In　this　paper,　we　consider　the　problem　of　option　pricing　when　return　rate　and　volatility　are　ambiguous.　Firstly　we　illustrate　how　to　describe　this　ambiguous　option　pricing　model　by　using　set-valued　differential　inclusion　and　how　to　change　the　discussion　of　pricing　bound　problems　of　options　into　that　of　maximal　and　minimal　conditional　expectations.　Secondly　we　discuss　the　properties　of　maximal　and　minimal　conditional　expectations,　especially　the　representation　theorem　of　maximal　and　minimal　expectations.　Finally　we　give　the　bounds　of　the　European　option　pricing　by　using　above　theorems.　©　2011　Springer-Verlag　Berlin　Heidelberg.