Optimal portfolio for HARA utility functions in a pure jump multidimensional incomplete market

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1 Optimal portfolio for HARA utility functions in a pure jump multidimensional incomplete market Giorgia Callegaro Scuola Normale Superiore I Pisa, Italy g.callegaro@sns.it, iziano Vargiolu Department of Pure and Applied Mathematics University of Padova I Padova, Italy vargiolu@math.unipd.it September 9, 2008 Abstract In this paper we analyse a pure jump incomplete market where the risky assets can jump upwards or downwards. In this market we show that, when an investor wants to maximize a HARA utility function of his/her terminal wealth, his/her optimal strategy consists in keeping constant proportions of wealth in the risky assets, thus extending the classical Merton result to this market. We finally compare our results with the classical ones in the diffusion case in terms of scalar dependence of portfolio proportions on the risk-aversion coefficient. 1 Introduction he problem of maximization of a utility function of the wealth of an investor operating in financial markets is a classical one. Very early (even before modern finance saw the light with the celebrated Black and Scholes formula), it 1

2 was discovered that, when the investor maximizes a utility function u belonging to a good class, then his optimal portfolio consists in allocating his wealth in constant proportion between the assets in the market: this result holds both in discrete [23 as in continuous time [17. hese good functions are the so-called HARA (Hyperbolic Absolute Risk Aversion) utility functions, characterized by the fact that u (x)/u (x) = δ/x, that is the so-called Pratt s absolute risk aversion [20, is an hyperbole. hese utility functions are thus u(x) = log x and u(x) = x γ /γ, with γ = 1 δ < 1 (the case γ = 0 corresponding to the logarithm). Notice that this result does not necessarily hold true if the utility function has another generic form. hese results, originally proved by the use of dynamic programming, have been recently generalized using convex duality, even in incomplete diffusion markets [11. In this paper, we solve the same problem in a market where a riskless asset and n risky assets are present, and the risky assets, driven by an m- dimensional Poisson process with independent components and constant intensities, can jump upwards or downwards in continuous time. his is an extension of the multinomial model, in the sense that the price of the risky assets can increase or decrease of fixed factors, but in this model the instants of these changes are not fixed but random. While market models which include jumps already appeared in literature in the field of utility maximisation problem dealing with only one risky asset [3, 10 and in other contexts [13, 14, 21, to the authors knowledge this is the first time that the utility maximisation problem is explicitly solved in a multidimensional model which includes jumps. Also, this model can also be obtained if one assumes that the price vector of the risky assets is driven by a multivariate Poisson process (see [21). his model can be significant when dealing (for example) with high-frequency data, where the evolution of the prices has not a continuous evolution but all the price movements are due to jumps. Also, by assigning to some of the intensities values much smaller or much greater than the others, one can as well represent models where different time scales play different roles. As these assets are modelled via m > n independent Poisson processes with constant intensities, this market is incomplete. he final result is that the optimal strategy of the investor is to keep constant proportions of his wealth in the market s assets, thus it is similar to the previous ones already present in literature about HARA utility functions. In order to derive our results, we use the by-now classical method of convex duality (see [24 for a survey). his method consists in transforming the original ( primal ) utility maximization problem in an equivalent ( dual ) problem, where we minimize the so-called conjugate function of u over the set of all equivalent martingale measures: it is well known that, in our situa- 2

3 tion, an equivalent change of measure corresponds to changing the intensities of the Poisson processes. We characterize this minimiser and show that the Poisson processes still have constant intensities under this optimal martingale measure. he advantages in using convex-duality approach instead of a direct method (see [6) are mainly that we explicitly obtain the optimal equivalent martingale measure Q, which is very useful in many situations, and that with this formulation Q can be clearly seen to be unique. By the use of this characterization, we find the optimal final wealth and show that it is admissible by proving that the so-called duality gap is zero. At last, we show that there exists a portfolio strategy which realizes the optimal portfolio, and by calculating it explicitly we find out that it corresponds to keeping fixed proportions of wealth at each time in the riskless and in the risky assets, these proportions being functions of the coefficients of the assets and of the utility function s parameters. his allows us to see a difference between our market and a diffusion market: in fact, in the latter case the optimal portfolio in the risky assets is proportional to a fixed vector of risky assets proportions of the total wealth, and this proportionality only depends on the risk-aversion coefficient γ [11. We show, by using a counterexample, that in our market this is no longer true. he paper is organized as follows: in Section 2 we present the market model and state the utility maximization problem. In Section 3 we characterize all the martingale measures of this market. In Section 4 we state and solve the dual problem. In Section 5 we characterize the optimal final wealth in terms of the optimal martingale measure, and show that it is admissible for the primal problem. In Section 6 we characterize the optimal portfolio strategy. In Sections 7 we analyze the complete market case and in Section 8 we obtain more explicit results for the case N = 1, M = 2. Finally, in Section 9 we show that the dependence of the optimal portfolio proportions on the risk-aversion coefficient is more general of a simple scalar dependence. he authors wish to thank Gianni Di Masi, Yuri Kabanov and Wolfgang Runggaldier for useful discussions at various stages of the work. 2 he market model and the primal problem We consider an extension of the multinomial model, in the sense that the price of the risky assets can increase or decrease of fixed factors, but the instants of these changes are not fixed but random. he financial market considered then consists of a money market account with price B t and n risky assets S i t, i = 1,..., n, whose dynamics are given, under the measure 3

