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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.

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