Why Futures and Forward Prices Match When Rates Are Certain

# Futures vs. Forwards Prices Intuition: why do futures settle using un-discounted payoffs?


# Futures vs. Forwards Prices Intuition: why do futures settle using un-discounted payoffs?












I've seen an argument that despite the differences in how they settle (futures are daily, forwards just settle at maturity), their prices are theoretically identical under the assumption of certain/known interest rates (and I think this is true more broadly for rates that are uncorrelated to the underlying). But I'm struggling to build intuition as to why this is the case, and it feels almost as though the daily settlement for futures is implemented "incorrectly". Here is what I mean:

Say we have a futures position opened at day 0, price $F(0,T)$, and we are checking on it at the end of the day, when the new futures price is $F(1,T)$. Technically, if we were to settle futures contracts daily, I believe we should settle using the discounted payoff $(F(1,T)-F(0,T))e^{-r(T-1)}$, because this is the change in VALUE of our position between day 0 and day 1 (and we should care about value because we effectively get reset to a contract with price $F(1,T)$ at the end of the day which has 0 value). But instead, we settle using just $F(1,T)-F(0,T)$ as a convention, which kind of seems wrong (you are receiving a cash flow earlier than you perhaps "should"). However, if interest rates are not correlated with gains and losses, then it doesn't matter because your gains feel better than they "should" and your losses hurt more than they "should", both to equal extents in expectation, so it all somehow balances out?

This feels very hand-wavy, and I'm having a tough time believing it. Any help? Thank you!

## Answer by user86198 (score 2)

https://quant.stackexchange.com/a/83667

(Posting this as an answer because it was too long for a comment.)

Here's a rephrasing of the original argument I saw which I think is good enough to buy: Consider an arbitrary forward contract with maturity at time $T$, bought at time $0$. We can replicate the the time $T$ payoff of this forward contract by instead buying (and closing out) a sequence of futures. That is, at each time t, buy $1/e^{r(T-t-1)}$ futures contracts and close it out at the end of the day. For each contract you buy and close, you receive the payoff at the end of the day, so its time $T$ value is equal to the payoff multiplied by $e^{r(T-t-1)}$.

The time $T$ payoff is then:

$\begin{align} &=(F(1,T)-F(0,T))/(e^{r(T-1)})*e^{r(T-1)} + (F(2,T)-F(1,T))/(e^{r(T-2)})*e^{r(T-2)} + \cdots + (F(T-1,T)-F(T-2,T))/(e^r)*e^r + (F(T,T)-F(T-1,T))\\ &= (F(1,T)-F(0,T)) + (F(2,T)-F(1,T)) + \cdots + (F(T-1,T)-F(T-2,T)) + (F(T,T)-F(T-1,T))\\ &= F(T,T)-F(0,T), \end{align}$

which is exactly the time $T$ payoff of going long a forward contract at time $0$. So if the forward contract at time $0$ has a different forward price, we would have arbitrage: you can replicate it using futures and collect the difference.

So, forward prices and futures prices have to be equal -- given known/certain interest rates, which is necessary in order to know how much of each futures contract you need to buy each day to cancel out the "amplication" caused by daily settlement. This also highlights the intuition that by settling daily using the undiscounted time $T$ payoff, futures "amplify" both your gains and losses so it winds up "balancing out", which you can see by scaling it back and reproducing a payoff identical to a forward.

## Answer by river_rat (score 0)

https://quant.stackexchange.com/a/83682

So roughly, the definition of a futures price process in continuous time for a terminal settlement value $f(X)$ is an adapted stochastic process $V$ that satisfies $V(T) = f(X)$ at the final expiry date $T$ and

$$ \mathbb{E}^Q \left( \int_{t}^{T} e^{-\int_{t}^{s} r(\eta) d\eta} \ dV_s \ \big| \ F_t \right) = 0 \quad \forall t \in [0, T] $$

i.e. The present value of all future variation margin payments is zero at every point in time. This is the defining feature of a futures contract, you can enter and exit at anytime with zero cost. With a little bit of Ito chasing you can then show that $V$ must be a martingale such that $$V(t) = \mathbb{E}^Q\left(f(X) \ \big| \ F_t \right)$$ In the case of geometric Brownian motion you get undiscounted Black76. We also have that if rates are determinist then the T-forward measure is the same as the risk-neutral measure and thus we see that futures prices must be the same as forward prices in that situation. Any difference between the futures price and the forward price we call the convexity adjustment.

So in short, the only way to have zero cost contract entry and exit at all times (with zero price alignment interest) is to settle the undiscounted risk-neutral value of the terminal payoff.