τ 9

30 Soluion. Rewrie hew payoff as V T ST K + I { K α ST } + αst I { K <ST } α so he no-arbirage price of he opions is given by he usual risk-neural formula C L S, τ; K, α e rτ Ẽ V T e rτ Ẽ ST K + I { K α ST } + αst I { K α <ST } e rτ Ẽ ST KI {K ST K α } + αst I { K α <ST } We seek o show ha CS, K, τ αc value of an opion wih payoff V T. Indeed: K CS, K, τ αc S, α, τ e rτ Ẽ,S ST KI {ST K} α S, K α, τ indeed represens he above ime- ST K I α {ST K α } e rτ Ẽ,S αst I {ST K α } + ST KI {ST K} ST K α + αk I α {ST K α } e rτ Ẽ,S αst I {ST K α } + ST K I {ST K} I {ST K α } e rτ Ẽ,S αst I {ST K α } + ST KI {K ST K α } and he las expression is precisely. Exercise 4.8. CM4, Exercise.9 A varian... Soluion. Exercise 4.9. CM4, Exercise. Assume he sock price {S} is a GBM wih consan volailiy σ and zero dividend in an economy wih consan ineres rae r. Le < T < T, i.e. T is an arbirary inermediae ime before expiry ime T, and consider a Europeansyle opion wih payoff a ime T given by V T min{st, ST } a Show ha he value V a ime T of his opion is given by V S N d + + e rt T N d, d ± r ± σ T σ Noe: he opion value is dependen on T T and does no depend on. b Wha is he posiion held in he sock a ime T in a self-financing replicaing sraegy? 3

31 Soluion. a We may wrie he payoff as V T min{st, ST } ST I {ST <ST } + ST I {ST <ST } and he opion price is given by he usual risk-neural pricing formula V, S e rt Ẽ V T e rt Ẽ ST I {ST <ST } + ẼST I {ST <ST } In order o compue hese expecaions we condiion firs on ST and use ieraed condiioning, namely Ẽ ST I{ST <ST } Ẽ E ST I{ST <ST } ST Ẽ E ST e r σ T T +σ T T ZI {Z< d } ST e r σ T T Ẽ ST E e σ T T Z I {Z< d } e rt T Ẽ ST N d + 3 e rt T e T SN d + e rt SN d + Here follows from he fac ha ST < ST if and only if Z < r σ T T σ T T r σ T T. σ In we used CM4, Formula A. o wrie E e σ T T Z I {Z< d } e σ T T N d + and in 3 we used ha e r S is a P -maringale. The firs summand in equaion is hus e rt Ẽ ST I {ST <ST } SN d + An idenical compuaion shows ha he second expecaion in equaion is ẼST I {ST <ST } Ẽ E ST I {ST <ST } ST Ẽ ST N d e rt SN d whence e rt ẼST I {ST <ST } e rt T SN d Adding up boh expressions we conclude ha V S N d + + e rt T N d as claimed. 3

32 b The Dela of his opion is which remains saic over ime. V S N d + + e rt T N d Exercise 4.. CM4, Exercise. Soluion. Exercise 4.. CM4, Exercise.5 Consider he discree geomeric averaging of a sock price process a evenly disribued discree imes j + jδ, j,,..., n, wih a ime sep δ T n ; n T is he ime of expiraion. Define he discreely moniored geomeric averaging by /k k G k S j, k,..., n j a Assuming ha he sock price follows a GBM process, show ha G n is a log-normal random variable. Find he mean and variance of ln G n. b Derive he risk-neural ime- prices of he fixed srike Asian call and pu opions wih respecive payoff funcions G n K + and K G n +, where K > is a srike price. Soluion. a Since S follows a GBM assume zero dividends we have S j S e r σ j +σw j, j,..., n whence n G n S j j /n S e r σ n n j j+ σ n n j W j 3 Recall ha from CM4, Exercise. we have ha n W j N, j n j n+ j d n j n+ j Z, Z N, j This follow from he ransformaion rule: if X, Y are a p-dimensional and m-dimensional random vecors, wih respecive covariance marices Σ X and S Y, and Y AX, hen Σ Y AΣ XA. In our case X W,..., W n and A,...,, so σ X i,j CovW i, W j min{i, j} and VarY AΣ XA n j j n+ j. 3 j

