# An expansion in the model space in the context of utility maximization

## Abstract

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian), an explicit second-order expansion formula for the power investor’s value function—seen as a function of the underlying market price of risk process—is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.

## Keywords

Continuous semimartingales Second-order expansion Incomplete markets Power utility Convex duality Optimal investment## Mathematics Subject Classification (2010)

91G10 91G80 60K35## JEL Classification

C61 G11## 1 Introduction

In an incomplete financial setting with noise governed by a continuous martingale and in which the investor’s preferences are modeled by a negative power utility function, we provide a second-order Taylor expansion of the investor’s value function with respect to perturbations of the underlying market price of risk process. We show that tractable models can be used to approximate highly intractable ones as long as the latter can be interpreted as perturbations of the former. As a by-product of our analysis, we explicitly construct first-order approximations of both the primal and the dual optimizers. Finally, we apply our approximation in two numerical examples.

There are two different ways of looking at our contribution: as a tool to approximate the value function and perform numerical computations, or as a stability result with applications to statistical estimation. Let us elaborate on these, and the related work, in order.

*An approximation interpretation.* The conditions for existence and uniqueness of the investor’s utility optimizers are well established (see [21, 27]). However, in general settings, the numerical computation of the investor’s value function and corresponding optimal trading strategy remain a challenging problem. Various existing approaches include the following:

1. In Markovian settings, the value function can typically be characterized by an HJB equation. Its numerical implementation through a finite-grid approximation is naturally subject to the curse of dimensionality. Many authors (see [23, 40, 7, 25, 33]) opt for affine and quadratic models for which closed-form solutions exist. Going beyond these specifications in high-dimensional settings by using PDE techniques seems to be very hard computationally. We refer to [26] and the references therein for recent advances on numerically solving the PDE stemming from the HJB equation.

2. In general (i.e., not necessarily Markovian) complete models, [10] and [12] provide efficient Monte Carlo simulation techniques based on the martingale method for complete markets developed in [9] and [20].

3. Other approximation methods are based on various Taylor-type expansions. The authors of [4] and [5] log-linearize the investor’s budget constraint as well as the investor’s first-order condition for optimality. Kogan and Uppal [24] expand in the investor’s risk-aversion coefficient around the log-investor (the myopic investor’s problem is known to be tractable even in incomplete settings). When solving the HJB equation numerically (using a Longstaff–Schwartz type of technique), Brandt et al. [3] expand the value function in the wealth variable to a fourth-degree Taylor approximation.

4. Based on the duality results in [21], Haugh et al. [14] provide an upper bound on the error stemming from using suboptimal strategies. Bick et al. [2] propose a method based on minimizing over a subset of dual elements. This subset is chosen such that the corresponding dual utility can be computed explicitly and transformed into a feasible primal strategy.

5. It is also important to mention the recent explosion in research in asymptotic methods in a variety of different areas in mathematical finance (transaction costs, pricing, ambiguity aversion, etc.). Since we focus on model expansion in utility maximization in this paper, we simply point the reader to some of the most recent papers, namely [1, 18, 16], and the references therein, for further information.

In our work, no Markovian assumption is imposed and we deal with general, possibly incomplete markets with continuous price processes. We consider the utility functions \(U(x) := x^{p}/p\) for \(x>0\). We note that while our results apply only to \(p<0\), it is possible to extend them to \(p\in(0,1)\) at the cost of imposing additional integrability requirements. We do not pursue such an extension because the parameter range \(p\in(0,1)\) which we leave out seems to lie outside the typical range of risk-aversion parameters observed in practice (see e.g. [39]).

*A stability interpretation.* As mentioned above, our contribution can also be seen as a stability result. It is well known (see e.g. [38]) that even in Samuelson’s model, estimating the drift is far more challenging than estimating the volatility. Larsen and Žitković [31] identify the kinds of perturbations of the market price of risk process under which the value function behaves continuously. In the present paper, we take the stability analysis one step further and provide a second-order Taylor expansion in an infinite-dimensional space of market price of risk processes. In this way, we not only identify the “continuous” directions, but also those features of the market price of risk process that affect the solution of the utility maximization problem the most (at least locally). Any statistical procedure which is performed with utility maximization in mind should therefore focus on those salient features in order to use the scarce data most efficiently.

Similar perturbations have been considered by [34], but in a somewhat different setting. [34] is based on Malliavin calculus and produces a first-order expansion for the utility indifference price of an exponential investor in an Itô-process driven market; some of the ideas used can be traced to the related work [11].

