# Asymptotics and Duality for the Davis and Norman Problem

###### Abstract.

We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in the seminal paper of Davis and Norman [Math. Operation Research, 15, 1990]. Similarly to Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by determining a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimization problem. However, we use a different parametrization, which facilitates computation and verification. Moreover, for small transaction costs, we determine fractional Taylor expansions of arbitrary order for the boundaries of the no-trade region and the value function. This extends work of Janeček and Shreve [Finance Stoch., 8, 2004], who determined the leading terms of these power series.

###### Key words and phrases:

Transaction costs, optimal consumption, shadow price, asymptotics###### 2000 Mathematics Subject Classification:

91B28, 91B16, 60H10## 1. Introduction

It is a classical problem of financial theory to maximize expected utility from consumption (cf., e.g., [15, 20] and the references therein). This is often called the Merton problem, because it was first formulated and solved in a continuous-time setting by Merton [18, 19]. More specifically, he found that – for logarithmic or power utility and one risky asset following geometric Brownian motion – it is optimal to keep the fraction of wealth invested into stocks equal to a constant , which is known explicitly in terms of the model parameters.

Magill and Constantinides [17] extended Merton’s setting to incorporate proportional transaction costs. In particular, they showed that – again for logarithmic or power utility – it is optimal to engage in the minimal amount of trading necessary to keep the fraction of wealth in stocks inside some no-trade region around .

Their somewhat heuristic derivation was made rigorous in the seminal paper of Davis and Norman [4], who also showed how to compute by solving a free boundary problem.

Using the theory of viscosity solutions, this was further generalized by Shreve and Soner [23]. Moreover, the same approach allowed Janeček and Shreve [12] to derive rigorous first-order asymptotic expansions of and the value function for small transaction costs (also cf. [2, 23, 21, 26] for related asymptotic results).

All of the above papers use methods from the theory of stochastic control. On the other hand, dating back to the pioneering papers of Jouini and Kallal [13] and Cvitanić and Karatzas [3], much of the general theory for markets with transaction costs is formulated in terms of duality theory leading to consistent price systems resp. shadow prices. These are frictionless markets evolving within the bid-ask spread of the original market with transaction costs, which lead to an equivalent dual optimization problem. For logarithmic utility, Kallsen and Muhle-Karbe [14] solved the Merton problem with transaction costs by simultaneously determining a shadow price and its optimal portfolio, i.e., by determining the dual and the primal optimizer at the same time (also compare [16] for an extension of this approach to a limit order market).

In the present article, we revisit this problem using a different parametrization in the spirit of [6]. We employ arguments that are tailor-made for sufficiently small transaction costs, i.e., which focus on asymptotic expansions and thus seem promising for tackling more complicated models. In particular, we determine rigorous asymptotic expansions both for the boundaries of the no-trade region and the corresponding value function, where terms of arbitrary order can be algorithmically computed. This extends the first-order expansions of [12]. However, we emphasize that we only consider the particularly simple case of logarithmic utility, unlike [12], who also deal with the more involved case of power utilities. An extension of the present approach to power utilities is subject of current research.

The remainder of this article is organized as follows. After introducing the Merton problem with transaction costs in Section 2, we present a heuristic derivation of our candidate shadow price process. Subsequently, in Section 4, we prove that this candidate is indeed well-defined and a shadow price process for sufficiently small transaction costs. Moreover, we derive asymptotic expansions for the boundaries of the no-trade region. Afterwards, we also determine an asymptotic expansion for the value function. Finally, in Section 6, we show that the shadow price corresponds to the solution of the dual problem.

## 2. The Merton Problem with transaction costs

We study the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the presence of proportional transaction costs, as in [4, 14, 23]. Fix a filtered probability space , whose filtration is generated by a standard Brownian motion . We consider a market with two investment opportunities, namely a bond and a stock. The ask price process of the stock is supposed to follow the Black-Scholes model

(2.1) |

and the bid price process is assumed to be given by for some ^{1}^{1}1This notation, also used in [25], turns out to be convenient in the sequel. It is equivalent to the usual setup with the same constant proportional transaction costs for purchases and sales (compare, e.g., [4, 12, 14, 23]). Indeed, set and . Then coincides with . Conversely, any bid-ask process with equals for and .. This means that one has to pay the higher
price when purchasing the stock at time , but only receives the lower price when selling it. The price of the bond is supposed to be constant and equal to . As in [4, 14], we make the following standing assumption, which ensures that the holdings in bond and stock remain positive at all times.

