Kinga Tikosi. Merton s Portfolio Problem

Transcription

1 Cenral European Universiy Deparmen of Mahemaics and is Applicaion Kinga Tikosi Meron s Porfolio Problem MS Thesis Supervisor: Miklós Rásonyi 2016 Budapes, Hungary

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3 iii Ackoledgemens I would like o express my graiude o my supervisor, Miklós Rásonyi, for his guidance and suppor while wriing his hesis. I am also very hankful o him for inroducing me o his heory. I am sincerely hankful o my parens for heir consan suppor and for always encouraging me o follow my dreams. Köszönenyilváníás Szereném megköszönni émavezeőmnek, Rásonyi Miklósnak, az érdekes émafelveés, számos épíő hozzászólás és szakmai anácso, amellyel hozzájárul a dolgoza elkészíéséhez. Továbbá szereném megköszönni a szüleimnek, hogy mindenben ámogaak és mindig báoríoak, hogy kövessem az álmaim.

4 Conens Inroducion 2 1 Background Theory and Formulaion of he Problem The Marke Model Sochasic Conrol Theory Uiliy Funcions Firs Example The Main Resul Uiliy of Consumpion Discouned Uiliy of Consumpion Simulaions Fads Models The Simulaion Model Finding he Bes Consan Allocaion Sraegy Meron-sraegy for he Modified Model Conclusion and Furher Exensions 28 References 30 Appendix 31 1

5 Inroducion Meron s opimal invesmen problem is a well known problem of coninuousime finance, which was named afer Nobel laureae Rober Meron s pioneering work in his area. I concerns finding he opimal invesmen sraegy for he invesor, who has only wo possible objecs of invesmen: a risk-less asse (e. g. a savings accoun), paying a fixed rae of ineres and a number of risky asses (e. g. sock, real esae) whose price is assumed o follow a geomeric Brownian moion. Assume ha he invesor lives for a finie period of ime, from presen unil ime T, he sars wih an iniial amoun of money, and he wans o decide how much of which securiy o hold a each ime in order o maximize he final wealh. Naurally, he invesor is risk averse o a cerain degree, which means ha he refrains from invesing in asses which have a high risk of loosing money even if i migh have high reurn. The invesor s aiude o risk and individual preferences can be characerized by he so called uiliy funcions, his way we do no maximize he expeced reurn of he invesmen, bu he expeced uiliy, making i possible no only o maximize he expeced value of he final wealh, bu also limi he risk of losing money a he same ime. We consider a coninuous-ime marke model, which means ha he invesor can re-balance his capial a any momen before he erminal ime, ha is moving capial from risk-less securiy o sock and vice versa, wihou any ransacion coss. The asses are supposed o be infiniely divisible, hey can be bough or sold in arbirary amouns anyime. The invesor has all he informaion abou he prices a presen, bu no addiional informaion. Meron formulaed and solved he problem (1969, 1971) [3], [4], for he case where here are no ransacion coss. Using he mehods of sochasic conrol, he derived a nonlinear parial differenial equaion (he so called Hamilon-Jacobi-Bellman equaion) for he value funcion of he opimizaion problem. For some special cases of uiliy funcions (i. e. logarihmic and power uiliy), closed form of soluion was found by him. For hese cases, i urns ou ha he opimal sraegy is o keep a consan fracion of he wealh in socks. 2

6 The aim of presen hesis o develop beer undersanding of he above presened problem. I consiss of hree chapers, firs chaper aims o give a brief inroducion, we presen he financial seing, inroduce a mehod of sochasic conrol called dynamic programming which we will use laer o solve he opimizaion problem. We also presen he soluion of he problem for he logarihmic uiliy funcion: his case can be reaed separaely due o he simpler form. In he second chaper we obain he heoreical resuls for power uiliy: firs we derive he opimal sraegy o maximize he uiliy of he consumpion hroughou he invesmen and also he final wealh for a finie ime horizon, hen for an infinie ime horizon we maximize he consumpion only. The resuls are indeed heoreical: keeping a consan fracion of he wealh in risky asse (and herefore also a consan fracion in he risk-less one) would mean coninuous rading, which is no very realisic. Also, o achieve he opimal resul we made he assumpion ha he asse prices follow a geomeric Brownian moion wih consan drif and consan volailiy. Meron in his original paper already quesioned he accuracy of his assumpion, however, his model is sill he mos frequenly used financial model. In he hird chaper we would like o check how his sraegy works in pracice. We ake a more general model dropping he assumpion of consan drif and check how he Meron sraegy works for his case. We would like o answer he following quesions: does he expeced uiliy decrease because of discreizaion? Will an invesor who wans o use a Meron-ype of sraegy, i.e. a consan allocaion sraegy, ge he same consan if we change he model a lile bi? Is i beer o inves less in he risky asse as uncerainy in he drif of he modified model grows or can an invesor profi from i? If we change he model bu use he allocaion sraegy which is opimal in he Black-Scholes case, can we sill reach he same expeced uiliy? We will come back o answer hese quesions in he end of he hesis. 3

7 Lis of Noaions and Abbreviaions W - Brownian moion 1 - he vecor (1, 1,..., 1) C 1,2 = {f(, x) f coninuously differeniable in, wice coninuously differeniable in x} Diag(V ) diagonal marix wih he enries of he vecor V in he diagonal E,x [Y ] = E[Y X = x], when i is clear which X we are alking abou E x [Y ] = E 0,x [Y ] D x f he gradien of f D xx f he Hessian of f J(, x, u) opimizaion crierion A(, x) admissible se V (, x) value funcion V (x 0 ) = V (0, x 0 ) 4

