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Continuous-time duality for superreplication with transient price impact. (English) Zbl 1444.91194
The authors consider a financial model with large investor who can invest in a riskless savings account bearing zero interest and whose trades into and out of a risky asset move bid- and ask prices that, in addition, are also driven by some exogenous noise, which is specified by a continuous, adapted process on a filtered probability space $$(\Omega,(\mathcal{F}_{t})_{t\ge0},\mathrm{P})$$. It is assumed that all $$(\mathcal{F}_{t})_{t\ge0}$$-martingales have a continuous version. The large investor’s trading strategy is described by his given initial holdings $$x_0\in\mathrm{R}$$ and a right-continuous, predictable process $$X=({X}_{t})_{t\ge0}$$ of bounded variation specifying the number of risky assets held at any time. Let $$X^{\uparrow}$$ and $$X^{\downarrow}$$ be the right-continuous predictable increasing and decreasing part resulting from the Hahn-decomposition of $$X_{t}=x_0+X^{\uparrow}_{t}-X^{\downarrow}_{t}$$, $$t\geq 0$$, $$X_{0-}=x_0$$, $$X^{\uparrow}_{0-}=X^{\downarrow}_{0-}=0$$. It is assumed that the half-spread follows the dynamics $$d\zeta_{t}^{X}=(dX^{\uparrow}_{t}+dX^{\downarrow}_{t})/\delta_{t}-r_{t}\zeta_{t}^{X}dt$$, $$\zeta_{0-}^{X}=\zeta_0\geq 0$$ for a given market depth process $$\delta$$ and resilience rate $$r$$. The authors assume that the market depth is continuous and adapted, the resilience rate is predictable and such that $$\delta$$ and $$\rho$$ are bounded away from zero and infinity, where $$\rho_{t}=\exp\left(\int_{0}^{t}r_{s}ds\right)$$, $$t\ge0$$, and $$\kappa_{t}=\delta_{t}/\rho^2_{t}$$ is strictly decreasing in $$t\geq 0$$. By time $$T\in (0,\infty)$$, the induced investor’s cash position will have evolved from its given initial value $$\xi_0\in\mathrm{R}$$ to the terminal cash position $$\xi_{T}^{X}=\xi_0-\int_{[0,T]}P_{t}^{X}\circ dX_{t}-\int_{[0,T]}\zeta_{t}^{X}\circ d(X_{t}^{\uparrow}+X_{t}^{\downarrow})$$, where $$P^{X}$$ is the mid-price. It is considered the classical superreplication problem for a cash-settled European contingent claim with $$\mathcal{F}_{T}$$-measurable payoff $$H\geq 0$$ at time $$T\geq 0$$. An exogenous payoff’s superreplication cost is defined by $$\pi(H)=\inf\{\xi_0\in\mathrm{R}:\ \xi_{T}^{X}\geq H$$ for some $$X\in\mathcal{X}$$ with $$X_{T}=0\}$$. The main result of this paper is following. Under previous assumptions the superreplication costs of contingent claim $$H\geq 0$$ have the dual description $$\pi(H)=\sup_{(Q,M,\alpha)}\left\{\mathrm{E}_{Q}[H]-\frac{1}{2}\Vert\alpha-\zeta_0\Vert_{L^2(Q\otimes\mu)}-M_0 x_0-\frac{1}{2}\iota x^2_0\right\}>-\infty$$, where the supremum is taken over all triples $$(Q,M,\alpha)$$ of probability measures $$Q<<\mathrm{P}$$ on $$\mathcal{F}_{T}$$, martingales $$M\in\mathcal{M}^2(Q)$$ and all optimal $$\alpha\in L^2(Q\otimes\mu)$$ which control the fluctuations of $$P$$ in the sense that $$\vert P_{t}-M_{t}\vert\leq\frac{\rho_{t}}{\delta_{t}}\mathrm{E}_{Q}\left[\int_{[0,T]}\alpha_{u}\mu(du)\vert\mathcal{F}_{t}\right]$$, $$0\leq t\leq T$$. Here $$\mu(dt)=\mathrm{1}_{(0,T)}(t)\vert d\kappa_{t}\vert+\kappa_{T}\mathrm{Dirac}_{T}(dt)$$, $$\iota$$ is some impact parameter. As an application, the authors show that in proposed transient price impact model the best way to superreplicate a call option is, under natural conditions, to buy and hold the asset until maturity, and it is proved a verification theorem for utility maximizing investment strategies.

##### MSC:
 91G10 Portfolio theory 91G20 Derivative securities (option pricing, hedging, etc.)
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