Bounds for the American perpetual put on a stock index (Q2731150)

Let \((S_i(t)) (t\geq 0)\) be the price process of stock \(i\) \((1\leq i\leq n)\), given by NEWLINE\[NEWLINEdS_i(t)= S_i(t)\bigl(rdt +\sigma_idW_i(t) \bigr), \quad t\geq 0.NEWLINE\]NEWLINE Here, \(r>0\) is the (common) interest rate and \(\sigma_i>0\) \((1\leq i\leq n)\) is the volatility of stock \(i\); \(W_i,\dots,W_n\) are (dependent) standard Wiener processes with quadratic covariation \([W_i,W_j]_t =\rho_{ij}t\), \(t\geq 0\). The index \(I_a\) is given by \(I_a(t)=a_1S_1(t) +\cdots+ a_nS_n(t)= a \cdot S(t)\) (scalar product with \(a=(q_1, \dots,a_n))\) for constants \(a_i>0\). The author studies the fair price of an American put (with strike price \(K)\) on \(I_a\) given by NEWLINE\[NEWLINEv_a(x)= \sup_\tau E_x\biggl[ e^{-r\tau} \bigl(K-I_a (\tau)\bigr)^+ \biggr]NEWLINE\]NEWLINE for \(x=(x_1, \dots,x_n)\), and \(x_i>0\) denoting the initial price of stock \(i\) (the supremum taken over all stopping times \(\tau)\). The author obtains a (rough) outer bound and an inner approximation of the early exercise region.