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American prices embedded in European prices. (English) Zbl 0982.60028
The authors analyze, in a Black-Scholes framework, relations between American options prices and European options prices. They exhibit a large class of payoffs $\psi$ for which the American option price $v^{\text{am}}_\psi(t, x)= \sup_{\tau\in{\cal T}} (0; t)$ has a closed-form expression. The main theorem is Theorem 3. Let $\widehat\varphi(x)= \inf_{t\ge 0} v_\varphi(t, x)$ where $v_\varphi(t, x)$ is the European option price for a payoff $\varphi$. Theorem 3 gives conditions on $\widehat\varphi$ to insure that $v^{\text{am}}_{\widehat\varphi}(t, x)= v_\varphi(t\vee\widehat t(x), x)$ where $\widehat t$ has to verify $v_\varphi(\widehat t(x), x)= \widehat\varphi(x)$. Examples verifying Theorem 3 hypotheses are given.

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 60G46 Martingales and classical analysis 91B70 Stochastic models in economics 91B28 Finance etc. (MSC2000)
##### Keywords:
American options; Black-Scholes framework; derivatives
Full Text:
##### References:
 [1] Crank, J.: Free and moving boundary problems. (1984) · Zbl 0547.35001 [2] Martini, C.: The UVM model and American options, rapport de recherche no 3697. (1999) [3] Musiela, M.; Rutkowski, M.: Martingale methods in financial modelling. (1998) · Zbl 0906.60001 [4] Revuz, D.; Yor, M.: Continuous martingales and Brownian motion. (1991) · Zbl 0731.60002