4 P, by the following stochastic differential equations: db t = rb t dt [ m dst i = S i t (e a ij 1)dN j t, i = 1,..., n (1) where we impose that the n m matrix A := (e a ij 1) i=1,...,n,,...,m has maximum rank, and (N t ) t = (Nt 1,..., Nt m ) t is an m-dimensional Poisson process, with m n, on a complete filtered probability space (Ω, F, P, (F t ) t ), where (F t ) t is the filtration generated by N augmented by all the P-null sets of Ω. We assume that the m components are independent and that their intensities, λ j, j = 1,..., m, are positive constants. Equivalently we have: B t = B 0 e rt [ m St i = S0 i exp a ij N j t, i = 1,..., n Such market is in general incomplete if m > n. If we furthermore suppose that there is no arbitrage possibility, this implies that there exists at least a martingale measure equivalent to P (possibly infinite many if the market is incomplete), which we call EMM (equivalent martingale measures) for short. Finally, let (α, β) = (α 1 t,..., α n t, β t ) t be an (n + 1)-uple of F t -predictable processes, representing the investment strategy at time t [0,, where α i t is the number of units of the i-th asset and β t is the number of units of the riskless asset which are held in the portfolio at time t. he processes α i t and β t have to satisfy the following integrability conditions with respect to the compensated processes M j t : t 0, i = 1,..., n, j = 1,..., m, t 0 α i s S i s λ j ds < t 0 β s λ j ds < P-a.s., (2) he value at time t of a portfolio corresponding to the strategy (α, β) is, then, the F t -measurable random variable V t = n αts i t i + β t B t i=1 he portfolio is self-financing if dv t = n αtds i t i + β t db t i=1 4

5 Notice that if the portfolio is self-financing and if we know V 0 = v and α = (αt 1,..., αt n ) t, we then also know V t and β t, t. In this case we often indicate the portfolio as V α. With these elements, we can formulate the primal problem for a generic utility function: given an initial wealth v and a fixed time horizon, maximize the expected value of the utility of the terminal value of the selffinancing portfolio max α E{u(V α)} α : self-financing strategy such that (3) E Q [B 1 V α v Q EMM When u is of the HARA class, then { x u(x) = γ /γ, γ < 1, γ 0 log x, γ = 0 Notice that if α is a self-financing strategy, the discounted value of the portfolio (Bt 1 Vt α ) t is a (Q, F t )-martingale Q EMM. For this reason the meaning of the constraint E Q [B 1 V α v Q in (3) is that the investor s initial wealth is less than or equal to v. he primal problem can be solved in two steps: SEP 1) determine V which solves the problem (without α) { max E{u(V )} V V v (4) where we define V v := {V r.v. : E Q [B 1 V v Q EMM} (5) SEP 2) determine the optimal investment strategy α such that V α = V a.s. 3 he set of all the EMMs Let us consider now the compensated Poisson processes under the measure P: M j t := N j t t 0 λ j du = N j t λ j t, j = 1,..., m (6) 5

6 which are (P, F t ) martingales. If we introduce the discounted prices S t i := St/B i t, the dynamics under the measure P are (see Eqs. (1)) { m } d S t i = d(st/b i t ) = S i t (e a ij 1) dn j t r dt (7) he Radon-Nikodym derivative for an absolutely continuous change of measure from P to Q, that implies a change of the Poisson intensities from λ j to λ j ψ j t, j = 1,..., m, is: dq dp = Z (ψ 1,..., ψ m ) (8) { } = exp (1 ψ j t )λ j dt + log ψ j t dn j t 0 where ψ j, j = 1,..., m, must be nonnegative predictable processes, satisfying 0 0 ψ j sλ j ds < P a.s. (9) Furthermore, we want the Radon-Nikodym derivative to give a probability measure, so the processes ψ j, j = 1,..., m, must be such that the following condition holds true: E P {Z t } = 1 t [0, (10) A sufficient condition for (10) to be valid can be found in [5, heorem VIII, 11. Defining the Poisson martingales M jq, j = 1,..., m, by dm j t Q = dn j t λ j ψ j t dt the dynamics of S i, i = 1,..., n, under Q become (see (7)), { d S t i = S m m } i t (e a ij 1) dm j Q t + (e a ij 1) λ j ψ j t dt rdt, i = 1,..., n (11) aking as numeraire, as usual, the money market account B t, we immediately see that Q is a martingale measure, i.e. a measure under which S t is a martingale, if and only if the ψ j 0, j = 1,..., m are chosen such that (e a ij 1)λ j ψ j t = r, P a.s. for all t (12) 6