8 Chaper 1 Background Theory and Formulaion of he Problem This chaper aims o give a brief inroducion, we presen he financial seing, and we inroduce a mehod of sochasic conrol called dynamic programming which we will use laer o solve he opimizaion problem. When presening he basic noions of sochasic finance, we will follow [2] and [8], he par abou sochasic conrol heory is based on [5]. 1.1 The Marke Model Le us consider a complee probabiliy space (Ω, F, P), wih he filraion (F ) [0,T ] saisfying he usual condiions, (i.e. F 0 conains all he measure 0 ses, and he filraion is righ-coninuous: F = s> F s, for all ) and le T > 0 be a non-random erminal ime. (W ) [0,T ] denoes a sochasic process called Brownian moion. Definiion 1. A process W = (W ) 0 is a P-Brownian moion (wih respec o (F )) if i is (F )-adaped and i saisfies 1. W is coninuous wih W 0 = 0 (a.s.), 2. W W s is independen of F s, 0 s <, 3. For any > 0, W N(0, ) under he probabiliy measure P, 0 s <. A higher dimensional Brownian moion is he vecor W = (W 1,..., W n ) where he W i are independen Brownian moions, all adaped o he same filraion F. 5

9 As he model for he financial marke we will use he Black-Scholes model: suppose ha he financial marke consiss of one bond (a riskless bank accoun paying a fixed ineres rae r) wih prices db = B rd, B 0 = 1, ha is B = e r, and one sock (risky asse) wih prices evolving like ds = S (µd + σdw ), S 0 = s 0 > 0, wih rend parameer µ R and volailiy σ > 0. Using Iô s formula, we can derive he explici soluion of he sochasic differenial equaion: S = S 0 exp((µ σ2 2 ) + σw ). A such kind of process is said o follow a geomeric Brownian moion. The wealh (he value of he porfolio) of he invesor wih iniial capial x 0 > 0 evolves like dx = N B db + N S ds, X 0 = x 0, where N B and N S denoe he number of he bonds and socks held by he invesor a ime. N B and N S are non-negaive, bu no necessarily ineger. Suppose ha he wealh is always non-negaive: X() 0 a.s., for 0 T. 1.2 Sochasic Conrol Theory In his secion we will inroduce he basic noions of sochasic conrol heory, following [5]. In his hesis we will consider conrols of Iô processes, which saisfy sochasic differenial equaions driven by Brownian moion. The SDE dx = b(, X )d + σ(, X )dw, where W denoes he m-dimensional Brownian moion, has a unique srong soluion called an Iô diffusion when he drif b : [0, ) R n R n and diffusion coefficien σ : [0, ) R n R n m saisfy he following condiions for all 0 s, and x, y R n b(s, x) b(, y) + σ(s, x) σ(, y) K( y x + s ) b(, x) 2 + σ(, x) 2 K 2 (1 + x 2 ) for some K posiive consan. 6

10 Definiion 2. An F-progressively measurable process (u ) [0,T ] wih values in some se U R p is called a conrol process. An n-dimensional process (Y ) [0,T ] conrolled by u if i is defined by dy = b(, Y, u ) + σ(, Y, u )dw, Y 0 = y 0, where b : [0, T ] R n U R n, σ : [0, T ] R n U R n m, (W ) [0,T ] denoes he m-dimensional Brownian moion. The opimizaion crierion is [ T J(, x, u) = E ψ(, X u, u )d + Ψ(T, XT u ) X u ] = x We denoe he admissible se of conrols by A(, x) and i conains all he conrols, which fulfil he following properies: 1. The conrol process u = (u s ) s [,T ] is progressively measurable wih values in U and E[ T u s 2 ds] <. 2. The SDE describing he conrolled process has a unique srong soluion (X s ) s [,T ] wih X = x and E,x [ sup X s 2 ] <. s T 3. The opimizaion crierion J(, x, u) is well defined. Wih he above noaions our aim is o maximize he value funcion of he he conrol problem, ha is defined by V (, x) = sup J(, x, u) u A(,x) and find he opimal value V (0, x 0 ) as well as he opimal conrol sraegy u for which his value is aained, ha is V (0, x 0 ) = J(0, x 0, u ) Dynamic programming The idea of dynamic programming is o break he opimizaion problem down o smaller sub-problems and use hese resul o ge he overall opimum. In his secion we briefly describe how his mehod works and laer we are going o use his o solve he porfolio opimizaion problem. In order o be able o use dynamic programming, he problem needs o have a specific opimal subsrucure. This propery is called he Bellman Principle: [ 1 ] V (, x) = sup E,x ψ(s, Xs u, u s )ds + V ( 1, X u 1 ) u A(,x) 7

11 This principle basically saes ha we can isolae a par of he opimizaion problem, an opimal conrol on an inerval [, 1 ] will say opimal if a 1 we coninue opimally. One way o solve an opimal conrol problem is o use he Bellman principle (i needs o be proved ha i holds). If he wealh process has sufficien smoohness properies, we can apply Iô formula o V ( 1, X u 1 ) and pu i back o he above equaion, yielding V (, x) = sup u A(,x) [ 1 E,x ψ(s, X s, u s )ds + V (, X ) V (s, X s )(D x V (s, X s )) b(s, X s, u s )ds (D x V (s, X s )) σ(s, X s, u s )dw s r((d xx V (s, V s )) σ(s, X s, u s )σ(s, X s, u s ) ] The sochasic inegral par 1 (D x V (s, X s )) σ(s, X s, u s )dw s is a maringale, so is expecaion is 0. Using he noaion for he diffusion marix we obain V (, x) = a(s, X s, u s ) = σ(s, X s, u s )σ(s, X s, u s ) sup u A(,x) 1 1 [ 1 E,x ψ(s, X s, u s )ds + V (, X ) V (s, X s )(D x V (s, X s )) b(s, X s, u s )ds ] r((d xx V (s, V s )) a(s, X s, u s ). Le us subrac V (, x) from he equaion and divide by ( 1 ) and le 1 end o. Now we have o check wheher aking limi and expecaion can be inerchanged, as well as aking supremum and limi. If so, since we ake condiional expecaion when X = x, i follows ha V (, X ) = V (, x). Then we ge ha { 0 = sup Ψ(, x, u) + V (, x) + (D x V (, x)) b(, x, u) u U + 1 } 2 r((d xxv (, x))a(, x, u)) We define a differenial operaor (which depends on u) for C 1,2 funcions L u f(, x) = V (, x) + (D x f(, x)) b(, x, u) r((d xxf(, x))a(, x, u)) 8