###### Standing Assumption 2.1.

The case can be dealt with using minor modifications of the present approach (cf. [6]). The degenerate case has to be treated separately, see [12, 23].

Since transactions of infinite variation lead to immediate bankruptcy (compare [1]), we confine ourselves to the following set of trading strategies.

###### Definition 2.2.

A trading strategy is an -valued predictable process of finite variation. Here, represents the initial endowment in bonds and stocks, whereas and denote the units held in bond and in stock at time , respectively. A (discounted) consumption rate is an -valued, adapted stochastic process satisfying a.s. for all . A pair of a trading strategy and a consumption rate is called portfolio/consumption process.

To capture the notion of a self-financing strategy, we use the intuition that no funds are added or withdrawn. As in [14], this leads to the following notion.

###### Definition 2.3.

A portfolio/consumption process is called self-financing, if

(2.2) |

where for increasing predictable processes which do not grow at the same time.

Note that since is continuous and is of finite variation, integration by parts yields that this definition coincides with the usual notion of self-financing strategies in the absence of transaction costs if we let .

The subsequent definition requires the investor to be solvent at all times. For frictionless markets, i.e., if , this coincides with the usual notion of non-negativity of wealth.

###### Definition 2.4.

A self-financing portfolio/consumption process is called admissible, if the liquidation wealth process

of is a.s. nonnegative.

We consider the following classical investment/consumption problem.

###### Definition 2.5 (Primal problem).

Given an initial endowment , an admissible portfolio/consumption process is called optimal if it attains the maximum in

(2.3) |

over all admissible portfolio/consumption processes . Here denotes a fixed given impatience rate.

Note that since , the value function of the Merton problem without transaction costs is finite by [4, Theorem 2.1]. This upper bound for the value function in the present setup with transaction costs then implies that the latter is finite, since it is also bounded from below by the finite utilities that can be obtained from liquidating at time zero and consuming at a sufficiently small constant relative rate.

###### Definition 2.6 (Dual problem).

Denote by the conjugate function of the utility function , i.e., the Legendre transform of . Then a pair is called dual optimizer, if it attains the minimum in

(2.4) |

Here runs through the set of all continuous semimartingales with values in the bid-ask spread and the density processes of corresponding martingale measures, i.e., the positive martingales with initial value for which is a martingale.

###### Remark 2.7.

The pairs correspond to the consistent price systems of [9, 22]. In the frictionless case there is only one of these, namely the stock price process itself and the corresponding density process determined by Girsanov’s theorem. In this case, the dual problem reduces to the usual notion for frictionless markets as in, e.g., [15, Theorems 3.9.11 and 3.9.12]. The only difference is the constant , which is added to let the optimal primal and dual values coincide (compare Theorem 6.1 below).

## 3. Heuristic derivation of the shadow price

The key to solving both the primal and dual problems above is the following concept:

###### Definition 3.1.

A shadow price process is a continuous semimartingale with values in the bid-ask spread , such that the maximal expected utilities in the frictionless market with price process and in the original market with bid-ask process coincide.

From an economic point of view, it is obvious that the maximal expected utility for any frictionless price process with values in the bid-ask spread is at least as high as for the original problem with transaction costs, since one can always buy at least as cheaply and sell at least as expensively. To achieve equality of the maximal expected utilities, the optimal number of stocks in the frictionless market with the shadow price process should increase (resp. decrease) only when (resp. ). Otherwise one could achieve higher utility than in the original market with transaction costs. This portfolio will then also be feasible in the original market with transaction costs and hence be optimal for this market as well.

It is well-known, dating back to [17], that there is no trading in the original market with transaction costs as long as the fraction

of total wealth in terms of the ask price invested in stocks lies inside some interval around the Merton proportion . Let us reparametrize this conveniently for the computations below (compare [6]): Set

(3.1) |

and define

Then is equivalent to . We now make the ansatz that the shadow price is given by

for some function . This is equivalent to assuming that is a function of the current ask price and the current fraction of wealth, in terms of the ask price , invested in stocks. However, using instead of turns out to be more convenient in the calculations below.

The function enjoys a scaling property inherited from the logarithmic utility function and its constant relative risk aversion:

This is because the optimal number of bonds changes to whereas the optimal number of stocks and the dimensionless constant remain the same when we rescale from to . Therefore, we can pass to the function

such that

We now heuristically derive a free boundary problem that determines the function as well as the constants and . With these quantities at hand, we then explain how to construct the process and in turn the shadow price .

