Matematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.
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1 Maemaisk saisik Tenamen: kl 8 13 Maemaikcenrum FMS17/MASM19 Prissäning av Derivaillgångar, 9 hp Lunds ekniska högskola Soluion. 1. In he firs soluion we look a he dynamics of X using Iôs formula. If X has no drif erm and if he Iô-inegral par of X has finie second momen we have ha X will be a Maringale. We should also check ha X is adaped o he some filraion. he easies choice here is he filraion generaed by he Brownian moions W 1 and W. The adapedness is sraigh-forward since X is jus a poinwise coninuous ransformaion of W 1 and W. Le f x 1, x = e x 1 cosx. Le X = f W 1, W = e W1 cosw. We can now calculae dx wih Iôs formula: dx = df W 1, W = 1 { f x 1, x dw x1 1 + f x 1, x dw 1 dw x 1 x } + f x 1, x dw 1 = x { f x1, x + x 1 =W 1,x =W x 1 dw 1 + f x 1, x x } dw { 1 ew 1 cosw 1 ew 1 cosw } d +e W 1 cosw dw 1 e W 1 sinw dw = e W 1 cosw dw 1 e W 1 sinw dw x 1 =W 1,x =W This process has no drif erm moreover we have ha he diffusion par has finie second momen using he Iô isomery, i.e. [ ] E e W1s cosw s dw 1 s e W1s sinw s dw s = E [ e W 1s cos W s + sin W ] ds = This gives ha X is a Maringale. Alernaive soluion: We can direcly calculae E[X F s ] for s < as E[X F s ] = E[e W 1 cosw F s ] E[e s ]ds = e 1/ <. = E[e W 1 W 1 s cosw W s + W se W 1s F s ] = E[e W 1 W 1 s cosw W s cosw se W 1s F s ] E[e W 1 W 1 s sinw W s sinw se W 1s F s ] 1
2 Taking ou wha is known = cosw se W 1s E[e W 1 W 1 s cosw W s F s ] sinw se W 1s E[e W 1 W 1 s sinw W s F s ] independence = cosw se W 1s E[e W 1 W 1 s ]E[cosW W s] sinw se W 1s E[e W 1 W 1 s ]E[sinW W s] = e W 1s cosw se s/ e s / + e W 1s sinw se s/ = e W 1s cosw s = X s. Finally we need o esablish ha E[ X ] < which is easily done since E[ X ] E[e W 1 ] = e / <.. According o he firs fundamenal heorem of asse pricing, a marke is free of arbirage if here exis a leas one MG-measure. According o he second fundamenal heorem of asse pricing: if a marke is free of arbirage hen he marke is complee if and only if he MG-measure is unique. Using he Girsanov heorem and ha he marke is driven by one Brownian moion we ge ha he marke is free of arbirage and complee if and only if he Girsanov kernel exiss and is unique. Applying he Girsanov heorem wih a Girsanov kernel g we ge he new dynamics for he asses S 1 and S as ds 1 = m 1 gs 1 S 1 d + s 1 S 1 dw, ds = m gs S d + s S dw, S 1 = s 1, S = s. We now wan ha S 1 and S should be maringales if we discoun hem. Therefore we mus have ha m 1 gs 1 = r m gs = r. These wo equaions will have a unique soluion if and only if m 1 r/s 1 = m r/s m 1 = m rs 1 /s + r m 1 = s 1 m + 1 s 1 s s r. The final equaion is exacly he condiion given in he problem. We should also check ha g saisfies he Novikov condiion, bu his is clearly rue since g is consan. 3. a Since he marke is free of arbirage we have ha all raded asses discouned by he bank accoun wih consan ineres rae should be maringales. We herefore ge ha E[Su F ] = Se ru, for s <. Due o he form of he payoff in he derivaive and ha we can change order of inegraion and aking expecaion a leas if we assume ha T is finie all we need o know is E[Su F ].Therefore we have ha he price will be uniquely deermined by he assumpions given in he problem.