33 and herefore σ n n W j d j σ n n j n+ j T Z no σ T Z T j wih Z N, and T n n j j. Wih regards o he drif erm, noe ha r σ n n j j so ha equaion 3 becomes r σ T r σ + σ σ T no r σ + δ T G n T S e r δ σ T + σ T Z and clearly ln G n T N ln S + r δ σ T, σ T There was no need o do all his o conclude normaliy, bu now we have G n wrien as a GBM wih consan dividend δ, so ha we can apply he Black-Scholes formula direcly in order o price fixed srike Asian calls and pus as hough hey were ordinary calls and pus on he underlying G n T. b Since from par a we have ha G n T follows a GBM G n T S e r δ σ T + σ T Z wih T n n j j, σ σ n T n j j n+ j, δ σ σ he Black-Scholes formula yields ha ime- value of a fixed-srike Asian call wih payoff G n T K + is C, S e δt SN d + e rt KN d wih d ± d ± e δ T S, T, d ± x, τ ln x r δ ± σ τ K σ τ The ime- value of a pu can be obained by pu-call pariy. 33

34 Exercise 4.. CM4, Exercise.6 Le he sock price process follow he GBM model ds Sr qd + σd W, ST Se r q σ T +σ W T W d Se r q σ T +σ T Z, Z N, Define he coninuously moniored geomeric average of S over a ime period, by G exp ln Su du. a Show ha he process ln G is Gaussian. b Find he mean and variance of GT condiional on G and S, for T. c Show ha GT can be wrien as GT G T T S T e µ+ σ Z for some µ, σ R, and where Z N, under measure P. Find he values of µ and σ. d Derive he risk-neural ime- pricing funcions for he fixed srike Asian call and pu opions wih respecive payoff funcions GT K + and K GT + for T. Express he pricing funcions in erms of he spo values G G >, S S >, and imes and T. e Esablish he pu-call pariy relaion for he fixed srike Asian call and pu opions. Soluion. a Since S Se r q σ +σw, we have ha ln G ln Su du ln S + r q σ u du + σ W u du From CM4, Proposiion. we know ha W u du N, 3 3 whence ln G N ln S + r q σ σ, 3 b Firs noe ha ln GT T T T ln G + T ln Su du T T ln Su du + T Now wrie Su Se r q σ u +σw u W whence T T ln Su du T T T ln Su du ln Su du 4 ln S + r q σ u du + σ T 34 T W u W du

35 For he firs inegral we have T T ln S + r q σ u and as for he second one T Noe ha T Var W u du W u W du W u du T du ln S T T W u du T Cov 3 T T T + r q T σ T W u du W T 3 T T W u du, W u du where he covariance compuaion can be found on CM4, Page 43. Since W u, u, T is independen of W we conclude ha T T Var W u W du Var W u du + Var W T T T + T T 3 3 Puing everyhing ogeher, from 4 we obain ln GT T ln G + ST T for Z N,. In conclusion ln GT ln G N ln G + ln ST T T c Exponeniaing 5 we obain where GT G T T S T no G T T S T + r q T σ + T 3 σ Z 5 T 3 T + r q σ T, exp r q σ T + 3 e µ+ σ Z µ r q σ T, σ σ T 3 3 T 35 σ T T 3 Z σ T 3 3 T

36 d In order find he ime- value of an Asian call wih payoff GT K + we proceed as usual: V, S, G e rt Ẽ,S,G GT K + e rt Ẽ,S,G G T T S T e µ+ σ Z K I G>K e µ rt G T T S T e σ ZI Ẽ,S,G Z>α,S,G Ke rt P,S,G Z > α σ e µ+ rt G T T S T N σ Ke rt N σ ln G T S T T K σ + ln G T S T T + µ K + µ where in we denoed α, S, G ln K T T ln G T σ ln S µ σ K µ + ln G T S T T e To derive a pu-call pariy relaion and he ime- value of a pu from i noe ha Discouning and aking expecaions yields GT K + K GT + GT K e rt Ẽ,S,G GT K + e rt Ẽ,S,G K GT + }{{}}{{} C,S,G P,S,G e rt rt Ẽ,S,G GT Ke whence and where P, S, G C, S, G e rt rt Ẽ,S,G GT + Ke e rt Ẽ,S,G GT e µ rt G T T S Ẽe σ Z T e µ+ σ rt G T T S T 36

