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Perfect hedging under endogenous permanent market impacts

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Abstract

We model a nonlinear price curve quoted in a market as the utility indifference curve of a representative liquidity supplier. As the utility function, we adopt a \(g\)-expectation. In contrast to the standard framework of financial engineering, a trader is no longer a price taker as any trade has a permanent market impact via an effect on the supplier’s inventory. The P&L of a trading strategy is written as a nonlinear stochastic integral. Under this market impact model, we introduce a completeness condition under which any derivative can be perfectly replicated by a dynamic trading strategy. In the special case of a Markovian setting, the corresponding pricing and hedging can be done by solving a semilinear PDE.

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Notes

  1. \(Z\) gives rise to a BMO martingale if \(E [ \int _{0}^{T}|Z_{s}|^{2} ds | \mathcal{F}_{t}] \leq K\) for all \(t\in[0,T]\) and a constant \(K\); see Barrieu and El Karoui [10].

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Correspondence to Masaaki Fukasawa.

Additional information

This work was supported by (i) Institute of Economic Research, Kyoto University Joint Usage and Research Center (Fukasawa), (ii) Japan Society for the Promotion of Science, KAKENHI Grant Numbers 17K05297, 25245046 and 24684006 (Fukasawa) and (iii) NWO VENI 2012 (Stadje).

Appendix: Convergence of Esscher measures

Appendix: Convergence of Esscher measures

Lemma A.1

Let \(\mu\) be a measure onwith

$$ \int(1+|x|)e^{yx} \mu(\mathrm{d}x) < \infty $$

for all \(y \in\mathbb{R}\). Denote

$$ \ell= \inf\mathrm{supp}(\mu) , \ \ r= \sup\mathrm{supp}(\mu), \ \ -\infty\leq\ell< r \leq\infty. $$

Define the Esscher measure \(\mu^{y}\) by

$$ \mu^{y}(\mathrm{d}x) = \frac{e^{yx}}{m(y)} \mu(\mathrm{d}x), \qquad m(y)= \int e^{yx} \mu(\mathrm{d}x), $$

and let \(\mathcal{J}\) be the set of all nondecreasing Borel functions \(\varphi: [\ell,r] \to[-\infty, \infty]\) with

$$ \int(1+|x|) |\varphi(x)| \mu^{y}(\mathrm{d}x) < \infty $$

for all \(y \in\mathbb{R}\). Then:

  1. 1)

    For any \(\varphi\in\mathcal{J}\),

    $$ y \mapsto\int\varphi(x)\mu^{y}(\mathrm{d}x) $$

    is nondecreasing.

  2. 2)

    If \(\ell> -\infty\), then \(\mu^{y}\) converges weakly to \(\delta _{\ell}\) as \(y \to-\infty\).

  3. 3)

    For any \(\varphi\in\mathcal{J}\) with \(\lim_{x \to\ell} \varphi(x)= -\infty\),

    $$ \lim_{ y \to-\infty} \int \varphi(x)\mu^{y}(\mathrm{d}x) = -\infty. $$
  4. 4)

    If \(r <\infty\), then \(\mu^{y}\) converges weakly to \(\delta_{r}\) as \(y \to\infty\).

  5. 5)

    For any \(\varphi\in\mathcal{J}\) with \(\lim_{x \to r} \varphi(x)= \infty\),

    $$ \lim_{ y \to\infty} \int \varphi(x)\mu^{y}(\mathrm{d}x) = \infty. $$

Here \(\delta_{\ell}\) and \(\delta_{r}\) are the delta measures in the points \(\ell\) and \(r\), respectively.

Proof

1) Note that

$$ \begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}y} \int\varphi(x) \mu^{y}(\mathrm{d}y) \\& = \frac{1}{m(y)}\int\varphi(x)xe^{yx}\mu(\mathrm{d}x) - \frac{1}{m(y)^{2}} \int\varphi(x) e^{yx}\mu(\mathrm{d}y) \int x e^{yx}\mu(\mathrm{d}x) \\ &= \int \varphi(x)x \mu^{y}(\mathrm{d}x) - \int\varphi(x)\mu^{y}(\mathrm {d}x) \int x \mu^{y}(\mathrm{d}x). \end{aligned} $$

The right-hand side sequence is nonnegative by the FKG inequality, or just because this is the covariance of comonotone random variables under the probability measure \(\mu^{y}\).

2) Denote

$$ a(y,u) = \int_{(-\infty,u]} e^{yx}\mu(\mathrm{d}x), \qquad b(y,u) = \int_{(u,\infty)} e^{yx}\mu(\mathrm{d}x). $$

Then for any \(y < 0\), \(u \in(\ell,r)\) and \(\epsilon\in(0,u-\ell)\),

$$ \frac{a(y,u)}{b(y,u)} \geq \frac{a(y,u-\epsilon)}{b(y,u)} \geq \frac{\int_{(-\infty,u-\epsilon]} e^{y(u-\epsilon)} \mu(\mathrm {d}x)}{\int_{(u,\infty)} e^{yu} \mu(\mathrm{d}x)} = e^{-y\epsilon} \frac {\mu((-\infty,u-\epsilon])}{\mu((u,\infty))}. $$

It follows then that \(a(y,u)/b(y,u) \to\infty\) as \(y \to- \infty\) for each \(u \in(\ell,r)\). This implies the convergence of the distribution function

$$ \mu^{y}\bigl((-\infty,u]\bigr) = \frac{a(y,u)}{a(y,u) + b(y,u)} \longrightarrow1 $$
(A.1)

as \(y\to-\infty\) for each \(u \in(\ell,r)\). Now assume \(\ell> -\infty\). Then \(\mu^{y}((-\infty,u]) = 0\) for all \(u < \ell\), and so \(\mu^{y} \to\delta_{\ell}\) weakly.

3) Let \(\varphi\in\mathcal{J}\) with \(\lim_{x\to\ell} \varphi (x) = -\infty\). Then for any \(n \in\mathbb{N}\), there exists \(\delta> 0\) such that for all \(x < \ell+\delta\), \(\varphi(x) < -n\). Therefore,

$$ \int\varphi(x) \mu^{y}(\mathrm{d}y) \leq -n \mu^{y}\bigl((-\infty,\ell+\delta]\bigr) + \int\varphi^{+}(x)\mu ^{y}(\mathrm{d}y), $$

where \(\varphi^{+}\) is the positive part of \(\varphi\). Since \(\varphi^{+} \in\mathcal{J}\), the second term is nondecreasing in \(y\) as we have already seen. Together with (A.1), this implies

$$ \limsup_{y\to-\infty} \int\varphi(x) \mu^{y}(\mathrm{d}y) \leq-n + \int\varphi^{+}(x) \mu^{0}(\mathrm{d}y). $$

Since \(n\) can be arbitrary, we conclude.

The proofs for 4) and 5) are similar to those for 2) and 3), respectively. □

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Fukasawa, M., Stadje, M. Perfect hedging under endogenous permanent market impacts. Finance Stoch 22, 417–442 (2018). https://doi.org/10.1007/s00780-017-0352-4

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