Optimal consumption and investment with Lévy processes

Optimal consumption and investment with Lévy processes^*

José Fajardo Barbachan

IBMEC Business School. E-mail: [e-mail]

ABSTRACT

Summary: 1. Introduction; 2. Model; 3. Lévy processes and equivalent martingale measures; 4. Optimization problem; 5. Choosing a measure; 6. Example; 7. Conclusions.

We study the intertemporal consumption and investment problem in a continuous time setting when the security prices follow a Geometric Lévy process. Using stochastic calculus for semimartingales we obtain conditions for the existence of optimal consumption policies. Also, we give a charaterization of the equivalent martingale measures.

Keywords: Lévy processes; incomplete markets.

JEL Codes: G11; C61.

RESUMO

Estudamos o problema do consumo e investimento intertemporal em tempo contínuo, quando os preços dos ativos seguem um processo de Lévy Geométrico. Usando cálculo estócastico para semimartingalas obtemos condições para a existência de políticas ótimas de consumo. Também, mostramos a caracterização das medidas martingalas equivalentes.

1 Introduction

The intertemporal optimal consumption and investment problem in a continuous time setting is a important task in the finance literature. The first to study this problem was Merton (1971), using stochastic dynamic programming he obtained explicit solutions for an economy with incomplete markets in which security prices follow a geometric Brownian motion, the endowments follow a Poisson process and the investor has a negative exponential utility with an infinite horizon.

Many other authors have tried to improve the above model adding income stream, transaction costs, borrowing constraints and other facts that make the model more real. In all these models the log-Normal hypothesis or a slight modification of it is assumed for stock returns. Unfortunately, it is very well known that this hypothesis did not hold for the majority of stocks, since they present stylized facts like: fat tails, asymmetry, autocorrelation, etc. For a survey about the principal results on stylized facts see Rydberg (1997).

For this reason, recently some results, assuming more general process to model stock returns, have appear in the literature. Kramkov and Schachermayer (1999) prove that if the asymptotic elasticity of the utility is strictly less than one, then there is a solution for the optimal consumption and investment problem.

Benth et al. (2001) have obtained a solution for the optimal consumption-investment problem assuming that the dynamic programming principle holds and that the stock returns are modelled by a pure-jump Lévy processes, to solve this problem they used a viscosity solution approach. Fajardo (2000) proves that when the stock returns follow a Hyperbolic Lévy motion and we choose a particular equivalent martingale measure (EMM) to construct the state price density, then it is possible to find conditions on the wealth process and strategies that guarantee the existence of optimal policies.

In this paper we extend the latter result considering a huge class of processes called geometric Lévy processes and all the set of EMM. To this end we characterize all the EMM that exist in the market. The paper is organized as follows: in section 2 we describe the model, in section 3 we address the existence of EMM in a Lévy market, that is, when stock prices follow a Geometric Lévy Motion. In section 4 we introduce the optimal problem and state the main result. In section 5 we show how to choose an EMM and in the last sections we present an example and the conclusions.

2 Model

We will consider a financial market consisting of 2 assets. The first is called bond (the riskless asset) and the second is called stock (the risky asset). We denote by B(t) and P(t) the bond's and stock's price at each time t Î [0,T], respectively. The evolution of these prices are modelled by the following equations:

In this model the sources of risk are modelled by a Lévy process Y (t),0 < t < T, i.e., a process with independent and stationary increments. Y is defined on a given complete probability space (W, , P) and denote by F = {(t), 0 < t < T} the P– augmentation¹ of the natural filtration generated by Y:

_Y (t) = s(Y (s),0 < s < t),0 < t < T

time horizon will be considered finite. The positiveness of the stock price will be analyzed in the next section. The interest rate {r(t):0 < t < T}, assumed finite, the appreciation rate {r(t),0 < t < T}, and the volatilities s(t),0 < t < T will be referred as the coefficients of the financial market . We assume that these coefficients are deterministic continuous functions. Now we introduce a small investor (his decisions does not affect the market prices), who will decide at each moment t Î [0,T]:

1. How much money p(t) he wants to invest in the stock;

2. His cumulative consumption C(t).

Of course, these decisions must be made without foreknowledge of future events, so C and p must be adapted processes. ² If we denote by X(t) the agent's wealth at time t, then the amount invested in the bond will be X(t) – p(t). From here, (1) and (2), we obtain the following equation for the wealth:

The solution of this linear stochastic differential equation with initial condition x Î , i.e. X(0) = x is:

for all 0 < t < T. Where

is the discount factor in . Now we give some definitions:

Definition 1

(i)An F– adapted process C = C(t), 0< t < T with nondecreasing, right-continuous paths, left-hand limits and C(0) = 0,C(T) < ¥ a.s is called a cumulative consumption process.

