September 2006 Q3 (iii)

[mange]

Hi all,

Does anyone know why to work out the equation below? I've looked at it for a long time and couldn't figure out how to get to the second line. It comes from the solution of Q3 (iii).

dV(t) = d(e^rt*Ft) = re^rt*Ftdt + e^rt dFt
= re^rt*Ftdt + e^rt *OtdSt + re^rt (Ft - Ot*Dt) dt = OtdSt + ψt dBt.

Your help is much appreciated.

Thanks.

 

[wannabeactuary]

This question's been posted a while ago but in case anyone's wondering the second line in the solution that mange refers to cannot be directly derived from the previous one (at least I cannot do so), but requires some intermediate steps as follows:

dV(t) = d(e^rt*Ft) = r*e^rt*Ftdt + e^rt dFt

= r*e^rt*Ftdt + e^rt Ot dDt .. from part (ii)
and since Dt = e^-rt * St this becomes

= r*e^rt*Ftdt + e^rt Ot (-r*e^-rt St dt + e^-rt dSt)

cancelling out the exp's we get

= r*e^rt*Ftdt + Ot dSt - r Ot St dt

= Ot dSt + r*e^rt (Ft dt - Ot Dt dt)

= Ot dSt + r*e^rt ψt dt

= Ot dSt + ψt dBt.

(apologies for the awkward notation)