12 and we can rewrie he above equaion as 0 = sup{ψ(, x, u) + L u V (, x)} u U This equaion is called Hamilon-Jacobi-Bellman equaion (or HJB equaion). Now we have derived a necessary condiion, since we have seen ha he value funcion V solves he Hamilon-Jacobi-Bellman equaion under cerain condiions. Now we wan o formulae a condiion which ensures ha he soluion found is indeed he value funcion and i provides an opimal conrol sraegy. Theorem 3. (Verificaion Theorem) Le σ saisfy he growh condiion σ(, x, u) 2 C σ (1 + x 2 + u 2 ), and le ψ be coninuous wih ψ(, x, u) 2 C ψ (1 + x 2 + u 2 ) for C σ, C ψ posiive consans for all R +, x R n, u U. 1. Le Φ C 1,2 ([0, T ) R n ) be coninuous on [0, T ] R wih Φ(, x) C Φ (1 + x 2 ) for a posiive consan C Φ and sup{φ(, x, u) + L u Φ(, x)} = 0, for [0, T ), x R n u U Φ(T, x) = Ψ(, x), x R n Then for every [0, T ], x R n Φ(, x) V (, x). 2. If here exiss a maximizer û(, x) of he funcion u ψ(, x, u) + L u Φ(, x) such ha u = û(, X ) is admissible, u = (u ) [0,T ], hen Φ(, x) = V (, x) for [0, T ), x R n and he opimal conrol sraegy is given by u. This means ha V (, x) = J(, x, u,x ), where u,x = (u s) s [,T ] and X denoes he soluion of he SDE describing he conrolled process, for conrol u s. Proof. Le us fix [0, T ], x R n and define he following hiing imes τ n = min{t, inf{s > : X s X n}}, his way we can argue for bounded processes. Using Iô formula for X we ge Φ(τ n, X τn ) = Φ(, x) + τn L us Φ(s, X s )ds + τn Φ x (s, X s ) σ(s, X s, u s )dw s 9

13 Finieness of E,x [ τ n Φ x (s, X s ) σ(s, X s, u s ) 2 ds follows from he coninuiy of Φ and boundedness of X, herefore Then E,x [ τ n Ψ saisfies he HJB, i.e. [ τ n ] E,x Φ x (s, X s ) σ(s, X s, u s )dw s = 0. ] ψ(s, X s, u s )ds + Φ(τ n, X τn ) [ τ n = E,x ψ(s, X s, u s )ds + Φ(, x) + τn ] L us Φ(s, X s )ds [ τ n ] = Φ(, x) + E,x (ψ(s, X s, u s ) + L us Φ(s, X s ))ds sup{φ(s, x, u) + L u Φ(s, x)} = 0, for [0, T ), x R n, u U herefore (ψ(s, X s, u s ) + L us Φ(s, X s )) 0 for all s [, T ], yielding E,x [ τ n ] ψ(s, X s, u s )ds + Φ(τ n, X τn ) Φ(, x). As n ends o infiniy τ n T. Then, using dominaed convergence, we ge E,x [ τ n ] ψ(s, X s, u s )ds + Φ(τ n, X τn ) J(, x, u), as n, as an inegrable dominaing funcion we can use he growh condiions on Φ and ψ o define τn T ψ(s, X s, u s )ds+φ(τ n, X τn ) C ψ (1+ X s 2 + u s 2 )ds+c Φ (1+ X T 2 ) L 1. Then we ge ha also and aking supremum in u J(, x, u) Φ(, x), V (, x) Φ(, x). To prove par (2.), le us observe ha if we manage o find an opimizer û(, x) and use he sraegy u = (, X ), hen we ge equaliy insead of inequaliy, so using similar argumens we ge V (, x) = Φ(, x) 10

14 To sum up, o solve he opimizaion problem we will proceed as follows: 1. Find an opimal û for he HJB equaion. 2. If we find a such û, i will depend on V, D x V, D xx V formally, so i will have he form û(, x) = û(, x, V, D x V, D xx V ). Then we plug in he value we go for u in he HJB equaion o ge a parial differenial equaion for V. Solving his equaion for he boundary condiion V (T, x) = Ψ(T, x) we ge a V which is our candidae for he value funcion. 3. Use he verificaion heorem o check ha he funcion V we go from he HJB and he maximizer û are indeed he value funcion opimal and he opimal conrol sraegy. 1.3 Uiliy Funcions In order o characerize he invesor s decisions and preferences we need he concep of uiliy funcions. Uiliy funcion basically expresses how saisfied he invesor is wih a cerain oucome of he invesmen, his way i is a funcion of he wealh or a funcion of he consumpion. We assume he invesor o be risk averse, meaning ha he will only accep invesmens which are "beer han fair game", his implies sric concaviy of he uiliy funcion. The second assumpion we make is ha he will always prefer more wealh o less, his can be referred o as non-saiaion of he invesor and i implies ha he uiliy funcion is sricly monoone increasing. Definiion 4. For a subse S R, U : S R is a uiliy funcion, if U is sricly increasing, sricly concave and coninuous on S. Definiion 5. For a uiliy funcion U : S R he absolue risk aversion coefficien is defined o be he raio A(x) = U (x) U (x), respecively he relaive risk aversion coefficien is he raio R(x) = xu (x) U (x) 11

