Optimal Investment

Ernst Eberlein; Jan Kallsen

doi:10.1007/978-3-030-26106-1_10

Mathematical Finance pp 461-535 | Cite as

Optimal Investment

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Ernst Eberlein
Jan Kallsen

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First Online: 04 December 2019

Part of the Springer Finance book series (FINANCE)

Abstract

As an investor in a securities market you have to decide how to arrange your portfolio. In this chapter we consider the natural situation that you want to maximise your profits. It is not entirely obvious how to formalise this goal because the payoff of investments is typically random.

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Appendix 1: Problems

The exercises correspond to the section with the same number.

10.1

Consider an Itō process model as in Example 9.3 where μ, σ are adapted to the filtration of W₂, …, W_n and hence independent of W₁. Determine the optimal portfolio for the expected utility of terminal wealth relative to power utility u(x) = x^1−p∕(1 − p).

Hint: Show that η₁ = 0 in Example 10.14.

10.2

Generalise Example 10.30 to the case when the asset price process S = (S₁, …, S_d) is a multivariate geometric Lévy process in the sense that

$$\displaystyle \begin{aligned}S_i(t)=S_i(0)\exp(X_i(t))=S_i(0)\mathscr{E}\big(\widetilde X_i\big)(t),\quad i=1,\dots,d\end{aligned}$$

for multivariate Lévy processes $X,\widetilde X$ with triplets (a^X, c^X, K^X) resp. $(a^{\widetilde X},c^{\widetilde X}, K^{\widetilde X})$.

10.3

Suppose that the discounted price process is a geometric Lévy process as in Example 9.2. Show that the martingale measure in Example 10.38 coincides with the minimal entropy martingale measure (MEMM).

10.4 (Time Consistency of Efficient Portfolios)

Let φ^⋆ be mean-variance efficient as in Sect. 10.4 and suppose that $\varphi ^\star \mathop {\bullet } S\leq m$ for m in Rule 10.43. This holds, for example, if S is continuous. Show that φ^⋆ is also conditionally mean-variance efficient in the sense that the following statements hold for any t ∈ [0, T].

1.
φ^⋆ maximises $E(\varphi \mathop {\bullet } S(T)|\mathscr {F}_t)$ among all self-financing strategies φ satisfying
$$\displaystyle \begin{aligned}\mathrm{Var}(\varphi\mathop{\bullet} S(T)|\mathscr{F}_t)\leq\mathrm{Var}(\varphi^\star\mathop{\bullet} S(T)|\mathscr{F}_t).\end{aligned}$$
2.
φ^⋆ minimises $\mathrm {Var}(\varphi \mathop {\bullet } S(T)|\mathscr {F}_t)$ among all self-financing strategies φ satisfying
$$\displaystyle \begin{aligned}E(\varphi\mathop{\bullet} S(T)|\mathscr{F}_t)\geq E(\varphi^\star\mathop{\bullet} S(T)|\mathscr{F}_t).\end{aligned}$$

10.5

Consider the market in Example 9.1 with proportional transaction costs of size εS(t), i.e. $ \underline S(t)=(1-\varepsilon )S(t)$ and $\overline S(t)=(1+\varepsilon )S(t)$ with constant ε > 0. The goal is to determine the optimal strategy/consumption pair for the utility function $u(t,x)=\log (x)e^{-\delta t}$ and infinite time horizon T. To this end, let the function $f:[\log (1-\varepsilon ),\log (1+\varepsilon )]\to \mathbb R$ denote the solution to the ordinary differential equation

$$\displaystyle \begin{aligned} f''(x)={}& \textstyle {2\delta\over\sigma^2}\big(1+e^{f(x)}\big) +\left({2\mu\over\sigma^2}-{4\delta\over\sigma^2}\big(1+e^{f(x)}\big)\right)f'(x) \\ &\textstyle{}+\left(-{4\mu\over\sigma^2}+{2\delta\over\sigma^2}\big(1+e^{f(x)}\big) +{1-e^{-f(x)}\over1+e^{-f(x)}}\right)(f'(x) )^2\\ &\textstyle{}+\left({2\mu\over\sigma^2}-{1-e^{-f(x)}\over1+e^{-f(x)}}\right)(f'(x) )^3 \end{aligned} $$

with $f'(\log (1-\varepsilon ))=-\infty =f'(\log (1+\varepsilon ))$, define

$$\displaystyle \begin{aligned} \widetilde\mu&:=-\mu+{\sigma^2\over2}\left({f'\over f'-1}\right)^2{1-e^{-f}\over1+e^{-f}},\\ \widetilde\sigma&:={\sigma\over f'-1}, \end{aligned} $$

C as the solution to the SDE

$$\displaystyle \begin{aligned}dC(t)=\widetilde\mu(C(t))dt+\widetilde\sigma(C(t))dW(t)\end{aligned}$$

for appropriate some C(0),

$$\displaystyle \begin{aligned} \widetilde S(t)&:=S(t)\exp(C(t)),\\ \widetilde\pi(t)&:={\mu+\widetilde\mu(C(t))\over(\sigma+\widetilde\sigma(C(t))^2}+{1\over2},\\ \widetilde V(t)&:=\widetilde V(0)\mathscr{E}\big(\widetilde\pi\mathop{\bullet}\mathscr{L}(\widetilde S)-\delta I\big)(t) \end{aligned} $$

for some appropriate $\widetilde V(0)$, and

$$\displaystyle \begin{aligned}\varphi(t):={\widetilde V(t)\widetilde\pi(t)\over\widetilde S(t)},\quad c(t):=\delta\widetilde V(t).\end{aligned}$$

Show that (φ, c) is optimal for the expected utility of consumption and $\widetilde S$ is a corresponding shadow price as in Rule 10.52.