7 for all i = 1,..., n. We have obtained infinitely many martingale measures characterized by the Radon-Nikodym densities and the Radon-Nikodym density processes dq ψ dp = Zψ Z ψ t = dqψ dp each one parameterized by the process ψ such that (9), (10) and (12) hold. he set of positive processes ψ, which parameterize the set of the EMM and make (9), (10) and (12) hold, will be denoted by N r. 4 he dual problem and its solution Using the theory of convex duality in [16 we now introduce the dual problem and we solve it in order to solve the primal problem. First of all, define the dual functional L(ψ, λ) := sup V = λv + sup V Ft {E [u(v ) λe Qψ [B 1 V + λv} {E [u(v ) λz ψ B 1 V } Recalling the definition of conjugate convex function ũ( ) associated to u( ): ũ(b) = sup {u(x) xb}, b > 0 x 0 we have L(ψ, λ) = λv + E [ũ(λz ψ B 1 ) So our dual problem is { min L(ψ, λ) ψ N r, λ > 0 (13) In our specific case of HARA utility functions, the conjugate convex function is b γ / γ, if γ < 1, γ 0, with γ := γ γ 1 ũ(b) = ( 1 ) log 1, if γ = 0 b In order to find the minimum of the dual problem, we first minimize L with respect to ψ, for all fixed λ R +, and obtain an optimum ψ, then we minimize with respect to λ and obtain an optimum λ. 7

8 4.1 he optimal ψ For all fixed λ > 0, the dual problem (13) is equivalent to min E [ũ(z ψ ) ψ N r (14) In fact, if γ < 1, γ 0 we have [ ( 1 L(ψ, λ) = λv E ) γ λ γ (λzψ B 1 = λv ( ) γ λ = λv + E [ũ(z ψ ) B B ) γ E while if γ = 0 we have [ L(ψ, λ) = λv 1 E log(λz ψ B 1 ( ) ) λ = λv 1 log E [log(z ψ ( ) B ) λ = λv log + E [ũ(z ψ ) B [ 1 ) γ γ (Zψ heorem 4.1 he solution to problem (14) is given by the positive process ψt ψ, where (ψ, λ) is an (m + n)-dimensional vector which is the unique solution to the algebraic system n λ i (e a ij 1) = (ψ j ) 1 γ 1 1, j = 1,..., m i=1 (15) (e a ij 1)λ j ψ j = r, i = 1,..., n Remark 4.2 Note that, by introducing the vectors λ = ( λ 1,..., λ n ) and Ψ = ((ψ 1 t ) γ 1 1,..., (ψ m t ) γ 1 1) and using the fact that the matrix A has maximum rank, the first m equations in (15) can be rewritten in the more compact form λ = ΨA (AA ) 1 Proof. We can see problem (14) as a stochastic control problem of a purejump process of the kind Φ(z, t) = inf ψ N r J (ψ) (z, t) = inf ψ N r E[ũ(Z ψ ) Zψ t = z (16) 8

9 where the control process is ψ N r and the controlled jump process is Z ψ t, whose dynamics is { { dzs ψ = Z ψ m s (1 ψj s)λ j ds + } m (ψj s 1)dNs j, t < s (17) Z ψ t = z As known, the solution to problem (16) is linked to the following HJB equation (see [18, h. 3.1) { inf [z (1 ψ j t )λ j φ ψ N r z (z, t) + m [φ(z + zy(ψ j t 1), t) R } φ(z, t)λ j δ 1 (dy) + φ (18) (z, t) = 0 t where λ j δ 1 (dy) = ν j (dy) is the Levy measure of N j, j = 1,..., n. If we try a solution of the kind φ(z, t) = 1 γ k(t)z γ for γ < 1, γ 0, with k a positive C 1 function, we get { [ m inf k(t) z γ (1 ψ j t )λ j 1 γ [ k(t) m } z γ ((ψ j ) γ t 1)λ j ψ N r 1 γ k (t)z γ = 0 which is equivalent to { m k(t)z γ inf (ψ j t 1)λ j 1 γ } ψ N ((ψj) γ t 1)λ j 1 γ k (t)z γ = 0 r An optimal ψ is then a solution to the problem ((ψ jt 1)λ j 1 γ ) ((ψjt ) γ 1)λ j inf ψ N r for all t [0, and so, to find the optimal EMM we have to solve the following constrained convex optimal problem (recall (12)) min ((ψ jt 1)λ j 1 γ ) ((ψjt ) γ 1)λ j ψ j t 0, j = 1,..., m, (19) (e a ij 1)λ j ψ j t = r, i = 1,..., n 9

