Difference between Price and Value in Forward and Futures

Question

Consider forward and futures contract on a zero-coupon bond.

Denote the time $t$ forward (contract) price on a $T_2$-maturity zero-coupon bond with delivery date $T_1$ as $F(t,T_1:T_2).$

Similarly, denote the time $t$ futures (contract) price on a $T_2$-maturity zero-coupon bond with delivery date $T_1$ as $G(t,T_1:T_2).$

I am a bit confused regarding the terminologies associated with these contracts.

My understanding is:

(1) Forward price is the pre-specified price that I buy the underlying zero-coupon bond on the delivery date, $T_1$, and this is similar to the futures prices.

(2) The payoff of forward contract is: $p(T_1,T_2)-F(t,T_1:T_2),$ where $p(t,T)$ is the time-$t$ zero-coupon bond that matures at $T$. This is similarly the case for futures except it marks to market everyday.

My questions are:

(A) Does the forward and futures prices change over time $t$ as we progress to $\{t+1,t+2,...\}$? My understanding is the little $t$ is equivalent to when the contract was written, which I take as $F(t,T_1:T_2)$ and $F(t+1,T_1:T_2)$ are two distinct forward contracts?

(B) My understanding of the value of the contract is equal to the payoff of that contract, so for forwards it would be:

$$V_F(t,T_1,T_2)=p(T_1,T_2)-F(t,T_1:T_2)$$

Clearly, the value of the forward contract depends on when it was written, and the underlying's maturity, the contract's delivery date. But when we look at a particular contract, we are fixing the little $t$, right, so in effect it becomes, which implies the value of any particular forward contract depends on the change in the delivery and underlying's maturity date?:

$$V_F(\bar t,T_1,T_2)=V_F(T_1,T_2)=p(T_1,T_2)-F(\bar t,T_1:T_2)$$

(C) I am confused when we mark-to-market with the futures contract. Consider the time interval $[t,t+1]$. At $t+1$, the futures contract is mark to market in the following sense:

$$G(t+1,T_1:T_2)-G(t+1,T_1:T_2).$$

My confusion is are you marking to market of the future contract written on $t$ with another futures contract with the same underlying and the same delivery date but written on $t+1$? But this doesn't change the "futures" price of your original contract, but it just generates a positive or negative cash flow to your portfolio, right? Is this distinction correct? Because the payoff of the futures contract is exactly similar to that of the forward contract, which is :

$$V_G(t,T_1,T_2)=p(T_1,T_2)-G(t,T_1:T_2)$$

The above tells that you are comparing the delivery date price of the underlying less the original futures prices, so the valuation of the futures contract written on $t$ is about the futures price on $t$. But my main confusion is that the Cash Flows generated along the mark-to-market process in futures is that everyday you are comparing the futures prices of the day before and today of some index $[t_j,t_j+1]$, where $t_0=t$, $t_j\not=t$ for $j>0$, so I am confused why we are obtaining cash flows from effectively distinct futures contract?

Answer 1

$\text{(A)}$ Exact, you write your forward contract at $t$ thus afterwards you can think of the delivery (or forward) price $F(t,T_1:T_2)$ as a fixed quantity. The time index $t$ is there to represent the fact that tomorrow, the day after, etc. you will not be able to write a new forward contract with the same delivery price: the forward price evolves in the market. So $F(t+1,T_1:T_2)$ is a price for a contract different than the one with delivery price $F(t,T_1:T_2)$. Again, once you've written a contract, the delivery price is locked.

$\text{(B)}$ Don't forget that you need to discount to get the value of the contract. A zero-coupon bond can be represented as (we assume deterministic rates for simplicity): $$p(T_1,T_2)=e^{-r(T_2-T_1)}$$ If $D(t,T_1)$ is your discount factor (equal to zero-coupon bonds when rates are deterministic): $$D(s,T_1)=e^{-r(T_1-s)}$$ you can treat your forward price $F(t,T_1:T_2)$ as a fixed quantity $K$ thus the value of the contract at $s \in[t,T_1]$ is: $$\begin{align} D(s,T_1)(p(T_1,T_2)-K)&=p(s,T_2)-D(s,T_1)K \\ &=p(s,T_2)-D(s,T_1)F(t,T_1:T_2) \end{align}$$

$\text{(C)}$ Let your Futures price be $G(t,T_1:T_2)$. The difference with a forward is that the mark-to-market value of your contract is settled daily: in a forward you receive the whole value at the end of the contract, in a futures you are receiving it little by little, each day. At the end of each day, the futures pays you: $$G(s+1,T_1:T_2)-G(s,T_1:T_2)$$ Hence at the end of the contract, assuming $n$ days between $t$ and $T_1$ with $s_0=t$ and $s_n=T_1$: $$\begin{align} \sum_{i=0}^{n-1}(G(s_{i+1},T_1:T_2)-G(s_i,T_1:T_2)) &=\sum_{i=1}^{n}G(s_{i},T_1:T_2)-\sum_{i=0}^{n-1}G(s_{i},T_1:T_2) \\ &=G(T_1,T_1:T_2)-G(t,T_1:T_2) \\[8pt] &=p(T_1,T_2)-G(t,T_1:T_2) \end{align}$$