From perpetual strangles to Russian options
Let \(\{S(t); t\geq 0\}\) be a geometric Brownian motion modeling the stock price process. Define the process \(M(t) = \max \{M, \max[S(u)\mid 0 \leq u \leq t]\}\), \(t \geq 0\), where \(M \geq S(0)\). \(M(t)\) is called the historical maximum of the stock prices up to time \(t\). The `Russian option' is a perpetual American option such that, if its owner exercises it at time \(t \geq 0\), he receives the amount \(M(t)\) at that time. The price at time 0 of the option is the supremum, over all stopping times \(T\), of \(E[\text{exp} (-rT) M(T)]\). The authors derive two pricing formulas by means of the optional sampling theorem. In particular, an alternative derivation of the Shepp-Shiryaev pricing formula for Russian option is provided.
geometric Brownian motion modeling the stock price process
scientific article; zbMATH DE number 768795