I have an FX process Xt=X0exp((rd−rf)t+σWt). Now clearly E[Xt]=FX0,t. i.e. a forward contract of the process X starting at time 0 and maturing at time t.
What if I want to look at a forward contract at a later date. i.e. FXa,b. Where $0 < a
Answer
[Question 1]
Let us define Xt=X0exp((rd−rf−12σ2)t+σWt)=X0exp((rd−rf)t)E(σWt) then, in that case E(Xt|F0)=X0exp((rd−rf)t)=FX(0,t) only because E(σWt) is a stochastic exponential (strictly positive martingale with mean 1).
Yet E(Xt|F0)≠FX(0,t) for Xt=X0exp((rd−rf)t+σWt) as you define it in your question.
[Question 2]
Under the domestic risk-neutral measure, the dynamics of the FOR/DOM (1 unit of foreign currency expressed in domestic currency) exchange rate Xt should write (to preclude arbitrage opportunities) dXtXt=(rd−rf)dt+σWt Applying Itô, to the function f(t,Xt)=ln(Xt) gives dln(Xt)=(rd−rf−12σ2)dt+σWt which one can easily integrate e.g. from t1 to t2 (assuming 0<t1<t2<T) to obtain ln(Xt2)−ln(Xt1)=(rd−rf−12σ2)(t2−t1)+σ(Wt2−Wt1) or equivalently Xt2=Xt1exp((rd−rf−12σ2)(t2−t1)+σ(Wt2−Wt1))
From the above the forward FOR/DOM exchange rate at t1 with maturity t2 computes as FX(t1,t2)=E[Xt2|Ft1]=Xt1exp((rd−rf)(t2−t1))
[Edit] E0[FX(t1,t2)]=E0[Xt1exp((rd−rf)(t2−t1))]=E0[Xt1]exp((rd−rf)(t2−t1))=F(0,t1)exp((rd−rf)(t2−t1))=X0exp((rd−rf)t1)exp((rd−rf)(t2−t1))=F(0,t2)
this is only normal since E[Xt2|F0]=F(0,t2)=E[E[Xt2|Ft1]|F0] by the tower integral property.
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