Model-independent no-arbitrage conditions on American put options (Q2800003)

The aim of this paper is to price American put options without knowing any underlying model but anyway, avoiding arbitrage. The authors provide a simple necessary condition to forbid arbitrage under any model which could underly the asset price process. There are neither dividend nor trading costs, but let \(r>0\) be a discount coefficient. The maturity is \(T\). The key of this necessary condition is the introduction of both functions: the function \(A\), an arbitrage free American price function and, the probability measure \(\mu\) satisfying that the initial price \(S_0=\int x\mu(dx)\), the function \(E: K\to\int(e^{-rT}K-x)^+\mu(dx)\). Then the necessary condition is NEWLINE\[NEWLINE\begin{gathered} A\text{ is increasing and convex},\;A'(K+)K-A(K)\geq E'(K+)K-E(K),\\ \sup(E(K), K-S_0)\leq A(K) \leq E(e^{rT}K).\end{gathered}\tag{1}NEWLINE\]NEWLINE Conversely, a sufficient condition for the existence of such a function \(A\) is restricted to the case where the functions \(A\) and \(E\) are given for a finite number of strikes, and extended by linear extrapolation to the real line. The sufficient condition is that these piecewise linear functions satisfy (1). Moreover, a numerical part provides the existence and calculation of the optimal stopping strategy for the American put option.