15 In his hesis we will use wo differen uiliy funcions, logarihmic uiliy U(x) = log(x), log denoing he naural logarihm, and power uiliy U(x) = xγ γ, where γ R, γ < 1, γ 0. In fac, he logarihmic case corresponds o γ = 0. These wo classes of funcions are he so called hyperbolic absolue risk aversion (HARA) or also referred o as consan relaive risk aversion (CRRA) class. Coming back o he porfolio opimizaion problem,a s conrol a ime we will use he fracion u of he oal wealh which should be invesed in risky asses. Then N B = (1 u )X, N S = u X B S hen we can rewrie he wealh process in he following way dx = (1 u )X rd + u X (µd + σdw ) = X ((r + u (µ r))d + u σdw ). 1.4 Firs Example In he case of logarihmic uiliy funcion we can derive he soluion in a simpler way for he case when here is no consumpion, meaning ha we only wan o maximize he expeced uiliy of he final wealh, wihou consuming money from he bank accoun hroughou he invesmen period. Our aim is o solve he following opimizaion problem: max E[U(X T ) X 0 = x 0 ] u Theorem 6. For U(x) = log x he opimal policy is u = µ r σ 2, for all [0, T ]. Proof. We need o solve he following SDE: To do so, guessing applying Iô-formula we ge dx = X ((r + u (µ r))d + u σdw ). X = x 0 exp{ 0 g s ds + 0 h s dw s }, dx = X g d + X h dw X g 2 d = X ((g h2 )d + h dw ) 12

16 hen comparing he coefficiens we ge g h2 = r + (µ r)u, h = σu. Thus we ge X = x 0 exp{ Our aim is o maximize over all admissible conrol sraegies in A(x 0 ) = {u : E 0 0 (r + (µ r)u s 1 2 σ2 u 2 s)ds + E[log(X u T ) X 0 = x 0 ] ( bu + σu 2 )d <, X u We have ha σu dw is a maringale, hus in paricular So for every u A(x 0 ) [ T ] E σu dw = σu s dw s } > 0, E[(log X T ) ] < }. [ T J(x 0, u) = log x 0 + E (r + (µ r)u s 1 ] 2 σ2 u 2 s)ds. We can check ha (r + (µ r)u s 1 2 σ2 u 2 s) is sricly concave, by aking derivaives u ((r + (µ r)u s 1 2 σ2 u 2 s)) = µ r σ 2 u 2 u ((r + (µ r)u 2 s 1 2 σ2 u 2 s)) = σ 2 < 0. Seing µ r σ 2 u = 0 we ge ha he unique opimal value for u is wih he corresponding value funcion u = µ r σ 2, V (0, x 0 ) = J(0, x 0, u ) = log(x 0 ) + (r + µ r 2σ 2 )T This means ha he opimal invesmen sraegy is o keep he same, fixed proporion of he oal wealh invesed in socks, e.g. if he opimal policy u = 0.2 hen he invesor should always keep 20% of he wealh is he risky asse. This sraegy is no doable in pracise: i would mean consan rading. In Chaper 3. we will see how close we can ge o he opimal expeced uiliy by discree rading. The simpliciy of his soluion is due o he fac ha he SDE had an exponenial soluion, hus aking he logarihm leads o a simple equaion which we 13

17 were able o maximize poinwise. This mehod would no work for anoher ype of uiliy funcion. The following sep would be he case, where here is no only one sock o choose. We basically ge he same resul for his case, as saed below, he opimal sraegy is keeping a consan fracion of he wealh in socks. Le us consider n socks wih prices (S ) [0,T ], S = (S 1,..., S n ), wih dynamics ds = Diag(S )(µd + σdw ), S 0 = s 0, where s i 0 > 0 for I = 1, 2,..., n, µ R n, and σ a non-singular volailiy marix in R n n. So sock i evolves like ( ds i = S i µ i d + n j=1 σ ij dw j ), i = 1,..., n. As conrol a ime, we will ake an n-dimensional process u, where u i denoes he fracion of wealh which is invesed in sock i. The opimal soluion is u = π = (σσ ) 1 (µ r), [0, T ]. 14

18 Chaper 2 The Main Resul Le us now assume ha he invesor does no only wan o maximize he expeced uiliy of he final wealh, bu also wans o make a living from he invesmen and he consumes money a rae c() from he bank accoun. This way, his objecive is o maximize he expeced uiliy of he consumpion hroughou he invesmen period. In his chaper we will follow [5]. 2.1 Uiliy of Consumpion We consider a similar financial marke consising of one bond wih prices evolving like db = B rd, B 0 = 1 ha is B = e r, and n socks wih prices ds = Diag(S )(µd + σdw ), S 0 = s 0, where W is an m-dimensional Brownian moion, m n, s i 0 > 0 for i = 1,..., n, µ R n, and σ R n m wih maximal rank, implying ha he marix σσ R n n is no singular.so sock i evolves like ( ds i = S i µ i d + m j=1 σ ij dw j ), i = 1,..., n. As conrols u = (π, c) we ake he vecor of fracions π = (π 1,..., π n ), where π i denoes fracion of he wealh invesed in sock i, and c sands for he 15

19 consumpion rae. Then he wealh process is evolving like dx u = n X u πds i /S i i + X u (1 π )db /B c d i=1 = ((r + π (µ r1))x u c )d + X u π σsw, The invesor assigns uiliy U 1 (c ) o he consumpion rae c and uiliy U 2 (X u T ) o he wealh a erminal ime T. U 1 and U 2 could be differen, bu we will consider power uiliy funcion for boh, i.e. U(x) = xα α, α < 1, α 0 and a discouning facor e β, β 0. Thus we wan o maximize J(, x, u) = E,s [ T ] e β U(c )d + e βt U(XT u ). Theorem 7. For U(x) = xα and he above presened opimizaion problem, he α opimal policy pair (c, π ) is c = e β 1 α h() 1 X π = 1 1 α (σσ ) 1 (µ r1) Proof. We wan o proceed using dynamic programming. The corresponding parial differenial operaor is L (π,c) v(, x) = v (, x) + ((r + π (µ r1))x c)v x (, x) π σσ πx 2 v xx (, x) wih he Hamilon-Jacobi-Bellman equaion sup π,c { e β cα α + L(π,c) V (, x) } = 0. When V is increasing and concave and x posiive, we ge he following maximizers c(, x) = (e β V x (, x)) 1 α 1 π(, x) = (σσ op ) 1 (µ r1) V x(, x) xv xx Puing his back o he Hamilon-Jacobi-Bellman equaion, we have o solve 1 α α β α e 1 α V α 1 x + V + rxv x 1 2 (µ r1) (σσ ) 1 (µ r1) V x 2 = 0 V xx a erminal ime we have boundary condiion V (T, x) = e βt xα α. Guessing 1 α xα V (, x) = h() α 16

