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Optimal investment and price dependence in a semi-static market

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Abstract

This paper studies the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be traded continuously in time and are modelled as locally bounded semimartingales. Using a general utility function defined on the positive half-line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.

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Notes

  1. In fact, one could take T to be a finite stopping time, as in Hugonnier and Kramkov [17], on which we rely.

  2. If X=X′−X″, take N=1+X″ to get X/N≥−1; vice versa, if X/N≥−1, choose X″=N to get X=X′−X″.

  3. We use the convention that the sup (inf) over an empty set takes the value −∞ (+∞).

  4. The duality in (3.2) is with respect to the variable y only, with p playing the role of a parameter.

  5. The interested reader can easily write down the alternative statement after comparing [26, Theorem 3] with [26, Theorem 4].

  6. As we show in Theorem 3.2, a sufficient condition for the solution of problem (2.7) to lie in \(\mathcal{K}\) is that the final value of the optimal portfolio is bounded below by a strictly positive constant.

  7. This follows from Theorem 3.3 and part 4 of Theorem 3.2.

  8. Actually, [19, Theorem 3.1] states that \(\nabla_{p} \tilde{u}=\tilde{q}\); the missing minus sign in front of \(\tilde{q}\) (which, in [19], is called λ ⋆) is a typo, whereas the term \(\partial_{x} \tilde{u}\) is missing because in this case \(\partial _{x} \tilde{u}=1\), as it follows from [19, Theorem 4.1].

  9. Although this is stated only for \((x,q)\in\mathcal{K}\), it is easy to see that this automatically implies that it holds for any \((x,q)\in \bar{\mathcal{K}}\) (see the proof of [36, Lemma 6.1]).

  10. Although stated only for finite convex functions, the theorem holds (with the same proof) for proper convex functions.

  11. Since 4(b) implies that \(-\nabla v(\tilde{y},\tilde{y} p)\) exists at all \(\tilde{y}>0\), but only for the one fixed p which we used in problem (2.7), and not for all p such that \((1,p)\in\mathcal{L}\).

  12. We can also give a more elegant proof that \(\tilde{u}(x,\cdot)\) is continuous, relying on a hard-to-prove theorem: since \(\tilde{v}\) is continuous and \(\tilde{u}(x)=\tilde{v}(\tilde{y})+x\tilde{y}\), where \(\tilde{y}= \partial_{x} \tilde{u}\), the continuity of \(\tilde{u}\) follows from the one of \(\tilde{y}\). To prove the latter, observe that the continuous bijection g of \((0,\infty)\times\mathcal{P}\) to itself given by \((y,p)\mapsto (-\partial_{y} \tilde{v}(y,p),p)\) has inverse \(g^{-1}(x,p)=(\tilde{y},p)\), and the map g is open, by Brouwer’s invariance domain theorem, so g −1 is continuous.

  13. It is enough to show this in dimension one, where a function is convex iff it is the integral of an increasing function; since a locally increasing functions is increasing, the thesis follows.

  14. Just remember to use the identities w 0=w λ −λ(w 1−w 0) and v λ w λ =1.

  15. Because the perspective function is continuous, and sends convex sets to convex sets (as proved in [2, Sect. 2.3.3]). Alternatively, convexity can also easily be proved directly from the definition of \(\mathcal{P}(x,q)\).

  16. This follows from ±L(x)=L(±εx)/ε≥o(±εx)/ε, taking limits for ε→0+.

  17. Indeed, consider the convex function given by \(g(x,y):=\max(1-\sqrt{x},|y|)\) for x≥0 and g(x,y)=∞ if x<0; then ∂g(1,1)=(−∞,0]×{1}, yet the function h(y):=g(0,y)=max(1,|y|) is not differentiable at y=1 even if (a,b)(0,1) is constant over (a,b)∈∂g(1,1).

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Acknowledgements

We thank Dmitry Kramkov for his valuable comments on a previous version of this paper, and Walter Schachermayer for the intuition behind the counterexample in Sect. 10.

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  1. Fakultät für Mathematik, Universitat Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Pietro Siorpaes

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Siorpaes, P. Optimal investment and price dependence in a semi-static market. Finance Stoch 19, 161–187 (2015). https://doi.org/10.1007/s00780-014-0245-8

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