*Mathematical challenges.* From a mathematical point of view, our approach is founded on two ideas. One of them is to extend the techniques and results of [31]; indeed, the basic fact that the dual minimizers converge when the market price of risk process does is heavily exploited. It does not, however, suffice to get the full picture. For that, one needs to work on the primal and the dual problems simultaneously and use a pair of bounds. The ideas used there are related to, and can be interpreted as, a nonlinear version of the primal–dual second-order error estimation techniques first used in [15] in the context of mathematical finance. The first-order expansion in the quantity of the unspanned contingent claim developed in [15] was generalized in [29] (see also [28]). The arguments in these papers rely on convexity and concavity properties in the expansion parameter (wealth and number of unspanned claims). This is not the case in the present paper; indeed, when seen as a function of the underlying market price of risk process, the investor’s value function is neither convex nor concave and a more delicate, local analysis needs to be performed.

*Numerical examples.* In Sect. 5, we use two examples to illustrate how our approximation performs under realistic conditions. First, we consider the Kim–Omberg model (see [23]) which is widely used in the financial literature. Under a calibrated set of parameters, we find that our approximation is indeed very accurate when compared to the exact values.

Our second set of examples belongs to the class of extended affine models introduced in [8]. The authors show that this class of models has superior empirical properties when compared to popular affine and quadratic specifications (such as those used e.g. in [33]). The resulting optimal investment problem for the extended affine models unfortunately does not seem to be explicitly solvable. Our approximation technique turns out to be easily applicable, and our error bounds are quite tight in the relevant parameter ranges. Furthermore, unlike numerical methods based on PDEs, our method’s computation time grows linearly in the number of underlying factors. Therefore, we can and do apply our theory to a high-dimensional extended affine model.

## 2 A family of utility maximization problems

### 2.1 The setup

We work on a filtered probability space \((\varOmega,{\mathcal{F}}, {\mathbb{F}}= ( {\mathcal{F}}_{t} )_{t\in[0,T]}, {\mathbb{P}})\) with a finite time horizon \(T>0\). We assume that the filtration \({\mathbb{F}}\) is right-continuous and that the \(\sigma\)-algebra \({\mathcal{F}}_{0}\) consists of all ℙ-trivial subsets of ℱ.

### 2.2 The utility maximization problem

### 2.3 The dual of the utility maximization problem

As is usual in the utility maximization literature, a fuller picture is obtained if one also considers an appropriate version of the optimization problem dual to (2.2). For that, we need to examine the no-arbitrage properties of the set of models introduced in Sect. 2.1 above.

*minimal*local martingale density, that is,

*conjugate*utility function \(V:(0,\infty)\to{\mathbb{R}}\) is defined by

*dual value function*\(v^{(\varepsilon)}:(0, \infty)\to{\mathbb{R}}\) by

### 2.4 A change of measure

## 3 The problem and the main results

We first provide first-order expansions and error estimates of the primal and dual value functions. Secondly, we provide an expansion of the optimal controls in the Brownian setting.

### 3.1 Value functions

### Theorem 3.1

*In the setting of Sect*. 2,

*we assume that*

*Then with*\(\Delta^{(0)}:= {\mathbb{E}}^{\tilde{{\mathbb{P}}}^{(0)}}[ \int_{0}^{T} \lambda'_{t}\, dR^{(0)}_{t}]\),

*where*\(\tilde{{\mathbb{P}}}^{(0)}\)

*is defined by*(2.6),

*we have*

### Theorem 3.2

*In the setting of Sect*. 2,

*we assume that*

*for some*\(\varepsilon_{0}>0\),

*where*\(\varPhi:= \int_{0}^{T} \hat{\pi} ^{(0)}_{t} \lambda'_{t}\, d\langle M \rangle_{t}\)

*and*\(\varPhi^{-} := - \min\{\varPhi,0\}\).

*Then there exist constants*\(C>0\)

*and*\(\varepsilon _{0}'\in(0,\varepsilon_{0}]\)

*such that for all*\(\varepsilon\in[0, \varepsilon'_{0}]\),

*we have*

### Remark 3.3

*logarithmic*derivatives \(p\Delta^{(0)}\) and \(q\Delta^{(0)}\), respectively, at \(\varepsilon=0\). Moreover, we have the small-\(\varepsilon\) asymptotics

*certainty equivalent*\(\mathrm{CE}^{(\varepsilon)}\) given by

2. A careful analysis of the proof of Theorem 3.2 below yields the following additional information:

(a) The proof of Proposition 4.3 reveals that \(\Delta ^{(0)}={\mathbb{E}}^{\tilde{{\mathbb{P}}}^{(0)}}[\varPhi]\).