37 5 Chaper 3: Risk-Neural pricing in a Muli-Asse Economy Exercise 5.. CM4, Exercise 3.4 A plain currency call opion on a foreign exchange rae has payoff C T XT κ + a Derive he pricing funcion C, x, < T for his call by evaluaing he risk-neural expecaion formula C, x e rt Ẽ XT κ + X x b Give he BSPDE for he pricing funcion C, x. Soluion. a Suppose ha he exchange rae process is a GBM driven by a d-dimensional Brownian moion, namely c.f. CM4, Equaion 3. XT Xe r rf σ X T +σx W T W d Xe r r f σ X T +σ X T Ẑ Ẑ σ X W T W σ X T where W W,..., W d and σ X σ X,,..., σ X,d. The pricing funcion C, x e rt Ẽ XT κ + X x e rt Ẽ Xe r rf + σ X T +σ X T Ẑ κ X x can be obained using he sandard Black-Scholes formula for he price of a call, wih ineres r r f, volailiy σ X, ime-o-expiry T and srike κ, namely C, x e XN rt d + X κ, T κn d X κ, T where as usual d ± x, τ ln x + r rf ± σ X τ σ X τ Exercise 5.. CM4, Exercise 3.5 Assume ha a foreign sock price process {S f } and he exchange rae {X} are correlaed geomeric Brownian moions wih respecive log-volailiy vecors σ S σ S, and σ X ρσ X, ρ σ X. Assume he domesic and foreign ineres raes r and r f are consans an ha foreign sock pays no dividend. a Derive a formula for he curren ime < T price C of a call opion on foreign sock denominaed in domesic currency wih domesic payoff + C T XT S f T K 37

38 b Similarly, derive a formula for he curren ime < T price P of a pu opion wih domesic payoff + C T K XT S f T c Derive a pu-call pariy formula relaing he call and pu prices C and P. Soluion. a We know c.f. CM4, Equaions 3. and 3.4 ha he process XS f saisfies he SDE dxs f XS f r q S d + σ X + σ S d W wih soluion where XT S f T XS f e r q S σ XS T +σx +σ S W T W XS f e r q S σ XS T +σ XS σ X σ X, σ S σ S, σ XS σ X + σ S T ˆX σ X + σ S + ρσ Xσ S and ˆX σ XS T σ X + σ S W T W N, The call opion can hus be priced by using he sandard Black-Scholes formula C BS S, K, τ; r, q, σ wih spo S XS f, srike K, ime o mauriy τ T, ineres rae r, no dividends q q S and volailiy σ σ XS, whence XS C, S f, X e S rt f f XN d +, T K XS f KN d, T K wih d ± x, τ ln x + r rf ± σ X τ σ X τ b Since we are using he sandard Black-Scholes formula, he sandard pu-call pariy relaion holds, namely C, S f, X P, S f, X S f rt X Ke Exercise 5.3. CM4, Exercise

39 Soluion. This call opion can be priced as a sandard call using he Black-Scholes formula. Exercise 5.4. CM4, Exercise 3.7 Consider he foreign equiy call sruck in foreign currency wih payoff + C T XT S f T K f. Assume he foreign sock price is a GBM wih consan log-volailiy σ S and having a dividend yield q S and he exchange rae is a GBM wih consan log-volailiy σ X. Derive is pricing funcion C, S, S. Soluion. This exercise is idenical o CM4, Example 3.4, wih he roles of S f T and XT exchanged. We will be reproducing mos of he argumen in CM4, Exercise 3.9c below, so we don duplicae i here. Exercise 5.5. CM4, Exercise 3.9 Assume ha a foreign sock price {S f }, a foreign exchange rae process {X} and a domesic asse price process {A} are all geomeric Brownian moions wih respecive consan log-volailiy vecors σ S, σ X and σ A : da A µ Ad + σ A dw, ds f S f µ Sd + σ S dw, dx X µ Xd + σ X dw W is a 3-dimensional sandard P -Bronwian moion in he physical measure P and he asses are correlaed, where σ S σ S, σ X σ X, σ A σ A, σ S σ A ρ SA σ S σ A, σ X σ A ρ XA σ X σ A, σ X σ S ρ XS σ X σ S. Assume a domesic and foreign economy wih respecive ineres raes r and r f as consans, zero dividends on all asses, and le A A, S f S and X X be he spo values. a Derive he ime < T pricing funcion for a domesic European opion wih payoff V T max{xt S f T, AT } b Derive he ime < T pricing funcion for a domesic European-syle opion wih payoff V T XT S f T I XT X + A T I XT <X where X X is a fixed posiive iniial exchange rae. c Derive he ime < T pricing funcion for a domesic European-syle opion wih payoff wih posiive consans a, b. V A + axt S f T bat Soluion. 39

