relationship between forward and futures prices? (p49)

[ajsa]

Hi David,

Could you explain why if the correlation is strongly positive: futures > forward? (I understand it is related to daily mark to market, but I do not understand why strongly positive correlation leads to "futures > forward".)

Also how is this related to convexity adjustment?

Thanks.

 

[David Harper CFA FRM]

Hi ajsa,

Note that p49 refers to the correlation between the underlying and interest rates, so you are exactly right, the key is daily settlement. Recall the cost of carry model implies an almost 1:1 relationship between change in spot and futures price (actually, delta of a forward is 1.0 but the delta of a future is slightly higher than 1 due to this daily settle!). So, the idea is, if the underling increases in price, then so must the futures price (with delta 1+) and this implies an *increase* in the margin account. The "excess" margin balance can be withdrawn and it will earn a higher interest rate (enter the correlation between spot and rates) and, if you are in the long future position, it is sort of a double-bonus.

But asymmetric because if spot goes down, so does future price. At some point, margin call which requires an investment of your funds. However, enter correlation, at a lower rate (funding cost to you). So, the correlation causes a dampening on the downside, it's not quite symmetrical.

Yes, this is pretty much the same concept as convexity adjustment except, to my knowledge, the "convexity adjustment" connotes, or tends to refer to the *special case* where we are talking about Eurodolllar futures versus FRA. Note, in this case, there is no need to specify that the underlying (an interest rate) is correlated to interest rates; in this case, we can assume very high positive correlation. (Such that, the Future > forward refers to a rate, not the settlement price; higher futures rate implies lower settlement price. So, there is an apparent confusion if you do not notice in this case the underlying is a rate, not a price)

David

 

[ajsa]

David, thanks for the convexity adjustment explaination. That is very clear.

 

[Tipo]

Im having trouble understanding this

For a positive correlation (future>forward)
I/r increases, S0 increases, r increases
I/r decreases, S0 decreases, r decreases

Negative correlation (forward>future)
I/r increases, S0 decreases, r increases
I/r decreases, S0 increases, r decreases

F0=S0*e^rt

Now if i were to plug the above into the equation, futures>forward only in Positive correlation(I/R increases)
What concept did i screw up?

Why does the zero look exactly the same as the O?

 

[David Harper CFA FRM]

Hi @Tipo

What's the source (is that our notes, I can't seem to locate)?

Your question is well-positioned as this is a reference to Hull's "argument" on page 112 (of Hull's book) in Chapter 5. But the cost of carry formula, F0=S0*e^rt, cannot by itself manage this distinction: this cost of carry formula is relatively simple and does not factor in the "convexity adjustment," which I describe above and accounts for the disparity between longer-term interest rate futures and interest rate forwards. What I mean is that Hull's cost of carry ....

F(0) = S(0)*exp[(c-y)]T, where c is generic and can include y = 0 and c = r such that: F(0) = S(0)*exp(rT)

... intends to model either a general forward or futures price because it does not itself explicate the convexity adjustment. I think it might be okay to "simulate" the point (i.e., positive correlation between r and S implies future > forward) by observing that a long interest rate futures contract holder can reinvest excess margin cash (this is the daily settlement that gives rise to the discrepancy) at a high rate when spot prices increase (i.e., positive correlation between S and r) such that he/she experiences effectively a higher (r) in the same F(0) = S(0)*exp(rT). But "future>forward" is a dynamic that plugs into the COC model directly, it's more akin to a liquidity factor that complicates the model. I hope that helps,

 

[Tipo]

Hi @Tipo

What's the source (is that our notes, I can't seem to locate)?

Your question is well-positioned as this is a reference to Hull's "argument" on page 112 (of Hull's book) in Chapter 5. But the cost of carry formula, F0=S0*e^rt, cannot by itself manage this distinction: this cost of carry formula is relatively simple and does not factor in the "convexity adjustment," which I describe above and accounts for the disparity between longer-term interest rate futures and interest rate forwards. What I mean is that Hull's cost of carry ....

F(0) = S(0)*exp[(c-y)]T, where c is generic and can include y = 0 and c = r such that: F(0) = S(0)*exp(rT)

... intends to model either a general forward or futures price because it does not itself explicate the convexity adjustment. I think it might be okay to "simulate" the point (i.e., positive correlation between r and S implies future > forward) by observing that a long interest rate futures contract holder can reinvest excess margin cash (this is the daily settlement that gives rise to the discrepancy) at a high rate when spot prices increase (i.e., positive correlation between S and r) such that he/she experiences effectively a higher (r) in the same F(0) = S(0)*exp(rT). But "future>forward" is a dynamic that plugs into the COC model directly, it's more akin to a liquidity factor that complicates the model. I hope that helps,

P1, Financial Markets & Products
Hull pg 67

If the risk-free rate, , is constant across all maturities, then the forward price should equal the Futures price (forward = Futures price). But this will vary where there is a correlation between the underlying asset (S) and interest rates
If the correlation is strongly positive: Futures > forward
If the correlation is strongly negative: Futures < forward

Could you elaborate on the above?

Tks

 

[David Harper CFA FRM]

Hi @Tipo

I copied the relevant Hull (5.8) below. This is due to the same dynamic that is called the convexity bias (and handled with the convexity adjustment) in interest rate futures. The difference arises due to a key difference between exchange-traded futures and OTC forwards: daily settlement. Daily settlement implies that, on a daily basis, the futures contract position might either receive excess margin cash or, on the other hand if a margin call occurs, need to "spend cash" into the margin account. This cash flow volatility, at the margin, can be favorable or unfavorable. I hope that helps!

Hull 5.8: "Are forward prices and futures prices equal? When interest rates vary unpredictably (as they do in the real world), forward and futures prices are in theory no longer the same. We can get a sense of the nature of the relationship by considering the situation where the price of the underlying asset, S, is strongly positively correlated with interest rates. When S increases, an investor who holds a long futures position makes an immediate gain because of the daily settlement procedure. The positive correlation indicates that it is likely that interest rates have also increased. The gain will therefore tend to be invested at a higher than average rate of interest. Similarly, when S decreases, the investor will incur an immediate loss. This loss will tend to be financed at a lower than average rate of interest. An investor holding a forward contract rather than a futures contract is not affected in this way by interest rate movements. It follows that a long futures contract will be slightly more attractive than a similar long forward contract. Hence, when S is strongly positively correlated with interest rates, futures prices will tend to be slightly higher than forward prices. When S is strongly negatively correlated with interest rates, a similar argument shows that forward prices will tend to be slightly higher than futures prices.
The theoretical differences between forward and futures prices for contracts that last only a few months are in most circumstances sufficiently small to be ignored. In practice, there are a number of factors not reflected in theoretical models that may cause forward and futures prices to be different. These include taxes, transactions costs, and the treatment of margins. The risk that the counterparty will default is generally less in the case of a futures contract because of the role of the exchange clearinghouse. Also, in some instances, futures contracts are more liquid and easier to trade than forward contracts. Despite all these points, for most purposes it is reasonable to assume that forward and futures prices are the same. This is the assumption we will usually make in this book. We will use the symbol F0 to represent both the futures price and the forward price of an asset today."