Almost-sure hedging with permanent price impact

Abstract

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.

Appendix

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.