20 we ge h(t ) = e βt 1 α where c = boundary condiion a erminal ime for h and e β 1 α + ch() + h () = 0, α ( 1 ) r + 1 α 2(1 α) (µ r1) (σσ ) 1 (µ r1). This is a linear ODE and for β (1 α)c 0 i has he soluion h() = e β 1 α e c(t ) + {e β (1 α)c 1 α for β (1 α)c = 0 (1 α)e c β (1 α)c h() = e c (1 + T ). e β (1 α)c T 1 α I is easy o see ha his soluion for h akes sricly posiive values for every. Then, since V (, x) = h() 1 α xα, if he equaion for he conrolled process α (for conrols c, π defined above) has a posiive unique soluion, hen he value funcion V is also posiive. This means ha V C 1,2, furhermore, monooniciy and concaviy of V follow from he sign of he derivaives V x (, x) = h() 1 α x α 1 > 0 V xx (, x) = (1 α)h() 1 α x α 2 < 0 }, Subsiuing V (, x) = h() 1 α xα α ino he equaions for c and π we ge c(, x) = e β 1 α h() 1 x π(, x) = 1 1 α (σσ ) 1 (µ r1) For he conrol defined by he policy pair (c, π ), where we ge ha he conrolled process X c = c(, x) = e β 1 α h() 1 X π = π(, x) = 1 1 α (σσ ) 1 (µ r1) is of form dx = ((r + π (µ r1))x u c )d + X π σsw = X ((r + π (µ r1) e β 1 α h() 1 )d + π σsw ), so he unique srong soluion for his SDE is a sochasic exponenial, hence sricly posiive. To use he verificaion heorem we need inegrabiliy condiions, which are saisfied because he soluion is srong. Admissibiliy and growh condiions can be checked using a a suiable se U of conrols, which, being parly difficul, is ou of he scope of his hesis. This means ha he value funcion is indeed how we defined i and (c, π ) is indeed he opimal conrol sraegy. 17

21 2.2 Discouned Uiliy of Consumpion We now wan o maximize he he discouned uiliy of consumpion in he same model, ha is J(x, u) = E x [ 0 e β cα α d ] over all admissible policy pairs u = (π, c) ha conrol he process dx = ((r + π (µ r1))x c )d + X π σdw, X 0 = x. Noe ha in his case we ake an infinie period of ime: final wealh has no uiliy, we only care abou he uiliy of consumpion. Theorem 8. For U(x) = xα and he above presened opimizaion problem, he α opimal policy pair (c, π ) is ( where A = α 1 α π = 1 1 α (σσ ) 1 (µ r1) c = A 1 α 1 X, ( )) α 1 β r 1 α 2(1 α) (σσ ) 1 (µ r1) σσ (σσ ) 1 (µ r1) Proof. Le β > 0, α (0, 1), iniial wealh x > 0, and we also assume ha P(X > 0) = 1 for every >0. The value funcion is V (x) = sup J(x, u), u A(x) and he corresponding Hamilon-Jacobi-Bellman equaion reads as { sup ((r + π (µ r1))x c)v x + 1 } u R n [0, ) 2 π σσ πx 2 V xx βv + cα = 0. α Supposing V x > 0 and V xx < 0, we ge he following opimum: π = ν V x(x) xv xx (x), where ν = (σσ ) 1 (µ r1), and c(x) = V x (x) 1 α 1. Puing hese values in he HJB equiaion, we ge he ODE 1 2 ν σσ ν V x 2 + rxv x βv + 1 α V xx α V α α 1 x = 0. 18

22 Guessing V (x) = A xα α for some A > 0 yields V x(x) = Ax α 1, V xx (x) = (1 α)ax α 2, so we need o solve 1 2(1 α) ν σσ ν + r β α + 1 α α A 1 α 1 = 0. for x > 0. So we need o assume ha β > α 2(1 α) ν σσ ν + αr, because A was assumed o be sricly posiive. Solving he equaion for A we ge A = ( α ( β 1 α α r 1 )) α 1. 2(1 α) ν σσ ν So we ge he following candidaes for he opimal policy π = 1 1 α (σσ ) 1 (µ r1) c = A 1 α 1 X and he wealh process, conrolled by his policy pair, is evolving like dx = 1 1 α X ((1 α)r + ν (σσ ν r1) (1 α)a 1 α 1 )d + ν σdw ). Solving he SDE gives us he unique srong soluion {(( X = X 0 exp 1 1 ) 1 α ν 1 r + 1 2α 2(1 α) 2 ν σσ ν A 1 α 1 ) α ν σw }. Since X 0 > 0, he soluion is also sricly posiive, hen i follows ha V x > 0 and V xx < 0. We wan o use he verificaion heorem o argue ha we indeed have found he opimum. V is clearly wice coninuously differeniable. The inegrabiliy condiions follow from he fac ha (X ) is an L 2 process. 19

23 Chaper 3 Simulaions As we have already seen in he previous chaper, he bes sraegy for he invesor is o keep a consan fracion of he wealh in he risky asse. To achieve his resul we made he assumpion ha he asse prices follow a geomeric Brownian moion. Meron in his original paper already quesioned he accuracy of he above assumpion, however, his model is sill he mos frequenly used financial model. In his chaper we would like o sudy how he Meron-sraegy works if we consider a more general marke model. 3.1 Fads Models The Fads-models were inroduced by Summers [9] and Shiller [6] in he eighies. These models can be considered as modified Black-Scholes models where we do no assume ha he drif is consan. For a posiive consan ε le Y ε be an Ornsein-Uhlenbeck process defined by he following SDE: dy ε () = 1 ε Y ε()d + dw () Y ε (0) = 0 Ornsein-Uhlenbeck processes, and his specific one in paricular, have mean revering propery: he drif par of he process depends on he curren value, and as he whie-noise erm given by dw () draws he process around, he drif pulls i back o he mean, 0 in his case. Plos below show ha he smaller ε is, he more he process is pulled back o 0. Above sochasic differenial equaion has he explici soluion Y ε () = 0 e ( s) 1 ε dw (s). 20

