Delta Hedging a Multi-Currency Derivative Under the T-Forward Measure

Summary

The document poses a question about deriving hedging dynamics for a multi-currency derivative whose inputs are domestic and foreign forward rates and a forward foreign-exchange rate. It sets up a portfolio holding the derivative and short positions in the underlying forwards, then asks how to apply Itô’s lemma, cancel stochastic terms through delta hedging, and determine the resulting drift.

The central issue is whether a hedged portfolio has zero drift under the T-forward measure because the relevant bond forwards are martingales, or whether the dynamics must first be expressed under a money-market measure. No answer, derivation, or numerical example is included, so the document frames a measure-consistency problem rather than resolving it. Its value is in identifying that the pricing numeraire and the measure used for portfolio dynamics must be handled carefully in a multi-currency setup.

Key ideas

• The proposed derivative depends on domestic and foreign forward rates and a forward exchange rate.
• The question sets up delta hedging by offsetting the derivative with positions in forward instruments.
• It asks whether a martingale property under the T-forward measure implies zero hedged-portfolio drift.
• The document does not derive the pricing PDE or resolve which measure should govern the hedged portfolio.

Tags

# Hedged portfolio dynamics under T-forward measure


# Hedged portfolio dynamics under T-forward measure












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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