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Any explanation I've found explaining why future and spot prices converge over time seem to only focus the explanation on why the spot and future price must be equal at maturity.

I understand that if the prices aren't the same at maturity, then an arbitrage opportunity will exist. But what's the explanation for the trend of convergence prior to maturity? I originally thought that this might be because the cost of carry decreases as time to maturity decreases, but doesn't this price behaviour hold true for non-commodity futures as well?

Thanks!

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The trend of convergence is also due to arbitrageurs.

Prior to maturity, when the spot is higher than futures, arbitrageurs would short the underlying asset and long futures contracts. This in effect reduces the spot price due to increase in supply while raising the futures price due to increase in demand.

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Let’s start with a forward contract on some asset $S_t$ with no carry. There, it is obvious that at any point in time $F_t = \mathbb{E}^{Q_t}[S_T] = S_t e^{r(T-t)}$. You can see that $F_t/S_t = e^{r(T-t)} \to 1$ as $t \to T$. If you have a future contract, then you have to start thinking about margin requirements and the possibility that the daily risk-free rate can be different from the rate at which you entered in the trade. Nonetheless this should be relevant only for futures on Fixed Income and FX.

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  • $\begingroup$ why do you consider a forward with no carry? $\endgroup$ – Permian Dec 4 '18 at 13:56
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    $\begingroup$ It does not really matter. Even with carry $y$, with the exception of pathological cases, it should be the case that $F_t/S_t = e^{(r-y)(T-t)} \to 1 \text{ as } t\to T$ $\endgroup$ – fni Dec 5 '18 at 13:24

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