40 a Recalling ha max{x, y} x y + + y we can rewrie he payoff as + V T max{xt S f T, AT } XT S f T AT + AT so see CM4, Equaion V, S, X, A e rt E,S,X,A XT S f T AT + e rt E,S,X,A AT and hese wo expecaions have been compued in he exchange opion example on CM4, Secion 3... wih XT S f T S T and AT S T. We simply need o esablish he correlaion coefficien beween boh processes. Recall ha he processes XS f and A saisfy XT S f T XS f e r σ XS T +σx +σ S W T W AT Ae r σ A T +σa W T W where σ XS σ X + σ S σ X + σ S + ρσ Xσ S The correlaion beween hese wo processes is given by σ A σ X + σ S σ A σ XS so we have he following equivalen expressions: ρ AX + ρ AS σ S σ XS no ρ XT S f T XS f e r σ XS T +σ XS T Ẑ XS AT Ae r σ A T +σ A where ẐXS and ẐA are independen sandard normal variables. T ρẑxs+ ρ Ẑ A Now we are in a posiion o apply he expression obained in he aforemenioned example: concreely, CM4, Eqauion 3.7 reads in our case XS XS C, S, X, A AN d + A, T + XSN d A, T where d ± x, τ ln x ± ν τ ν, ν σ A σ S + σ X τ σ A + σ XS ρσ Aσ XS c By he risk-neural pricing formula we have ha V, X, S f, A e rt Ẽ XT S f T I XT X + e rt Ẽ AT I XT <X We compue each expecaion separaely using an appropriae change of numraire. For he firs one, using XS f as numraire we are reduced o e rt Ẽ XT S f T I XT X XS f Ê I XT X XS f ˆP XT X 4

41 In order o compue he laer probabiliy i suffices o represen XT as a GBM random variable involving he ˆP -BM, exacly as i is done in CM4, Example 3.4. Recall ha by CM4, Eqauion 3. we have XT Xe r rf σ X T +σx W T W Recall ha he Brownian incremens in he risk-neural measure and he measure under numraire g are relaed by W T W W g T W g + σ g T no Ŵ T Ŵ + σxsf T Subsiuing his in he represenaion for XT above we obain XT Xe r rf σ X T +σx Ŵ T Ŵ +σ XSf T Xe r rf +σ X σ S + σ X Xe r rf +σ X σ S + σ X T +σ X T +σx Ŵ T Ŵ T Ẑ where recall ha σ XSf σ X + σ Sf and where Ẑ N,. Therefore we have ha XT X Xe r rf +σ X σ S + σ X T +σ X T Ẑ X Ẑ ln X X r rf + σ X σ S + σ X T σ X T In conclusion, he firs expecaion we needed o compue is e rt Ẽ XT S f T I XT X XS f ˆP XT X ln X XS f X N + r r f + σ X σ S + σ X T σ X T The second expecaion we compue analogously, bu now using A as numraire, namely: e rt Ẽ AT I XT <X A Ẽ A no IXT <X A ˆP XT < X Proceeding as above, we may relae he Brownian moion incremens in he risk-neural measure and our new measure ˆP in numraire A o obain we skip he deails whereby XT Xe r rf + σ X T +σx Ŵ T Ŵ e rt Ẽ AT I XT <X A ˆP XT < X ln X X AN r rf + σ X T σ X T 4

