Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation

Abstract

We consider a singular version with state constraints of the stochastic target problems studied in Soner and Touzi (SIAM J. Control Optim. 41:404–424, 2002; J. Eur. Math. Soc. 4:201–236, 2002) and more recently Bouchard et al. (SIAM J. Control Optim. 48:3123–3150, 2009), among others. This provides a general framework for the pricing of contingent claims under risk constraints. Our extended version perfectly fits the market models with proportional transaction costs and the order book liquidation issues. Our main result is a direct PDE characterization of the associated pricing function. As an example application, we discuss the valuation of VWAP-guaranteed-type book liquidation contracts, for a general class of risk functions.

Appendix: Proof of Theorem 2.8

For the reader’s convenience, we provide here the main elements of the proof of Theorem 2.8. It follows line by line the arguments in [10], which in turn are based on [24]. One difference is that we formulate our result in terms of a family of stopping times $\{\theta^{\phi},\;\phi\in \mathcal {A}\} \subset\mathcal {T}_{[t,T]}$ and not in terms of a fixed one θ. This does not change anything, except at the level of notation. Another one is that the assumption A1 (pasting property) in [24] is not “formally” satisfied by elements of $\mathcal{L}$. This is however easily handled.

Let t∈[0,T] be fixed. We must show that $V(t)=\bar{V}(t)$, where

$$\bar{V}(t):= \Big\{ z \in\mathbb{R}^{d+1}: \exists\phi\in\mathcal {A} \mbox{ s.t. } \big( Z^\phi_{t,z}(\tau\wedge{\theta^{\phi}}) \in O \bigoplus^{\tau,{\theta ^{\phi}}} V\ \mbox{for\ all}\ \tau\in\mathcal{T}_{[t,T]}\big)\Big\}. $$

In the following, we work on the d-dimensional canonical space endowed with the Wiener measure ℙ. The filtration is given by the augmentation of the raw filtration generated by W. The canonical path is denoted by ω=(ω _t ) _t∈[0,T]. We use the notation ω ^s:=(ω _t∧s ) _t∈[0,T] and δ _s ω:=(ω _t∨s −ω _s ) _t∈[0,T], s∈[0,T].

We split the proof in several lemmas.

Lemma A.1

$V(t) \subset\bar{V}(t)$.

Proof

Fix z∈V(t) and $\phi\in \mathcal {A}$ such that $Z^{\phi}_{t,z}(s) \in O(s)$ for all s∈[t,T] ℙ-a.s. By the flow property, we have $Z^{\phi}_{{\theta^{\phi}},\xi}(s \vee {\theta^{\phi}}) \in O(s \vee{\theta^{\phi}})$ for all s∈[t,T] ℙ-a.s., where $\xi:=Z^{\phi}_{t,z}({\theta^{\phi}})$. Hence

It follows that for ℙ-a.e. ω∈Ω,

where $\tilde{\phi}_{\omega}$ is the map $\varOmega\ni\tilde{\omega}\mapsto\phi(\omega^{\theta(\omega)},\delta _{\theta(\omega)}\tilde{\omega})$, with ϕ viewed as a map on Ω. Using the right-continuity of $Z^{\tilde{\phi}_{\omega}}_{{\theta^{\phi }}(\omega),\xi(\omega)}$ and the Standing Assumption 3, we then deduce that for ℙ-a.e. ω∈Ω, $Z^{\tilde{\phi}_{\omega}}_{{\theta^{\phi}}(\omega),\xi(\omega)}(s \vee {\theta^{\phi}}(\omega) )(\tilde{\omega}) \in O(s\vee{\theta^{\phi }}(\omega) )$ for all s∈[θ ^ϕ(ω),T] for ℙ-a.e. $\tilde{\omega}\in\varOmega$. This shows that $\xi=Z^{\phi}_{t,z}({\theta ^{\phi}})\in V({\theta^{\phi}})$ ℙ-a.s. Since we already know that $Z^{\phi}_{t,z}(\tau) \in O(\tau)$ for all $\tau\in\mathcal{T}_{[t,T]}$, this shows that $z\in\bar{V}(t)$. □

It remains to prove the opposite inclusion. In the following we equip $\mathcal{A}=\mathcal{U}\times\mathcal{L}$ with the distance

$$d\big((\nu,L),(\nu',L')\big):=\mathbb {E}\bigg[{\int_{0}^{T}\big|\nu_{s}-\nu '_{s}\big|^{2} \,ds + \sup_{s\le T }\big|L_{s}-L'_{s}\big|^{2}}\bigg ]^{\frac{1}{2}} $$

and denote by $\mathcal{B}_{\mathcal{A}}$ the induced Borel σ-field.

Lemma A.2

$\bar{V}(t) \subset V(t)$.

Proof

We now fix $z \in\bar{V}(t)$ and $\phi=(\nu,L)\in\mathcal {A}$ such that for all $\tau\in\mathcal{T}_{[t,T]}$,

(A.1)

(1) We fix an arbitrary $\tau\in\mathcal{T}_{[t,T]}$ and first work on the event {θ ^ϕ<τ}. On this set, we have $Z^{\phi}_{t,z}({\theta^{\phi}}) \in V({\theta^{\phi}})$ and therefore

$$\big({\theta^{\phi}},Z^\phi_{t,z}({\theta^{\phi}})\big) \in\mathcal {D}:= \big\{ (t,z) \in[0,T]\times \mathbb {R}^{d+1}: z \in V(t)\big\}. $$