(ii) An F– progressively measurable, càdlàg,

– valued process p = {p(t),0 < t < T} with

is called a portfolio process.

(iii) For a given x Î and (p, C) as above, the process X(t) = X^x,p,C(t) of (3), (5) is called the wealth process corresponding to initial capital x, portfolio p, and cumulative consumption process C.

It is important to notice that (X(t) – p(t)) and p(t) can take negative values, in other words short-sales of stock and borrowing at interest rate r(·) are allowed. So we need to impose some restriction on portfolios.

Definition 2: We say that a given portfolio process p(·) is tame, if the associated discounted gain process:

is a.s. bounded from below by some real constant:

Condition (9) is necessary to avoid doubling strategies, that is, portfolios that attain arbitrary large values of wealth with probability one at t = T, starting with zero initial capital at t = 0.³

Definition 3 A tame portfolio that satisfies:

is called an ''arbitrage opportunity'' (or free lunch). We say that a market

is arbitrage free if no such portfolio exist.

The free lunch interpretation of (10) is clear: starting with zero initial capital and using the strategy p(·) at the end of the period t = T. Since g(T) = B(T)^–1, we have X(T) = X^0,p,0(T) = B(T)M^p(T), then X(T) > 0 a.s, i.e. no risk and positive probability of gain P[X(T) > 0] > 0.

So we need conditions for precluding these arbitrage opportunities. We know that the existence of an EMM in general context rule out this opportunities. In our context Eberlein and Jacod (1997) show that when r(t) = r, "t Î [0,T], there would be EMM and Chan (1999) shows analogous result with r(·) being a deterministic continuous function. We will show that these existence imply that our market is arbitrage free. An important consequence of our extension is that many market models are incomplete then we have not an unique EMM. The criterion to choose one of these EMM will be also presented.

3 Lévy Processes and Equivalent Martingale Measures

In this section we characterize all the equivalent martingale measures in the model introduced in section 2. We know that all the infinitely divisible distributions⁴ (Y_t) admit the following Lévy-Khintchine representation:

where f is the characteristic function of the infinitely divisible distribution, a is the drift, c is the quadratic variation coefficient and G is a positive measure with òminx²,1G(dx) < ¥. This measure is called the Lévy measure and describes the jumps of the process.

We also know that all Lévy process must be a linear combination of a standard Brownian Motion (W_t) and a quadratic pure jump process⁵ (N_t) which is independent of the Brownian Motion W_t, then

Y_t= cW_t + N_t

The process N_t has a Lévy decomposition: Let L(dt, dx) be a Poisson measure on ⁺ × \0 with expectation (or compensator) measure dt × G ⁶, then:

Now assume that⁷

E[exp(–bY₁)] < ¥"b Î (-b₁, b₂)

and

e^–bxdG(x) < ¥"b Î (–b₁, b₂)

where 0 < b₁, b₂< ¥ . The first assumption said that Y_t has all moments finite and the second is technical and will let us separate integrands. With this in mind we can return to the jumps and transform the equation (12) into:

N_t = (L((0, t], dx) – tG(dx)) + tEN₁

It is easy to see that the process

M_t = (L((0, t], dx) – tG(dx))

is a martingale. Then N_t = M_t + at, with a = EN₁, as a consequence the original process can be written as

Before passing to characterize the absolutely continuous measures with respect to the original measure, let us introduce some elements of stochastic calculus: for any measurable function f(t, x) we have

and for any C² function f, we have the Generalized Itô's formula for càdlàg semimartingales X¹, ..., Xⁿ:

with , and [Xⁱ, X^j]^c the continuous part of the mutual variation⁸ of Xⁱ and X^j.

