FRM Fun 18 (normal backwardation), P1

[David Harper CFA FRM]

P1 only.

Today a great question was asked by sammyjny on our youtube channel:

I can't understand why is there a difference between futures price and expected spot price in future. From arbitrage theory of pricing forwards/futures, shouldn't the future price be exactly equal to the expected spot price in future? Then what is causing the difference?

The question is posted at this video, about two hours ago (not that you want or need to view the video. Yikes, I recorded this five years ago!):

So (my paraphrase of) the question is:

How is F(0) <> E[S(t)] compatible with a no-arbitrage assumption;
does not no-arbitrage insist that forward price F(0) = E[S(t)]?

 

[LL]

Wow ! The turtle on bottom right in the video above is all groomed in 5 years !

The post out here is good for understanding too:
http://www.bionicturtle.com/how-to/article/contango_backwardation_expected_future_price

David, correct me if I am wrong but this is how I understand this works:

A futures contract is priced assuming there is no arbitrage. But I believe that changes in the expected spot price in future is not a violation of this assumption.
The convergence of the futures price to spot price is governed by the arbitrage and law of supply and demand.

We wouldn't have any futures contracts if the expected spot price in futures are same as these contracts !
This whole business is due to:
'CURRENT' futures contract price being different than the 'EXPECTED SPOT PRICE IN FUTURE (no contracts out here.. just the expectation)'

Example to show convergence (simple one... ignoring cost of carry / storage / etc.):

Spot price = $50
Future Price (6 months delivery) = $60

Arbitrage:
Seller of contract: Buys the commodity for delivery @ Spot Price = $50
Seller of contract: Sells contract at futures price = $60
Profit made = $10

Supply and Demand:
Sellers selling the fuures contract => Drop on futures prices because of increase in supply
Sellers buying of commodity to deliver => Demand Increase => Increase in Spot Price

 

[ShaktiRathore]

Futures price, F0=E(St)
Also that F0=S0 as the futures price at any time is equal to spot price otherwise an arbitrage is possible.
which implies that E(St)=S0
now due to contango and backwardation phenomenon, (incl. cost of carry, convenience yield, interest cost etc.)
we can say St=S0*e^(r+c-y)
=> St <> S0 or that F0 <>E(St).

thanks

 

[David Harper CFA FRM]

Thank you both for your great answers! I do not pretend to have a definitive, authoritative answer to this provocative question. Here's my opinion:

• I think the answer presupposes a theory (model) of futures prices, and therefore different answers may be valid (this is not my idea, I learned this in Kolb)
• The first issue is with E[S(t)]: unlike F(0) or S(0), E[S(t)] is unobserved and, I think, cannot be measured (an unobserved market consensus). My view is that F(0) = Function[E[S(t)]], including the special case of F(0) = E[S(t)] is really a function expressing a theory about E[S(t)]
• In any case, the FRM studies Hull who employs the classic model of normal backwardation; i.e., future hedgers pay profit-seeking speculators to assume the price risk. In this theory, the short (hedger) incurs an expected loss, transferred to the long (speculator's) expected gain.
• The functional manifestation of this theory of normal backwardation is (Hull 5.20): F(0) = E[S(t)]*[exp(r-k)*T].
  This formula says: if the asset has any systemic risk, then F(0) must be <> E[S(t)].
• My interpretation/resolution is therefore: unlike an arbitrage which exploits the cost of carry, there is not a true arbitrage available with respect to currently unobserved E[S(t)]: the so-called arbitrage depends on the realization of the E[S(t)]. But realized S(t) is uncertain. The formula shows that the long futures position profits, under normal backwardation, not by an arbitrage, but because they are compensated for incurring a systematic risk (i.e., the excess of k over r).
• So my view is that no-arbitrage actually does not apply here. Arbitrage conditions apply only in the special case when r = k and the asset is riskless. Others, E[S(t)] is reasonably different from F(0) due to risk aversion (Kolb's term).