42 In conclusion, adding up boh expecaion we have shown ha V, X, S f, A e rt Ẽ XT S f T I XT X + e rt Ẽ AT I XT <X ln X XS f X N + r r f + σ X σ S + σ X T σ X T ln X X + AN r rf + σ X T σ X T d This exercise is jus CM4, Exercise 3.6 wih XT S f T S T and AT S T so we may jus apply he formulas obained herein afer compuing he correlaion beween he processes XS f and A as in par a. Exercise 5.6. CM4, Exercise 3. Soluion. Exercise 5.7. CM4, Exercise 3. Assume as in CM4, Exercise 3.5 ha a foreign sock price process {S f } and he exchange rae {X} are correlaed geomeric Brownian moions wih respecive log-volailiy vecors σ S σ S, and σ X ρσ X, ρ σ X and wih ime- spo values S f S, X X. Assume he domesic and foreign ineres raes r and r f are consans an ha foreign sock pays no dividend. Derive he ime- pricing formula as a funcion of S, T, X, for a domesic European opion having payoff a mauriy T > given by V T XT S f T I {MT <K} where MT is he maximum realized value of he exchange rae up o ime T : MT max T X Soluion. Using g XS f as numraire, we need o compue e rt Ẽ XT S f T I {MT <K} F XS f ˆP MT < K F and i hence suffices o obain he PDF of he sampled maximum MT in measure ˆP. Recall ha by CM4, Equaion 3. we have XT Xe r rf σ X τ+σx W T W The Brownian moion incremens W T W and Ŵ T Ŵ are relaed via Ŵ T Ŵ W T W σ XS T, σ XS σ X + σ S 4

43 so he expression for X becomes XT Xe r rf +σ X σs + σ X τ+σx Ŵ T Ŵ d Xe r r f +σ X σs + σ X τ+σ X τ Ẑ where Ẑ σ X τ σ X Ŵ T Ŵ N,. We can furher wrie X Xe σ XX where X µ + Ẑ, µ r rf + σ X σ S + σ X σ X and M X max u Xu XeσM X, Similarly, since we will be condiioning on F we have { } M X T max Xu max M X, Xe σm X τ, u T M X max u X u M X τ max u T X u We are now in he conex of CM4, Secion.3., whereby he join densiy funcion of M X τ and Xτ in measure ˆP is given by CM4, Equaion.8, namely ˆf M X τ,xτw, x w x τ πτ e µ τ+µτ w x τ, µ r rf + σ X σ S + σ X σ X Our expecaion is hence eiher if M X K or else is given by XS f ˆP MT < K F XS f ˆP Xe σ XM X τ < K F XS f ˆP M X τ < ln K σ X X F XS f ln σ X K X w ˆf M X τ,xτw, x dx dw Exercise 5.8. CM4, Exercise 3.3 Consider a domesic economy wih consan ineres rae r and wo correlaed GB sock price processes given by S T S e µ σ τ+σ τz, S T S e µ σ τ+σ τρz + ρ Z where Z, Z are independen sandard normal random variables. le S S, S S, T, be he sock spo values.derive he pricing funcion V, S, S for a European chooser max call wih payoff V T max{s T, S T } K + Soluion. Re-wrie he payoff as Λ T S T K + I {S T >S T } + S T K + I {S T >S T } 43

44 Then V, S, S e rτ Ẽ S T K + I {S T >S T } + S T K + I {S T >S T } We illusrae how o compue Ẽ S T K + I {S T >S T } We have ha Ẽ S T K + I {S T >S T } Ẽ S T KI {S T >K,S T >S T } e r q σ τ Ẽ e σ τz KI {Z >α,fz >Z } fx e σ τx K η x, y; ρ dy dx α where α ln K S r q σ τ σ τ fz η x, y; ρ π ρ e x +y ρxy/ ρ Exercise 5.9. CM4, Exercise 3.5 Consider hree socks wih GBM price dynamics as in CM4, Exercise 3. wih consan ineres rae r and consan dividend yield q i on each sock i,, 3. a Derive he pricing funcion V, S, S, S 3, < T for a European opion wih payoff V T S 3 T I {S3 T >S T,S 3 T >S T } b Derive he pricing funcion for a European opion wih payoff V T max{s T, S T, S 3 T } Soluion. For a derivaion using he change of numraire echnique, see OW6, Secion 6.. b Par b is sraighforward once we have esablished a, since max{s T, S T, S 3 T } S T I {S T >S T,S T >S 3 T } + S T I {S T >S T,S T >S 3 T } + S 3 T I {S3 T >S T,S 3 T >S T } 44

