I'm looking to find the hedging PDE for a multi-currency derivative $u(F_d, F_f, X,t, T)$ under the T-forward measure, using the delta-hedging argument (F - forward rate, X - forward FX rate).

Assuming $\Pi_t = u_t - \left(\Delta_t^{(1)} F_d(t, T) + \Delta_t^{(2)} F_f(t, T) X(t, T)\right)$, the standard boilerplate for the delta-hedging argument is to first use Ito's lemma to break down $\mathrm{d}\Pi_t$ in terms of the individual processes, remove stochastic components and then set this equal to the the process dynamics for $\mathrm{d}\Pi_t$ - which under the money market measure for bonds is $\mathrm{d}\Pi_t = r\Pi_t\mathrm{d}t$, corresponding to the dynamics of a risk-free investment in the money market.

My question is, given that under the Black and Bachelier model, forward bond dynamics are martingales, does that mean $\mathrm{d}\Pi_t$ = 0 under the T-forward measure? If so, how do I explain the portfolio dynamics?

Alternatively, do I need to go back and convert all the dynamics for $F_d, F_f, X$ to the money market measure and then work backwards?



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