Optimal Portfolio Positioning on Multiple Assets Under Ambiguity

Abstract

This paper determines the optimal financial portfolio in the multidimensional setting when the investor exhibits ambiguity aversion. We consider the Maccheroni et al. (Econometrica 74(6):1447–1498, 2006) framework which includes both the Gilboa and Schmeidler’s (J Math Econ 18(2):141–153, 1989) multiple priors preferences and the (American Econ Rev 91:60–66, 2001) multiplier preferences. We determine the optimal portfolio profile under ambiguity when the investors can invest on various risky assets. We investigate in particular the CRRA case while introducing an ambiguity index based on the relative entropy criterion. Such result extends Ben Ameur and Prigent (Econ Model 34:89–97, 2013) when there is only one risky asset. Indeed, we show for example how the ambiguity on the correlations between the risky assets crucially modify the optimal payoff. Such results have important practical applications in structured portfolio management when investing on multiple financial indices and basket options.

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Notes

  1. 1.

    See Campbell and Viceira (2002) and Prigent (2007) for an overview of such results.

  2. 2.

    Recall that the relative entropy \(R(\mathbb {P},\mathbb {P}_{0})\) of the probability distribution \(\mathbb {P}\) with respect to the probability distribution \(\mathbb {P}_{0}\) is defined by:

    $$\begin{aligned} R(\mathbb {P},\mathbb {P}_{0})=E_{\mathbb {P}_{0}}\left[ \left( \frac{d\mathbb {P }}{d\mathbb {P}_{0}}\right) Log\left( \frac{d\mathbb {P}}{d\mathbb {P}_{0}} \right) \right] . \end{aligned}$$

    Note in particular that \(R(\mathbb {P}_{0},\mathbb {P}_{0})=0\). The relative entropy has been introduced in both the probability and information theories to measure the difference between the probability distributions. It corresponds to the Kullback-Leibler divergence. It quantifies the information gain when we revise beliefs from the prior probability distribution to the posterior probability distribution.

  3. 3.

    This corresponds for example to the distribution of a geometric Brownian motion \((S_{t})_{t}\) in a continuous-time framework, satisfying

    $$\begin{aligned} S_{t}=S_{0}exp\left[ (\mu -1/2\sigma ^{2})t+\sigma W_{t}\right] , \end{aligned}$$

    with \(d=(\mu -1/2\sigma ^{2})\) and where W denotes the standard Brownian motion.

  4. 4.

    Minimax Theorem (Sion 1958): (Saddle-point ) Let C and K be two closed convex sets in two topological vector spaces X and Y respectively. Let further F(xy) : \( C\times K\rightarrow R\) be a function which is quasiconcave in x and quasiconvex in y. If F is upper (or lower) semi continuous in x and lower semi continuous in y, while K is compact, then the function F(xy) possesses a saddle-value on \(C\times K\) and

    $$\begin{aligned} \sup _{x\in C}\inf _{y\in K}F(x,y)=\inf _{y\in D}\sup _{x\in K}F(x,y). \end{aligned}$$
  5. 5.

    Function g is the Radon-Nikodym density \(d\mathbb {Q}/d\mathbb {P}\).

  6. 6.

    We examine this case because it is the most common in practice. Other cases can also be considered if the Log return of the risky asset is no longer Gaussian.

  7. 7.

    See e.g. Prigent (2007).

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Appendix

Appendix

Proof of Proposition 13

In what follows, we use the following notations (see Jacod and Shiryaev 2002). For any two semimartingales X and Y, \(\left[ X,Y\right] \) denotes the quadratic variation of the processes X and Y. The process \( \left\langle X,Y\right\rangle \) denotes the predictable compensator of these processes. Recall that we have:

$$\begin{aligned} \left[ X,Y\right] _{t}=\left\langle X,Y\right\rangle _{t}+\sum _{s\le t}\Delta X_{s}\Delta Y_{s}, \end{aligned}$$

where \(\Delta X_{t}\) and \(\Delta Y_{t}\) denote the jumps of the processes and \(\left\langle X,Y\right\rangle _{t}\) is defined from the following condition: The respective martingales parts \(M^{X}\) and \(M^{Y}\) of X and Y are such that \(\left( M_{t}^{X}M_{t}^{Y}-\left\langle X,Y\right\rangle _{t}\right) _{t}\) is a (local)martingale. We have also: (integration by part formula)

$$\begin{aligned} d\left( XY\right) =XdY+YdX+d\left[ X,Y\right] . \end{aligned}$$