If is a shadow price, it is well-known that for logarithmic utility the optimal consumption rate at time in the frictionless market with price process is given by

where denotes the discount factor in the optimization problem (cf., e.g., [20]). Consequently, we have

(3.2) |

in the no-trade region, where we have inserted our ansatz and the definition . Now consider the process

which represents the fraction of wealth in stocks (in terms of the ask price ) over wealth in bonds, normalized by . In view of the dynamics (2.1) of and (3.2) of and , Itô’s formula yields

(3.3) |

Again by Itô’s formula, the dynamics of the shadow price are

during each excursion into the no-trade region. Due to Merton’s rule for logarithmic utility [19], the Merton ratio for must equal the proportion of total wealth invested in stocks in terms of the shadow price , i.e., we must have

After rearranging, this yields the following ODE for :

(3.4) |

Note that in the limit , this is precisely the ODE from [6]. By (3.1), the process takes values in . Hence it is natural to complement the ODE (3.4) with some boundary conditions at and as in [6]. Since we have a second order ODE, and the parameters and are also yet to be determined, we need two boundary conditions at each endpoint. The shadow price should equal the bid price at the selling boundary , where , resp. the ask price at the buying boundary , where . Hence we impose

(3.5) |

Moreover, we add the smooth pasting conditions

(3.6) |

They can be derived by looking at the ratio , which should stay in , and setting its diffusion coefficient equal to zero at and (compare [6]). These four boundary conditions should now determine the function as the solution to our second-order ODE as well as the two free variables and , which in turn identify the boundaries and of the no-trade region.

With at hand, it now remains to extend this construction from one excursion into the no-trade region to the entire positive real time line. To this end notice that, even when trading in the stock does happen, the process must always remain within , in order to keep the corresponding fraction of wealth in stocks in . However, the diffusion coefficient in the SDE (3.3) is bounded away from zero when approaches either or . Therefore we need to complement the SDE (3.3) for with instantaneous reflection at both endpoints of . Then will by definition remain in and follow the claimed dynamics during each excursion into the no-trade region. With the process at hand, we can then define the candidate shadow price .

## 4. Construction of the shadow price

We begin with an existence result for the free boundary value problem from the preceding section. As in [14, Proposition 4.2], it would be possible to obtain a similar result for any relative width of the bid-ask spread. However, we restrict ourselves to the case of sufficiently small here. This is all we need to derive asymptotic expansions in terms of below and seems much better suited to the analysis of more complicated models, since it requires only a much coarser inspection of the involved equations.

###### Lemma 4.1.

If is sufficiently small, then there are constants and as well as a function satisfying the ODE

(4.1) |

with boundary conditions

(4.2) |

###### Proof.

We first ignore the free boundary and consider only the ODE (4.1) with initial conditions , with as free parameter. This problem can be written as

where , and

Since , the function (and of course also ) is analytic in a complex neighbourhood of . By a standard existence and uniqueness result for analytic ODEs [11, Theorem 1.1], we obtain a neighbourhood of with a unique analytic solution . The coefficients of its Taylor expansion

can be calculated recursively. Note that for , where is the Kronecker delta, which follows from the initial conditions .

From this analytic function of two variables we will now construct the solution of the free boundary value problem (4.1)–(4.2), by supplying appropriate quantities and (depending on ). These should satisfy

(4.3) |

We divide the latter equation by , reflecting the fact that we are not interested in the solution (which exists because of the initial conditions ). A Taylor expansion yields

(4.4) |

for some real, explicitly computable coefficients . It turns out that , while is non-zero, so that we can apply the implicit function theorem for multivariate analytic functions [10, Theorem I.B.4] to obtain an analytic function

(4.5) |

such that (4.4) vanishes if is substituted for . Here and in what follows, the coefficients of the power series symbolized by the -s can be algorithmically computed. Now we insert the function into the right hand side of the first equation in (4.3):

where . If we insert also on the left hand side, the first equation in (4.3) becomes

Dividing by , we obtain

Let us denote the coefficients of the series on the left hand side by :

Now take the third root:

(Note that the coefficients of powers of power series can be algorithmically calculated [8].) Since is non-zero, we can apply the implicit function theorem, so that the latter equation has a unique solution for small . It is an analytic function of :

The coefficients in this power series expansion can be calculated with the Lagrange inversion formula:

(4.6) |

where the operator extracts the -th coefficient of a power series. (This is analogous to the proof of Proposition 6.1 in [6].) With this value of , the function , together with , is the desired solution of the free boundary value problem. To see that it maps to , note that otherwise the equation would have a solution in the open interval . This is impossible, since is the unique zero of (4.4).

