BEA400 Microeconomics. Lecture 11:

Transcription

1 BEA4 Miroeonomis Leure BEA4 Miroeonomis Leure Module 5: Choie Over ime ih Unerin Leure : Sohsi Proesses, Io s Lemm nd Sohsi Opiml Conrol Sohsi Proesses he Pure Weiner Proess or Bronin Moion Sling he vrine of he Proess Weiner Proess or Bronin Moion ih Drif Mrkovin Proess Generl Weiner Proess or Bronin Moions Sohsi Inegrion Io s Lemm Demonsrion of Io s Lemm Some Properies Io s Lemm Derivion of Io s Lemm Appliions of Io s Lemm A Sohsi Re of Inflion Rel Re of Reurn ih Sohsi Inflion Blk-Sholes Opion Priing Sohsi Opiml Conrol heor Hmilon-Jobi-Bellmn (HJB Equion Opiml Exrion of n Unerin Non-Reneble Resoure Consumpion-Svings Deision ih Risk Inome Consumpion-Svings Deision ih opionl Risk Asse Logrihmi Uili HARA Uili Reding: I hven found ex h I m hpp ih for his pr of he ourse, so leure noes should suffie. You ould r: Mlliris, A.G. nd W.A Brok, Sohsi Mehods in Eonomis nd Finne, Norh- Hollnd, 98. Chpers, 3 nd 4. Universi of smni - CRICOS PROVIDER CODE 586B

2 BEA4 Miroeonomis Leure Sohsi Proesses. he Pure Weiner Proess or Bronin Moion A Pure or Bsi Weiner Proess or Bronin Moion refers o oninuous ime x x hih evolves over sohsi proess here rndom vrible ime, = smll inervl of ime d, ording o sohsi differenil equion, dx = z d here z is sndrd norml ih men zero nd sndrd deviion of. Sine z is normll disribued i follos h dx ill be normll disribued ih n expeed vlue of zero nd sndrd deviion of d. hus x is hnging rndoml b dx sine i depends on z d dx ~ N (, d h is dx is be normll disribued ih E( dx = nd Vr ( dx = d (nd SD( dx = d he reson h dx is sled ih d is h n oher hoie for he mgniude of dx ould led o problem h is eiher meningless or rivil hen e onsider h hppens he limi hen d. Also if dx ere no sled in his, he vrine of he rndom lk ould hve limiing vlue of or. No le us onsider dx over o ver smll bu onseuive ime periods. Consider o ime periods nd, suh h = + ih orresponding vlues of x, x ( = x nd x ( = x nd hnges dx ( = dx nd dx ( = dx Universi of smni - CRICOS PROVIDER CODE 586B

3 BEA4 Miroeonomis Leure Wih he hnge over boh periods s dx = dx + dx hen he expeed vlue of dx over boh periods is E dx = E dx + E dx = + = nd vrine = + + (, Vr dx Vr dx Vr dx Cov dx dx If e ssume h he vlues of z re independen over ime hen dx in n period of ime is independen of dx in ll oher periods nd he vrine is = + Vr dx Vr dx Vr dx = d + d = d More generll over long ime period, E( dx = nd vrine Vr ( dx = d = d nd SD( dx= d h is he vrine of he Weiner proess over he ime period, is equl o he sum of he hnges in ime periods. his llos us o onsider he vlue x over disree hnges in ime d = Consider sring poin s = hen ime, fer hnges of d = hen he vrine of dx Vr dx = d = d = = s= s= A he sring poin s =, so ih xs = x he hnge in x over, dx is dx = x x Universi of smni - CRICOS PROVIDER CODE 586B 3

4 BEA4 Miroeonomis Leure nd x = x + dx If he proess srs zero, x = x =, hen he vlue of x in n period > s >, is given b s x = x + dx = dx hus from dx ~ N (, d E( dx = nd vrine Vr ( dx = d SD( dx = d ih he ssumpions of s =, x = x = nd d = s e obin h x in n fuure period is x ~ N, E( x = nd vrine Vr ( x = SD( x = nd for n ime period in beeen s nd, x N(,. Sling he vrine of he Bsi Weiner Proess he Bsi Weiner Proess n be enhned o sle he sndrd deviion of he rndom vrible x b σ dx = σ z d. In hih se x ~ N(, σ Noe h σ n be modelled o depend on ime,, nd he vlue of x suh h ( x, σ = σ Universi of smni - CRICOS PROVIDER CODE 586B 4