45 a Par a is a generalizaion of CM4, Equaion 3.7, whereby if S T S e µ σ τ+σ WT W, S T S e µ σ τ+σ WT W wih σ i σ i, σ σ ρσ σ, ν σ σ σ + σ ρσ σ hen E S T I {S T >S T } S S, S S ln S S e µ τ S N + µ µ + ν τ ν τ 6 We firs show his -dimensional case and we will hen indicae how o exend i o 3 socks even hough compuaions are edious. Sar by wriing S T S e µ σ τ+σ τz, S T S e µ σ τ+σ τρz + ρ Z where Z, Z are independen sandard normal random variables. Then we have E S T I {S T >S T } S S, S S S e µ σ τ E e σ τz I Z >fz where fz ln S S + µ µ σ σ τ + σ τ ρ Z σ ρσ τ We now use ieraed condiioning E S T I {S T >S T } S S, S S S e µ σ τ E E e σ τz I Z >fz Z Recalling ha for X N, we have Ee BX I X>A e B / N B A we have E S T I {S T >S T } S S, S S S e µ τ EN σ τ fz ln S S e µ τ S E N + µ µ + ν τ σ τ ρ Z σ ρσ τ Recalling ha for X N, we have EN AX + C N C +A we conclude ha E ln S S T I {S T >S T } S S, S S S e µ τ S N + µ µ + ν τ ν τ In dimension 3 we sar by compuing he lower Cholesky facorizaion U of he correlaion marix ρ ρ 3 ρ ρ 3 UU, U ρ ρ ρ 3 ρ 3 ρ 3 α β 45

46 where Wrie α ρ 3 ρ ρ 3, β ρ ρ 3 ρ 3 ρ ρ 3 ρ S T S e µ σ τ+σ τz, S T S e µ σ τ+σ τρ Z + ρ Z, S 3 T S 3 e µ 3 σ 3 τ+σ 3 τρ3 Z +αz +βz 3 where Z, Z, Z 3 are independen sandard normal random variables. Now we proceed as in he -dimensional case by ieraed condiioning e rτ E,S,S,S 3 S3 T I {S3 T >S T,S 3 T >S T } e rτ S 3 e µ 3 σ 3 τ E e σ 3 τρ3 Z +αz +βz 3 I {Z3 >fz,z 3 >fz } e rτ S 3 e µ 3 σ 3 τ E e σ 3 τρ3 Z +αz E e βz 3 I {Z3 >fz,z 3 >fz } Z, Z e rτ S 3 e µ 3 σ 3 τ E e σ 3 τρ3 Z +αz E e βz 3 I {Z3 >max{fz,fz }} Z, Z The inner expecaion E e βz 3 I {Z3 >max{fz,fz }} Z, Z can be compued using once again ha Ee BX I X>A e B / N B A for X N,... Alernaively: if we insead use S 3 as numraire, so are reduced o compuing e rt Ẽ S3 T I {S3 T >S T,S 3 T >S T } S3 ˆP > S T S 3 T, > S T S 3 T and i now suffices o express S T S 3 T and S T S 3 T as GBM s generaed by wo correlaed sandard normal variables and o inegrae he corresponding bivariae normal densiy funcion. Again, by CM4, Eqauion 3. we have S i T S 3 T S i S 3 e σ i σ 3 T +σ i σ 3 Ŵ T Ŵ S i S 3 e ν i T +ν i T Ẑ i for i,, where ν i σ i σ 3 and Ẑ i σ i σ 3 Ŵ ν T Ŵ, ˆρ : CorrẐ, Ẑ σ σ 3 σ σ 3 i T ν ν Moreover noe ha S i T S 3 T S i S 3 e ν i T +ν i T Ẑ i < Ẑi < ln S 3 S i + ν i T no d i, i, ν i T 46