Let $\mathcal{B}_{\mathcal{D}}$ denote the collection of Borel subsets of $\mathcal{D}$. Applying Lemma A.3 below to the measure induced by $({\theta ^{\phi}},Z^{\phi}_{t,z}({\theta^{\phi}}))$ on [0,T]×ℝ^d+1, we can construct a measurable map $\bar{\phi}:(\mathcal{D},\mathcal{B}_{\mathcal{D}}) \to(\mathcal {A},\mathcal{B}_{\mathcal{A}})$ such that

$$Z^{\bar{\phi}({\theta^{\phi}},Z^\phi_{t,z}({\theta^{\phi}}))}_{{\theta ^{\phi}},Z^\phi_{t,z}({\theta^{\phi}})}(\vartheta) \in O(\vartheta) \quad \mbox{for\ all}\ \vartheta\in\mathcal{T}_{[{\theta^{\phi}},T]}. $$

Since $\mathcal{A}$ is a separable metric space, we can then find a progressively measurable process ϕ ₁=(ν ¹,L ¹) such that $\phi_{1}=\bar{\phi}({\theta^{\phi}},Z^{\phi}_{t,z}({\theta^{\phi}}))$ on 〚θ ^ϕ,T〛 dt×dℙ-a.e.; see Lemma 2.1 in [24]. We then define $\hat{\phi}=(\hat{\nu},\hat{L})$ by

Note that ν ¹, and therefore $\hat{\nu}$, need not be square-integrable, but that both are at least ℙ-a.s. square-integrable and take values in U. Similarly, $\hat{L}_{T}$ need not belong to L ²(Ω), but $\hat{L}$ is continuous and nondecreasing. Since the dynamics of $Z^{\phi_{1}}_{\theta^{\phi},Z^{\phi}_{t,z}({\theta^{\phi}})}$ only depends on the increments of L ¹ on [θ ^ϕ,T], the above defined control satisfies

$$Z^{\hat{\phi}}_{{\theta^{\phi}},Z^\phi_{t,z}({\theta^{\phi}})}(\tau) \in O(\tau) \quad\mbox{on}\ \{ {\theta^{\phi}} < \tau\}. $$

(2) Let $\hat{\phi}$ be defined as above and note that by (A.1), we also have

$$Z^{\hat{\phi}}_{t,z}(\tau)=Z^\phi_{t,z}(\tau) \in O \bigoplus^{\tau ,{\theta^{\phi}}} V=O(\tau) \quad \mathrm{on}\ \{ \tau\le{\theta ^{\phi}} \}. $$

(3) Combining the two above steps implies that

$$Z^{\hat{\phi}}_{t,z}(\tau) \in O (\tau). $$

Since $\tau\in\mathcal{T}_{[t,T]}$ is arbitrary, our Standing Assumption 2 implies that $Z^{\hat{\phi}}_{t,z}(s)\in O(s)$ for all s∈[t,T] ℙ-a.s. Although the control $\hat{\phi}$ might not satisfy the integrability condition imposed on elements of $\mathcal {A}$, we know from step (1) above that $\hat{\nu}$ is at least ℙ-a.s. square-integrable and that $\hat{L}$ is continuous and nondecreasing. Our Standing Assumption 4 thus allows one to conclude that $\mathcal {A}_{t,z}\ne\emptyset$, i.e., z∈V(t). □

It remains to prove the following result which was used in the previous proof.

Lemma A.3

For any probability measure μ on [0,T]×ℝ^d+1, there exists a Borel-measurable function $\bar{\phi}:(\mathcal{D},\mathcal{B}_{\mathcal{D}}) \to(\mathcal {A},\mathcal{B}_{\mathcal{A}})$ such that

$$\bar{\phi}(t,z) \in \mathcal {A}_{t,z} \quad\mbox{\textit{for}}\ \mu\mbox{-\textit{a.e.}}\ (t,z) \in\mathcal{D}. $$

Proof

Set $B:=\{ (t,z,\phi) \in[0,T]\times \mathbb {R}^{d +1} \times\mathcal {A} : \phi\in\mathcal{A}_{t,z} \}$. It follows from our Lipschitz-continuity assumptions that the map

$$(t,z,\phi) \in[0,T]\times \mathbb {R}^{d+1 } \times\mathcal{A} \to Z^\phi_{t,z}(r) \in L^{2}(\varOmega) $$

is continuous, and therefore Borel-measurable, for any r≤T. Then for any bounded continuous function f, the map $\psi^{r}_{f} : (t,z,\phi) \in[0,T]\times \mathbb {R}^{d+1 } \times\mathcal{A} \to \mathbb {E}[f(Z^{\phi}_{t,z}(r))]$ is Borel-measurable. Since O(r) is a Borel set, the map 1 _O(r) is the limit of a sequence of bounded continuous functions (f ⁿ) _n . Therefore $\psi^{r}_{\mathbf {1}_{O(r)}}=\lim_{n \to\infty}\psi^{r}_{f^{n}}$ is a Borel function. This implies that

is a Borel set. Appealing to the Standing Assumption 3 and the right-continuity of $Z^{\phi}_{t,z}$, we then deduce that B=⋂ _{r≤T, r∈ℚ} B ^r. This shows that B is a Borel set and therefore an analytic subset of $[0,T]\times \mathbb {R}^{d+1 } \times\mathcal{A}$; see [4].

Applying the Jankov–von Neumann theorem (see [4] Proposition 7.49), we then deduce that there exists an analytically measurable function $\bar{\phi}: \mathcal {D} \to\mathcal{A}$ such that

$$\bar{\phi}(t,z) \in\mathcal{A}_{t,z} \quad\mbox{for\ all}\ (t,z) \in \mathcal{D}. $$

Since an analytically measurable map is also universally measurable, the required result follows from Lemma 7.27 in [4]. □