(b) The condition involving \(\varPhi\) in (3.3) is needed only for the upper bound in (3.4) and the lower bound in (3.5). The other two bounds hold for all \(\varepsilon \geq0\), even if (3.3) holds with \(\varepsilon_{0}=0\).

(c) The constants \(C\) and \(\varepsilon_{0}'\) depend (in a simple way) on \(\varepsilon_{0}\), \(p\) and the \({\mathbb{L}}^{2(1-p)}( \tilde{{\mathbb{P}}}^{(0)})\)- and \({\mathbb{L}}^{1}( \tilde{{\mathbb{P}}}^{(0)})\)-bounds of the random variables in (3.3). For two one-sided bounds, explicit formulas are given in Propositions 4.2 and 4.3. The other two bounds are somewhat less informative; so we do not compute them explicitly.

(d) Even though we cannot claim that the functions \(u^{(\varepsilon)}\) and \(v^{(\varepsilon)}\) are convex or concave in \(\varepsilon\), it is possible to show their local *semiconcavity* (see [6, Definition 1.1.1]). This can be done via the techniques from the proof of Theorem 3.2.

### 3.2 Optimal controls

### Theorem 3.4

*In the above Brownian setting*,

*we assume*

*as well as the existence of a constant*\(\varepsilon_{0} >0\)

*such that*

*for all*\(\varepsilon\in(0,\varepsilon_{0})\).

*Then we have*

*as*\(\varepsilon\searrow0\).

*In*(3.9)

*and*(3.10),

*we have defined*

*where the processes*\(\gamma^{B}\)

*and*\(\gamma^{W}\)

*are given by the representation*(3.6).

### Remark 3.5

4. As illustrated in the examples in Sect. 5.3 below, the exponential moment condition (3.8) can often be made to hold by imposing some smallness condition on either \(T>0\) or \(\varepsilon _{0}>0\).

## 4 Proofs of the main theorems

### 4.1 Proof of Theorem 3.1

The proof of Theorem 3.1 is based on the stability results of [31] and the following lemma.

### Lemma 4.1

*Let*\((K^{(\varepsilon)})_{\varepsilon\geq0}\)

*be a family of positive random variables such that*

- 1.
\({\mathbb{E}}[ Z^{(\delta)}_{T} K^{(\varepsilon)}]\leq1\)

*for all*\(\varepsilon,\delta\geq0\); - 2.
\(K^{(\varepsilon)}\to K^{(0)}\)

*in probability as*\(\varepsilon\searrow 0\).

*Then under the conditions of Theorem*3.1,

*we have*

### Proof

### Proof of Theorem 3.1

### 4.2 Remaining proofs

### Proposition 4.2

*Suppose that*\(\eta\in{\mathbb{L}}^{2(1-p)}\)

*and*\(\varLambda, \varLambda \eta\in{\mathbb{L}}^{1-p}\).

*Then for all*\(\varepsilon\geq0\),

*we have*

*where*\(C_{v}= |q|\|\eta\|_{{\mathbb{L}}^{2(1-p)}}^{1/2} + \|\varLambda \|_{{\mathbb{L}}^{1-p}}\)

*and*\(C'_{v} =|q|\|\eta\varLambda\|_{{\mathbb{L}} ^{1-p}}\).

### Proof

Unfortunately, the same idea cannot be applied to obtain a similar lower bound. Instead, we turn to the primal problem and establish a lower bound for it.

### Proposition 4.3

*Let*\(\varPhi:= \int_{0}^{T} \hat{\pi}^{(0)}_{t} \lambda'_{t}\, d \langle M \rangle_{t}\),

*let*\(\varLambda\)

*be defined by*(4.2)

*and*\(\tilde{{\mathbb{P}}}^{(0)}\)

*by*(2.6).

*Given*\(\varepsilon_{0}>0\),

*assume that*\(\varLambda\in{\mathbb{L}}^{1-p}\)

*and*\(\varPhi^{2} e^{ \varepsilon_{0} \vert p \vert \varPhi^{-}}\in{\mathbb{L}}^{1}(\tilde{{\mathbb{P}}}^{(0)})\).

*Then*

*where*\(C_{u}(\varepsilon) := \frac{1}{2}p^{2} |u^{(0)}| {\mathbb{E}} ^{\tilde{{\mathbb{P}}}^{(0)}}[ \varPhi^{2} e^{\varepsilon\vert p \vert \varPhi^{-}}]\).