10 he solution ψ to problem (19) is unique and given by the solution of system (15), as it will be shown in Lemma 4.3. By putting k (t) := e r (t ), where r := γ (ψ 1)λ j ((ψ ) γ 1)λ j we have that φ(z, t) := 1 γ k (t)z γ solves the HJB equation (18) with final condition φ(z, ) = 1 γ z γ. hen, by [18, heorem 3.1, ψ solves our problem. he case γ = 0 is analogous (the proof in this case is even simpler recalling the definition of intensity of a Poisson process (see [5)). Lemma 4.3 he solution ψ to the constrained convex optimization problem (19) is unique and it is given by the unique solution (ψ, λ ) to system (15). Proof. Our aim is to prove that, in order to determine the optimal ψt, we can consider and easily solve a problem equivalent to (19), where the admissible region is compact. he admissible region is not empty because we have assumed absence of arbitrage on the market, and so (by the First Fundamental heorem of Asset Pricing) there exists at least one equivalent martingale measure, i.e. the constraint in (19) holds for at least one vector (ψ t ) t. Now, consider one such point ψ t = ( ψ t 1,..., ψ t m ) for a fixed time t [0,. We now evaluate the objective function in ψ t and define m := [( ψ jt 1)λ j 1 γ (( ψ jt ) γ 1)λ j =: f( ψ t ). Since we are looking for the minimum of f, this will be equal to or lower than m. We also have lim ψ t + f(ψ t) = + his means that the original problem is equivalent to a problem with the same value function f and the compact admissible region { } C m := ψ ψj 0, j = 1,..., M; f(ψ) m; (e a ij 1)λ j ψ j = r, i = 1,..., n he set C m is closed, convex and bounded. hus we have a continuous function defined on a compact set: it admits minimum and problem (19) has 10

11 a solution, which, furthermore, is unique because of the strict convexity of f. Introducing the Lagrangian function with Lagrange multipliers λ 1,..., λ n we can finally solve the problem using first order necessary conditions, which are the first m equations in (15). By Remark 4.2, the uniqueness of λ is evident. Remark 4.4 Since the process ψ is constant, Equations (9) and (10) are satisfied and furthermore heorem VIII, 11 of [5 is verified: Z ψ t is a real Radon-Nikodym derivative and so Q ψ = Q is actually an EMM. 4.2 he optimal λ In order to solve the dual problem it remains now to find the optimal λ. For γ < 1, γ 0, then we have to solve min λ>0 L(ψ, λ) = λv 1 γ (λ) γ (B 1 ) γ E[(Z ψ ) γ By differentiating L(ψ, λ) with respect to λ and considering the equation we have: λ = λ L(ψ, λ) = 0 ( v(b ) γ E[(Z ψ ) γ ) 1 γ 1 If γ = 0 we have to minimize with respect to λ the dual functional (20) L(ψ, λ) = λv 1 log λ E [log (Z ψ B 1 ) he optimal value λ is easily obtained setting its first derivative with respect to λ equal to zero and so the optimal value λ is λ = 1 v. (21) Notice that this is a particular case of equation (20), as γ = 0 implies γ = 0. We have finally obtained the optimal solution (ν, λ ) to the dual problem. 11

12 5 he relation between primal and dual optimal solutions: the optimum V (λ, ψ ) We now want to obtain a relation between the optimal solution of the primal problem { max E{u(V )} V V r.v. : E Q [B 1 V (22) v Q EMM and the optimal solution of the dual problem (which we have just obtained in the previous section) { min L(ψ, λ) (23) ψ N r, λ > 0 Proposition 5.1 Let (ψ, λ ) be the optimal solution of the dual problem and define ( ) 1 V = λ B 1 γ 1 Zψ (24) [V hen V is admissible for the primal problem, EQψ v, and so it is the optimal solution of the primal problem (22). B 1 = EQ [V B 1 = In order to prove the above proposition, it is important to recall the notion of duality-gap and its connection with the optimal solution of an optimization problem. In general, when we deal with a primal and a dual optimization problem, if both the admissible regions are non-empty, then the following result is always true: the values that the objective function of the max-problem has in its admissible region are less than or equal to the analogous values of the min-problem. In our case we have (see problems (22) and (23)): E[u( V ) L( ψ, λ) for each V admissible for the primal problem and for each pair ( ψ, λ) admissible for the dual one. It follows that sup E[u(V ) inf L(ψ, λ) (25) V V v ψ N r,λ>0 Furthermore, we define the duality-gap associated to the primal admissible value V and the pair of dual admissible values ( ψ, λ) as E[u( V ) L( ψ, λ) 0 12

