Hull.02.16 - Doubt?

[hmehrotra]

David,

Please see the Q&A given below. I have a basic doubt - If we are saying that F1 and S (at time 1) are different, then doesn't that imply arbitrage opportunity? So, if F1 = S (at time 1, then the payoff should be zero. Ditto for the F2 contract. Am I missing something?

Thanks,
CandidFRM

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On July 1, 2006, a US company enters into a forward contract to buy 10 million GBP on January 1, 2007. On September 1, 2006, it enters into a forward contract to sell 10 million GBP on January 1, 2007. Describe the profit or loss the company will make in dollars as a function of the forward exchange rates on July 1, 2006 and September 1, 2006.

Answers:

Suppose F1 and F2 are the forward exchange rates for the contracts entered into July 1, 2006 and September 1, 2006, and S is the spot rate on January 1, 2007. (All exchange rates are measured as dollars per pound). The payoff from the first contract is 10(S – F1) million dollars and the payoff from the second contract is 10(F2 – S) million dollars. The total payoff is therefore 10(S – F1) + 10(F2 – S) = 10(F2 – F1) million dollars.

 

[David Harper CFA FRM]

Hi CandidFRM,

Say the spot S0 (today, Oct) = $10 and F1 (Jan 2010) = $14, that doesn't give you necessarily an arbitrage
you can buy the spot commodity (-$10) but then you sell forward b/c you think the $14 is "rich"
...the problem is that you have to (cost of) carry the commodity over time, so it depends on the financing/storage cost
...put another way, S0 = the PV, but your profit is only if the $14 in the future is greater than the future cumulative cost of the spot
(here is XLS illustrating the arbitrage, simplified to the point where the only issues are interest rate and transactions:
http://www.bionicturtle.com/premium/spreadsheet/3.a._arbitrage/
i personally find the cash and carry, reverse cash and carry frustratingly difficult, I need to do the XLS to see the dynamics ...maybe you will find useful...i seriously have an much easier time with calculus )

so i think the challenging part concerns S0 versus St (i.e., the spot in the future)
for the first forward contract ("a US company enters into a forward contract to buy 10 million GBP on January 1, 2007")
if we are starting from S0 (July 1st) and the forward price is, say, $14, so:
On July 1st, S0 = $10 and F1 = $14
entering the long forward is just a promise to pay $14 in exchange for the commodity
...need to go foward in time to January
... then see what is St. If St = $16, then pay $14 as promised and receive something worth $16

so mabye i am wordy but i think it's the time dimension that complicates the arbitrage...David

 

[hmehrotra]

David,

Thanks for the quick reply. I understand the cash and carry trade but I am thinking from a 'zone of convergence' perspective i.e, if S0 (Current Spot) = 10 and F1= 14 then shouldn't S1 = 14 as spot converges to the forward price at the time of delivery. If not, then this will make the Spot expensive (say S1 = 16) and Forward cheap resulting in a reverse cash and carry trade.

Please clarifiy where I am going wrong.

Thanks,
CF

 

[David Harper CFA FRM]

CF,

Maybe you are forgetting the delivery price (K) is fixed throughout and will not converge?
When the contract is entered, the delivery price equals the forward price, but going forward the delivery price is constant.

At time T0 (July): S0 = $10 and F1 (January forward) = $14
At this time T0, you go long the forward, then your delivery price (K) = F4 = $14
(i.e., you promise to buy @ $14)

...now march forward in time to December:
per zone of convergence, spot is coverging to forward
maybe Spot(dec) = $12 and Jan forward is $13
or maybe Spot(dec) = $14 and Jan forward is $15
but your delivery price remains $14 (your delivery price does not converge)

In January, the forward price will roughly converge, but that could be at $11, $17
(neither the original forward nor spot needs to maintain)

viewed another way, as time marches forward, the forward price on the Jan contract is converging (changing price) but you are not "realizing" that change unless you *roll* the contract

Hope that helps, David