Almost-sure hedging with permanent price impact


We consider a financial model with permanent price impact. Continuous-time trading dynamics are derived as the limit of discrete rebalancing policies. We then study the problem of superhedging a European option. Our main result is the derivation of a quasilinear pricing equation. It holds in the sense of viscosity solutions. When it admits a smooth solution, it provides a perfect hedging strategy.

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Corresponding author

Correspondence to Bruno Bouchard.

Additional information

Research supported by ANR Liquirisk and Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047).



We report here the measurability property that was used in the proof of Proposition 3.3.

In the following, \({\mathcal {A}}_{k}\) is viewed as a closed subset of the Polish space \({\mathbf {L}}_{\lambda}^{2}\) endowed with the usual (strong) norm topology \(\|\cdot\|_{{\mathbf {L}}_{\lambda}^{2}}\).

We consider an element \(\nu\in{\mathcal {U}}_{k}\) as a measurable map \(\varOmega\ni\omega\mapsto\nu(\omega) \in {\mathcal {M}} _{k}\), where \({\mathcal {M}}_{k}\) denotes the set of nonnegative Borel measures on \({\mathbb{R}}\times[0,T]\) with total mass at most \(k\), endowed with the topology of weak convergence. This topology is generated by the norm

$$ \|m\|_{{\mathcal {M}}}:=\sup\bigg\{ \int_{{\mathbb{R}}\times[0,T]} \ell(\delta,s) m(d\delta,ds): \ell\in\mathrm{Lip}_{1}\bigg\} , $$

in which \(\mathrm{Lip}_{1}\) denotes the class of 1-Lipschitz-continuous functions bounded by 1; see e.g. [4, Proposition 7.2.2 and Theorem 8.3.2]. Then \({\mathcal {U}}_{k}\) is a closed subset of the space \(\mathbf{M}_{k}^{2}\) of \({\mathcal {M}}_{k}\)-valued random variables. \(\mathbf{M}_{k}^{2}\) is made complete and separable by the norm

$$ \|\nu\|_{\mathbf{M}^{2}}:=\mathbb{E}\big[\|\nu\|_{{\mathcal {M}}}^{2}\big]^{\frac{1}{2}}; $$

see e.g. [9, Chap. 5]. We endow the set of controls \(\varGamma_{k}\) with the natural product topology

$$ \|\gamma\|_{{\mathbf {L}}_{\lambda}^{2}\times\mathbf{M}^{2}}:=\| \vartheta\|_{{\mathbf {L}} _{\lambda}^{2}}+\|\nu\|_{\mathbf{M}^{2}},\;\mbox{ for } \gamma =(\vartheta,\nu). $$

As a closed subset of the Polish space \({\mathbf {L}}_{\lambda }^{2}\times\mathbf{M} _{k}^{2}\), \(\varGamma_{k}\) is a Borel space, for each \(k\ge1\). See e.g. [3, Proposition 7.12].

The following stability result is proved by using standard estimates. In the following, we use the notation \(Z=(X,Y,V)\).

Proposition A.1

For each \(k\ge1\), there exists a real constant \(c_{k}>0\) such that

$$ \|Z^{t_{1},z_{1},\gamma_{1}}_{T}-Z^{t_{2},z_{2},\gamma_{2}}_{T}\| _{{\mathbf {L}}^{2}}\le c_{k}\Big(|t_{1}-t_{2}|^{\frac{1}{2}}+|z_{1}-z_{2}|+\| \gamma_{1}-\gamma_{2}\|_{{\mathbf {L}}_{\lambda}^{2}\times\mathbf {M}^{2}} \Big) $$

for all \((t_{i},z_{i},\gamma_{i})\in\mathrm{D}\times\varGamma_{k}\), \(i=1,2\).

A direct consequence is the continuity of \(\mathrm{D}\times\varGamma _{k}\ni (t,z,\gamma) \mapsto Z^{t,z,\gamma}_{T}\), which is therefore measurable.

Corollary A.2

For each \(k\ge1\), the map \(\mathrm{D}\times\varGamma_{k}\ni(t,z,\gamma)\mapsto Z^{t,z,\gamma}_{T} \in{\mathbf {L}}^{2}\) is Borel-measurable.

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Bouchard, B., Loeper, G. & Zou, Y. Almost-sure hedging with permanent price impact. Finance Stoch 20, 741–771 (2016).

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  • Hedging
  • Price impact

Mathematics Subject Classification

  • 91G20
  • 93E20
  • 49L20

JEL Classification

  • G13
  • G12