13 If there exist V, ψ, λ such that the duality-gap is zero, Eq. (25) is satisfied as an equality relation and this implies that V, ψ, λ are the optimal solutions of our problems (see [1 or [16). Now we prove Proposition 5.1. Proof. o prove the proposition we must show: 1) the primal admissibility of V 2) that the duality-gap is zero if in the primal and dual objective functions we substitute V and (ψ, λ ) respectively. First of all we observe that if V is given by Equation (24), since in this case λ is given by Equation (20) we have E Q [B 1 V = E[Z ψ [ = E B 1 (λ Z ψ (Z ψ ) γ (B 1 ) γ B 1 ) γ 1 = (26) v(b ) γ E[(Z ψ ) γ We start from point 2). he optimal value of the dual objective function, if we use formulas (24) and (26), is given by L(ψ, λ ) = λ v 1 γ E[(λ ) γ (B 1 ) γ (Z ψ ) γ = λ E Q [B 1 V 1 γ E[(λ ) γ (B 1 = λ E[Z ψ B 1 (λ ) γ 1 (B 1 1 γ E[(λ ) γ (B 1 ) γ (Z ψ ) γ = (1 1 γ )E[(λ ) γ (B 1 ) γ (Z ψ ) γ = 1 γ E[(V ) γ = E[u(V ) = v ) γ (Z ψ ) γ ) γ 1 (Z ψ ) γ 1 which is thus the optimal value of the primal objective function, as the duality-gap is equal to zero, if point 1) holds. In order to prove this, we have to show that E Q [B 1 V v Q EMM that is, using (24) and (20): E[Z ψ B 1 = E (λ ) γ 1 (B 1 ) γ 1 (Z ψ [ Z ψ B v(b ) γ E[(Z ψ ) γ ) γ 1 = ) γ 1 (Zψ (B ) γ 1 = E [ v Zψ Z ψ (Z ψ ) γ E[(Z ψ ) γ v ψ N r 13

14 If we now define a new measure Q using the Radon-Nikodym derivative (Z ψ ) γ /E[(Z ψ ) γ with respect to P, the proof of the admissibility of V reduces to proving the following relation [ Z ψ Ẽ 1 (27) Z ψ where Ẽ denotes the expected value under the measure Q, which is proved by the following Proposition 5.2. Proposition 5.2 Let ψ be the optimal solution of the dual problem. hen Equation (27) holds ψ N r. Proof. First of all we note that using the Radon-Nikodym derivative Z := d Q dp = ) γ (Zψ E[(Z ψ ) γ for all j = 1,..., m, the intensity of the Poisson process N j t changes and, under the measure Q, becomes λ j (ψ j ) γ. In fact (see Equation (9)): Z = exp{ γ m (1 ψj )λ j } m (ψj ) γn j exp{ γ m (1 ψj )λ j } E{ m (ψj ) γn j } and now, using the independence of the N j and recalling that if X Po(λ), then E[c X = e λ(c 1), we obtain Z = m [(ψ j ) γ N j P m e λj [1 (ψ j ) γ, (28) As Z is a Radon-Nikodym derivative, it implies a change of the intensities of the Poisson processes from λ j to λ j (ψ j ) γ, j = 1,..., m. Using Itô s Formula and recalling Equation (17) we have: ( Z ψ ) t d Z ψ t = Zψ t Z ψ t { m [(1 ψ j t )λ j + (ψ j 1)λ j dt + } ( ψ j ) t 1 dn j ψ j t. Since we have to calculate the expected value of Z ν /Zν under the measure Q, it is useful to introduce the Q-martingales M j t, j = 1,..., m, with dynamics d M j t = dn j t λ j (ψ j ) γ dt 14

15 So we have ( Z ψ ) t d Zt ν = Zψ t Z ψ t +(ψ j 1)λ j + ( ψ j ) t 1 d ψ M j j t + Zψ t [(1 ψ j Z ψ t )λ j t ( ψ j ) t 1 λ j (ψ j ) γ dt ψ j and finally, under suitable assumptions on the coefficients of d M j t, j = 1,..., m, { [ Z ψ Ẽ = 1 + Z Ẽ Z ψ [ m } t λ j (ψ j ψ t ψ j )((ψ j ) γ 1 1) dt (29) 0 [ Z ψ Ẽ Z ψ Z ψ t 0 Z ψ t o prove Equation (27), we will now show that the integrand random variable in (29) is null, so that the expected value under the measure Q is equal to 1. he optimal ψ j, j = 1,..., m, as shown, are the unique solution to the algebraic system (15) and then they satisfy both the first group of equations and the second of (15), while a generic ψ j, j = 1,..., m satisfy only the second group. Using conditions in Equation (15) we now show that the integrand random variable in Equation (29) is null and so the proposition is proved. In fact, we have { = 1 + Ẽ Z ψ [ m } n t λ j (ψ j t ψ j ) λ i (e a ij 1) dt = 1 + Ẽ = 1 + Ẽ { 0 { 0 Z ψ t Z ψ t Z ψ t Z ψ t [ n λ i i=1 m i=1 } λ j (ψ j t ψ j )(e a ij 1) dt } [ n λ i (r r) dt = 1 i=1 6 he optimal investment strategy α From the previous section we know that the optimal solution of the primal problem is V = (λ B 1 Zψ ) 1 (Z ψ ) γ 1 γ 1 = vb E[(Z ψ ) γ 15

