Mean variance hedging in a general jump market (Q2786037)

The authors consider a financial market in which the discounted price process \(S\) is an \(R^{d}\)-valued semimartingale with bounded jumps, and the variance-optimal measure \(Q^{opt}\) is only known to be a signed measure. It is presented a backward semimartingale equation and it is shown that the density process \(Z^{opt}\) of \(Q^{opt}\) with respect to \(P\) is a possibly non-positive stochastic exponential if and only if this backward semimartingale equation has a solution. For a general contingent claim \(H\) the authors consider the market, where there is an agent who is given a family of admissible strategies \(Adm\). With initial wealth \(w\) and the admissible strategy \(\pi\in Adm\), \(X^{w,\pi}:=w+\pi\cdot S\) is the corresponding wealth process. The following generalized version of the classical mean-variance hedging problem \(\min_{\pi\in Adm}E[(X_{\tau}^{w,\pi})^2I_{\{\tau\leq T\}}+| H-X_{T}^{w,\pi}|^2I_{\{\tau> T\}}]\), where \(\tau=\inf\{t>0|\;Z^{opt}_{t}=0\}\) is considered. The authors obtain an explicit form of the optimal strategy and the optimal cost by the solution of the corresponding backward semimartingale equation.