Now with the above results we study the solution of equation (2):

dP(t) = P(t^–)[r_tdt + s_tdY(t)] = (as_t + r_t)P_t–dt + s_tP_t–(cdW_t + dM_t)

When the coefficients r_t and s_t are deterministic continuous function the solution of this equation is given by the Doléans-Dade exponential:⁹

with (13) we obtain:

to ensure that P_t > 0, a.s. "t Î [0, T], we need that

1 + s_tDM_t > 0, "t Î [0, T]

If we assume the convention 's > 0', we only need that the jumps of N_t be bounded from below¹⁰, i.e., DN_t > –, it means that we consider only ''semi-fat tailed'' distributions as Poisson, Gamma, Hyperbolic and Normal Inverse Gaussian and we eliminate processes with heavy tails, it is worth noting that the stable distributions (without including the Gaussian Case) were eliminated when we supposed that Y has all moments finite.

The following step consist in characterize all the measures that are absolutely continuous with respect to P, to this end let:

(dt,) = L(dt, dx) – dtG(dx)

then

Now two useful results:¹¹

Lemma 1Let R_t and K (t,x) be a previsible and a Borel previsible processes¹² respectively. Suppose that

and K > 0, K(t,0) = 1 "t Î . Let k(t, x) be another Borel previsible process such that

Define a process

_t by

then is a local martingale with ₀ = 1 and is positive if and only if K > 0.

Proof From the fact

_t – _t– = _t–( K(t, DN_t) – 1)

and applying Itô's formula and ( 14), we can obtain:

This expression is a local martingale.

Teorem 1 Let Q be a measure which is absolutely continuous with respect to P on

_T. Then

where is as in the lemma 1, for some R, K and k for which E

_T = 1. Moreover under Q, the process

is a Brownian Motion and the process N_t is a quadratic pure jump process with compensator measure given by dt(dx) with

(dx) = K(t, x)G(dx)

and constant part given by

Then under Q the process N_t can be represented as

with

This process is a Q-martingale and it is easy to see that D = DM_t. Now let

be the discounted price process. Replacing the processes W_t and M_t in the equation (15) by their respective Q-versions, we obtain

A necessary and sufficient condition for be a Q-martingale is the existence of R and K > 0 a.s. for which:

and E

_t = 1, "t > 0. Since the process

is a Q-martingale. Now we can state the following

Theorem 2If there exist R and K > 0 for which E

_t = 1 "t and the market price of risk is given by

Then the market

is arbitrage free.

ProofTake a tame portfolio p and suppose that P(M^p(T) > 0) = 1, by definition we have

With equation (13) and (19) we obtain

Now using (16) and (17)

We have that M^p is a Q- local martingale bounded from below, since p is tame, then a supermartingale. Hence

E^QM^p(T) < 0

then Q (M^p(T) > 0) = 0, so there are not arbitrage opportunities.

From these result we know that our market is arbitrage free. Now we can introduce the investor problem.

4 Optimization Problem

In this section we formalize the individual problem of the investor, give some definitions and finally we present the main result.

Definition 4A pair (p,C) of portfolio consumption process is called admissible for the initial capital x > 0. If

The class of all such pairs will be denoted by (x).

Now take any equivalent martingale measure Q and define the following state price density:

where

using (5) and the generalized Itô's lemma for F((t), g(t)X(t)) = (t) X(t), we obtain:

Now with the decomposition (13), we obtain

d((t)X(t)) = – (t^–)dC(t) + (t^–)p(t)s(t)[cdW (t) + dM(t)]

+ g(t) X(t^–)d(t) + dD(t)

where D(t) is given by

In order to formulate our optimization problem we need the concept of utility function.

Definition 5We say that a function u:(0, ¥) ® R is a utility function if it is strictly increasing, strictly concave, continuously differentiable and

Examples of utility function are u(x) = log x and u(x) = ,d Î (–¥, 1)0.

We will denote by I(·) the inverse of the derivative u¢(·), both these functions are continuous, strictly decreasing, and map (0, ¥) onto itself with I(0₊) = u¢(0₊) = ¥, I(¥) = u¢(¥) = 0. We shall consider also de convex dual

Of u(·): a convex decreasing function, continuously differentiable on (0, ¥) and satisfies

Remarks on Utility Functions

• Observe that (23) is the usual Inada's condition.

• Negative wealth is not allowed to consider utilities that are economically relevant, as is the case of the u(x) = ln(x). But, we could take u(0) = –¥ and make u:[0, ¥) ® RÈ{–¥}

• In general we must have that marginal utility from consumption tends to infinite when wealth tends to the infimum of its allowed values.