16 where λ is given by Equation (20) and ψ is given by the solution of system (15). he aim of this section is to determine the optimal strategy α = (α 1 t,..., α n t, β t ), i.e. the one which satisfies V = V α P a.s. We shall solve this hedging problem using a martingale representation method: in particular we will take advantage of the property that the discounted value of a self-financing portfolio is a martingale under any martingale measure. It will also be convenient to work under the optimal martingale measure Q := Q ψ. We shall define a martingale corresponding to Ṽ at time and then we will use the dynamics of both these martingales in order to obtain relations between α and the coefficients in the dynamics of the new martingale. We have n Ṽ t = αt i S n t i + β t, dṽt = α t d S t i and i=1 Ṽ = v E[(Z ψ ) γ i=1 (Z ψ ) γ 1 We now introduce the (Q, F t )-martingale M: M t := E Q [Ṽ v F t = E[(Z ψ ) γ E Q [(Z ψ ) γ 1 F t in order to compare the dynamics of the two (Q, F t )-martingales M t and Ṽt (note that M 0 = v and M = Ṽ ). It is useful to recall that under the measure Q the intensities of the Poisson processes are λ j ψ j, j = 1,..., m. Recalling Equation (28) we have ( ) E[(Z ψ ) γ = exp γ (1 ψ j )λ j + λ j ((ψ j ) γ 1) and E Q [ (Z ψ ) γ 1 F t = E Q [ e ( γ 1) P m (1 ψj )λ j m (ψ j ) ( γ 1)N j F t = [ m = e ( γ 1) P m (1 ψj )λ j E Q (ψ j ) ( γ 1)(N j N j t +N j t ) F t = = e ( γ 1) P m (1 ψj )λ j m (ψ j ) ( γ 1)N j t 16 m E Q [ (ψ j ) ( γ 1)(N j N j t )

17 where the last equality holds because for all t and for every j = 1,..., m, the random variables (N j N j t ) are independent and each one is also independent of the σ-algebra F t. If we now recall that under Q (N j N j t ) Po(λ j ψ j ( t)), j = 1,..., m, we finally have: E Q [(Z ψ ) γ 1 F t = e ( γ 1) P m (1 ψj )λ j m (ψj ) ( γ 1)N j t e ( t) P m λj ψ j ((ψ j ) γ 1 1) (30) After some calculations we then have m M t = v (ψ j ) ( γ 1)N j t e t P m λj ψ j ((ψ j ) γ 1 1) = (31) = M { m 0 exp t 0 log (ψ j ) γ 1 dn j s + t 0 } (1 (ψ j ) γ 1 )λ j ψ j ds (recall that M 0 = v). In order to obtain the optimal investment strategy α we now have to compare the differentials of the two martingales M t and Ṽ t. he differential of Mt is easily obtained from (32) and, under the optimal martingale measure Q, is given by { m } d M t = M t ((ψ j ) γ 1 1)dN j t + (1 (ψ j ) γ 1 )λ j ψ j dt { = M m t ((ψ j ) γ 1 1)dM j t If we compare the following two dynamics } Q dṽ t = n m i=1 αi t Si t (ea ij 1)dM j t d M t = M m t ((ψj ) γ 1 1)dM j Q t we find that the optimal investment strategy has to satisfy the following system of m equations n i=1 αt i Si t (ea ij 1) = M t ((ψ j ) γ 1 1), j = 1,..., m. Proposition 6.1 he optimal investment strategy exists and it is uniquely determined by αt i := λ i M t, i = 1,..., n, where λ = ( λ S 1,..., λ n ) is the vector i t found in heorem Q