5 BEA4 Miroeonomis Leure 3. Weiner Proess or Bronin Moion ih Drif he Bsi Weiner Proess n lso be enhned b dding drif or groh erm, µ o he sohsi differenil equion. dx = µ d + σz d In hih se x ~ N( µ, σ Noe h s ih σ, µ n be modelled o depend on ime,, nd he vlue of x suh h µ µ ( x, = 4. Mrkovin Proess A Mrkovin proess is one here he probbili vlues of fuure vlues of x ondiionl on being ime, onl depend upon he urren vlue of x nd no oher informion. ( P x, = P x, x = x While his migh seem rher resriive i n be modified o llo fixed moun of ps informion. he Generl Weiner Proess desribed belo is n exmple of Mrkovin proess. 5. Generl Weiner Proesses or Bronin Moions Generl Weiner Proess or Bronin Moion refers o oninuous ime sohsi proess here rndom vrible x = x evolves over ime,, ording o some sohsi differenil equion: (, (, dx = µ x d + σ x z d his lso lled n Io Proess. Universi of smni - CRICOS PROVIDER CODE 586B 5

6 BEA4 Miroeonomis Leure Sohsi Inegrion Sohsi Inegrion s developed b Io(944 ho generlised he sohsi inegrl firs inrodued b Weiner (93 Consider for hih sohsi proess dx (, x, z ih deerminisi omponen nd rndom omponen hih follos Sndrdised Weiner proess z. (, (, dx = µ x d + σ x z d Sohsi Inegrion rnsforming he bove ino n inegrl equion e ge, ( µ (, σ (, x = x + s x ds+ s x z ds s s We hve enounered inegrls of he form ( sx, s σ ih ( sx, s µ z d hih does no exis? Io s Lemm! ds before bu ho do e ope Universi of smni - CRICOS PROVIDER CODE 586B 6

7 BEA4 Miroeonomis Leure Io s Lemm Suppose x follos Bronin Moion ih = µ (, + σ (, dx σ (, x z d s dx x d x z d nd = d = nd = dz dz d nd = d If he dnmis of x ( n be rien b n Io Proess, hen he dnmis of ellbehved funion of x ( h desribe is disribuion, = F(, x ill lso be desribed b n Io Proess. Io s Lemm gives: d = F dx + F d + F dx x xx ( d = F + F µ + F σ d + F σdz x xx x nd expeion [ ] = ( + xµ + xxσ + xσ [ ] = ( F + Fxµ + F xxσ Ed/ d F F F F Edz/ d s Edz= [ ] [ ] = ( xµ xxσ xσ = ( F + Fxµ + F xxσ d ( ( Ed E F F F d F dz E d = E F + Fxµ + Fxxσ d + Fxσ F + Fxµ + Fxxσ dzd + Fxσ dz ( xµ xxσ xσ = F + F + F d + F E dz [ ] σ = x Vr d F E dz = F σ x s = Edz Universi of smni - CRICOS PROVIDER CODE 586B 7

8 BEA4 Miroeonomis Leure Demonsrion of Io s Lemm Suppose x follos Bronin Moion ih dx = µ xd + σ xdz nd dx σ x dz σ x d = = s d =, dz d dz = nd = d Consider = ln x Io s Lemm d = F dx + F d + F dx x xx Noe h d = dx + dx x x dx = σ x dz = σ x d, d = ( µ xd + σxdz σ x d x x = µ σ d + σdz hih n be rien hih gives = + ( + µ σ d σ dz = + ( µ σ + σ z No subsiuing bk = ln x b using e = x nd x = ( exp ( µ σ + σ x x z = e gives Some Properies Io s Lemm ( + = + σ σ dz σ dz σ dz E dz E d σ = σ if σ E s ds < Universi of smni - CRICOS PROVIDER CODE 586B 8