Now consider an small investor who has an initial capital x > 0, and he wants to choose a portfolio p(·) and consumption processes {c(t), 0 < t < T} in order to maximize his expected utility from the terminal wealth X^x,p,C(T) and from consumption.

Given the utility functions g and u(t, ·), as in the above definition, we define the following classes:

remember f^–(x) = max – f(x), 0.

Then our small investor will have to maximize the expected utility from consumption and terminal wealth over the following class:

The value function will be

Now to solve this optimization problem consider, in the context of the market model described before, a contingent claim ¹³ x and a consumption process C that satisfy

Then if there exist a portfolio process p(·), such that (p, C) Î (x) and X^x,p,C(T) = x; we could conclude that the optimal problem is in some sense equivalent to the following problem:

over all pairs (x, c) of contingent claims and consumption rate process that satisfy the constraint (29) .

Now with y > 0 (Lagrange multiplier) and with (24):

the equality holds if and only if

then in the constraint (29), we define

and if we consider x Î (0, ¥) that is (y) < ¥, "0 < y < ¥. This function maps (0, ¥) onto itself and is continuous, strictly decreasing with

If we denote by (·) = (·). Then the lagrange multiplier y > 0 is uniquely determined by

y = (x)

Now we can state the main result

Theorem 3Suppose x Î (0, ¥) and (x) < ¥, "x Î (0, ¥) For any x > 0, consider the optimization problem with value function (x) as in (??) and define x

and c(·) as in (31). Then if

a) there is a portfolio process p(·) such that (p, C) Î (x) and

b) the process D_t is a Local martingale and D₀ = 0.

Then (p, c) are the solutions of the optimal problem and the value function is given by

where

and the convex dual of (·) is

Proof By construction x and c satisfy (29) and using the following inequality

f (I_f(y)) > f (x) + y[I_f (y) – x]

for every utility function f, we obtain

therefore

Since EH(t) < e^rT,0 < t < T, where r = sup_{0 < t < T}r(t) < ¥ and by (a) we have that there exist a portfolio process p with (p, C) Î (x) (also in ₀(x) thanks to (34)) and a.s.).

Now take an arbitrary x > 0, (p, C) Î ₀(x) and y > 0 from (b) we have that the following process is a bounded (from below) local martingale:

then a supermartingale and with (30) we have

where

in particular follows

V_Z(x) < Q(y) + xy, "x > 0

hence

On the other hand the inequality (34) holds as equality if and only if y = (x) and (p, c) = (p, c) then

now

and also

For every y > 0, and in conjunction with (35) we obtain Q(y) = (y), "y > 0.

Remarks on the Assumptions

• Assumption (a) is necessary and sufficient for the optimality of consumption. For the existence of the portfolio financing this optimal consumption we need to solve the following equation:

• Assumption (b) is a sufficient condition which allow us to control the jumps on wealth and depending on the process Y considered, it can be easy verified as we will see in section 6 with an example. Condition D₀ = 0 is just to simplify the proof. We can prove the same with D₀¹ 0.

As we have seen in section 3, we obtained a characterization of the EMM. But it can be argued that choosing an EMM can be a hard task to solve, for these reason we show how to choose an EMM in the following section.

5 Choosing a Measure

There exists some approaches to choose one EMM, we concentrate our attention in the approach introduced by Gerber and Shiu (1994), define the new probability

where j(q) = Ee^qY1. When the stock price process has constant coefficients, Gerber and Shiu (1994) prove that for a given constant r it is possible to find a solution q of the following:

Then we can verify that the process = e^–rtP_t is a martingale under P^q,i.e. P^qÎ (set of EMM). Moreover the process is a Lévy Process under this probability and is called the Esscher transform¹⁴ of the original process. In our model we consider time dependent functions, then we consider the generalized Esscher Transform:

Now we can choose q_s in order to satisfy equation (18), since this is the case of K(s, x) = exp(–q_sx), k(s, x) = –q_sx and R_s = –cq_s. In fact with these expressions we obtain

It easy to verify that this equation has an unique solution for which j(q_s) < ¥ and q_sÎ (–b₁, b₂) "s. Then we can get a EMM, and important fact of this measure is that this is the measure of minimum relative entropy with respect to P, to see this remember the definition of entropy:

Where Q is any absolutely continuous measure with respect to P, with lemma 1 we have

where Q depends on the choice of K and R, and these functions have to satisfy equation (18). We can show ¹⁵ that this minimum is obtained when K = exp (–xsl) and R = –csl, where l is the lagrange multiplier associated to the constraint (18), this can justify the choice of the measure associated to q = sl.