18 Proof. By defining αt i Si t M t := λ i, i = 1,..., n, (32) the system above is equivalent to the system given by the first m equations in (15), which admits a unique solution. Remark 6.2 Notice that, if we introduce the fractions h i t, i = 1,..., n, of wealth invested at time t in S i t, i = 1,..., n, h i t := αi ts t V t = αi S t t, i = 1,..., n, Ṽ t so that h i t = (αt i Si t )/Ṽ t, noting that Ṽ t = M t, from (32) we find that the optimal fractions of wealth to invest in each risky asset are constant over time: h i t h i, i = 1,..., n, and they are equal to the λ i, i = 1,..., n, of heorem 4.1. Remark 6.3 From heorem 4.1 and from the previous remark, substituting the ψ j t, j = 1,..., m, from the first m equations in (15) into the remaining n and recalling that the optimal h i t are equal to the λ i, i = 1,..., n, we find that the h i t must satisfy [ n λ j (e a ij 1) h i (e a ij 1) + 1 γ 1 = r (33) i=1 which is the same result obtained in [6, Section 4, solving directly the primal problem. 7 he complete market case In the case when m = n the market is complete and there exists only one EMM, which we will denote by Q. It can be easily seen that system (12) becomes a system of m equations in m unknowns, with matrix A having maximum rank m, and so it admits a unique solution ψ = (ψ 1,..., ψ M ). In this case it is not necessary to consider and solve the dual problem, since the primal problem (4) has only one constraint and becomes { max E{u(V )} E Q [B 1 V v 18

19 and can be solved using the Lagrange multiplier technique. By introducing Z := dq and the Lagrange multiplier λ, our problem becomes dp max V [E{u(V )} λe Q {B 1 V } = max E{u(V ) λz B 1 V }. (34) V We now notice that, thanks to the properties of the utility function u( ), the inverse function of its derivative exists and we will denote it by I( ) := (u ( )) 1. On the other hand, maximizing the expectation on the right hand side of (34) is equivalent to maximizing its argument for each ω Ω. A necessary condition, then, for V to be optimal is that it satisfies u (V ) = λz B 1, V = I(λZ B 1 ) (35) with the Lagrange multiplier λ satisfying the budget equation E Q [B 1 I(λZ B 1 ) = v = E[Z B 1 I(λZ B 1 ) =: V (λ). Having defined V ( ), whenever it is invertible, we finally find λ = V 1 (v) and V = I(V 1 (v)z B 1 ). In the specific setting of HARA utility functions, we have and so, from (35), Furthermore v = E[Z B 1 I(y) = y 1 γ 1 V = (λz B 1 ) 1 γ 1 (λz B 1 = y γ 1 = (λz B 1 ) γ 1. ) γ 1 = λ γ 1 (B 1 ) γ E[(Z ) γ = V (λ) and so and V λ = ( = (λ Z B 1 v(b ) γ E[(Z ) γ ) 1 γ 1 ) γ 1 = vb (Z ) γ 1 E[(Z ) γ 19

20 i.e. we have obtained again Equations (20) and (24), with the difference that in this setting there is only one EMM and so the optimal Q trivially coincides with the unique EMM. o determine the optimal investment strategy, finally, we work as in Section 6, under the measure Q and we compare, as usual, the two dynamics of Ṽ t and M t. he optimal investment strategy α t i, i = 1,..., N, is again αt i Ṽ t = λi S i t where λ = ( λ 1,..., λ n ) is given by λ = ΨA 1, with Ψ = ((ψ 1 ) γ 1 1,..., (ψ n ) γ 1 1) (by Remark 4.2). 8 he one-dimensional case Let us now assume that we are allowed to trade in a single risky asset S. For a realistic model, we require that this asset can go both up and down, so that m = 2 Poisson processes are required to describe it. As m = 2, n = 1, the market is incomplete and so all the results up to Section 6 hold true, with the advantage that in this case the optimal EMM and the optimal investment strategy have an explicit form. For simplicity, we use the notation (N t ) t [0, = (N + t, N t ) and ds t = S t [(e a 1)dN + t + (e b 1)dN t with a > 0, b > 0. As the market is incomplete, we obtain infinitely many martingale measures, and Equation (12) reduces to the condition (e a 1)λ + ψ + t + (e b 1)λ ψ t = r P a.s. t (36) where λ + and λ are the intensities of the Poisson processes N + and N under the original measure P, respectively, and the process ψ = (ψ +, ψ ) stands on a half-line in the first quadrant, which can be parameterized as follows ψ t =: ν t 0, ψ + t = r (e b 1)λ ν t (e a 1)λ +. Because of the introduction of (ν t ) t, the Radon-Nikodym densities and the Radon-Nikodym density processes will be denoted by dq ν dp = Zν, Zt ν = dqν dp. Ft 20

