Here is the statements of future price in Shreve's book Stochastic Calculus for Finance II page 244 to proof the value of cash flow is zero. enter image description here But I have problem that here we use the fact $D(t_{k+1})$ is $\mathcal{F}(t_k)$-measurable, since the definition of $D(t_{k+1})$ in the discrete interest rate case as following: enter image description here

but, in the continuous interest rate case, we can never guarantee $D(t_2)$ is $\mathcal{F}(t_1)$-measurable when $t_2>t_1.$ Does the statement still hold?

  • $\begingroup$ @Quantuple yeah I agree with the infinitesimal case, but if just for two discrete time $t_1, t_2,$ the value of discounted cash flow is not zero, right? $\endgroup$ – A.Oreo Aug 4 '17 at 14:31
  • $\begingroup$ @Quantuple For any $t<t_1 < t_2,$ we can't get $E_t[D(t_2)(F(t_2) - F(t_1))] = 0,$ instead we only have $E_t[\int^{t_2}_{t_1}D(s)dF(s)] = 0?$ $\endgroup$ – A.Oreo Aug 4 '17 at 14:39
  • $\begingroup$ That's right in my opinion. More specifically, the futures contract can be seen as an asset with zero value but paying a dividend given by the difference between the quoted futures prices. Provided that $D(s)$ verifies the usual conditions (so that $\int_{t_1}^{t_2} D(s) dF(s,T)$ is indeed an Itô integral) you still have that $F(s,T)$ is a martingale. $\endgroup$ – Quantuple Aug 4 '17 at 15:03
  • $\begingroup$ @Quantuple sorry, I think it's s Riemann integral, since $F$ is not random, even $D(t)$ may be random $\endgroup$ – A.Oreo Aug 4 '17 at 15:30
  • $\begingroup$ $F(t,T)$ is a stochastic process. $\endgroup$ – Quantuple Aug 4 '17 at 15:31

In practice, a futures contract can be seen as a margined forward contract. More specifically, futures are marked to the market at the end of each business day by accounting for the change in value of an associated futures price process $(\mathcal{F}(t,T))_{t \in [0,T]}$ verifying $\mathcal{F}(T,T)=S_T$. This MtM variation is settled each day by the exchange by crediting/debiting the holder's margin account. This daily margining means that the value of the futures contract is effectively reset to zero each day.\

If we were to model this in continuous time, we should therefore interpret a futures contract as a financial instrument

  • which can be entered (or unwinded) at zero cost at any time such that its $t$-value $V_t \equiv 0, \forall t$
  • paying a cash dividend $dD_t = \mathcal{F}(t+dt,T)-\mathcal{F}(t,T)$ over each infinitesimal period $[t,t+dt[$ where $\mathcal{F}(t,T)$ figures the future price process of maturity $T$ and $D_t$ the cumulated dividend process.
  • such that the future price process should verify $\mathcal{F}(T,T) = S_T$ almost surely.

REM: In these definitions, it is crucial not to confuse the value of a futures contract, which is zero by definition, with the futures price which is not.

For any dividend-paying asset $V_t$, it is well-known that the self financing strategy consists in fully reinvesting all contributions of its dividend process $D_t$, so that overall there are no exogenous cash withdrawal or infusion as time passes. This leads to the following self-financing portfolio wealth evolution starting from $X_0=0$ \begin{align*} dX_t &= dV_t + (X_t - V_t) r dt + dD_t \end{align*} and arbitrage free pricing theory tells us that any self-financing strategy, here $X_t$, should be a martingale under the risk-neutral measure $\Bbb{Q}$ associated to the money market account $B_t$ numéraire. Here, this suggests that $B_t^{-1}X_t$ should be a $\Bbb{Q}$-martingale. Now, \begin{align} d(B_t^{-1} X_t) &= B_t^{-1} \left( dX_t + X_t r dt \right) \\ &= B_t^{-1} \left( dV_t + dD_t - rV_t dt \right) \\ &= B_t^{-1} \left( B_t(d(B_t^{-1}V_t)) + dD_t \right) \\ &= d(B_t^{-1}V_t) + B_t^{-1} dD_t \end{align} such that \begin{equation} B_t^{-1} X_t = B_t^{-1} V_t + \int_0^t B_s^{-1} dD_s \end{equation}

For futures, $V_t \equiv 0$ and the $\Bbb{Q}$-martingale property can further be written as $$ \Bbb{E}^\Bbb{Q}_0 \left[ B_t^{-1} X_t \right] = B_0^{-1} X_0 = 0 $$ hence $$ \Bbb{E}^\Bbb{Q}_0 \left[ \int_0^t B_s^{-1} dD_s \right] = \Bbb{E}^\Bbb{Q}_0 \left[ \int_0^t B_s^{-1} d\mathcal{F}(s,T) \right] = 0 $$

which, assuming some light conditions on the adaptability and regularity of $(B_s)_{t \geq 0}$, is verified if the future price process $\mathcal{F}(t,T)$ itself emerges as a $\Bbb{Q}$-martingale (Itô integral), which along with the terminal condition mentioned earlier gives us $$ \mathcal{F}(t,T) = \Bbb{E}^\Bbb{Q}_t \left[ F(T,T) \right] = \Bbb{E}^\Bbb{Q}_t \left[ S_T \right] $$

  • $\begingroup$ I agree with the above deduction, but why can't understand the future price as the sum of discounted cash flow is zero? $\endgroup$ – A.Oreo Aug 7 '17 at 7:56
  • $\begingroup$ Can you please be more clear? $\endgroup$ – Quantuple Aug 7 '17 at 8:14
  • $\begingroup$ @ intuitively, we should have $E[\sum D(t_{j+1})(F(t_{j+1}) - F(t_j))] = 0,$ but when $t_{j+1} - t_j\rightarrow 0,$ this is not compatible with $E[\int D(s)d F(s)] = \lim E[\sum D(t_j)(F(t_{j+1}) - F(t_j))] = 0.$ unless we can show $\lim E[\sum D(t_j)(F(t_{j+1}) - F(t_j))] = \lim E[\sum D(t_{j+1})(F(t_{j+1}) - F(t_j))],$ which is something like a Riemann int. $\endgroup$ – A.Oreo Aug 7 '17 at 8:27
  • $\begingroup$ The last relationship is the difference between an Itô integral and a Stratonovich integral, which are not equal in general. IMO intuition fails when talking about the limiting behaviour of discrete time processes and reciprocally for the continuous counterpart. This is not an easy topic. Anyway, as soon as you assume that $D(t)$ is predictable (which is also the assumption used by the author), then you'll be able to fall back on your feet. Again I recommend that you think in terms of dividend process and link that to what you would do with an equity to make things clearer. $\endgroup$ – Quantuple Aug 7 '17 at 9:02

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