The process \(\mathcal {E}(X)\) denotes the Dade-Doléans stochastic exponential, defined from the stochastic differential equation (SDE):

$$\begin{aligned} d\mathcal {E}(X)=\mathcal {E}(X)dX. \end{aligned}$$

Note that, for continuous semimartingales X, we have \(\left[ X,Y\right] _{t}=\left\langle X,X\right\rangle _{t}\) thus we get:

$$\begin{aligned} \mathcal {E}(X_{t})=\mathcal {E}(X_{0})\exp \left[ X_{t}-\frac{1}{2}\left\langle X,X\right\rangle _{t}\right] . \end{aligned}$$

The result is established by using the Girsanov’s theorem. We parallel the approach of Buhlmann et al. (1996). Each of the d basic assets \(S_{i,t}\) must satisfy the following condition: when they are discounted by the nominal money market account C, they must be martingales with respect to the risk-neutral probability \(\mathbb {Q}\). This is equivalent to the fact that, when they are multiplied by the Radon-Nikodym density \(\eta \) and divided by C, they must be martingales with respect to the historical probability \(\mathbb {P}\). Note that we have:

$$\begin{aligned} S_{i,t}\eta _{t}/C_{t}=\mathcal {E}\left[ \left( \mu _{i}-r\right) t+\sigma _{i}W_{i,t}\right] \mathcal {E}\left[ -\sum _{j=1}^{d}\lambda _{j}W_{j,t} \right] , \end{aligned}$$

which is also equal to (Yor’s formula):

$$\begin{aligned} \mathcal {E}\left[ \left( \mu _{i}-r\right) t+\sigma _{i}W_{i,t}-\sum _{j=1}^{d}\lambda _{j}W_{j,t}-\sigma _{i}\left( \sum _{j=1}^{d}\lambda _{j}\rho _{i,j}\right) t\right] . \end{aligned}$$

Note that we have:

$$\begin{aligned} \left\langle \sigma _{i}W_{i,t},\sum _{j=1}^{d}\lambda _{j}W_{j,t}\right\rangle =\sigma _{i}\left( \sum _{j=1}^{d}\lambda _{j}\rho _{i,j}\right) t. \end{aligned}$$

The fact that the processes \(\left( S_{i,t}\eta _{t}/C_{t}\right) \) are martingales with respect to \(\mathbb {P}\) is equivalent to the following property: Their bounded variation components are equal to 0. This later condition implies the d following equalities: for all \(i=1,\ldots ,d\),

$$\begin{aligned} \left( \mu _{i}-r\right) -\sigma _{i}\left( \sum _{j=1}^{d}\lambda _{j}\rho _{i,j}\right) =0, \end{aligned}$$

which leads to Eq. (19).

Proof of Corollary 14

We have:

Proof of Proposition 16

In what follows, we prove that, for the fundamental example, the optimal portfolio value satisfies:

$$\begin{aligned} V_{t}^{*}= & {} P_{t}+C_{t}, \\&\text {with} \\ P_{t}= & {} e^{-r(T-t)}pV_{0}\text { (the floor),} \\ C_{t}= & {} \psi (t)\prod \limits _{i=1}^{d}\left( S_{i,t}/S_{i,0}\right) ^{m_{i}^{*}}\text { (the cushion),} \end{aligned}$$

- Indeed, as seen in (23), the portfolio value \(V_{T}^{*}\) at maturity T is given by:

$$\begin{aligned} V_{T}^{*}= & {} pV_{0}+\widetilde{\psi }(T)\prod \limits _{i=1}^{d}\left( S_{i,t}/S_{i,0}\right) ^{\left( \frac{1}{\phi }\frac{\lambda _{i}}{\sigma _{i}}\right) } \\&\text {with} \\ \widetilde{\psi }(t)= & {} c\exp \left[ 1/2\frac{1}{\phi }\left( \sum _{i=1,j=1}^{d}\lambda _{i}\lambda _{j}\rho _{i,j}\right) t\right] \prod \limits _{i=1}^{d}\exp \left( -\frac{1}{\phi }\frac{\lambda _{i}}{\sigma _{i}} (\mu _{i}-1/2\sigma _{i}^{2})t\right) \end{aligned}$$

- From the martingale property, the portfolio value \(V_{t}\) at any time t satisfies:

$$\begin{aligned} V_{t}^{*}=e^{-r(T-t)}\mathbb {E}\left[ V_{T}^{*}\left| \mathcal {F }_{t}\right. \right] . \end{aligned}$$