9 BEA4 Miroeonomis Leure Derivion of Io s Lemm (No exminble A lor series expnsion of F, x = round gives d he expeed vlue of d n be ompued (noing h Edz= [ ] s [ ] E = F x E x x F x E x x + + higher order erms sine E ( x x = µ nd [ ] E x x = V x x + µ = σ + µ [ ] µ ( σ µ E = F x + F x + + higher order erms ih he vrine of ( dz = d nd dz d = ignoring higher order erms nd using d =, so b seing = llos he hnge in o be given b σ d = F ( x µ + F ( x d + F ( x σdz or if F lso depends on hen ( d = F + F µ + F σ d + F σdz x xx x = ( + µ + σ + σ F F F F z x xx x nd finll dividing b d nd king he expeion (noing Edz= [ ] gives [ ] = ( + xµ + xxσ + xσ [ ] = ( F + Fxµ + F xxσ Ed/ d F F F F Edz/ d Universi of smni - CRICOS PROVIDER CODE 586B 9

10 BEA4 Miroeonomis Leure Appliions of Io s Lemm A Sohsi Re of Inflion dp Suppose he re of inflion is given b bsi Bronin Moion = µ d + σdz P We n esil find he men nd vrine of suh series, dp E = µ E[ d] + σe[ dz] P dp E = µ d P he verge proporione hnge in pries is expeed o be µ muliplied b he ime period over hih i is onsidered. his n resul n be re-expressed s he dp / d E = µ P dp& E = µ P E π = µ [ &] hus he expeed oninuos re of hnge in inflion is µ dp dp dp Vr = E E P P P dp Vr = µ + µ σ + σ E d E d dz E dz µ E d P dp = σ [ ] = [ ] = Vr d s E dz, E d dz nd d = dz P [ ] { [ ]} he vrine in he proporione hnge in pries is σ muliplied b he ime period over hih i is onsidered. dp / P σ = Vr d d σ Vr [ & π ] = d σ SD[ & π ] = s dz = d dz Universi of smni - CRICOS PROVIDER CODE 586B

11 BEA4 Miroeonomis Leure While he Bronin Moion n redil provide he men nd vrine for he proporione hnge in pries if e n o kno somehing bou he prie level in n priulr period e ill need o solve he differenil equion dp = µ Pd + σ Pdz. he presene of P, hih depends on ime, in he drif erm is no problem s e hve lern o solve suh differenil equions previousl, bu is presene ih he dz erm requires he use Io s Lemm in order o solve for P. Le lnp = hen b Io s Lemm d = F dx + F d + F dx x xx d = dp + dp P P = µ Pd + σpdz σ P d P P P = µ σ d + σdz All erms re independen of ime nd so n esil be inegred o provide = µ σ + σz ( exp ( µ σ σ P = e = P + z Rel Re of Reurn ih Sohsi Inflion If e lso hve n sse reurning nominl reurn of Q period h is groing r hen dp = µ Pd + σpdz nd dq = rqd hen defining Q q = he rel reurn hen Io s lemm gives P q q q q q dq d dp dq dp dq q = dqdp P Q P Q Q P Sine ( d =, ( dz Whih llos = d nd, dqdp nd dq equl zero. Universi of smni - CRICOS PROVIDER CODE 586B

12 BEA4 Miroeonomis Leure Q Q dq = dp + dq + P σ d 3 P P P Q Q = ( µ Pd + σpdz + rqd + σ d P P P dq = ( r µ + σ d σdz q sine q = Q P Blk-Sholes Opion Priing Suppose here re hree sses. he firs ih men re of reurn µ p nd SD σ p, he seond ih men re of reurn µ p nd SD σ p nd he hird risk free sse h erns re of reurn r per period. he nominl vlue of he porfolio is = + + v n p n p np 3 = µ + σ nd = µ + σ dp p d p dz p p dv = n dp + n dp + dnp 3 ( µ p σ ( p µ p σ p np ( µ p σ p ( µ p σp dp p d p dz p p = n p d + dz + n p d + dz + r np d 3 dv np np = d + dz + d + dz + r d 3 v v v v Defining, nd 3 s he shres he hree sses from of he porfolio s vlue np np np3 so h =, = nd 3 = = so h he bove equion v v v n be rien dv v ( µ p σ p ( µ p σ p ( - = d + dz + d + dz + r d Defining, for riskless porfolio hen vr dv = σpdz + σp dz = v σ p σ pdz = σ p dz = σ p dv = µ pd + µ p d + r - d v Universi of smni - CRICOS PROVIDER CODE 586B