24 Figure 3.1: 5 plos of Ornsein-Uhlenbeck process for ε = 0.01 Figure 3.2: 5 plos of Ornsein-Uhlenbeck process for ε = 1 We can now inroduce a modified Black-Scholes model as follows: ds() = S()((µ + Y ε ())d + σρdw () + σ 1 ρ 2 db()), where B() and W () are wo independen Brownian moions and 0 < ρ < 1. For ε = 0 his is exacly he original Black-Scholes model, bu bu ε 0 i clearly does no have consan drif. 3.2 The Simulaion Model We will use he mehod of discreizaion o simulae he soluion rajecories of he sochasic differenial equaions. These rajecories will be he asse prices. Le T be fixed and le us ake N discreizaion seps, 21

25 0 = 0 < 1 <... < N = T, d = T/N. We know he analyic soluion of SDEs describing boh he geomeric Brownian moion X and Ornsein-Uhlenbeck Y, so hese processes can be modeled as follows: yielding he discreized version: X = X 0 exp((µ σ2 2 ) + σw 1()) n 1 X( n ) = X 0 (exp( (µ 1 2 σ2 )d + σ dz 1 (j))). Y ε, = j=0 0 Y ε ( n ) = e n ε e ( s) 1 ε db(s), n 1 j=1 e j ε dz2 (j) Figure 3.3: 20 plos of geomeric Brownian moion for parameers µ = 0.12, σ = 0.4, X 0 = 100 wih T = 1 and 200 discreizaion seps For he soluion rajecories of he Fads model we have: ds() = S()((µ + Y ε ())d + σρdw () + σ 1 ρ 2 db()), which can be simulaed by discreizaion as follows: S( n+1 ) = S( n )(µ + Y ε ())d + σρ dz 3 (n) + σ 1 ρ 2 dz 2 (n), 22

26 where W 1, W an B are independen Brownian moions, hus Z 1, Z 2, Z 3 are vecors of independen N(0, 1) random variables. Once he asse prices are simulaed, we can ge he wealh process for a rading sraegy. 3.3 Finding he Bes Consan Allocaion Sraegy In his secion we sudy which is he bes consan allocaion sraegy for he Fads-model for differen values of ε. The raio of he wealh invesed in sock is a value beween 0 and 1, so we ake a mesh of he [0, 1] inerval and compue he expeced uiliy of final wealh using his consan rading sraegy. This case resembles an invesor, who is aware of he fac ha he marke is no a Black-Scholes marke, sill he wans o use a Meron-ype of sraegy, i.e. a consan allocaion sraegy Logarihmic uiliy The following able conains he simulaion resuls for logarihmic uiliy, iniial capial V 0 = 500, iniial sock price S 0 = 100, ρ = 0.3, MR and OV being he heoreically obained Meron-raio and opimal value, Sim. MR and Sim. OV respecively he values obained by simulaion. For hese resuls we ook N = 1000 discreizaion seps (his means also re-balancing he wealh 1000 imes) and M = rajecories of each process. Model µ σ r ε MR Sim. MR OV Sim OV BS Fads Fads Fads BS Fads Fads Fads From he firs row we can see ha ha in he case of Black-Scholes model, we ge back he Meron-raio we obained heoreically, bu he simulaed expeced uiliy of he final wealh will always be a lower esimaion of he rue opimal value, since coninuous rading is no possible. 23

27 Now looking a he consans we go for he bes allocaion sraegies, we can see ha i slighly decreases as ε grows (we can also hink of he Black-Scholes case as he Fads-model wih ε = 0). The inerpreaion for his phenomenon would be ha, alhough an invesor wih logarihmic uiliy funcion is no very risk-averse, as he uncerainy in he drif grows, he will inves less and less in he risky asse in order o conrol he downside risk Power uiliy The following able conains he simulaion resuls for power uiliy funcion, wih exponen γ, iniial capial V 0 = 500, iniial sock price S 0 = 100, ρ = 0.3, MR and OV being he heoreically obained Meron-raio and opimal value, Sim. MR and Sim. OV respecively he values obained by simulaion. For hese resuls we ook N = 1000 discreizaion seps (his means also re-balancing he wealh 1000 imes) and M = rajecories of each process. Model µ σ r γ ε MR Sim. MR OV Sim OV BS Fads Fads Fads BS Fads Fads Fads BS Fads Fads Fads We can see ha in all of he above cases i is beer for he invesor o inves more in he risky asse as uncerainy in he drif grows. No only he Meron-raio grows, bu we can see a significan growh in he expeced uiliy as well. Noe ha hese invesors are even less risk averse han he previous one, e.g. while he invesor using logarihmic uiliy funcion inves 30% of he wealh in he risky asse for he parameers µ = 0.12, σ = 0.4, r = 0.07, ε = 1, he invesor using power uiliy funcion wih parameer γ = 0.5 will keep all his money in he risky asse. The relaive risk aversion of an invesor using logarihmic uiliy 24