Thus, we have:

$$\begin{aligned} V_{t}^{*}=e^{-r(T-t)}pV_{0}+\widetilde{\psi }(T)\prod \limits _{i=1}^{d}\left( S_{i,t}/S_{i,0}\right) ^{\left( \frac{1}{\phi }\frac{ \lambda _{i}}{\sigma _{i}}\right) }\mathbb {E}\left[ \prod \limits _{i=1}^{d}\left( S_{i,T}/S_{i,t}\right) ^{\left( \frac{1}{\phi }\frac{ \lambda _{i}}{\sigma _{i}}\right) }\left| \mathcal {F}_{t}\right. \right] . \end{aligned}$$

It means that the portfolio value \(V_{t}^{*}\) is a function of the basic assets \(S_{i,t}\) at time t, given by:

$$\begin{aligned} V_{t}^{*}= & {} e^{-r(T-t)}pV_{0}+\psi (t)\prod \limits _{i=1}^{d}\left( S_{i,t}/S_{i,0}\right) ^{\left( \frac{1}{\phi }\frac{\lambda _{i}}{\sigma _{i}}\right) } \\&\text {with} \\ \psi (t)= & {} \widetilde{\psi }(t)\mathbb {E}\left[ \prod \limits _{i=1}^{d}\left( S_{i,T}/S_{i,t}\right) ^{\left( \frac{1}{\phi }\frac{ 1}{\gamma }\frac{\lambda _{i}}{\sigma _{i}}\right) }\left| \mathcal {F} _{t}\right. \right] . \end{aligned}$$

from which we deduce the result.

- We examine now the term \(\mathbb {E}\left[ \prod \limits _{i=1}^{d}\left( S_{i,T}/S_{i,t}\right) ^{\left( \frac{1}{\phi }\frac{\lambda _{i}}{\sigma _{i}}\right) }\left| \mathcal {F}_{t}\right. \right] \):

We have:

$$\begin{aligned} \left( S_{i,T}/S_{i,t}\right) ^{m_{i}^{*}}= & {} \exp \left[ m_{i}^{*}\left( (\mu _{i}-1/2\sigma _{i}^{2})(T-t)+\sigma _{i}\left( W_{i,T}-W_{i,t}\right) \right) \right] \\= & {} \exp \left[ m_{i}^{*}(\mu _{i}-1/2\sigma _{i}^{2})(T-t)+1/2\left( m_{i}^{*}\right) ^{2}\sigma _{i}^{2}(T-t)\right] \\&\times \exp \left[ -1/2\left( m_{i}^{*}\right) ^{2}\sigma _{i}^{2}(T-t)+m_{i}^{*}\sigma _{i}\left( W_{i,T}-W_{i,t}\right) \right] . \end{aligned}$$

Thus:

$$\begin{aligned}&\mathbb {E}\left[ \prod \limits _{i=1}^{d}\left( S_{i,T}/S_{i,t}\right) ^{m_{i}^{*}}\left| \mathcal {F}_{t}\right. \right] \\&=\exp \left[ \sum _{i=1}^{d}m_{i}^{*}(\mu _{i}-1/2\sigma _{i}^{2})(T-t)+1/2\left( m_{i}^{*}\right) ^{2}\sigma _{i}^{2}(T-t)\right] . \end{aligned}$$

Proof of Proposition 17

In what follows, we prove that, the Delta hedging strategies are given by:

$$\begin{aligned} \Delta _{i,t}^{*}=\frac{e_{i,t}^{*}}{S_{i,t}}=\frac{m_{i}^{*}C_{t}}{S_{i,t}}=\left( \frac{1}{\phi }\frac{\lambda _{i}}{\sigma _{i}} \right) \frac{\psi (t)}{S_{i,t}}\left( S_{i,t}/S_{i,0}\right) ^{\left( \frac{ 1}{\phi }\frac{\lambda _{i}}{\sigma _{i}}\right) }\prod \limits _{j=1,j\ne i}^{d}\left( S_{j,t}/S_{i,0}\right) ^{\left( \frac{1}{\phi }\frac{\lambda _{j}}{\sigma _{j}}\right) }. \end{aligned}$$

To determine the optimal portfolio shares, first we determine the SDE satisfied by the optimal portfolio value. We have:

$$\begin{aligned} dV_{t}^{*}=dP_{t}+\psi (t)d\left( \prod \limits _{i=1}^{d}\left( S_{i,t}/S_{i,0}\right) ^{m_{i}^{*}}\right) +\prod \limits _{i=1}^{d}\left( S_{i,t}/S_{i,0}\right) ^{m_{i}^{*}}\psi ^{\prime }(t)dt \end{aligned}$$

