Arbitrage opportunity and Cost of Carry

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12. The 3-month futures contract of a certain index is priced at $1,020. Its underlying is valued at $1,010 and pays a continuous dividend rate of 1%. If the current risk-free rate is 2.75%, the arbitrage profit opportunity is closest to:

a. $7.50
b. $5.57
c. $10.00
d. $1.75


The correct answer is b) $1,020 - $1014.47, where $1014.47 = (1010)e^(r-q)(t).However, the book (Kaplan) goes further to say that this future price is "overvalued." Best arbitrage strategy is to sell the future -- and buy purchase the idex.

I just want to make sure that I am understanding this correctly intuitively:
(1) we use TODAY's quoted underlying price to derive a projected futures price.
(2) our futures price is LESS than the given futures price, which means that the given futures is OVERVALUED.
(3) hence, our best arbitrage strategy is to BUY the underlying (i.e. purchase the index) and sell the future? I'm still not quite sure I have this correctly, can you please explain?

Thanks,
Eva

 

[David Harper CFA FRM]

Hi Eva,

Yes, I agree with your (1) to (3) except minor edit to (1)

First please note there are three concepts:
i. spot price, S0 (underlying @ 1010);
Ii. Observed MARKET futures price F0 ($1,020) -- that is observed today!
Iii. MODEL futures price (1014.47) -- that is neither observed nor projected; it is our estimate of what the F0 should be under the COC model

COC model tells us that the observed futures (market) price is "trading rich;" i.e., greater than our COC model-based price tell us it should be.

So I would rephrase your (1) to: we use TODAY’s quoted underlying spot to derive a COC model-based estimate of the "fair value" futures price.

Then the arbitrage is always to BUY the cheap thing and SELL the expensive thing. Here the futures contract is expensive/rich, it should be shorted. Then buy the other (the spot)

David

 

[[email protected]]

THank you David.

My follow-ups:

(1) How is this linked -- if at all, to the fact that the convenience yield (y) is less than the rfr? (Since future price is greater than spot).

(2) Since this is a case of contengo (futures price greatre than spot) -- wouldn't we know that the arbitraguer will take advantage by shorting the futures and buying the spot anyhow? given that futures = $1020 and spot = $1010? Why would we need to calculate the MODELED futures price at all? Is it because the ACTUAL profit ($5.57) is less than if we were to subtract $1020 - $1010? Due to the fact that we had to observe CAPM/ COC, etc?

(3) Finally -- the description of Fo > Soe^(rt) is that arbitragers will proftit by selling the forward, buying the asset (spot) with borrowed funds? Why is this BORROWER funds?

Thanks!

 

[David Harper CFA FRM]

Hi shi,

1) This is an investment not consumption commodity. No convenience yield assumption
2) It is contango (F0> S0) but that does not imply an arbitrage: whoever buys the spot must CARRY it (in this case, incur financing costs which are offset/reduced by dividend). The basis of the arbitrage depends on the future relationship, which depends on F0 and ST.
3) The complete arbitrage assumes the spot is purchased with cash that is borrowed; even if it isn't borrowed, it has an opportunity cost

David