28 is 1, while for he γ-power uiliy i is 1 γ, ha is 0.5 for his case. 3.4 Meron-sraegy for he Modified Model In his secion we ake a look a how he opimal sraegy obained by using he Black-Scholes model works for he generalized model. This case resembles an invesor who wans o use he Meron-sraegy alhough he knows ha he Black-Scholes model is no accurae Logarihmic uiliy Table below conains he heoreically obained values for he Meron raio (MR) and expeced uiliy (OV - opimal value) of his rading sraegy in he case of Black-Scholes model and logarihmic uiliy funcion, and he expeced uiliy (OV) given by he same rading sraegy for he modified model. For hese resuls we ook N = 1000 discreizaion seps, M = , he iniial wealh is 500, he iniial sock price S 0 = 100. The expeced uiliy of he final wealh for he Black-Scholes model denoed by OV, and respecively he opimal policy (MR) are calculaed by he formulas: E[log(X T )] = log(v 0 ) + (r + µ r 2σ 2 ) u = µ r σ 2 Model µ σ r ε ρ MR OV Sim OV BS Fads Fads Fads BS Fads Fads Fads We have already seen in he previous secion ha he bes consan allocaion sraegy for he logarihmic uiliy does no always agree wih he Meron-raio, and we could no ge bigger expeced uiliy for any consan allocaion sraegy 25

29 in he case of Fads-models han wha we go for he Black-Sholes model, using he Meron-raio. According o his, he Meron-raio for he modified model gives slighly worse expeced uiliy for he final wealh han in he Black-Scholes case Power uiliy Table below conains he heoreically obained values for he Meron raio (MR) and expeced uiliy (OV - opimal value) of his rading sraegy in he case of Black-Scholes model and power uiliy funcion wih exponen γ, and he expeced uiliy (OV) of he final wealh, given by he same rading sraegy for he modified model. For hese resuls we ook N = 1000 discreizaion seps, M = , µ denoes he drif, σ he volailiy of he processes, r = 0, he iniial wealh is se o be 500 and he iniial sock price S 0 = 100. The expeced uiliy of he final wealh for he Black-Scholes model denoed by OV, and respecively he opimal policy (MR) are calculaed by he formulas: [ X γ ] T E = V γ ( 0 γ γ exp µ 2 γ ) 2σ 2 (1 γ) u = µ σ 2 (1 γ) Model µ σ ε γ MR OV Sim. OV BS Fads Fads Fads BS Fads Fads Fads BS Fads Fads Fads BS Fads Fads Fads

30 Looking a he resuls we can see ha for γ = 0.01, when we are "very close" o he case of logarihmic uiliy, he expeced uiliy of he final wealh decreases as we modify he model, bu for he less risk-averse invesors, for γ = 0.1 and γ = 0.5, he invesor can acually profi from he uncerainy in he drif and ge even beer expeced uiliy. 27

31 Conclusion and Furher Exensions In his hesis we sudied Meron s classical porfolio problem. Firs we derived heoreically he opimal allocaion sraegy using he mehods of sochasic conrol heory, hen we esed his resul in pracice in discree ime simulaion scenario. The answers for he quesions formulaed in he inroducion would be: Does discreizaion decrease he expeced uiliy of he final wealh? Yes i does. The maximal value obained heoreically requires coninuous rading, a discree approximaion of coninuous rading leads o a lower esimae for he uiliy. Will an invesor who wans o use a Meron-ype of sraegy, i.e. a consan allocaion sraegy, ge he same consan if we change he model a lile bi? For he case when we modified he model jus a lile bi (i.e. ε = 0.01 in he Fads model), we go he same/almos he same consan allocaion sraegy as in he Black-Scholes model. Is i beer o inves less in he risky asse as uncerainy in he drif of he modified model grows or can an invesor profi from i? I is also up o he risk aversion of he invesor: we have seen ha he invesor using logarihmic uiliy funcion is more careful: is his case is beer o slighly decrease he amoun of money invesed in he risky asse, however, less risk-averse invesors can ge higher expeced uiliy by invesing more in sock. If we change he model bu use he allocaion sraegy which is opimal in he Black-Scholes case, can we sill reach he same expeced uiliy? We can reach i or a leas we can ge close o i. In he logarihmic case, we canno reach i, however, since he opimal consan allocaion sraegy for he modified model is very close o he Meron-raio, he expeced uiliy almos agrees. For he power-uiliy, we can reach he same expeced uiliy, 28

32 bu in his case we have seen, ha even higher expeced uiliy is possible using anoher consan allocaion sraegy. There are several ways how his problem can be generalized for furher sudy and research: Non-random/random income can be added. Transacion coss can be added (e.g. proporional or small fixed ransacion coss) [1]. Oher ypes of uiliy funcion can be used. A more general marke model can be used (e.g. sochasic variance, sochasic drif). Bankrupcy can be reaed [7]. 29

33 Bibliography [1] Mark HA Davis and Andrew R Norman. Porfolio selecion wih ransacion coss. Mahemaics of Operaions Research, 15(4): , [2] Ioannis Karazas and Seven E Shreve. Mehods of mahemaical finance, volume 39. Springer Science & Business Media, [3] Rober C Meron. Lifeime porfolio selecion under uncerainy: The coninuous-ime case. The review of Economics and Saisics, pages , [4] Rober C Meron. Opimum consumpion and porfolio rules in a coninuousime model. Journal of economic heory, 3(4): , [5] Jörn Saß. Sochasic conrol: Wih applicaions o financial mahemaics. Appuni delle lezioni, RICAM JKU Linz, 2oo6, , 6, 15 [6] Rober J Shiller, Sanley Fischer, and Benjamin M Friedman. Sock prices and social dynamics. Brookings papers on economic aciviy, 1984(2): , [7] Luz Rocío Soomayor and Abel Cadenillas. Explici soluions of consumpioninvesmen problems in financial markes wih regime swiching. Mahemaical Finance, 19(2): , [8] J Michael Seele. Sochasic calculus and financial applicaions, volume 45. Springer Science & Business Media, [9] Lawrence H Summers. Does he sock marke raionally reflec fundamenal values? The Journal of Finance, 41(3): ,