6 Example

Take s = r = c = 1 and r = 0. Now assume that , where N_i is a Poisson process with rate 1, so:

• G =

• a = EN_t = 0

Now we choose k(x) = and K(x) = 1 – k(x), replacing in condition (18) we obtain R_s = -, from the fact that DN_s = 0, we have

It is easy to verify that E

_T = 1, in other words it is a EMM density. So we have find one EMM. Now we apply the approach presented in last section to find the EMM that minimize entropy: as s = 1, then q = l. We obtain the parameter q that satisfy equation (38):

reducing this expression, we have:

the solution of this equation is q^*» 0,6626. In equation (5):

we have also Y_t = W_t + N_t and

where the first equality is due to the independence of W_t and N(t), the second is due to the fact that N₁(t) and N₂(t) are independent and identically distributed, and the third is the expected value of a Log-normal variable. From here we have

and this measure is obtained when K(s, x) = e^–0,6626x, k(s, x) = -0,6626x and R_s = –0,6626.

Now as N_t and W_t are martingales condition (b) in the main theorem is satisfied. Then if we take u(t, x) = g(x) = log(x), then I_u(x) = I_g(x) = and

but

then y = and

So depending on the state price density, i.e., on the EMM we choose, we have the path of the optimal policies. More precisely, if we desire to choose the EMM closest to P in terms of its information contents, since P contains information about the behavior of the market, we have to choose the EMM given by (39).¹⁶

7 Conclusions

In this paper we studied the optimal investment and consumption problem assuming that the risky asset is driven by a geometric Lévy process which allow us to obtain more realistic results. Another important issue addressed in this paper is the characterization of the EMM in a Lévy market and how to choose one of them using a minimal entropy criteria.

An important and difficult issue not considered in the paper is the availability of the portfolio processes, since many of the market models considered in the paper can be incomplete, we need to ensure that the financing portfolio is available in the market. For a discussion on this type of problem see Cvitani and Karatzas (1992).. Also from the existence of EMM we obtained the arbitrage free economies, with this in mind some equilibrium analysis in continuous time can be done.

This paper was received in Nov. 2002 and approved in Feb. 2003.

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*

I want to thank Maria Eulália Vares for valuable comments and IMPA-Brazil, where this project was partially developed. I also acknowledge financial support from CNPq, Brazil.

1

The augmented filtration

F is defined by

(

t) = s(

_Y(

t) È

), where

= {

E Ì W : $

G Î

with

E Í

G,

P (

G) = 0} denotes the set of

P- null events.

2

We said that a process {

X_t} is adapted with respect to

F if for all

t Î [0,

T],

X_t is an

(

t)-measurable random variable.

3

See Karatzas and Shreve (1998) for an example.

4

An infinite divisible distribution generates a Lévy processes, for abuse of notation we use

Y_t for both. For more details see Sato (1999) or Shiryaev (1999).

5

A process

X is said to be a quadratic pure jump process if

^cº 0, where

^c is the continuous part of its quadratic variation

. Remember that

is the process such that (

N_t)

² –

_t is a martingale.

6

"

B Î

×

\0,

L(

B) has Poisson distribution with parameter (

dt ×

G)(

B)

7

E(·) denote the expectation with respect to P

8

For more details see Shiryaev (1999).

9

See Jacod and Shiryaev (1987)

10

Observe that from

N_t =

M_t +

at we have D

M_t = D

N_t.

11

See Chan (1999)

12

a process

K

_w(

t,

x) is said to be a Borel previsible function or process if the process

t ®

K

_w(

t,

x) is a previsible function for fixed

x and the function

x ®

K

_w(

t,

x) is Borel-measurable for fixed

t.

13

A contingent claim x is a random variable

_T–measurable.

14

For more details on Esscher transforms see Buhlmann et al. (1996)

15

See Chan (1999)

16

For more details on how to construct EMM with different types of Lévy processes see Fajardo (2003).

Publication Dates

• Publication in this collection
  14 June 2004
• Date of issue
  Dec 2003

History

• Received
  Nov 2002
• Accepted
  Feb 2003