### Proof

### Remark 4.4

If one is interested in an error estimate which does not feature the optimal portfolio \(\hat{\pi}^{(0)}\) (through \(\varPhi\)), one can adopt an alternative approach in the proof (and the statement) of Proposition 4.3. More specifically, by using \({\tilde{X}}= \hat{X} ^{(0)}{\mathcal{E}}( \varepsilon\lambda'\cdot R^{(\varepsilon)})\) as a test process (instead of \({\mathcal{E}}( \hat{\pi}^{(0)}\cdot R ^{(\varepsilon)})\)), one obtains a constant \(C_{u}(\varepsilon)\) which depends only on the primal and dual optimizers \(\hat{X}^{(0)}\) and \(\hat{Y}^{(0)}\), in addition to \(\lambda'\), \(\eta\) and \(\varLambda\).

### Proof of Theorem 3.2

### Proof of Theorem 3.4

## 5 Examples

### 5.1 First examples

### Example 5.1

(Small market price of risk)

### Example 5.2

(Deviations from the Black–Scholes model)

### Example 5.3

(Uniform deviations)

### 5.2 The Kim–Omberg model

The Kim–Omberg model (see [23]) is one of the most widely used models for the market price of risk process. Because it allows explicit expressions for all quantities involved in CRRA utility maximization, it serves as an excellent test case for the practical implementation of our main results. Appendix A contains all technical details.

#### 5.2.1 Exact computations

Certainty equivalents for the zeroth-, first- and second-order approximations and the exact values in the Kim–Omberg model with \(\beta:=\varepsilon\) and unit initial wealth. The model parameters used are \(\gamma=0.04395\), \(\kappa= 0.0404\), \(\theta= 0.117\), \(p=-1\) and \(T=10\)

| \(\lambda_{0}\) | \(\mathrm{CE}(u^{(0)}) \) | \(\mathrm{CE}( \delta^{(0)})\) | \(\mathrm{CE}( \delta^{(00)})\) | \(\mathrm{CE}(u^{(\varepsilon)})\) |
---|---|---|---|---|---|

−0.01 | 0.1 | 1.046 | 1.047 | 1.048 | 1.048 |

−0.05 | 0.1 | 1.046 | 1.054 | 1.081 | 1.084 |

−0.10 | 0.1 | 1.046 | 1.063 | 1.181 | 1.206 |

−0.01 | 0.5 | 1.614 | 1.647 | 1.648 | 1.649 |

−0.05 | 0.5 | 1.614 | 1.794 | 1.850 | 1.846 |

−0.10 | 0.5 | 1.614 | 2.020 | 2.339 | 2.272 |

#### 5.2.2 Monte Carlo-based computations

One of the advantages of our approach is that it lends itself easily to computational methods based on Monte Carlo (MC) simulation. For the Kim–Omberg model, we use the standard explicit Euler scheme from MC simulation to compute the involved quantities of interest. In other words, we do not rely on the availability of exact expressions for the value functions or the correction terms \(\Delta^{(0)}\) and \(\Delta ^{(00)}\).

\(95\%\)-confidence intervals for certainty equivalents for the upper and lower bounds as well as the base model optimizer \(\hat{\pi}^{(0)}\) for the Kim–Omberg model. The true exact values for the \(\varepsilon\)-model are included in the last column for comparison. Except for the last column, the numbers are based on MC simulation using an Euler scheme with one million paths each with time-step size 0.001. The model parameters are the same as in Table 1

| \(\lambda_{0}\) | \(\text{CE}^{(\varepsilon)}(\hat{\pi}^{(0)}) \) | LB | UB | \(\text{CE}(u^{(\varepsilon)})\) |
---|---|---|---|---|---|

−0.01 | 0.10 | [1.047,1.048] | [1.048,1.049] | [1.048,1.049] | 1.048 |

−0.05 | 0.10 | [1.052,1.053] | [1.083,1.084] | [1.083,1.085] | 1.084 |

−0.10 | 0.10 | [1.057,1.058] | [1.200,1.201] | [1.204,1.208] | 1.206 |

−0.01 | 0.50 | [1.644,1.649] | [1.647,1.653] | [1.646,1.657] | 1.649 |

−0.05 | 0.50 | [1.760,1.764] | [1.844,1.850] | [1.843,1.857] | 1.846 |

−0.10 | 0.50 | [1.868,1.871] | [2.248,2.256] | [2.266,2.286] | 2.272 |

In Table 2, we note the significant difference between the performance of the base-model optimizer \(\hat{\pi}^{(0)}\) and its second-order improvement \(\tilde{\pi}^{(\varepsilon)}\), especially for larger values of \(\varepsilon\). Furthermore, the lower and upper bounds appear to be quite tight.

