Sensitivity analysis for expected utility maximization in incomplete Brownian market models

Abstract

We examine the issue of sensitivity with respect to model parameters for the problem of utility maximization from final wealth in an incomplete Samuelson model and mainly, but not exclusively, for utility functions of positive-power type. The method consists in moving the parameters through change of measure, which we call a weak perturbation, decoupling the usual wealth equation from the varying parameters. By rewriting the maximization problem in terms of a convex-analytical support function of a weakly-compact set, crucially leveraging on the work (Backhoff and Fontbona in SIAM J Financ Math 7(1):70–103, 2016), the previous formulation let us prove the Hadamard directional differentiability of the value function with respect to the market price of risk, the drift and interest rate parameters, as well as for volatility matrices under a stability condition on their Kernel, and derive explicit expressions for the directional derivatives. We contrast our proposed weak perturbations against what we call strong perturbations, where the wealth equation is directly influenced by the changing parameters. Contrary to conventional wisdom, we find that both points of view generally yield different sensitivities unless e.g. if initial parameters and their perturbations are deterministic.

Appendix A

We provide the proof of a version of the envelope or Danskin’s theorem (see [8]), adapted to our purposes. First, we recall the notion of Hadamard differentiability. Given two Banach spaces $(\mathcal {X}, \Vert \cdot \Vert _{\mathcal {X}})$ and $(\mathcal {Z}, \Vert \cdot \Vert _{\mathcal {Z}})$ a map $f: \mathcal {X}\rightarrow \mathcal {Z}$ is directionally differentiable at x if for all $h\in \mathcal {X}$ the limit in $\mathcal {Z}$

$$\begin{aligned} Df(x,h):= \lim _{\tau \downarrow 0} \frac{f(x+\tau h)-f(x)}{\tau }, \end{aligned}$$

exists. If in addition, for all $h\in \mathcal {X}$ the following equality in $\mathcal {Z}$ holds

$$\begin{aligned} Df(x,h)= \lim _{\tau \downarrow 0, \; h'\rightarrow h} \frac{f(x+\tau h')-f(x)}{\tau } , \end{aligned}$$

then we say that f is directionally differentiable at x in the Hadamard sense. An important property of Hadamard differentiable functions is the chain rule. More precisely, if $(\mathcal {V}, \Vert \cdot \Vert _{\mathcal {V}})$ is another Banach space, $g: \mathcal {V}\rightarrow \mathcal {X}$ is directionally differentiable at v and f is directionally differentiable at g(v) in the Hadamard sense, then the composition $f\circ g$ is directionally differentiable at v (see e.g. [4, Proposition 2.47]) and $D(f\circ g)(v, v')= Df(g(v),Dg(v,v'))$ for all $v'\in \mathcal {V}$. If in addition, g is is also Hadamard directionally differentiable at v, then $f\circ g$ is directionally differentiable at v in the Hadamard sense.

Now, suppose that $K\subseteq \mathcal {X}$ is a weakly compact set. Let us consider the problem:

$$\begin{aligned} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \sup _{Z\in X} \langle d, Z \rangle \quad \text{ s.t. } \; Z\in K, \qquad \qquad \qquad \qquad \qquad \qquad \qquad {(AP_{d})} \end{aligned}$$

where $d\in \mathcal {X}^{*}$ and $\langle \cdot , \cdot \rangle$ denotes the bilinear pairing between $\mathcal {X}$ and $\mathcal {X}^{*}$. Let us define $v: \mathcal {X}^{*} \rightarrow \mathbb {R}$ as the optimal value of problem $(AP_{d})$ and $\mathcal {S}(d)$ the set of optimal solutions of $(AP_{d})$, i.e.

$$\begin{aligned} v(d):= \sup _{Z\in K} \langle d, Z \rangle , \quad \mathcal {S}(d):= \left\{ Z\in K \quad v(d)= \langle d, Z \rangle \right\} . \end{aligned}$$

Note that v is well defined, it is a Lipschitz function and $\mathcal {S}(d)\ne \emptyset$. In fact,

$$\begin{aligned} | v(d_1)- v(d_2) | \le \Vert d_1- d_2\Vert _{\mathcal {X}^{*}}\sup _{Z\in K} \Vert Z\Vert _{\mathcal {X}}. \end{aligned}$$

(A.1)

The proof of the following result is a simple modification of the proof in [4, Theorem 4.13].

Lemma A.1

For any $\bar{d}\in \mathcal {X}^{*}$, the following assertions hold true

(i) The set $\mathcal {S}(\bar{d})$ is weakly compact.

(ii) The function v is directionally differentiable in the Hadamard sense and its directional derivative is

$$\begin{aligned} Dv(\bar{d}, \Delta d)= \sup _{Z\in \mathcal {S}(\bar{d})} \langle \Delta d, Z \rangle \quad \hbox {for} \,\hbox {all} \Delta d\in \mathcal {X}^{*}. \end{aligned}$$

(A.2)

Proof

The first assertion follows directly from the weak-continuity of $\langle \bar{d}, \cdot \rangle$, which implies the weak closedness of $\mathcal {S}(\bar{d})$. Now, in view of [4, Proposition 2.49] and (A.1) it suffices to show that v is directionally differentiable. Let $\bar{Z}\in S(\bar{d})$ be such that $\langle \Delta d, \bar{Z} \rangle = \sup _{Z\in \mathcal {S}(\bar{d})} \langle \Delta d, Z \rangle$ and for $\tau >0$ set $d_{\tau }:= \bar{d}+ \tau \Delta d$. By definition

$$\begin{aligned} v(d_{\tau })- v(\bar{d}) \ge \langle d_{\tau }-\bar{d}, \bar{Z}\rangle = \tau \langle \Delta d, \bar{Z} \rangle , \end{aligned}$$

which implies that

$$\begin{aligned} \textstyle \liminf _{\tau \rightarrow 0} \frac{ v(d_{\tau })- v(\bar{d})}{\tau } \ge \langle \Delta d, \bar{Z} \rangle = \sup _{Z\in \mathcal {S}(\bar{d})} \langle \Delta d, Z \rangle . \end{aligned}$$

(A.3)

Analogously, let $Z_{\tau } \in S(d_{\tau })$. Then

$$\begin{aligned} \textstyle v(\bar{d})- v(d_\tau ) \ge - \langle d_{\tau }-\bar{d}, Z_{\tau }\rangle = - \tau \langle \Delta d, Z_\tau \rangle . \end{aligned}$$

(A.4)

On the other hand, using (A.1) we get that $v(d_{\tau })\rightarrow v(\bar{d})$ as $\tau \downarrow 0$, which implies, since $d_{\tau }\rightarrow \bar{d}$ strongly in $\mathcal {X}^*$, that any weak limit point of $Z_{\tau }$ belongs to $\mathcal {S}(\bar{d})$. Thus, (A.4) yields

$$\begin{aligned} \textstyle \limsup _{\tau \rightarrow 0} \frac{ v(d_{\tau })- v(\bar{d})}{\tau } \le \limsup _{\tau \rightarrow 0} \langle \Delta d, Z_{\tau } \rangle \le \sup _{Z\in \mathcal {S}(\bar{d})} \langle \Delta d, Z \rangle . \end{aligned}$$

(A.5)

Therefore, (A.2) is a consequence of (A.3) and (A.5).