21 In this case the solution to system (15) can be made explicit: in fact we have λ(e a 1) = (ψ t + ) 1 γ 1 1 λ(e b 1) = (ψt ) 1 γ 1 1 (e a 1)λ + ψ t + + (e b 1)λ ψt = r and we find that the optimal ν, recalling the parametrization (36), is the unique solution to the following equation (ν) γ 1 (e b 1) (e a 1) ( r (e b 1)λ ν (e a 1)λ + ) γ 1 = 1 (e b 1) (e a 1) If γ = 0 = γ (log-utility case), to determine ν we have to solve the algebraic second degree equation ν 2 λ k(e b 1) ν rk + ν (e a 1)(e b 1)(λ + + λ ) r (e a 1) = 0 where k := [(e b 1) (e a 1) < 0. Since λ k(e b 1) r(e a 1) > 0, there is only one positive solution, given by where ψ t = ν t rk (ea 1)(e b 1)(λ + + λ ) + 2λ k(e b 1) t [0,, = [rk (e a 1)(e b 1)(λ + + λ ) 2 + 4rkλ (e a 1)(e b 1) > 0 he optimal λ is thus obtained as in Section 4.2 and the relation between primal and dual optimal solutions is, obviously, the one in Section 5. As concerns the determination of the optimal investment strategy, by using Proposition 6.1, it is sufficient to find the optimal λ: from Equation (15), we have that λ = (ψ+ ) γ 1 1 e a 1 = (ψ ) γ 1 1 e b 1 so that the optimal investment strategy is = (ν ) γ 1 1 e b 1, α t = M t S t (ψ+ ) γ 1 1 e a 1 = M t S t (ψ ) γ 1 1 e b 1 = M t S t (ν ) γ 1 1 e b 1. 21

22 9 Mutual fund theorem A peculiar result about generic utility functions in continuous time in both complete [17 and incomplete markets [11 is that the optimal portfolio in the risky assets is proportional to a fixed vector of risky assets proportions of the total wealth, and this proportionality only depends on the risk-aversion coefficient γ. In more detail, if the risky assets follow a dynamics of the kind [ ds i t = S i t b i dt + σ ij dw j t, i = 1,..., n, with the assumptions that W is a m-dimensional Brownian motion, m n and that σσ has maximum rank, then the optimal portfolio proportions in the risky assets are given by h t = (h 1 t,..., h n t ) := µ f (t)(σσ ) 1 (b r1 n ) where 1 n = (1,..., 1) R n, r is the risk-free interest rate and µ f (t) > 0 is a scalar stochastic process depending on V. his is known as Merton s mutual fund theorem (see [4 for details). If this result holds, then the optimal portfolio h is qualitatively the same (stay long / short / not invest in the i-th asset) and proportional to the vector (σσ ) 1 (b r1 n ) (37) In the case of HARA utility functions, µ f = (µ f (t)) t turns out to be constantly equal to ). his means that the optimal proportions are pro- 1 1 γ portional to the fixed vector (37) which in particular does not depend on γ. In our context of a pure jump market, Merton s mutual fund theorem would imply that the optimal portfolio proportions (h i ) i in Remark 6.3 would be proportional to a fixed vector. A first consequence of our results is that, in the simple case when only one risky asset is available in the market, this proportionality trivially holds. heorem 9.1 If n = 1, then the optimal proportion (h t ) t is constant over time and depends on γ. Proof. he result in the theorem is a direct consequence of Remark 6.3: in fact, by putting n = 1 we obtain that the optimal h must satisfy the algebraic nonlinear equation ( γ 1 λ j (e a 1j 1) h (e a 1j 1) + 1) = r 22

23 From the results of Section 6, we know that there exists a unique solution h to this equation, which is the optimal portfolio fraction invested in the risky asset, and it is constant over time. he non-trivial situation is of course the case when n > 1. In this case, we find out that, even in the simplest complete market case m = n = 2, the mutual fund theorem does not always hold, thus making a difference between market where assets follow pure diffusion processes and markets where assets can jump. he following is a counterexample where it is shown that the optimal proportions h have a dependence on γ which can not be brought back to a proportionality of a fixed vector. Example 9.2 ake m = n = 2, λ 1 = 1, λ 2 = 0.8, r = 0.05, and the multiplicative jumps ( ) (e a ij ) i,j = By using the results in Section 7 and solving numerically Equation (33) and the following for γ ( 0.3; 0.3), we obtain that the parametric curve obtained by seeing the optimal proportions h = (h 1, h 2 ) as a function of γ is the one in the following Figure 1: Being this the situation when the market is complete, we argue that a simple scalar dependence of h on γ is not met also in incomplete markets, unless possibly in few specific cases. [INSERIRE NOA ALLA LUCE DI SS References [1 M. Avriel, Nonlinear programming: analysis and methods, Prentice- Hall Series in Automatic Computation (1976) [2 D. P. Bertsekas, Dynamic Programming and Stochastic Control, Mathematics in Science and Engineering, vol. 125, Academic Press, London (1976) [3 N. Bellamy, Wealth optimization in an incomplete market driven by a jump-diffusion process, J. Math. Econ. 35, No. 2 (2001) pp [4. Björk, Arbitrage theory in Continuous ime, Oxford University Press (2004) 23

24 Figure 1: Graph of the function γ (h 1 (γ), h 2 (γ)), with h 1 in the orizontal axis and h 2 in the vertical axis 24

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