13 BEA4 Miroeonomis Leure sine dv rd v = for he riskless porfolio ( ( µ p ( µ p r r d + = = ( µ p r ( µ p r Combining he o σ ( µ p r p = = σ p ( µ p r ( µ p r ( µ p r σ = σ p p nd so hus he re of reurn per uni of risk mus be equl for he o risk sses. If sse is sok opion hose prie is deermined b p F ( p dp = µ d + σ dz p p = nd, Using Io s Lemm nd h he re of reurn per uni of risk mus be equlised = + + dp F d F dp F dp p pp ( dp = F + µ F + F σ p d + F σ p p pp p p p p pp σ p dp = F + rp F rf + F p If he opion n onl be exerised erminl ime ih exerise prie p he boundr ondiion F(, s = s, s = - nd F( p, = mx[, p p ] hen Blk nd Sholes (973 nd Meron (973 demonsred h rs (,,, σp = φ + e φ p sp r p d d p σ p d= ln + r + s p σ p s d= d σ s p s / φ ( = e ds π Noe h φ ( is he umulive norml disribuion Universi of smni - CRICOS PROVIDER CODE 586B 3

14 BEA4 Miroeonomis Leure Sohsi Opiml Conrol heor Previousl our equion of moion s rien, = (,, f u = is he No onsider sohsi differenil equion here f ( u,, deerminisi (non-rndom omponen nd (,, ih ( u,, σ udzis he sohsi omponen σ being deerminisi funion nd dz being Bronin moion inremen. he problem hen beomes V(, = Mx E F(,, u d u subje o (,, σ (,, d = f u d + u dz he opiml vlue of he mximised problem n be rien s + V (, = Mx E F (,, u d F (,, u d u + + u ( ( = Mx E F,, u + V +, + (, Mx { (,, + V( +, + } V E F u u Noe h folloing our previous seion on Io s Lemm he bove sohsi differenil equion n be rerien s, dv = V + V d + V d (,, σ (,, σ (,, = V+ Vf u d+ V u d+ V u dz Assuming h V(, is ie differenible e expnd he funion on he righ round (, b lor Series expnsion: V +, + = V, + V + V + V + V + V + ho.. Universi of smni - CRICOS PROVIDER CODE 586B 4

15 BEA4 Miroeonomis Leure Insering = f (,, u + σ (,, u z, using ( =, ( z nd hen simplifing gives. = nd z = { σ } (, Mx (,, V(, (,, σ (,, V E F u + + V + Vf u + V u + V z u No le he expeion of he bove, he onl sohsi erm is expeion is E( z =. hen subr (, z nd is V from boh sides nd divide hrough b nd le. u { F( u V Vf u V σ ( u } = Mx,, + +,, +,, ( σ V = Mx F,, u + V f,, u + V,, u u Hmilon-Jobi-Bellmn (HJB Equion his is he Hmilon-Jobi-Bellmn (HJB equion of sohsi onrol heor ( σ V = Mx F,, u + V f,, u + V,, u u Noe h he o-se vrible is V so h ih respe o he se vrible gives (,, V (,, V ( equion n be rien u = V = = V = =, differeniing so h he HJB σ V = Mx F,, u + f u,, + u,,. % % is If he rnsformed Hmilonin funion H = H( u,,, (,, (,, σ (,, H% = F u + f u + u hen V = Mx H% u H% Assuming = u insering ino he HJB ( V = H % *,,. n be solved for he opiml hoie u* u* (,, = hen Universi of smni - CRICOS PROVIDER CODE 586B 5

16 BEA4 Miroeonomis Leure sine u* = u* (,, (,,, σ(,,, d = f d + dz hen sine H% * = f (,,, d = H% * d + dz σ I n lso be shon b using he definiion of = V, he sohsi equion of moion for he se vrible nd Io s lemm h d = H% * d + σ dz hen opiml HJB ondiions re: H% = u Equion for opiml hoie u. d = H% * d + σ dz Equion of moion for he o-se vrible. σ d = H% * d + dz Equion of moion for he se vrible. (, = Endpoin resriion. Universi of smni - CRICOS PROVIDER CODE 586B 6