We have to identify the factors that multiply the terms \(dS_{i,t}/S_{i,t}\). For this purpose, we note that:

$$\begin{aligned} d\left( \prod \limits _{i=1}^{d}S_{i,t}^{m_{i}^{*}}\right) =\sum _{i=1}^{d}\left( m_{i}^{*}\left[ \prod \limits _{j=1,j\ne i}^{d}S_{j,t}^{m_{i}^{*}}\right] S_{i,t}^{m_{i}^{*}-1}dS_{i,t}\right) +\vartheta _{t}, \end{aligned}$$

where \(\vartheta \) is a bounded variation process. Therefore we have also:

$$\begin{aligned} d\left( \prod \limits _{i=1}^{d}S_{i,t}^{m_{i}^{*}}\right) =\sum _{i=1}^{d}\left( m_{i}^{*}\left[ \prod \limits _{i=1}^{d}S_{i,t}^{m_{i}^{*}}\right] \frac{dS_{i,t}}{S_{i,t}} \right) +\vartheta _{t}. \end{aligned}$$

Since the portfolio value satisfies:

$$\begin{aligned} V_{t}^{*}-P_{t}=\psi (t)\prod \limits _{i=1}^{d}\left( S_{i,t}/S_{i,0}\right) ^{m_{i}^{*}}, \end{aligned}$$

we deduce that:

$$\begin{aligned} dV_{t}^{*}=\sum _{i=1}^{d}m_{i}^{*}\left[ V_{t}^{*}-P_{t}\right] \frac{dS_{i,t}}{S_{i,t}}+\varkappa _{t}, \end{aligned}$$

where \(\varkappa \) is a bounded variation process. Finally, by identifying the factors of \(\frac{dS_{i,t}}{S_{i,t}}\), we get:

$$\begin{aligned} e_{i,t}^{*}=m_{i}^{*}\left[ V_{t}^{*}-P_{t}\right] . \end{aligned}$$

The Delta hedging strategies are immediately deduced from the equality \( \Delta _{i,t}^{*}=\frac{e_{i,t}^{*}}{S_{i,t}}\).

Proof of Lemma 26

The Radon-Nikodym density \(\frac{d\mathbb {P}}{d\mathbb {P}_{0}}\) is determined as follows:

We denote by \(f_{S_{T}}\) the probability distribution function (pdf) of asset \(S_{T}=\left( S_{T}^{(1)},S_{T}^{(2)}\right) \) with respect to \( \mathbb {P}\) and by g the ratio \(f_{S_{T}}/f_{0,S_{T}}\).

Denote:

$$\begin{aligned} \alpha _{T}^{(1)}= & {} \left( \mu ^{(1)}-0.5\sigma ^{(1)2}\right) T; \alpha _{T}^{(2)}=\left( \mu ^{(2)}-0.5\sigma ^{(2)2}\right) T, \\ \widetilde{\Sigma }= & {} \left[ \begin{array}{cc} \sigma ^{(1)2}T &{} \sigma ^{(1)}\sigma ^{(2)}\rho T \\ \sigma ^{(1)}\sigma ^{(2)}\rho T &{} \sigma ^{(2)2}T \end{array} \right] . \end{aligned}$$

We have:

