Why Futures Prices Are Risk-Neutral Expectations Under Daily Margining

# Zero value of cash flow for future in Shreve's book


# Zero value of cash flow for future in Shreve's book












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.` 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:

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?

## Answer by Quantuple (score 1, accepted)

https://quant.stackexchange.com/a/35487

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 $$ \mathcal{F}(t,T) = \Bbb{E}^\Bbb{Q}_t \left[ F(T,T) \right] = \Bbb{E}^\Bbb{Q}_t \left[ S_T \right] $$