17 BEA4 Miroeonomis Leure Consumpion-Svings Deision ih Risk Inome Suppose h eh period he onsumer reeives n inome ( per period i gros onsn re of sndrdised Weiner proess. d = µ d + σ dz d = µ Y bu ih rndom omponen σ dz Y here dz is We ould inlude s se vrible, ih orresponding o-se vrible nd ppl he rules of Sohsi Opiml Conrol. Hoever i ill be muh simpler for us o use Io s Lemm on d firs (see sohsi re of inflion exmple nd repling ih: = ( µ σ + σ exp z so h he elh evoluion no expliil inludes he vrine of inome. d = r d + d d exp ( µ σ σ = r d + + z d d ( exp ( µ σ E = r + sine E z = ih erminl ondiions =, =. Proeeding in his mnner e n simpl use he non-sohsi version of he Hmilonin, hih for his problem is ( δ ( µ σ σ H,,, = e ln + r + exp + z ih Hmilonin Condiions H: H: H δ = e = H d = & = = r d r ( µ σ r r e = + e d e d = δ e Universi of smni - CRICOS PROVIDER CODE 586B 7

18 BEA4 Miroeonomis Leure H3: & d H = = = r d ( = ( e r ( µ σ r δ + e e r = ( r o go ih = e δ nd ( µ σ δ ( e r r r r ( µ σ r δ = e e ( e + e e δ r µ σ ( for he onrol vrible, onsumpion nd se vrible elh. o give he soluion We n see h inome groh dds o iniil onsumpion in h i effeivel redues he re of disouning on fuure inome b is groh. While inome unerin redues iniil onsumpion in h i effeivel inreses he re of disouning on presen vlue of fuure inome due o is unerin. If e hd proeeded ihou solving for d firs, e ould hve hd o use he sohsi version of he Hmilonin, hih in his problem ould be δ H,,,, = e ln + r + µ + σ nd is firs order ondiions. Universi of smni - CRICOS PROVIDER CODE 586B 8

19 BEA4 Miroeonomis Leure Consumpion-Svings Deision ih Opionl Risk Asse Logrihmi Uili Ignoring inome for he momen, ssume h here is risk sse ih re of reurn r nd SD of σ hen evoluion of elh beomes. d = r r + d + σ dz = r + r r d + σ dz he Mximum Vlue Funion is ( δ V, = Mx e ln d so h σ V = Mx F,, u + V f,, u + V,, u onrol problem is, u he sohsi opiml, { ( } δ ln σ V = Mx e + V r + r r + V Or repling V = nd defining he sohsi Hmilonin s δ ( H%,,,,, = e ln + r r r + + σ { } δv = Mx %,,,,, H[ ], Exmine he Hmilonin Condiions (removing ime from he vribles for lri. H% ( r r HA: = r r+ σ = * = σ H: H% e δ = = = e δ H implies H% e δ = = Universi of smni - CRICOS PROVIDER CODE 586B 9

20 BEA4 Miroeonomis Leure F F = V F = V F = V F V / = F V / F V / = F V / = = V V here = is he Arro-Pr Mesure Relive Risk Aversion RRA of V elh / he elsii of elh for onsumpion / F F F F = he ineremporl elsii of subsiuion hus he ineremporl elsii of subsiuion is equl o he produ of Arro-Pr Mesure Relive Risk Aversion for elh nd he elsii of elh for onsumpion. Or hus he Arro-Pr Mesure Relive Risk Aversion is equl o he produ of nd he elsii of onsumpion for elh nd he ineremporl elsii of subsiuion. Sine V = he Arro-Pr Mesure Relive Risk Aversion he opiml V proporion invesed in he risk sse is ( ( r r r r * = = σ RRA σ Universi of smni - CRICOS PROVIDER CODE 586B