$$\begin{aligned}&f_{S_{T}}(s^{(1)},s^{(2)})=\frac{1}{2\pi }\frac{1}{s^{(1)}s^{(2)}\sqrt{\det \left( \widetilde{\Sigma }\right) }}\\&\exp \left[ -\frac{1}{2}\left[ \ln \left[ s^{(1)}\right] -\alpha _{T}^{(1)},\ln \left[ s^{(2)}\right] -\alpha _{T}^{(2)}.\Sigma ^{-1}\left( \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{T}^{(1)} \\ \ln \left[ s^{(2)}\right] -\alpha _{T}^{(2)} \end{array} \right) \right] \right] \\&f_{0,S_{T}}(s^{(1)},s^{(2)})=\frac{1}{2\pi }\frac{1}{s^{(1)}s^{(2)}\sqrt{ \det \left( \widetilde{\Sigma }_{0}\right) }}\\&\exp \left[ -\frac{1}{2}\left[ \ln \left[ s^{(1)}\right] -\alpha _{0,T}^{(1)},\ln \left[ s^{(2)}\right] -\alpha _{0,T}^{(2)}.\Sigma _{0}^{-1}\left( \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{0,T}^{(1)} \\ \ln \left[ s^{(2)}\right] -\alpha _{0,T}^{(2)} \end{array} \right) \right] \right] \\&g(s^{(1)},s^{(2)})=\frac{\sqrt{\det \left( \widetilde{\Sigma }_{0}\right) }}{ \sqrt{\det \left( \widetilde{\Sigma }\right) }}\\&\qquad \times \exp \left[ -\frac{1}{2}\left[ \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{T}^{(1)},\ln \left[ s^{(2)}\right] -\alpha _{T}^{(2)}.\Sigma ^{-1}\left( \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{T}^{(1)} \\ \ln \left[ s^{(2)}\right] -\alpha _{T}^{(2)} \end{array} \right) - \\ \ln \left[ s^{(1)}\right] -\alpha _{0,T}^{(1)},\ln \left[ s^{(2)}\right] -\alpha _{0,T}^{(2)}.\Sigma _{0}^{-1}\left( \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{0,T}^{(1)} \\ \ln \left[ s^{(2)}\right] -\alpha _{0,T}^{(2)} \end{array} \right) \end{array} \right] \right] \end{aligned}$$

Thus, we get:

$$\begin{aligned}&\log \left[ g(s^{(1)},s^{(2)})\right] =\ln \left( \frac{\sqrt{\det \left( \widetilde{\Sigma }_{0}\right) }}{\sqrt{\det \left( \widetilde{\Sigma } \right) }}\right) \\&-\frac{1}{2}\left[ \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{T}^{(1)},\ln \left[ s^{(2)}\right] -\alpha _{T}^{(2)}.\Sigma ^{-1}\left( \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{T}^{(1)} \\ \ln \left[ s^{(2)}\right] -\alpha _{T}^{(2)} \end{array} \right) - \\ \ln \left[ s^{(1)}\right] -\alpha _{0,T}^{(1)},\ln \left[ s^{(2)}\right] -\alpha _{0,T}^{(2)}.\Sigma _{0}^{-1}\left( \begin{array}{c} \ln \left[ s^{(1)}\right] -\alpha _{0,T}^{(1)} \\ \ln \left[ s^{(2)}\right] -\alpha _{0,T}^{(2)} \end{array} \right) \end{array} \right] . \end{aligned}$$

Therefore, we deduce:

$$\begin{aligned} \delta _{T}= & {} \int _{0}^{+\infty }\int _{0}^{+\infty }g(s^{(1)},s^{(2)})\log \left[ g(s^{(1)},s^{(2)})\right] f_{0,S_{T}}(s^{(1)},s^{(2)})ds^{(1)}ds^{(2)}\\= & {} \ln \left( \frac{\sigma _{0}^{(1)}\sigma _{0}^{(2)}\sqrt{1-\rho _{0}^{2}}}{ \sigma ^{(1)}\sigma ^{(2)}\sqrt{1-\rho ^{2}}}\right) -1\\&+\frac{1}{2\left( 1-\rho _{0}^{2}\right) }\left( \begin{array}{c} \frac{\sigma ^{(1)2}}{\sigma _{0}^{(1)2}}+\frac{\sigma ^{(2)2}}{\sigma _{0}^{(2)2}}+\left( \frac{\alpha _{T}^{(1)}-\alpha _{0,T}^{(1)}}{\sigma _{0}^{(1)}\sqrt{T}}\right) ^{2}+\left( \frac{\alpha _{T}^{(2)}-\alpha _{0,T}^{(2)}}{\sigma _{0}^{(2)}\sqrt{T}}\right) ^{2} \\ -2\frac{\rho _{0}}{\sigma _{0}^{(1)}\sigma _{0}^{(2)}T}\left[ \sigma ^{(1)}\sigma ^{(2)}\rho T+\left( \alpha _{T}^{(1)}-\alpha _{0,T}^{(1)}\right) \left( \alpha _{T}^{(2)}-\alpha _{0,T}^{(2)}\right) \right] \end{array} \right) . \end{aligned}$$

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Ben Ameur, H., Boujelbène, M., Prigent, J.L. et al. Optimal Portfolio Positioning on Multiple Assets Under Ambiguity. Comput Econ 56, 21–57 (2020). https://doi.org/10.1007/s10614-019-09894-y

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Keywords

  • Portfolio optimization
  • Ambiguity
  • Multiple assets
  • Structured portfolio

JEL Classification

  • C61
  • G11
  • L10