Weighted norm inequalities and hedging in incomplete markets
This paper is concerned with questions which arise from a number of optimization problems in financial applications. That's why let us explain the backgrounds and financial interpretation of obtained results before we turn to describe them very briefly. The authors start with a \(d\)-dimensional stochastic process \(X= (X_i)_{0\leq t\leq T}\), defined on a probability space \((\Omega,{\mathfrak F},\mathbb{F}, P)\) with filtration \(\mathbb{F}=({\mathfrak F}_t)\) with filtration \(\mathbb{F}= ({\mathfrak F}_t)\), which describes the discounted price evolution of risky assets in a financial market containing also some riskless asset with discounted price \(Y\equiv 1\). One of the central problems in financial mathematics in such a framework is the pricing and hedging of contingent claims (like European call option on some asset \(i\) with expiration data \(T\) and strike price \(R\), say) by means of dynamic trading strategies based on \(X\). More generally, a contingent claim will here simply be an \({\mathfrak F}_t\)-measured random variable \(H\) describing the net payoff at \(T\) of the financial instrument we want to consider. The problems of pricing and hedging \(H\) can then be formulated as follows: What price should the seller \(S\) of \(H\) charge the buyer \(B\) at time \(0\)? And having sold \(H\), how can the seller \(S\) insure himself against the upcoming random loss at time \(T\)? A natural way to approach these questions is to consider dynamic portfolio strategies of the form \((\theta,\eta)= (\theta_t, \eta_t)_{0\leq t\leq T}\), where \(\theta_t\) describes the number of units of \(d\) assets and \(\eta_t\) is the amount invested in the riskless asset, helt at time \(t\). The value of portfolio \((\theta_t,\eta_t)\) is then given by \(V_t= \theta_t' X_t+ \eta_t\), the cumulative gains from trade up to time \(t\) are \(G_t(\theta)= \int^t_0\theta_s dX_s=: (\theta\cdot X)_t\) (it is here assumed that \(X\) is a semimartingale and \(G_t(\theta)\) is then the stochastic integral of \(\theta\) with respect to \(X\)), and the cumulative costs up to time \(t\) incurred by using \((\theta,\eta)\) are given by \(C_t= V_t- \int^t_0 \theta_s dX_s= V_t- G_t(\theta)\). A strategy is called self-financing if its cumulative cost process \(C\) is constant in time, and this is equivalent to saying that its value process \(V\) is given by \[ V_t= c+ \int_0 \theta_s dX_s= c+ G_t(\theta),\tag{1} \] where \(c= V_0= C_0\) denotes the initial cost to start the strategy. Now fix a contingent claim \(H\) and suppose that there exists a self-financing strategy \((c,\theta)\) whose terminal value \(V_T\) equals \(H\) with a probability 1. If our market model does not allow arbitrage opportunity, it is immediately clear that the price of \(H\) must be given by \(c\), and that \(\theta\) furnishes a hedging strategy against \(H\). This basic insight leading to the celebrated Black-Scholes formula for option pricing is now generalized to notion of the so-called complete market (we speak of a c.m. if every contingent claim is attainable, i.e., if there exists a self-financing trading strategy whose terminal value equals \(H\) with probability 1). But this notion seems very rigid to the authors of the paper. That's why they propose on using only the self-financing constraints (1). The possible final outcomes of such strategies are of the form \(c+ G_T(\theta)\) for some initial capital \(c\in\mathbb{R}\) and some strategy component \(\theta\) in the set \(\Theta\), say, of all integrands allowed in (1). By definition, a non-attainably claim \(H\) is not of this form, and so it seems natural to look for a best approximation of \(H\) by the terminal value \(c+ G_T(\theta)\) of some pair \((c,\theta)\). To find such a mean-variance optimal strategy, one therefore has to project \(H\) in \({\mathfrak L}^2(P)\) on the space \(\mathbb{R}+ G_T(\Theta)\) of attainable claims. In particular, this raises the question whether the space \(G_T(\Theta)\) of stochastic integrals in closed in \({\mathfrak L}^2(P)\). This is the main problem studied in this paper. The authors provide necessary and sufficient conditions for the closedness of \(G_T(\Theta)\) in \({\mathfrak L}^2(P)\), thus characterizing the existence of mean-variance optimal hedging strategies for arbitrary contingent claims \(H\). Moreover, they also provide new results on the existence and continuity of the Follmer-Schweizer decomposition, thus ensuring the existence of locally risk-minimizing hedging strategies.
mean-variance optimal hedging strategies
scientific article; zbMATH DE number 1210555