Hi all,I have a technical question : let's assume I have a FX spot [$]S_t^{d/f}[$] following a stochastic local vol model under a (local) martingale measure [$]\mathbf{Q}^d[$], where domestic and foreign rates [$]r_t^d[$] and [$]r_t^f[$] are stochastic also, something quite general that is. All processes are adapted to a brownian filtration [$]\left( \mathscr{F}_t \right)_{t\in {\mathbf{R}}_{+}}[$].Assume having two strike [$]K_2,K_3[$] and three dates [$]T_1,T_2,T_3[$] such that [$]T_1 \leq T_2 \leq T_3[$].I would to know if the following is true or not : do we have [$]A \Rightarrow B[$] where[$]A[$] is the following assertion :[$]\mathbf{E}^{\mathbf{Q}^d}\left[ \left. e^{-\int_{T_1}^{T_2} r_s^d ds} \left( S_{T_2}^{d/f} - K_2 \right)_{+} \right| \mathscr{F}_{T_1}\right] \geq \mathbf{E}^{\mathbf{Q}^d}\left[ \left. e^{-\int_{T_1}^{T_3} r_s^d ds} \left( S_{T_3}^{d/f} - K_3 \right)_{+} \right| \mathscr{F}_{T_1}\right], \mathbf{Q}^d \textrm{ almost surely}[$]and where [$]B[$] is the following assertion :[$] \left( S_{T_2}^{d/f} - K_2 \right)_{+} \geq \mathbf{E}^{\mathbf{Q}^d}\left[ \left. e^{-\int_{T_2}^{T_3} r_s^d ds} \left( S_{T_3}^{d/f} - K_3 \right)_{+} \right| \mathscr{F}_{T_2}\right] , \mathbf{Q}^d \textrm{ almost surely}[$]?Obviously we have [$]B \Rightarrow A[$] by tower property of the conditional expectation. For the reverse implication [$]A \Rightarrow B[$], I would say it is somehow dued to the absence of arbitrage opportunities, but I'm quite unsure...Any idea ?Thx a lot !

A will hold if [$]K_3 >> K_2[$] and B will fail if [$]S_{T_2} < K_2[$], so "not true".

Yeah, I should have been precise --> under which conditions would A => B be true ? (Typically, I will never have K_3 >> K_2, the K_i being part of a sequence of quite close rates...) I nevertheless feel that this question is desperate not-numerically-wise... Sorry !QuoteOriginally posted by: AlanA will hold if [$]K_3 >> K_2[$] and B will fail if [$]S_{T_2} < K_2[$], so "not true".