# Mark to market forward contract

(All prices are in $) Say that at time$t=0$,$A$goes long a forward contract with maturity$T$on an underlying asset$X$with forward price 100 \$, that is, $A$ agrees to buy $X$ for 100 \$at time$T$. At initialisation, the actual value of the forward contract is equal to$0$. Suppose, we are now at time$t$where$0<t<T$, and we want to determine the fair value of the forward contract. Say that at time$t$, the forward price for the same underlying with same maturity is for example 110 \$. This means that $A$ has gained since the market tells that $X$ can now be bought for 110 \$at time$T$instead of 100\$. Therefore $A$ would actually make a profit of 10 \$at time$T$and discounting back to time$t$, the contract value is$10e^{-r(T-t)}$where$r$is the risk-free rate for period$T-t$. Generally speaking, if$F_0$is the forward price at$t=0$, then the forward contract's value at time$t$is$S_t-F_0e^{-r(T-t)}$. So basically the time$t$value of the forward contract is$(F_t-F_0)e^{-r(T-t)}$where$F_t$is the time$t$forward price. Now, why is this not exactly the same than mark-to-market? If you mark-to-market at time$t$, then party$B$(the party that shorted the contract) has to give 10 \$ to A and the new contract will be as if it were a forward with forward price 110 \$. I would expect that$B$has to pay$10e^{-r(T-t)}$to$A$rather than 10 \$, because the actual loss on the forward contract is reflected on maturity and has thus to be discounted?

Say that the forward price keeps increasing over the life of the contract and that $A$ always gets a positive amount added to it's margin. For example, the forward price was 100 (day 0), 110 (day 1), 120 (day 2) and 130 (day 3 of maturity, so 130 is the spot price of $X$). If there was no concept of margin, then $A$ would have a pay-off of $130-100 = 30$. With the concept of margin, $A$ has gained 10 \$over the last three days, but the 10 \$ gained on the margin after day 1 has in the mean time accrued at the risk-free rate so I would expect the pay-off with the margining to result in a slightly higher result then if it was not margined?

Can someone explain me where I mix up the concept?

• Exactly my problem. Thank you, I'm not going mad. Mar 19, 2021 at 23:26

There are two types of contract (a) a forward contract and (b) a futures contract. In (a) there is no payment of margin on a daily basis. Its value is $(F_1-F_0)e^{-r(T-t)}$ as you describe. In (b) there is a direct payment of $F_1-F_0$ on each day and the value of the contract is always zero at the close of business. Does that answer your question?
• Thanks for pointing out the difference, however I still feel that I miss the understanding of this margining. Say that you would similarly reset the forward contract value to zero at the close of each day. You would then pocket the contract values $(F_t - F_0)e^{-r(T-t)}$ over the forward's life. If you accumulate these cash flows at maturity then it would exactly give you the forwards pay-off $S_T-F_0$, which makes sense. This is how I see it for the futures as well which is apparently not the case since the discount factor is not there. Mar 22, 2018 at 9:24
• The result of the futures margining is that 1 futures contract is a larger trade than 1 forward contract, by a factor of $e^{-r(T-t)}$. If you start with $e^{-r(T-t)}$ futures contacts, and maintain that balance throughout the life as t goes from 0 to T, you can make the same argument.