21 BEA4 Miroeonomis Leure hus he proporion invesed in he risk sse is inversel reled o he Relive Risk Aversion of elh nd posiivel reled o sses reurn over he risk free sse nd negivel o is vrine. hus if RRA is onsn hen * ill lso be onsn over life. F = V F / F = V / V For ln( he ineremporl elsii of subsiuion is F F = = = F F = RRA / / sine = = RRA hus he Arro nd so Pr Mesure Relive Risk Aversion for logrihmi uili is RRA( = nd Reurning o he oher Hmilonin Condiions H: H3: H d = & = = r + r r d & d H = = = r + ( r r σ d ( r r Insering HA: * = ino H nd H3 σ H: ( r r ( σ & = r r r ( r r = r σ Universi of smni - CRICOS PROVIDER CODE 586B

22 BEA4 Miroeonomis Leure Sine V RRA( = v = = & r ( r r = + σ No H3: H3: ( ( ( ( r r r r & = r + r r σ σ σ r r r r = r + σ σ & = r hus = r = e nd H is no ( r e δ r ( = e δ so = hus r r r & = r e δ + σ r r Denoe l = + σ ( r & r = l e δ ( = & l r e r e δ Using f x f( x = e e dx ' + A f ( x δs δ o inegre he RHS e ds = e + A gives δ s= Universi of smni - CRICOS PROVIDER CODE 586B

23 BEA4 Miroeonomis Leure r δ s e ( & r = l e ds s= r e = l e + A A δ r δ e = A+ l e δ δ seing = A = l δ e e δ δ r = l ( seing = e e δ r δ = l ( = r ( e l δ δ ( e l l Noe >, < r σ hus iniil onsumpion r <, > σ Consumpion groh is he sme = ( r e δ r While he PV of elh e = ( e δ nd loer σ. l ill be higher for higher r δ Sine he ineremporl elsii of subsiuion (Arro-Pr degree of relive risk version for onsumpion is onsn nd equl o one. In RRA( = here is no prmeer o desribe degree of degree of risk version for elh ih he logrihmi uili funion oher hn he elh nd onsumpion. Universi of smni - CRICOS PROVIDER CODE 586B 3

24 BEA4 Miroeonomis Leure Hperboli Absolue Risk Aversion (HARA lss of uili funion Hoever if e hoose differen funionl form for uili he piure is differen. If e onsider Hperboli Absolue Risk Aversion (HARA lss of uili funion. θ θ ( = u For his HARA uili funion he ineremporl elsii of subsiuion is θ ( θ F F = = = θ θ F F θ = RRA / / sine = = RRA hus he nd so θ Arro-Pr Mesure Relive Risk Aversion for simple HARA uili is RRA = ( θ he orresponding sohsi Hmilonin is H%,,,,, = e + r + r r + θ δ θ ( ( σ HA: H% = r r+ σ = ( r r * = σ H: H% e δ θ = = δ = e θ H3: ( ( ( ( r r r r & = r + r r σ σ σ r r r r = r + σ σ & = r hus r = e nd H is no ( r δ = e θ so = θ hus = e ( r δ θ Universi of smni - CRICOS PROVIDER CODE 586B 4

25 BEA4 Miroeonomis Leure Sine V RRA( = v ( θ = = ( θ hen subsiuing = ( θ ( r r & = r θ σ ino & o give ( r r & = r + σ ( θ And subsiuing = e ( r δ θ ( r r θ r δ & = r ( e + σ ( θ r r Denoe l = + σ ( θ & r = l e ( r δ θ ( ( r δ r r θ & = l e r e Using f x f( x = e e dx ' + A f ( x ( r δ ( r δ s θ θ θ o inegre he RHS e ds = e + A gives r δ s= Universi of smni - CRICOS PROVIDER CODE 586B 5

26 BEA4 Miroeonomis Leure ( r δ θ r ( & = r s l s= e r e ds ( r δ θ r r θ = l + e e A A rθ δ ( r δ θ r r θ = l e A e rθ δ seing = θ A = + l rθ δ rθ δ r θ θ e = + l e rθ δ Noe h if θ= he logrihmi se e e δ δ r = l ( If = hen e n use he erminl ondiion for elh o remove from he problem. If here s no erminl ondiion on elh e ould need o use he rnsversli ondiions, (suh s = if he erminl se is free ogeher ih he equion of moion of he o-se vrible o provide nd so. rθ δ r θ θ e = + l e rθ δ r ( e = rθ δ θ θ l e rθ δ Universi of smni - CRICOS PROVIDER CODE 586B 6