Static arbitrage bounds on basket option prices
Let \(p\in \mathbb R^m_+\), \(K_0\in \mathbb R_+\), \(w_0\in \mathbb R^n_{++}\) and \(K_i \in \mathbb R_+\), \(w_i\in \mathbb R^n_+\) for \(i=1,\dots,m\). The authors consider the problem of computing upper (resp. lower) bounds on the price of a European basket call option with maturity \(T\), strike \(K_0\) and weight vector \(w_0\): \[ \max/\min E_\pi[(w_0^Tx-K_0)_+]\text{ subject to }E_\pi[(w_i^Tx-K_i)_+]= p_i,\;i=1,\dots,m, \tag{*} \] over all probability distributions \(\pi\) on the asset price vector \(x\), consistent with a given set of observed prices \(p_i\) of options on other baskets. Here, (*) is thought of as an integral transform inversion problem. The authors detail a relaxation technique providing upper (resp. lower) bouns on the solution to (*). This technique has the advantage of being polynomial-time in the number of assets and constraints. In some particular cases, the authors provide exact solutions to (*) that have polynomial complexity on the number of assets and constraints.
scientific article