Henceforth, we consider sufficiently small transaction costs for Lemma 4.1 to be valid and write resp. for the corresponding function resp. constants defined therein.

The existence of the process as the solution to the SDE (3.3) with reflection at and is a consequence of a classical result of Skorokhod [24].

###### Lemma 4.2.

Let . Then there exists a strong solution to the Skorokhod SDE

(4.9) |

with instantaneous reflection at and , that is, a continuous, adapted, -valued process and nondecreasing adapted processes and increasing only on the sets and , respectively, such that

holds for all .

###### Remark 4.3.

In the limit , the process is just a geometric Brownian motion instantaneously reflected at and . Hence it coincides with the process from [6]. Let us also point out that is related to the process from [14, Lemma 4.3] by a nonlinear transformation. However, the present parametrization evidently leads to simpler dynamics. Moreover, the additional process from [14, Lemma 4.4] is no longer necessary.

###### Proof of Lemma 4.2.

It was shown in [6, Proposition 2.1] that, for , the process is an Itô process, i.e., does not involve local time. We now deduce from Lemma 4.2 that this still holds for arbitrary . The present proof only uses Itô’s formula and the smooth pasting conditions (4.2) for . In particular, it allows to avoid the somewhat ad hoc localization procedures used in the proof of [6, Proposition 2.1].

###### Proposition 4.4.

For and as in Lemma 4.2, is a positive Itô process of the exponential form

with

and takes values in the bid-ask spread .

###### Proof.

Using Itô’s formula, we obtain

and

Hence integration by parts yields

Since and only increase on resp. , where the integrand is zero by the smooth pasting conditions for , the last term vanishes. The Itô process decomposition of now follows by inserting the ODE (4.1) for .

For the second part of the assertion, notice that , because . Hence it follows from a comparison argument that on . As this implies that the derivative of is non-positive, and yield that is indeed -valued. ∎

Using standard results for frictionless markets, we can now determine the optimal portfolio/consumption pair for .

###### Lemma 4.5.

Set

(4.10) |

For and as in Lemma 4.2 and Proposition 4.4, respectively, define

as well as

Then

(4.11) |

and is an optimal portfolio/consumption pair for the frictionless market with price process . The corresponding wealth process and the optimal fraction of wealth invested in stocks in terms of are given by and , respectively.

Note that the first resp. second cases in (4.10) occur if the initial fraction of wealth in stock lies below resp. above . In this case, there is a jump from the initial position , which moves the initial fraction of wealth to the nearest boundary of the interval . Since this initial bulk trade involves the puchase resp. sale of stocks, the initial value of is chosen so as to match the initial ask resp. bid price in this case.

###### Proof.

Since is an Itô process with bounded coefficients and is bounded away from zero, its optimal portfolio/consumption pair is characterized by standard results for frictionless markets, cf., e.g., [7, Theorem 3.1]. In particular, the optimal fraction of wealth (in terms of ) invested in stocks is given by Merton’s rule, i.e., equals

as claimed. Moreover, the optimal wealth process, the optimal consumption rate and the optimal numbers of bonds resp. stocks are indeed given by , , , and . Now Itô’s formula and the ODE (4.1) for imply

Moreover, by Yor’s formula, we have

Taking into account that the process is continuous and of finite variation, integration by parts finally yields the claimed representation for . ∎

The representation (4.11) and the definition of via (4.10) imply that stocks are only purchased (resp. sold) when (resp. ). Hence the optimal portfolio/consumption process for is also admissible for the bid-ask process . Since shares can be bought at least as cheaply and sold as least as expensively when trading instead of , it follows that the maximal expected utilities coincide, i.e., is a shadow price. Made precise, this is the content of the following analogue of [14, Theorem 4.6].

###### Theorem 4.6.

The portfolio/consumption process from Lemma 4.5 is also optimal for the bid-ask process . In particular, is a shadow price in this market.

###### Proof.

This follows verbatim as in the proof of [14, Theorem 4.6]. ∎

In view of Lemma 4.5 and the definition of , the optimal proportion of stocks in terms of the shadow price takes values in the interval . Translating this into terms of the ask price , we find that the optimal fraction in terms of is kept within , i.e., the lower resp. upper boundaries of the no-trade region are given by resp. . The following series expansions are immediate consequences of (4.7) and (4.8) (upon taking one additional term in both expansions).

###### Corollary 4.7.

The lower and upper boundaries of the no-trade region in terms of the ask price have the expansions