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Hedging with temporary price impact. (English) Zbl 1409.91226

This paper deals with the problem of hedging a European contingent claim in a Bachelier model with temporary price impact. It is considered an agent who is trading in a financial market consisting of a risky asset. The number of shares the agent holds at time \(t\in [0,T]\) of the risky stock is defined as \(X_{t}^{u}=x+\int_0^{t}u_{s}ds\), where \(x\in R\) denotes her given initial holding. The real-valued stochastic process \((u_{t})_{0\leq t\leq T}\) represents the speed at which the agent trades in the risky asset. It is assumed to be chosen in the set \(U=\left\{u:\;u\;\text{progressively\;measurable\;s.t.}\;E\int_{0}^{T}u_{t}^2 dt<\infty \right\}\). Given a real-valued predictable process \((\xi_{t})_{0\leq t\leq T}\) in \(L^1(P\otimes dt)\) and a fixed constant \(\kappa>0\), the agent’s objective is to minimize the performance functional \(J(u)=E\left[(1/2)\int_0^{T}(X_{t}^{u}-\xi_{t})^2 dt+(1/2)\kappa\int_0^{T}u_{t}^2 dt\right]\). This leads to the optimal stochastic control problem \(J(u)\to\min_{u\in U}\). Let us denote \(\tau^{\kappa}(t)=(T-t)/\sqrt{\kappa}\), \(0\leq t\leq T\). One of the main results of paper is following. The optimal stock holding \(\hat X\) of considered stochastic optimal control problem with unconstrained terminal position satisfy the linear ordinary differential equation \(d\hat X_{t}=\frac{\tanh(\tau^{\kappa}(t))}{\sqrt{\kappa}}(\hat\xi_{t}-\hat X_{t})dt\), \(\hat x_0=x\), where, for \(0\leq t<T\), we let \(\hat\xi_{t}=E\left[\int_{t}^{T}\xi_{u}K(t,u)du|{\mathcal F}_{t}\right]\) with kernel \(K(t,u)=\cosh(\tau^{\kappa}(u))/(\sqrt{\kappa}\sinh(\tau^{\kappa}(t)))\), \(0\leq t\leq u<T\). The minimal costs are given by \(\inf_{u\in U}J(u)=\frac{1}{2}\sqrt{\kappa}\tanh(\tau^{\kappa}(0))(x-\hat\xi_0)^2+\frac{1}{2}E\left[ \int_0^{T}(\xi_{t}-\hat\xi_{t})^2dt\right]+ \frac{1}{2}E\left[\int_0^{T}\sqrt{\kappa}\tanh(\tau^{\kappa}(t))d\langle\hat\xi\rangle_{t}\right]<\infty\). The similar result is obtained for the constrained problem with given agent’s terminal position \(X_{T}^{u}=\Theta_{T}\).

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91G10 Portfolio theory
91G80 Financial applications of other theories
93E20 Optimal stochastic control
60H30 Applications of stochastic analysis (to PDEs, etc.)
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