### 5.3 Extended affine models

The two models considered in this section do not have closed-form expressions for the value functions \(u\) and \(v\). We consider models belonging to the class of extended affine specifications of the market price of risk process developed in [8]. The second example is based on a high-dimensional underlying Markov process which prevents PDE methods from being applicable. Appendix B contains all technical details. In particular, as we explain in Appendix B, in these affine models, the integrability conditions (3.7) and (3.8) hold provided that \(T>0\) is sufficiently small.

#### 5.3.1 One-dimensional extended affine model

^{1}However, for \(\varepsilon=0\), the resulting model is covered by the analysis in [25] (see Theorem B.1 in Appendix B which is from [25]). Therefore, we choose the constant market price of risk process

\(95\%\)-confidence intervals for certainty equivalents for the upper and lower bounds as well as the base model optimizer \(\hat{\pi}^{(0)}\) for the extended affine model. The parameter values are \(\kappa= 5\), \(\theta= 0.0169\), \(\beta= -0.1\), \(\gamma=0.1744\), \(p=-1\) and \(T=10\). The numbers are based on MC simulation using an Euler scheme with one million paths each with time-step size 0.001

| \(F_{0}\) | \(\text{CE}^{(\varepsilon)}(\hat{\pi }^{(0)})\) | LB | UB | \(\text{CE}^{(0)}(\hat{\pi}^{(0)})\) |
---|---|---|---|---|---|

0.10 | 0.01 | [1.724,1.726] | [10.159,10.399] | [10.226,10.481] | 1.043 |

0.05 | 0.01 | [1.342,1.343] | [2.141,2.151] | [2.131,2.149] | 1.043 |

0.01 | 0.01 | [1.097,1.098] | [1.118,1.119] | [1.117,1.120] | 1.043 |

0.10 | 0.05 | [1.728,1.729] | [9.660,9.877] | [9.766,10.000] | 1.045 |

0.05 | 0.05 | [1.344,1.345] | [2.105,2.115] | [2.102,2.120] | 1.045 |

0.01 | 0.05 | [1.099,1.100] | [1.119,1.121] | [1.117,1.121] | 1.045 |

Perhaps even more than in the Kim–Omberg model, the numbers in Table 3 above illustrate the superiority of the second-order approximations (columns 4 and 5) over its first-order version (column 3) as well as the zeroth order values (column 6). Again, the bounds in Table 3 appear quite tight when compared to the first-order approximations for moderate values of \(\varepsilon\).

#### 5.3.2 High-dimensional extended affine model

\(95\%\)-confidence intervals for certainty equivalents for the upper and lower bounds as well as the base model optimizer \(\hat{\pi}^{(0)}\) for the model (5.10), (5.11). The factor process parameter values are taken from Table 8 in Sect. 5 in [8], \(p=-1\) and \(T=10\). The numbers are based on MC simulation using an Euler scheme with one million paths each with time-step size 0.001

| \((F^{(i)}_{0})_{i=1}^{3}\) | \(\text{CE}^{(\varepsilon )}(\hat{\pi}^{(0)})\) | LB | UB | \(\text{CE}^{(0)}(\hat{\pi}^{(0)})\) |
---|---|---|---|---|---|

0.05 | 0.01 | [2.420,2.427] | [6.600,6.646] | [6,682,6.802] | 2.313 |

0.01 | 0.01 | [2.382,2.392] | [2.474,2.485] | [2.471,2.495] | 2.313 |

0.005 | 0.01 | [2.347,2.356] | [2.371,2.382] | [2.361,2.383] | 2.313 |

0.05 | 0.05 | [2.423,2.429] | [6.618,6.664] | [6.776,6.896] | 2.315 |

0.01 | 0.05 | [2.382,2.391] | [2.479,2.491] | [2.474,2.489] | 2.315 |

0.005 | 0.05 | [2.354,2.364] | [2.370,2.381] | [2.376,2.398] | 2.315 |

## Footnotes

## Notes

### Acknowledgements

The authors would like to thank the anonymous referees, the anonymous Associate Editor, the Editor Martin Schweizer, Milica Čudina, Claus Munk, Mihai Sîrbu, and Kim Weston for useful comments.

During the preparation of this work, the first author has been supported by the National Science Foundation under Grant No. DMS-1411809 (2014–2017). The second author has been supported by the National Science Foundation under grant No. DMS-1600307 (2015–2018). The third author has been supported by the National Science Foundation under Grant No. DMS-1107465 (2012–2017) and Grant No. DMS-1516165 (2015–2018). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

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