Hints:

1.
Apply Rule 10.28 in order to show that (φ, c) is optimal for the expected utility of consumption in the frictionless market with price process $\widetilde S$.
2.
Use Itō’s formula in order to obtain that
$$\displaystyle \begin{aligned}1_{\{\widetilde S(t)\neq S(t)\}}d\varphi(t)=0dt+0dW(t).\end{aligned}$$
Put differently, the otherwise constant stock holdings φ are adjusted only when $\widetilde S$ or, equivalently, $\widetilde \pi $ or φ reaches some no-trade bounds. What are these bounds?

Appendix 2: Notes and Comments

Monographs and textbooks on or covering optimal investment include [29, 187, 226, 227, 234, 236, 250, 287].

Classical references to expected utility maximisation in a dynamic context include [217, 218, 222, 258]. The dual approach goes back to [27, 136, 137, 189]. We refer to [24, 197] for versions of Rule 10.3 in a general semimartingale setup and to [188] in the consumption case. Rules 10.4 and 10.26 are literally true only in simple finite models, cf. [167]. For the connection to distances or divergences between martingale measures in Rule 10.5, we refer to [21, 127]. Rules 10.6, 10.28 are taken from [125, 126]. Rule 10.7 can be found in [187, Theorem 3.10.1] and [126, Lemma 5.3]. For mutual fund theorems we refer as in Chap. 7 to [218, 260]. Rule 10.7 is taken from [187, Theorem 3.10.1] and [126, Lemma 5.3]. The numeraire portfolio plays a key role in the benchmark approach to Mathematical Finance of [236]. Strictly speaking, S∕V_φ is only a local martingale (for locally bounded processes) or even a supermartingale in the general case, cf. [197] for details.

For disillusions along the lines of Example 10.10 see [248]. The concept of an opportunity process is introduced in [55] for quadratic utility and extended to power utility in [225]. Its characterisation in Rules 10.11, 10.17, 10.29 is inspired by the parallel statement in Rule 10.45. Examples 10.13, 10.19, 10.30 are based on [165], whereas [243] considers the case with claim in Example 10.22. For BSDE characterisations similar to Example 10.14, 10.20, 10.31 we refer to [146, 253]. The PDEs in Examples 10.15, 10.21, 10.32 are related but not identical to the classical Hamilton–Jacobi–Bellman PDEs for the value function in a Markovian setup, cf. [218]. The measure P^H of (10.44) is considered, for example, in [180].

Utility maximisation of P&L has been introduced in [163, 164]. As an optimisation concept over in some sense infinitesimal time horizons it is related to local risk minimisation in the sense of [266, 267].

Portfolio choice based on mean and variance goes back to [213]. It is mostly studied in a one-period context. Its dynamic counterpart starts to be considered in [6, 244, 268]. We follow here the reasoning of [55], where the notion of an opportunity process as in (10.83) and the subsequent rules can be found. The terms adjustment process and variance-optimal martingale measure are taken from [269]. The adjustment process of Example 10.48 is determined in [148]. PDE characterisations of the opportunity process similar to Example 10.50 can be found in [205, 235].

References showing the duality of Rules 10.51, 10.52 include [22, 65, 67, 68, 161, 175, 208]. Example 10.53 goes back to [176, 177].

The discussion of book values for P&L in the sense of Definition 10.54 is new. A rigorous uniqueness proof remains to be given. Moreover, it is theoretically conceivable that the maximal utility of P&L in a model is increased by replacing the “true” price process with a positive bid-ask spread, i.e. by artificially considering a liquid market as illiquid. It is an open question to what extent this undesirable phenomenon really occurs in reasonable market models such as those in Chap. 8.

Utility maximisation for P&L is considered in [214, 215] in the presence of small proportional transaction costs. However, they consider profits and losses relative to the mid-price process instead of some book value. Transaction costs are deducted separately, essentially by adjusting the drift part of the wealth process. In spite of this slightly different setup, they obtain the same no-trade region as in Example 10.56.

Problem 10.1 is based on [174]. Problems 10.2 and 10.3 are solved in [165].

It is often claimed that dynamic mean-variance portfolio selection is time inconsistent in the sense that one later regrets the decision made at earlier times, cf. e.g. [13, 35, 66]. Problem 10.4 indicates that this may be primarily due to how the problem is phrased.

Problem 10.5 is treated in [173].

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Ernst Eberlein
- 1
Jan Kallsen
- 2

1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
2.Department of MathematicsKiel UniversityKielGermany

Optimal Investment

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