3 b Using wha have from a we ge ha [ 1 T ] P = e rt E Q Ss ds S F T = e rt 1 T = e rt 1 T = e rt 1 T T E Q [ Ss F ] ds Se rt Ss ds + e rt 1 T Ss ds + e rt 1 rt T e rs rt S ds Se e rt 1 rt S Se The naural poin in ime o price his conrac is of course =. So puing = we obain e rt S 1 e rt 1 rt. rt We see ha his price is non-negaive for all r and all T. 4. Using Feynman-Ka cs represenaion formula we obain f, x = e rt E[IX T > K X = x] where X has he following dynamics for u T dx u = rx u du + sx u dw u, X = x. So he soluion is in fac he price of a binary call opion wih srike K and mauriy T for he case where he underlying asse follows he sandard Black-Scholes model. Using his we see ha X T = xe r s /T +sw T W d = xe r s /T +s T G, where G is sandard Gaussian random variable. We hus obain ha where f, x = e rt = e rt dx, y Ixe r s /T +s T y > K dy e y dy dx, Symmery = e rt e y dy = e r Ndx,, dx, = lnx/k + r s /, = T. s To check ha he soluion saisfies he PDE i is easier o work wih he expression in he equaion *. f, x = rf, x + dx, e y dy e r 3
4 s x rx x = rf, x + e = rf, x + e = rf, x + e dx, f, x = e r = e dx, r dx, r dx, r dx, r rx xs r s dx, s x f, x = e r x x s e = e dx, r dx, r Puing all his ogeher we ge ha = e =. xs xs d, x r s / + lnx/k + r s // s lnx/k r s / s lnx/k + r s / s s + lnx/k + r s / s f, x + rx x + s x / rf, x x dx, r lnx/k r s / + r s lnx/k r s / s 5. a Using he risk neural valuaion formula we ge ha P = e rt E Q [X F ], where X = K S T IK 1 < S T < K wih S T = S expr s /T + sw T W = d S expr s / + s G, where = T and where G is a sandard Gaussian random variable. Using his we obain P = e r E[K S e r s /+s G IK 1 < S e r s /+s G < K ] = e r K S e r s /+s x IK 1 < S e r s /+s x < K e x / / dx = e r d S d 1 S = e r K d S d 1 S K S e r s /+s x e x / / dx e x / / d S s dx S 4 d 1 S s e x / / dx
5 where = e r K Nd S Nd 1 S S Nd S s Nd 1 S s, d 1 S = lnk 1/S r s / s, d S = lnk /S r s /. s b To calculae he porfolio weighs h S, h B for he sock and bank accoun respecively we use ha h s = P, h B = P h S S. S B To calculae h S i will be more convenien o use expression **. We can see ha S is in hree differen places in he expression, upper limi, lower limi and in he inegrand. However since he inegrand is zero a he upper limi we only have o deal wih he wo las cases. This gives ha h S = P S = e r K S e r s /+s d 1 S e d 1S / / 1 S s Nd S s Nd 1 S s = e r K K 1 e d 1S / / 1 S s Nd S s Nd 1 S s. h B = e K s Nd r S Nd 1 S + K K 1 e d 1S / /. s We see ha jus as for he ordinary pu-opion we shor-sell he sock and keep a nonnegaive amoun on he bank-accoun. 6. a We use ha p, u = exp u f, s ds. Now le Y = u f, s ds which gives ha p, u = expy and hus we have using Iô s formula where dp, u = p, u dy + p, udy /. u dy = f, d df, s ds u v, s = r d d s s v, s dw ds u v, s u = r d ds d v, s ds dw s s = r v, u v, / d + v,, u v, dw = r v, u / d + v,, u dw. Plugging his ino he previous expression and using ha dw = d, d = ddw = 5
6 we obain dp, u = p, u dy + p, udy / = p, ur v, u / d + p, uv,, u dw + p, u v, u / d = rp, u d + p, uv, u dw We here see ha he drif is rp, u which i should be since he ZCB is a raded asse and he discouned price process should herefore be a MG. b Under he forward measure F T we have ha X s is a raio beween a raded asse and he numeraire i should herefore be a MG. So all we need is o calculae he volailiy par since he drif should be zero, we hus ge 1 dx s = ps, T ps, vs, ps, ps, T ps, T vs, T dw s FT = X svs, vs, T dw FT s, where W FT is a F T sandard d-dimensional Brownian moion. c According o he risk neural valuaion formula we have ha price a ime ime is given by P = p, T E FT [maxx X, ], where X according o b is given by X = X exp 1 T1 vs, vs, T ds + d = X exp 1 S + SG, where Using his we obain he price as T1 S = T1 vs, vs, T ds. P = p, T E [maxx FT X, ] [ = p, T X E max exp 1 S + SG vs, vs, T dw FT s ] 1, = p, max exp 1 e S y + Sy 1, dy = p, = p, = p, S/ S/ = p, N = p, N S/ S S exp 1 e S y + Sy 1 dy e S 1 Sy+y dy p, T1 e y dy p, T1 S N S 1. 6 e y S/ e y S/ dy dy
7 d By using ha p, s = exp s f, u du and he definiion of X we obain pt1, maxx X, = max p, T p, T 1 p, T, = max 1 exp f, u du exp T f, u du exp, T f, u du T T = max exp f, u du exp f, u du,, which is he prescribed expression in he problem. 7
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