In this question, you will verify that the value of a long futures position to be held over an interval

of time is always $0.$ First the futures price is defined as

$F_S{t}_S(t,T)=E^Q[S(T)|F_t],0tT.$

$($a$)$ Verify that $F_S(t,T)$ is a martingale w$.$r$.$t filtration $F_t$ under $Q.$ Hence $($no proof$),$ by the

martingale representation theorem, there exists a process widetilde$()(t),$ adapted to $F_t,$ such that

$Fu{t}_S(t,T)=F_S(0,T)+{\displaystyle \underset{0}{\overset{t}{}}}$widetilde$()(u)dwidetilde(W)(u),0tT.$

In SDE form,

$dFu{t}_S(t,T)=$widetilde$()(t)dwidetilde(W)(t).$

$($b$)$ Let $t{t}_0{t}_1(t)R(t)x(t)dx(t)=R(t)x(t)dt+(t)dFu{t}_S(t,T).d(D(t)x(t))=D(t)(t)widetilde()(t)dwidetilde(W)(t)t_{0}x(t_0)=0E^Q[D(t_1)x(t_1)|F_{t_0}]=0.x(t)t_{0}x(t_1)(u)-=1(u)-=-10t_0$ and consider $an$ agent who $at$ times $t$ between $t_0$ and $t_1$ holds $(t)$ futures

contracts. $It$ costs nothing $to$ change the position $in$ futures contracts, but because the futures

contracts generate cash flow, the agent may have cash $to$ invest $or$ need $to$ borrow $in$ order $to$

execute this strategy. $H\frac{e}{S}he$ does this investing $an\frac{d}{o}r$ borrowing $at$ the interest rate $R(t).$

Let $x(t)be$ the value $(profi\frac{t}{l}$oss$)of$ this agent's portfolio. Then

$dx(t)=R(t)x(t)dt+(t)dFu{t}_S(t,T).$

Use $It\text{'}s$ lemma $to$ verify that

$d(D(t)x(t))=D(t)(t)widetilde()(t)dwidetilde(W)(t)$

$(c)$ Assume that $at$ time $t_0$ the agent's profit $isx(t_0)=0.$ Verify that

$E^Q[D(t_1)x(t_1)|F_{t_0}]=0.$

You have just shown that $by$ treating $x(t)as$ a bond, the value $at$ time $t_{0}of$ a payment $x(t_1)$

$is0.$ Since a futures contract must $be$ held $as$ one unit, choosing $(u)-=1$ for the long and

$(u)-=-1$ for the short are enough.