34 Appendix MATLAB code f u n c i o n BlackScholes = BlackScholes (N,M,mu, sigma,t, S0 ) %Consrucion o f Black S c holes r a j e c o r i e s %N d i s c r e i z a i o n seps, M r a j e c o r i e s, mu d r i f, %sigma v o l a i l i y, T ime horizon, S0 i n i i a l wealh d = T/N; z = randn (N,M) ; BS = (mu 0.5 sigma ^2) d+sigma s q r ( d ) z ; BS = S0 exp (cumsum(bs ) ) ; BlackScholes = [ S0 ones (1,M) ; BS ] ; end f u n c i o n Fads = Fads (N,M, e p s i l o n, sigma,mu, ro,t, S0 ) %Consrucion o f Fads r a j e c o r i e s %N d i s c r e i z a i o n seps, M r a j e c o r i e s, mu d r i f, %sigma v o l a i l i y, T ime horizon, S0 i n i i a l wealh %e p s i l o n consan f o r he O U p r o c e s s %ro c o r r e l a i o n o f Brownian moions %s i m u l a i o n o f O U p r o c e s s d = T/N; = 0 : d :T; w1 = randn (N,M) ; w2 = randn (N,M) ; c = exp( 1/ e p s i l o n. ) ; OU = z e r o s (N+1,M) ; f o r m = 1 :M OU( :,m) = diag ( c ) cumsum( diag ( exp (1/ e p s i l o n ) ) [ 0 ; s q r ( d ) w1 ( :,m) ] ) ; 31

35 end end %s i m u l a i o n o f FADS model S = z e r o s (N+1,M) ; S ( 1, : ) = S0 ones (M, 1 ) ; f o r m = 1 :M f o r n = 1 :N S ( n+1,m) = S (n,m) + S (n,m) ( (mu+ou(n,m)) d+ sigma ro s q r ( d ) w1(n,m)+sigma s q r (1 ro ^2) s q r ( d ) w2(n,m) ) ; end end Fads = S ; f u n c i o n mer = mer (S, mr, V0,N,, r ) %r e u r n s he f i n a l wealh, when he sock p r i c e s %f o l l o w he p r o c e s s S, mr Meron r a i o, %r i n e r e s rae, V0 i n i i a l wealh B = exp ( r. ) ; %value o f bond a ime %(1 mr) V0 wealh i n bond a 0, %mr V0 wealh in sock a ime 0 NB = (1 mr) V0/B( 1 ). ones (1,N+1); %i n i number o f bonds held a i NS = mr V0/S ( 1 ). ones (1,N+1); %i n i number o f s o c k s held a i f o r i = 1 :N wealh = NB( i ) B( i +1)+NS( i ) S( i +1); % o a l wealh ar ime i +1 NB( i +1) = (1 mr) wealh /B( i +1); %r e b a l a n c i n g NS( i +1) = mr wealh /S( i +1); %r e b a l a n c i n g end mer = NB(N) B(N)+NS(N) S(N) ; end 32

36 f u n c i o n [ ov, op ] = v l o g (mu, sigma, r, V0) %C a l c u l a i n g he rue maximal expeced u i l i y ( ov ) %and he Meron r a i o ( op ) f o r l o g u i l i y bea = (mu r )/ sigma ; op = bea / sigma ; ov = l o g (V0)+( r + (mu r )/(2 sigma ^ 2 ) ) ; end f u n c i o n [ ov, op ] = vpow (mu, sigma, r, V0, alpha ) %True v a l u e s f o r he maximal expeced u i l i y ( ov ) %and Meron r a i o ( op ) f o r power u i l i y op = (mu r ) / ( ( 1 alpha ) sigma ^2); cons = mu^2 alpha /(2 sigma^2 (1 alpha ) ) ; ov = ( (V0^alpha )/ alpha ) exp ( cons ) ; end f u n c i o n ExpULog = ExpULog (S,mu, sigma,m,n, V0,, r ) %Compuing he expeced u i l i y o f f i n a l wealh using %he s r a e g y given by he Meron r a i o achieved % h e o r e i c a l l y f o r he FADS model f o r l o g u i l i y [ ov, op ] = v l o g (mu, sigma, r, V0 ) ; f o r m = 1 :M x (m) = mer ( S ( :,m), op, V0,N,, r ) ; end ExpULog = mean( l o g ( x ) ) ; end f u n c i o n ExpUPow = ExpUPow(S,mu, sigma,m,n, V0,, r, alpha ) %Compuing he expeced u i l i y o f f i n a l wealh using %he s r a e g y given by he Meron r a i o achieved % h e o r e i c a l l y f o r he FADS model f o r power u i l i y [ ov, op ] = vpow (mu, sigma, r, V0, alpha ) ; f o r m = 1 :M x (m) = mer ( S ( :,m), op, V0,N,, r ) ; end ExpUPow = mean ( ( x.^ alpha )/ alpha ) ; end 33

37 f u n c i o n [ val, op ] = OpConsLog (S,N,M, V0,, r ) %Finds he opimal consan a l l o c a i o n s r a e g y %S sock p r i c e r a j e c o r i e s, N d i s c r e i z a i o n seps, M % r a j e c o r i e s simulaed, V0 i n i i a l wealh, ime, r %i n e r e s r a e u = z e r o s ( , 1 ) ; f o r i = 1:1000 x = z e r o s (M, 1 ) ; f o r m = 1 :M x (m) = mer (S ( :,m), i /1000,V0,N,, r ) ; end u ( i ) = mean( l o g ( x ) ) ; end [ val, op ] = max( u ) ; end f u n c i o n [ val, op ] = OpConsPow (S,N,M, V0,, r, alpha ) %Finds he opimal consan a l l o c a i o n s r a e g y and %he opimal value %S sock p r i c e r a j e c o r i e s, N d i s c r e i z a i o n seps, %M r a j e c o r i e s simulaed, V0 i n i i a l wealh, ime, %r i n e r e s r a e end u = z e r o s ( , 1 ) ; f o r i = 1:1000 x = z e r o s (M, 1 ) ; f o r m = 1 :M x (m) = mer (S ( :,m), i /1000,V0,N,, r ) ; end u ( i ) = mean ( ( x.^ alpha )/ alpha ) ; end [ val, op ] = max( u ) ; 34