47 and hence our expecaion is S 3 ˆP > S T S 3 T, > S T d d S 3 n x, x ; ˆρ dx dx S 3 T where η x, x ; ˆρ π ˆρ exp x ˆρ x x + x ˆρ Exercise 5.. CM4, Exercise 3.6 Consider an exchange opion on wo socks having payoff V T as T bs T +, a, b >. Assume he socks are GB; processes S T S e µ σ τ+σ τz, S T S e µ σ τ+σ τρz + ρ Z where Z, Z are independen sandard normal random variables. Derive he ime- price V for his opion by explicily using one of he socks as numraire asse and by implemening he risk-neural pricing CM4, Formula Noe: he derivaion is similar o ha in CM4, Example 3.3. Soluion. This exercise is idenical o CM4, Example 3.3. Working wih numraire g S and denoing Ê ẼS, he ime- value of he opion is given by VT V S Ê S T F S Ê a S T + S S T b T F as Ê S T b + F a no as Ê Y T b a + F where in he las equaliy we denoed Y T S T S T. I is easy o see ha he process Y S S is a ˆP -maringale As in CM4, Example 3.3 we can wrie where Y S S Y E σ σ Ŵ Y Y e X X σ σ + σ σ Ŵ ν + ν Ẑ Eiher see CM4, Equaion 3. or direcly compue he sochasic differenial d S S. 47

48 wih Ẑ σ σ Ŵ ν N, and ν : σ σ σ +σ ρσ σ. We can hus compue V S Ê Y T b + F S Ê a Y e ν T +ν T Ẑ b + F, Ẑ N, a by using he Black-Scholes formula wih zero ineres rae and dividend, volailiy ν, ime o mauriy T, spo Y and srike b a. In oher words: V T, S, S S Y N d + Y, T b a N d Y, T S N d + Y, T b a S N d Y, T Exercise 5.. CM4, Exercise 3.7 Consider a domesic economy wih consan ineres rae r and wo domesic sock price processes S T S e µ σ τ+σ τz, S T S e µ σ τ+σ τρz + ρ Z where Z, Z are independen sandard normal random variables. Derive he ime- pricing funcion V V T, S, S in he spo variables S S, S S, for a European pahdependen opion wih payoff a mauriy T given by S V T S T min T S Soluion. Recall he numraire invarian form of he risk-neural pricing formula: for an aainable payoff V T and numraire g we have VT V gê gt F, Ê no Ẽg Picking g S as numraire we hus have V S Ê min T S S F I is easy o see 3 ha he process Y S S is a ˆP -maringale As in CM4, Example 3.3 we can wrie Y S S Y E σ σ Ŵ Y Y e X 3 Eiher see CM4, Equaion 3. or direcly compue he sochasic differenial d S S. 48

49 where X σ σ + σ σ Ŵ ν + ν Ẑ wih Ẑ σ σ Ŵ ν N, and ν : σ σ σ + σ ρσ σ. We are hus reduced o compuing he expecaion V S Ê min Y T S Ê S Ê min Y T e ν +ν Ẑ min S Ê T Y ex min T Y eνz no S Ê Y e ν mz where Z Ẑ ν ν and m Z min u Zu see CM4, Secion.3.. The las expecaion can be compued by inegraing agains he join densiy funcion of he sampled minimum m Z and Z c.f. CM4, Equaion.9, namely Ê Y e ν mz Y x w f m Z,Zw which is a bi edious o ype. τ πτ w e νw f m Z,Zw, x dx dw, e ν +νx x w τ 49

50 6 Chaper 4: American opions Exercise 6.. CM4, Exercise 4. Prove CM4, Proposiion 4.3 for an arbirary American opion wih a differeniable payoff funcion Λ. In paricular show he following. a A any poin, S of he early-exercise boundary, he American opion pricing funcion V saisfies he smooh pasing condiion V, S S Λ S SS b The opion value V saisfies he zero ime-decay condiion on he early-exercise domain V, S, S D {, s : V, S ΛS Soluion. a We proceed as in CM4, Secion 4... Suppose he American opions has no been exercised a ime and le B denoe he collecion of all possible early-exercise boundaries defined by coninuous funcions b :, T R +, so ha for each b B here is an exercise policy T b. Then V, S sup V, S; b, b B V, S; b Ẽ,S e rtb ΛST b The oal derivaive of he funcion V, S; b along he boundary is given by dv V, S; b S V, S; b db S b + Sb b Along he curve S b i is clear ha S b When b S, we have by opimaliy. Sb Therefore V,S;b b dv db V, S; b S SS On he oher hand, he opion value is equal o he payoff funcion when S b, ha is V, b; b Λb whence V, S S SS V, S; S S def SS by 7 dv, b; b db b The oal derivaive of V, S wih respec o ime is bs Sb dλb db Λ S bs 7 dv d V, S + V, S S S 5